Andromeda Galaxy: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
 
en>Coconutporkpie
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
{{Distinguish|Taurus (disambiguation)}}
== ZbkTLk Jordan 6 World Cup Brazil Pog1NBqj ==
{{About|the surface and mathematical concept of a torus}}


[[File:Torus.png|thumb|A torus]]
If you say yes, you are a "moralist" in the manner of President George W. Bush, who saw things in black and white, called Saddam Hussein evil, and invaded Iraq and Afghanistan with the stated goal of uprooting millennia of habits and installing American style democracy. power  and the world's demand for it.<br><br>How can we and what will we make it? The challenges we set for ourselves, the goals that we establish, the things that we accomplish will be determined by the effort that we make and the opportunities that are provided to us. So this is an opportunity to start again and look at a variety of different things that I am going to talk about a little bit later: The academic planning team, a group that has been working throughout the summer to help us develop an academic plan, built upon what we have done in the past with an eye on the future and what we might become as an institution. <br><br>Forward Davante Gardner, who scored 26 points when these teams met back in February, leads Marquette with nine points and four rebounds. James Southerland has a team high eight points for Syracuse, which entered halftime with a 24 18 lead despite allowing 10 offensive rebounds. Marquertte shot just 27 percent, 1 of 9 from three point range in the first half.<br><br>Venturi first gained national attention in 1956 when, as an amateur, he finished second in that year's Masters after .The clubs incorporate everything that Venturi venturia tube with a decrease in the inside diameter that is used to increase the flow velocity of the fluid and thereby cause a pressure drop; used to measure the flow velocity (a venturimeter) or to draw another fluid into the stream. <br><br>The second oldest hotel in [http://www.ibewlocal58.org/answerss.php??p=59042-buy-authentic-outlet-Nike-Shox-R4-alry-White-Yellow-online-2014.html buy authentic outlet Nike Shox R4 alry White Yellow online 2014] the United States, the Equinox has enjoyed a rich history that adds to its historic [http://www.ibewlocal7.com/themess.php??p=37680-cheap-authentic-Nike-Air-Max-90-White-Blue-Black-running-sneaker.html cheap authentic Nike Air Max 90 White Blue Black running sneaker] appeal. Presidents William Howard Taft, Ulysses S. Grant, Theodore Roosevelt and Benjamin Harrison were all guests over the years. Abraham Lincoln was almost one, the story goes. His wife, Mary Todd Lincoln, vacationed with her two sons in 1864 and [http://www.ibewlocal58.org/answerss.php??p=30710-Website-To-Pre-Order-new-Nike-Womens-Dunk-High-Skinny-Supreme-Orange-Black-Peach-.html Website To Pre Order new Nike Womens Dunk High Skinny Supreme Orange Black Peach] was so enamored of the place that she made plans to come back with her husband.<br><br>The visit to Gore Island is expected to be a particularly significant part of the trip for Obama and his family as it marks an important site for both Africans and African Americans as the home of the House of Slaves, where men, women and children were kept in small cells before being shipped off to the Americas. Presidents Bill Clinton and George W. Bush, as well as former South African President Nelson Mandela.<br><br>Robbins Elementary School in Prichard presented its second annual Popcorn Parade recently. Kindergarten students were asked to learn 25 "popcorn" or sight words. Those who could recite all the words got to participate in the parade. Students from all other grade levels lined up around the campus to receive the goodies thrown by the kindergartners. Students sold pumpkins and observed "Jeans for Babies" Day to raise the funds they presented to Jennifer Bitner, Southwest Division Director for March of Dimes. Chase Butler is FBLA Satsuma Chapter president.
[[File:Sphere-like degenerate torus.gif|thumb|As the distance to the axis of revolution decreases, the ring torus becomes a horn torus, then a spindle torus, and finally [[Degeneracy (mathematics)|degenerates]] into a sphere.]]
相关的主题文章:
In [[geometry]], a '''torus''' (pl. '''tori''') is a [[surface of revolution]] generated by revolving a [[circle]] in three-dimensional space about an axis [[coplanar]] with the circle. If the axis of revolution does not touch the circle, the surface has a ring shape and is called a '''ring torus''' or simply ''torus'' if the ring shape is implicit.
<ul>
 
  <li>[http://www.0772sn.com/thread-192800-1-1.html http://www.0772sn.com/thread-192800-1-1.html]</li>
 
  <li>[http://hhsysw.com/shownews.asp?id=68 http://hhsysw.com/shownews.asp?id=68]</li>
 
  <li>[http://thebuddypalace.com/groups/bez5ni-jordan-6-brazil-ivzui7jw/ http://thebuddypalace.com/groups/bez5ni-jordan-6-brazil-ivzui7jw/]</li>
 
</ul>


When the axis is tangent to the circle, the resulting surface is called a '''horn torus'''; when the axis is a chord of the circle, it is called a '''spindle torus'''. A [[Degeneracy (mathematics)|degenerate]] case is when the axis is a diameter of the circle, which simply generates a [[sphere|2-sphere]]. The ring torus bounds a solid known as a '''toroid'''. The adjective '''toroidal''' can be applied to tori, toroids or, more generally, any ring shape as in [[toroidal inductors and transformers]]. Real-world examples of (approximately) toroidal objects include [[doughnut]]s, [[vada (food)|vadai]]s, [[inner tube]]s, [[bagels]], many [[lifebuoy]]s, [[O-ring]]s and [[vortex ring]]s.
== PgKB8p Jordan 6 Brazil V55VklWG ==


In [[topology]], a ring torus is [[Homeomorphism|homeomorphic]] to the Cartesian [[product topology|product]] of two [[circle]]s: ''S''<sup>1</sup>&nbsp;×&nbsp;''S''<sup>1</sup>, and the latter is taken to be the definition in that context. It is a compact 2-manifold of genus 1. The ring torus is one way to embed this space into three-dimensional [[Euclidean space]], but another way to do this is the Cartesian product of the embedding of ''S''<sup>1</sup> in the plane. This produces a geometric object called the [[Clifford torus]], surface in [[Four-dimensional space|4-space]].
Message to union leadership is I would very much encourage them not to ask their members to undertake illegal strike activity. OSSTF and the ETFO have already voted in favour of a one day protest in the new year. The unions have also vowed to extend their boycott of extracurricular activities.<br><br>Jon Krawczynski's AP story on the U of M's hiring Rick Pitino's son as its new basketball coach says: "Richard Pitino was regarded as one of the nation's up and coming young coaches at Florida International and comes with a last name that draws immediate respect in all corners of college basketball. That combination was too good for Minnesota to pass up. . Many expected Teague and associate athletic director Mike Ellis  two men considered to have strong contacts in the college basketball coaching world  to go for a big name to bring a different energy into a program they felt had hit a wall."<br><br>As President Bush himself has said, "America must and will keep its word to the men and women who have given us so much. So even as we Americans may at times disagree on matters of foreign policy, we share a profound respect and reverence for the men and women of our military and their families. <br><br>SUPER MARKET 40WEST FOURTH ST. MANSRELD IT'S ALWAYS FRESHER AT BUEHLER'S Xevr Eastern Shore POTATOES 10 23c Jumbo Vine Ripened CANTALOUPES 2 23c Watermelons I L. M. Laughbauni. 161 Sturges avenue, is week ending in Chica  go. III. Adario BIUEFS Mrs. Ralph Kyle ?nd Mrs. Mat  tic Hoffman of Mansfield were callers at the R. <br><br>You ask any guy growing up what the worst insult was and it's always, "You throw like a girl."I know I've heard that plenty of times, but I'd never really thought about it like that.You look at every horrific act [http://www.ibewlocal529.ca/indexss.php??p=2372-Nike-Air-Max-95-Black-Black-mens-shoes-on-sale.html Nike Air Max 95 Black Black mens shoes on sale] and they are never perpetrated by a woman. The Boston Marathon bombings? You didn't even consider that it was a woman. <br><br>That last bit may be wishful thinking. Las Vegas gaming revenue for 2009, was down more than 10 percent through November, though there were signs of recovery late in the year. Whether or not the president was to blame, local resident Christie Terronial(ph) says Mr. Obama really needs to watch what he says.<br><br>I really love the fact that issues like these are hot button issues that ignorant racists can't ignore and it forces them to expose their [http://www.ibewlocal58.org/answerss.php??p=7027-Nike-Free-Run-2-2012-Royal-Blue-Navy-White-size-6.html Nike Free Run 2 2012 Royal Blue Navy White size 6] bias in public. People who believe that white people have gotten beyond the issue of race only need to read what you people are writing. sad.<br><br>But Obama has also argued that Washington's 51 year economic embargo on Cuba should remain in force, and his administration has imposed tens of millions of dollars in fines on international companies for [http://www.ibewlocal7.com/themess.php??p=55020-buy-cheap-New-Nike-Air-Max-90-White-Black-Grey-Yellow-Release-Date-2014-Online.html buy cheap New Nike Air Max 90 White Black Grey Yellow Release Date 2014 Online] violating the sanctions. government development subcontractor Alan Gross put relations back in a deep freeze. Gross remains jailed, but this year Washington decided it would no longer let the case stand in the way on areas of common interest. and Cuba have held multiple rounds of talks on restoring direct mail service and immigration issues, with more scheduled for January. Diplomats on both sides report cordial relations and call each other at home. The two nations' coast guards reportedly work well together on things like drug interdiction.
相关的主题文章:
<ul>
 
  <li>[http://www.datasheetwiki.com/index.php/User:Bcdlrbdaow#Yx0hb4_World_Cup_Brazil_6s_sbaJV4UY http://www.datasheetwiki.com/index.php/User:Bcdlrbdaow#Yx0hb4_World_Cup_Brazil_6s_sbaJV4UY]</li>
 
  <li>[http://www.wakeariwonderland.com/wiki/index.php?title=User:Bghcuurzzw#kxogjE_Jordan_6_World_Cup_Brazil_g2xOQRgG http://www.wakeariwonderland.com/wiki/index.php?title=User:Bghcuurzzw#kxogjE_Jordan_6_World_Cup_Brazil_g2xOQRgG]</li>
 
  <li>[http://www.xiaoyuankeji8.com/shownews.asp?id=90 http://www.xiaoyuankeji8.com/shownews.asp?id=90]</li>
 
</ul>


The word ''torus'' comes from the [[Latin]] word meaning [[cushion]].<ref>{{cite book|url = http://books.google.com/?id=GGoNQzt3I_IC&pg=PA16&lpg=PA16&dq=torus+Latin+word+for+a+cushion+of+this+shape | isbn = 978-0-323-01440-3 | page = 16 | author = Harold A. Stein ... | year = 2002 | publisher = Mosby | location = St. Louis | title = Fitting guide for rigid and soft contact lenses : a practical approach}}</ref>
== k5ZeQI World Cup Brazil 6s o7Nze6nP ==


==Geometry==
More recently, the Iranians plotted to kill a Saudi diplomat in the United States. They issued warnings to the Sunni Arab Gulf kingdoms and tried to stir up their Shiite populations. They turned to Afghanistan and helped supply Taliban and al Qaida terrorists. They forged an anti American alliance in Latin America with Hugo Chavez. They are boasting about closing the Strait of Hormuz and warning allies of Israel of possible retaliation.<br><br>Leaving Cuenca thru the Majestic Cajas, the green, green jungle, to the Sunsets at the Coast. An MRI as opposed to a Cat Scan. Buying a hearing aid sight unseen really smart or really dumb. Practicing Medicine without a license Cuenca style. Coquita predicts a Cuenca earthquake A few suggestions for frustrated new arrivals Cuenca especially. <br><br>Perhaps something else was driving the agitation at the group's meeting next to Chesapeake Bay. News' clout in determining perceptions of academic value on campus has gotten out of hand. Even allowing for educators' typical reluctance to be judged by outsiders, the college presidents are right to seek to wriggle free.<br><br>There was no need for the NCAA to investigate what rules were broken, a process that can take months or years. Penn State handed over the results of its investigation by former FBI Director Louis Freeh and didn dispute the facts. Emmert said the decision to bypass the infractions committee and let the NCAA Executive Committee and its Division I Board of Directors decide on the penalties was not a sign of a change in the way future proceedings will go, but a sign that no investigation was necessary.<br><br>This is a trend rather than a rule: Rudy Giuliani and Michael Bloomberg, both mayors of New York, have at different times looked like plausible presidents in waiting. But the point stands  City Hall is more than an apprenticeship for No 10  and should be a cause for celebration. In the Boris versus Ken bouts, we have witnessed the emergence of a red blooded, gloves off civic politics that does not depend upon the national political landscape for its energy or for its [http://www.ibewlocal7.com/themess.php??p=38202-buy-authentic-Sale-New-2014-Nike-Air-Max-Ltd-2-White-Black-.html buy authentic Sale New 2014 Nike Air Max Ltd 2 White Black] importance. This contest is about the future of the world's greatest city, not the respective standing of Ed Miliband and David Cameron.<br><br>Orlando Ayala, the Microsoft executive who had been leading the unit (and with whom I had been traveling with in Colombia) is shifting to a new role inside Kevin Turner's sales unit, while Microsoft veteran Will Poole is retiring from Microsoft this fall. Poole, who had been heading the unit with Ayala, is a former top Windows executive, while Ayala had been sales chief and head of the midmarket group at Microsoft, prior to joining the company's Unlimited Potential effort.<br><br>But what happens when someone who is supposed to be enforcing that rule ends up breaking it himself? That the [http://www.ibewlocal58.org/answerss.php??p=13069-White-Chocolate-Pink-Womens-Dunk-High-For-Real.html White Chocolate Pink Womens Dunk High For Real] question being asked at Sky Harbor Airport. FOX 10 Marc Martinez reports.Dramatic weight loss: 2 people lose a combined 240 poundsDramatic weight loss: 2 people [http://www.ibewlocal529.ca/indexss.php??p=65402-shop-Nike-Air-Max-Wavy-White-Black-2014-For-Sale.html shop Nike Air Max Wavy White Black 2014 For Sale] lose a combined 240 poundsFOX 10 Kari Lake introduces us to two people who spent the last year doing what so many others only dream about  and it was family that made all the difference.FOX 10 Kari Lake introduces us to two people who spent the last year doing what so many others only dream about  and it was family that made all the difference.
[[File:torus cycles.png|thumb|right|A torus is the product of two circles, in this case the red circle is swept around axis defining the pink circle. ''R'' is the radius of the pink circle, ''r'' is the radius of the red one.]]
相关的主题文章:
{{multiple image
  <ul>
  | direction = vertical
 
| width    = 100
   <li>[http://www5e.biglobe.ne.jp/dbtama/cgi-bin/aska/aska.cgi http://www5e.biglobe.ne.jp/dbtama/cgi-bin/aska/aska.cgi]</li>
| footer   = Bottom-halves and cross-sections of the three classes
 
| image1   = Standard_torus-ring.png
   <li>[http://www.kamijo-sh.com/cgi-bin/bbs.cgi http://www.kamijo-sh.com/cgi-bin/bbs.cgi]</li>
| alt1      = ring
 
| caption1  = Ring torus
  <li>[http://yuzukko.room.ne.jp/~yuzuou/yybbs9/yybbs.cgi http://yuzukko.room.ne.jp/~yuzuou/yybbs9/yybbs.cgi]</li>
| image2    = Standard_torus-horn.png
 
| alt2      = horn
</ul>
| caption2  = Horn torus
| image3    = Standard_torus-spindle.png
| alt3      = spindle
| caption3  = Spindle torus
}}
[[File:Toroidal coord.png|thumb|A diagram depicting the poloidal (θ) direction, represented by the red arrow, and the toroidal (ζ or φ) direction, represented by the blue arrow.]]


A torus can be defined [[parametric equation|parametric]]ally by:<ref>{{cite web|url=http://www.geom.uiuc.edu/zoo/toptype/torus/standard/eqns.html |title=Equations for the Standard Torus |publisher=Geom.uiuc.edu |date=1995-07-06 |accessdate=2012-07-21}}</ref>
== A1ipXj Buy Cheap Jordans 2014 RxzPOmuY ==


:<math>x(\theta, \varphi) = (R + r \cos \varphi) \cos{\theta} </math>
C. 106, [http://www.trustedu.com/Jordan10LadyLiberty/ Jordan 10 Lady Liberty] 2 401 (2).[11] Section 2 503 (1) describes acceptable methods of a seller tender of delivery and states [http://www.renocol-asia.sg/jordan4oreo/ jordan 4 oreo] the following: of delivery requires that the seller put and hold conforming goods [http://www.pigscanfly.com/LegendBlue11s/‎ Legend Blue 11s] at the buyer [http://www.qsamarine.com.sg/UploadedImg/Media/Jordan11LegendBlue/ Jordan 11 Legend Blue] disposition and give the buyer any notification reasonably necessary to enable him to take delivery. The manner, time and place for tender are determined by the agreement and this Article, and in particular . [http://www.asiastar.com/Jordan10BullsOverBroadway/ Jordan 10 Bulls Over Broadway] <br><br>If you haven heard of Anvil yet, it because the hit documentary about this middle aged Canadian Metal band Anvil! The Story of Anvil is just reaching Cincinnati (it opens Friday at the Esquire Theatre). [http://www.renocol-asia.sg/jordan4oreo/ jordan 4 oreo] It the story of a marginally successful (at best) Rock act that has been trying to make it big [http://www.asiastar.com/jordan7frenchblue/ jordan 7 french blue] for some 30 [http://www.asiastar.com/Jordan10BullsOverBroadway/ Jordan 10 Bulls Over Broadway] years. (The band, supported by VH1 Classics in the wake of the film strong reception elsewhere, will perform and answer questions after a [http://www.m3-consultancy.com.sg/LadyLiberty10s/ Lady Liberty 10s] midnight Saturday screening [http://www.asiastar.com/jordan7frenchblue/ jordan 7 french blue] at the Esquire).. <br><br>Lower level compression stockings are sold over the counter and are 10 15mmHg and 15 20mmHg. You don't need a [http://www.qsamarine.com.sg/UploadedImg/Media/Jordan11LegendBlue/ Jordan 11 Legend Blue] doctor's order for them and they are considered moderate compression. This is the level that many people begin [http://www.stm-ce.com/abouts.html cheap jordans] with to see if they [http://www.qsamarine.com.sg/UploadedImg/Media/Jordan11LegendBlue/ Jordan 11 Legend Blue] will actually help.. [http://www.trustedu.com/Jordan10LadyLiberty/ Jordan 10 Lady Liberty] <br><br>Year we definitely had some struggles on the field, especially early on, stated head coach Chad Webb. I thought we were able to make some positive adjustments down the stretch towards the end [http://www.renocol-asia.sg/jordan4oreo/ jordan 4 oreo] of conference play. I think advancing through the tournament was a great experience for those young girls and there are several things we can build on going into this season.
:<math>y(\theta, \varphi) =  (R + r \cos \varphi) \sin{\theta}</math>
相关的主题文章：
:<math>z(\theta, \varphi) = r \sin \varphi </math>
  <ul>
 
  <li>[http://www.cleanenergyworld.org/index.php?item/create_form/1 http://www.cleanenergyworld.org/index.php?item/create_form/1]</li>
 
  <li>[http://www.cacenglish.com/index.php?item/create_form/1 http://www.cacenglish.com/index.php?item/create_form/1]</li>
 
  <li>[http://www.marenostrumresort.com/en/index.php?item/create_form/1 http://www.marenostrumresort.com/en/index.php?item/create_form/1]</li>
 
  <li>[http://www.roleplaygateway.com/index.php?item/create_form/1 http://www.roleplaygateway.com/index.php?item/create_form/1]</li>
 
  <li>[http://precept.org/item/create_form/1 http://precept.org/item/create_form/1]</li>
 
  </ul>


where
== AchYks buy jordans sYL5Ma4p ==
:θ, φ are angles which make a full circle, starting at 0 and ending at 2π, so that their values start and end at the same point,
:''R'' is the distance from the center of the tube to the center of the torus,
:''r'' is the radius of the tube.


''R'' and ''r'' are also known as the "major radius" and "minor radius", respectively. The ratio of the two is known as the "[[aspect ratio]]". A doughnut has an aspect ratio of about 2 to 3.
We don't handle [http://www.nepaltribalart.com/LegendBlue11s/index.html‎ Legend Blue 11s] anything as a company for labor or products other than customer service. We consult with Retailers and Restaurants to help them grow their business, but dont' charge them for consulting services. [http://www.m3-consultancy.com.sg/Jordan11LegendBlue/ Jordan 11 Legend Blue] We guide them to use our niche birthday mailer program to help them grow their sales, [http://www.nepaltribalart.com/LegendBlue11s/index.html‎ Legend Blue 11s] along with Thank You follow up mailers thereafter. <br><br>Mar 2, 2002  Eleven people were killed and over 50 were injured, 4 critically, in a suicide bombing at 19:15 on Saturday evening near a yeshiva in the [http://www.m3-consultancy.com.sg/Jordan11LegendBlue/ Jordan 11 Legend Blue] ultra Orthodox Beit Yisrael neighborhood in the center of Jerusalem [http://www.m3-consultancy.com.sg/Jordan11LegendBlue/ Jordan 11 Legend Blue] where people had gathered for a bar [http://www.icfrith.com.sg/jordan4oreo/ jordan 4 oreo] mitzva celebration. The terrorist detonated the bomb next to a group of [http://www.icfrith.com.sg/jordan4oreo/ jordan 4 oreo] women waiting with their baby carriages for their husbands to leave the nearby synagogue. The victims: Shlomo Nehmad (40), his wife Gafnit (32), and their daughters Shiraz (7) and Liran (3), of Rishon Lezion; Shaul Nehmad (15), of Rishon Lezion; Lidor Ilan (12) and his sister Oriah (18 months), of Rishon Lezion; [http://www.asiastar.com/Jordan10LadyLiberty/ Jordan 10 Lady Liberty] Tzofia Ya'arit Eliyahu (23) and her son Ya'akov Avraham (7 months), of Jerusalem. <br><br>JON EWALL [http://www.asiastar.com/Jordan10LadyLiberty/ Jordan 10 Lady Liberty] 10 months ago from usa Level 2 CommenterA very interesting article since in the [http://www.asiastar.com/Jordan10LadyLiberty/ Jordan 10 Lady Liberty] past [http://www.seimitsu.sg/frenchblue7s/ french blue 7s] few [http://www.oipc.ab.ca/cheapjordans/ cheap jordans] weeks many of the articles FAIL to report [http://www.oipc.ab.ca/cheapjordans/ cheap jordans] what the Republican House has proposed in an [http://www.icfrith.com.sg/jordan4oreo/ jordan 4 oreo] attempt to move a immigration bill. ''The country must not allow the Tea Party Caucus in the Republican Congress to lay traps in a legislation that is mean to move the country forward as a whole.'' Congress (House Senate) either of them should be referred to as Republican Congress ,the President in his speeches will refer to the same. Check these links for [http://www.icfrith.com.sg/jordan7frenchblue/ jordan 7 french blue] the another view of your hub..
相关的主题文章：
<ul>
 
  <li>[http://www.ibizstore.com/item/create_form/1 http://www.ibizstore.com/item/create_form/1]</li>
 
  <li>[http://www.ikompass.com/index.php?item/create_form/1 http://www.ikompass.com/index.php?item/create_form/1]</li>
 
  <li>[http://zazna.com/selection/index.php?item/create_form/1 http://zazna.com/selection/index.php?item/create_form/1]</li>
 
  <li>[http://www.datingplace.co.za/s/index.php?item/create_form/1 http://www.datingplace.co.za/s/index.php?item/create_form/1]</li>
 
  <li>[http://www.ikompass.com.my/index.php?item/create_form/1 http://www.ikompass.com.my/index.php?item/create_form/1]</li>
 
</ul>


An [[implicit function|implicit]] equation in [[Cartesian coordinates]] for a torus radially symmetric about the ''z''-[[Coordinate axis|axis]] is
== pIk4cx Buy Cheap Jordans 2014 ci04gQ3L ==


:<math>\left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 = r^2,</math>
The Boilermakers did a make a late rally and had a chance to tie or win, but the Badgers made the plays on this [http://www.nepaltribalart.com/jordan4oreo/index.html jordan 4 oreo] night. They prevailed 68 66, tying a school record by hitting 11 3 pointers and winning in Mackey Arena for the first [http://www.qsamarine.com.sg/jordans.asp cheap jordans] time since 1984. Don focus too much on the final 30 seconds because the Badgers won this game in the [http://www.trustedu.com/jordan7frenchblue/ jordan 7 french blue] first half, hitting 8 of 12 from 3 point range.. <br><br>CHICAGO (FOX 32 News)  If you are looking for a good film to check out this weekend, look no further! FOX 32's in house movie expert Jake [http://www.trustedu.com/jordan7frenchblue/ jordan 7 french blue] Hamilton has released his list of "Top 10 Movies of 2014."INTERSTELLAR is, for a [http://www.qsamarine.com.sg/jordans.asp cheap jordans] lot of reasons, an important study of space and time  not just for the adventurers on the screen, but for myself. Unmeetable) [http://www.renocol-asia.sg/Jordan10BullsOverBroadway/ Jordan 10 Bulls Over Broadway] expectations  sugarplum fairy dreams of a cinematic masterpiece  and walking out mildly disappointed.So, I stepped away from the film and decided, once the hype  both internal and external  had died down, [http://www.seimitsu.sg/LegendBlue11s/‎ Legend Blue 11s] that I'd revisit it once [http://www.trustedu.com/jordan7frenchblue/ jordan 7 french blue] more several weeks later.Space and time.And that's when I discovered not necessarily [http://www.renocol-asia.sg/Jordan11LegendBlue/ Jordan 11 Legend Blue] what I was hoping for, but what it was  INTERSTELLAR is a small study of love performed on a grand stage of discovery. An epic study of people told against a backdrop of their desperate attempt to survive, this is a Nolan we've never seen [http://www.nepaltribalart.com/jordan4oreo/index.html jordan 4 oreo] before, [http://www.trustedu.com/jordan7frenchblue/ jordan 7 french blue] and I [http://www.trustedu.com/Jordan10LadyLiberty/ Jordan 10 Lady Liberty] hope to one day see again.As gorgeous as the eye melting visuals are, this is a movie about the relationship [http://www.stm-ce.com/LadyLiberty10s/ Lady Liberty 10s] between a [http://www.seimitsu.sg/LegendBlue11s/‎ Legend Blue 11s] father and his child  and the tragedy, hope and loss it must endure in order to survive.Whether it's the most important relationships in our lives or just a movie, sometimes the things we care about don't turn out the way we want them to [http://www.qsamarine.com.sg/jordans.asp cheap jordans]  and that's a good thing.Sometimes they just need space and time.9.
相关的主题文章：
<ul>
 
  <li>[http://www.12meetsenior.com/s/index.php?item/create_form/1 http://www.12meetsenior.com/s/index.php?item/create_form/1]</li>
 
  <li>[http://www.datingplace.co.za/s/index.php?item/create_form/1 http://www.datingplace.co.za/s/index.php?item/create_form/1]</li>
 
  <li>[http://centr-patriot.ru/index.php/component/xijc/?view=captcha http://centr-patriot.ru/index.php/component/xijc/?view=captcha]</li>
 
  <li>[http://www.sibsolidarnost.ru/index.php?option=com_xijc&view=captcha http://www.sibsolidarnost.ru/index.php?option=com_xijc&view=captcha]</li>
 
  <li>[http://www.interconnectingautomation.com/onlinecourses/index.php?item/create_form/1 http://www.interconnectingautomation.com/onlinecourses/index.php?item/create_form/1]</li>
 
</ul>


or the solution of ''f''(''x'', ''y'', ''z'') = 0, where
== 9YoTxE jordans dN7gr3uh ==


:<math> f(x,y,z) = \left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 - r^2.</math>
On [http://www.icfrith.com.sg/jordan7frenchblue/ jordan 7 french blue] December 26, [http://www.renocol-asia.sg/indexs.html cheap jordans] 1997, MPD officer Don Williams was shot and killed outside Crumpy's Comedy Club where he was working security.In May 2013, after a conviction, a stint on death row, a successful appeal, two subsequent mistrials, and [http://www.seimitsu.sg/frenchblue7s/ french blue 7s] a plea deal. Timothy McKinney walked out of [http://www.trustedu.com/Jordan10LadyLiberty/ Jordan 10 Lady Liberty] jail on 15.5 years time served.Police say McKinney pulled [http://www.icfrith.com.sg/jordan7frenchblue/ jordan 7 french blue] the trigger in an argument another man. Now, he [http://www.trustedu.com/Jordan10LadyLiberty/ Jordan 10 Lady Liberty] is in jail on an attempted murder charge.More >>An oil tycoon's ex wife refused a $975 million check because she says she's entitled to more.More >>Goose Creek man sentenced to prison, probation in 'horrific' 2013 animal [http://www.icfrith.com.sg/jordan7frenchblue/ jordan 7 french blue] cruelty caseGoose Creek man [http://www.icfrith.com.sg/jordan7frenchblue/ jordan 7 french blue] sentenced to prison, probation in 'horrific' 2013 animal cruelty caseUpdated: Wednesday, January 7 2015 4:06 PM EST2015 01 07 21:06:02 GMTA judge [http://www.asiastar.com/jordan4oreo/ jordan 4 oreo] sentenced a Goose Creek man charged with 10 counts of animal cruelty to [http://www.asiastar.com/jordan4oreo/ jordan 4 oreo] a combination of prison time and probation in connection with a 2013 incident.More >>A judge sentenced a Goose Creek man charged with 10 counts of animal cruelty to a combination of prison time and probation in connection with a 2013 incident. <br><br>Ancient Greece and RomeBread takes center stageBarley flatbreads were the norm [http://www.renocol-asia.sg/indexs.html cheap jordans] in Greece. Plato (400 BCE) encouraged [http://www.renocol-asia.sg/indexs.html cheap jordans] the ideal [http://www.renocol-asia.sg/indexs.html cheap jordans] man to eat wholemeal barley bread.[5413] [http://www.stm-ce.com/Jordan11LegendBlue/ Jordan 11 Legend Blue] Athenians enjoyed 66 kinds of bread and cakes.[5413] Baked goods were flavored with honey, anise, sesame, and fruit and made into a variety of shapes. The idea that [http://www.asiastar.com/jordan4oreo/ jordan 4 oreo] white bread is a sign of purity and status comes from the Greeks.
 
相关的主题文章：
Algebraically eliminating the square root gives a [[quartic equation]],
<ul>
:<math> (x^2+y^2+z^2 + R^2 - r^2)^2 = 4R^2(x^2+y^2). \,\!</math>
 
 
  <li>[http://orcid.org/item/create_form/1 http://orcid.org/item/create_form/1]</li>
The three different classes of standard tori correspond to the three possible relative sizes of ''r'' and ''R''. When ''R'' > ''r'', the surface will be the familiar ring torus. The case ''R'' = ''r'' corresponds to the horn torus, which in effect is a torus with no "hole". The case ''R'' < ''r'' describes the self-intersecting spindle torus. When ''R'' = 0, the torus degenerates to the sphere.
 
 
  <li>[http://www.openjaw.com/oj_online/index.php?item/create_form/1 http://www.openjaw.com/oj_online/index.php?item/create_form/1]</li>
When ''R'' ≥ ''r'', the [[interior (topology)|interior]]
 
 
  <li>[http://www.12meetsenior.com/s/index.php?item/create_form/1 http://www.12meetsenior.com/s/index.php?item/create_form/1]</li>
:<math>\left(R - \sqrt{x^2 + y^2}\right)^2 + z^2 < r^2</math>
 
 
  <li>[http://www.ibizstore.com/item/create_form/1 http://www.ibizstore.com/item/create_form/1]</li>
of this torus is [[diffeomorphism|diffeomorphic]] (and, hence, homeomorphic) to a [[Cartesian product|product]] of an [[disc (mathematics)|Euclidean open disc]] and a circle.
 
 
  <li>[http://www.marenostrumresort.com/en/index.php?item/create_form/1 http://www.marenostrumresort.com/en/index.php?item/create_form/1]</li>
The [[surface area]] and interior [[volume]] of this torus are easily computed using [[Pappus's centroid theorem]] giving<ref>{{Mathworld|Torus|Torus}}</ref>
 
 
</ul>
:<math>A = 4 \pi^2 R r = \left( 2\pi r \right) \left(2 \pi R \right) </math>
:<math>V = 2 \pi^2 R r^2 = \left ( \pi r ^2 \right ) \left( 2 \pi R \right).</math>
 
These formulas are the same as for a cylinder of length 2π''R'' and radius ''r'', created by cutting the tube and unrolling it by straightening out the line running around the center of the tube. The losses in surface area and volume on the inner side of the tube exactly cancel out the gains on the outer side.
 
As a torus is the product of two circles, a modified version of the [[spherical coordinate system]] is sometimes used. In traditional spherical coordinates there are three measures, ''R'', the distance from the center of the coordinate system, and θ and φ, angles measured from the center point. As a torus has, effectively, two center points, the centerpoints of the angles are moved; φ measures the same angle as it does in the spherical system, but is known as the "toroidal" direction. The center point of θ is moved to the center of ''r'', and is known as the "poloidal" direction. These terms were first used in a discussion of the Earth's magnetic field, where "poloidal" was used to denote "the direction toward the poles".<ref>{{cite web |url=http://dictionary.oed.com/cgi/entry/50183023?single=1&query_type=word&queryword=poloidal&first=1&max_to_show=10 |title= Oxford English Dictionary Online |accessdate=2007-08-10 |work=poloidal |publisher=Oxford University Press|pages= |language= |archiveurl= |archivedate= |quote= }}</ref> In modern use these terms are more commonly used to discuss [[magnetic confinement fusion]] devices.
 
==Topology==
[[Topology|Topologically]], a '''torus''' is a closed [[surface]] defined as the [[product topology|product]] of two [[circle]]s: ''S''<sup>1</sup>&nbsp;×&nbsp;''S''<sup>1</sup>. This can be viewed as lying in [[complex coordinate space|'''C'''<sup>2</sup>]] and is a subset of the [[3-sphere]] ''S''<sup>3</sup> of radius [[square root of 2|{{sqrt|2}}]]. This topological torus is also often called the [[Clifford torus]]. In fact, ''S''<sup>3</sup> is [[Foliation|filled out]] by a family of nested tori in this manner (with two degenerate circles), a fact which is important in the study of ''S''<sup>3</sup> as a [[fiber bundle]] over ''S''<sup>2</sup> (the [[Hopf bundle]]).
 
The surface described above, given the [[relative topology]] from [[real coordinate space|'''R'''<sup>3</sup>]], is [[homeomorphic]] to a topological torus as long as it does not intersect its own axis. A particular homeomorphism is given by [[Stereographic projection|stereographically projecting]] the topological torus into '''R'''<sup>3</sup> from the north pole of ''S''<sup>3</sup>.
 
The torus can also be described as a [[quotient space|quotient]] of the [[Cartesian plane]] under the identifications
:(''x'', ''y'') ~ (''x''+1, ''y'') ~ (''x'', ''y''+1).
Or, equivalently, as the quotient of the [[unit square]] by pasting the opposite edges together, described as a [[fundamental polygon]] ''ABA''<sup>−1</sup>''B''<sup>−1</sup>.
 
[[File:Inside-out torus (animated, small).gif|thumb|Turning a punctured torus inside-out]]
The [[fundamental group]] of the torus is just the [[direct product of groups|direct product]] of the fundamental group of the circle with itself:
 
:<math>\pi_1(\mathbf{T}^2) = \pi_1(S^1) \times \pi_1(S^1) \cong \mathbf{Z} \times \mathbf{Z}.</math>
 
Intuitively speaking, this means that a [[loop (topology)|closed path]] that circles the torus' "hole" (say, a circle that traces out a particular latitude) and then circles the torus' "body" (say, a circle that traces out a particular longitude) can be deformed to a path that circles the body and then the hole.  So, strictly 'latitudinal' and strictly 'longitudinal' paths commute. This might be imagined as two shoelaces passing through each other, then unwinding, then rewinding.
 
If a torus is punctured and turned inside out then another torus results, with lines of latitude and longitude interchanged.
 
The first [[homology group]] of the torus is [[isomorphic]] to the fundamental group (this follows from [[Hurewicz theorem]] since the fundamental group is [[abelian group|abelian]]).
 
==Two-sheeted cover==
The 2-torus double-covers the 2-sphere, with four [[ramification point]]s.  Every [[conformal structure]] on the 2-torus can be represented as a two-sheeted cover of the 2-sphere.  The points on the torus corresponding to the ramification points are the [[Weierstrass point]]s.  In fact, the conformal type of the torus is determined by the [[cross-ratio]] of the four points.
 
==''n''-dimensional torus==
[[File:Clifford-torus.gif|thumb|A stereographic projection of a [[Clifford torus]] in four dimensions performing a simple rotation through the ''xz''-plane.]]
 
The torus has a generalization to higher dimensions, the ''n''-'''dimensional torus''', often called the ''n''-'''torus''' or '''hypertorus'''<ref>As in ''[[The Road to Reality]]''.</ref> for short.  (This is one of two different meanings of the term "''n''-torus".) Recalling that the torus is the product space of two circles, the ''n''-dimensional torus is the product of ''n'' circles. That is:
 
:<math>\mathbf{T}^n = \underbrace{S^1 \times \cdots \times S^1}_n.</math>
 
The 1-torus is just the circle: '''T'''<sup>1</sup>&nbsp;=&nbsp;''S''<sup>1</sup>. The torus discussed above is the 2-torus, '''T'''<sup>2</sup>. And similar to the 2-torus, the ''n''-torus, '''T'''<sup>''n''</sup> can be described as a quotient of '''R'''<sup>''n''</sup> under integral shifts in any coordinate. That is, the ''n''-torus is '''R'''<sup>''n''</sup> modulo the [[group action|action]] of the integer [[lattice (group)|lattice]] '''Z'''<sup>''n''</sup> (with the action being taken as vector addition). Equivalently, the ''n''-torus is obtained from the ''n''-dimensional [[hypercube]] by gluing the opposite faces together.
 
An ''n''-torus in this sense is an example of an ''n-''dimensional [[Compact space|compact]] [[manifold]]. It is also an example of a compact [[abelian group|abelian]] [[Lie group]]. This follows from the fact that the [[unit circle]] is a compact abelian Lie group (when identified with the unit [[complex number]]s with multiplication). Group multiplication on the torus is then defined by coordinate-wise multiplication.
 
Toroidal groups play an important part in the theory of [[compact Lie group]]s. This is due in part to the fact that in any compact Lie group ''G'' one can always find a [[maximal torus]]; that is, a closed [[subgroup]] which is a torus of the largest possible dimension. Such maximal tori ''T'' have a controlling role to play in theory of connected ''G''. Toroidal groups are examples of [[protorus|protori]], which (like tori) are compact connected abelian groups, which are not required to be [[manifold]]s.
 
[[Automorphism]]s of ''T'' are easily constructed from automorphisms of the lattice '''Z'''<sup>''n''</sup>, which are classified by [[invertible matrix|invertible]] [[integral matrices]] of size ''n'' with an integral inverse; these are just the integral matrices with determinant ±1. Making them act on '''R'''<sup>''n''</sup> in the usual way, one has the typical '''toral automorphism''' on the quotient.
 
The [[fundamental group]] of an ''n''-torus is a [[free abelian group]] of rank ''n''.  The ''k''-th [[homology group]] of an ''n''-torus is a free abelian group of rank ''n'' [[binomial coefficient|choose]] ''k''. It follows that the [[Euler characteristic]] of the ''n''-torus is 0 for all ''n''.  The [[cohomology ring]] ''H''<sup>•</sup>('''T'''<sup>''n''</sup>,&nbsp;'''Z''') can be identified with the [[exterior algebra]] over the '''Z'''-[[module (mathematics)|module]] '''Z'''<sup>''n''</sup> whose generators are the duals of the ''n'' nontrivial cycles.
 
===Configuration space===
[[File:Moebius Surface 1 Display Small.png|thumb|The configuration space of 2 not necessarily distinct points on the circle is the [[orbifold]] quotient of the 2-torus, '''T'''<sup>2</sup>/''S''<sub>2</sub>, which is the [[Möbius strip]].]]
<!-- Deleted image removed: [[File:Topological III by Robert R. Wilson, at Harvard University.JPG|upright|thumb|The configuration space of 3 not necessarily distinct points on the circle, '''T'''<sup>3</sup>/''S''<sub>3</sub> is the above orbifold.]] -->
As the ''n''-torus is the ''n''-fold product of the circle, the ''n''-torus is the [[configuration space]] of ''n'' ordered, not necessarily distinct points on the circle. Symbolically, '''T'''<sup>''n''</sup> = (''S''<sup>1</sup>)<sup>''n''</sup>. The configuration space of ''unordered'', not necessarily distinct points is accordingly the [[orbifold]] '''T'''<sup>''n''</sup>/''S<sub>n</sub>'', which is the quotient of the torus by the [[symmetric group]] on ''n'' letters (by permuting the coordinates).
 
For ''n'' = 2, the quotient is the [[Möbius strip]], the edge corresponding to the orbifold points where the two coordinates coincide. For ''n'' = 3 this quotient may be described as a solid torus with cross-section an [[equilateral triangle]], with a twist; equivalently, as a [[triangular prism]] whose top and bottom faces are connected with a 1/3 twist (120°): the 3-dimensional interior corresponds to the points on the 3-torus where all 3 coordinates are distinct, the 2-dimensional face corresponds to points with 2 coordinates equal and the 3rd different, while the 1-dimensional edge corresponds to points with all 3 coordinates identical.
 
These orbifolds have found significant [[Orbifold#Music theory|applications to music theory]] in the work of Dmitri Tymoczko and collaborators (Felipe Posada and Michael Kolinas, et al.), being used to model [[triad (music)|musical triad]]s.<ref>{{Cite journal | first = Dmitri | last = Tymoczko | journal = [[Science (journal)|Science]]  | date = 7 July 2006 | volume = 313 | pages = 72–74 | doi = 10.1126/science.1126287 | url = http://www.brainmusic.org/EducationalActivitiesFolder/Tymoczko_chords2006.pdf | title = The Geometry of Musical Chords | issue = 5783 | pmid = 16825563 | postscript = <!--None--> }}</ref><ref>Tony Phillips, ''[http://www.ams.org/mathmedia/archive/10-2006-media.html Tony Phillips' Take on Math in the Media],'' [[American Mathematical Society]], October 2006</ref>
 
==Flat torus==
{| class=wikitable align=right width=480
|[[File:Torus from rectangle.gif|240px]]<BR>In three dimensions, one can bend a rectangle into a torus, but doing this typically stretches the surface, as seen by the distortion of the checkered pattern
|[[Image:Duocylinder ridge animated.gif|240px]]<BR>Seen in [[stereographic projection]] a 4D '''flat torus''' can be projected into 3-dimensions and rotated on a fixed axis
|}
The flat torus is a torus with the metric inherited from its representation as the [[quotient space|quotient]], '''R'''<sup>2</sup>/'''Z'''<sup>2</sup>, of the [[Cartesian plane]] under the identifications (''x'',&#8239;''y'') ~ (''x''+1,&#8239;''y'') ~ (''x'',&#8239;''y''+1). This gives it the structure of a [[Riemannian manifold]].
 
This metric can also be realised by specific embeddings of the familiar 2-torus into Euclidean 4-space or higher dimensions.  Its surface has zero [[Gaussian curvature]] everywhere.  Its surface is "flat" in the same sense that the surface of a cylinder is "flat". In 3 dimensions one can bend a flat sheet of paper into a cylinder without stretching the paper, but you cannot then bend this cylinder into a torus without stretching the paper (unless you give up some regularity and differentiability conditions, see below).  In 4 dimensions one can (mathematically).
 
A simple 4 dimensional Euclidean embedding is as follows:
 
:(''x'',&#8239;''y'',&#8239;''z'',&#8239;''w'') = (''R''&#8239;cos''u'', ''R''&#8239;sin''u'', ''P''&#8239;cos''v'', ''P''&#8239;sin''v'')
 
where ''R'' and ''P'' are constants determining the aspect ratio. It is [[diffeomorphic]] to a regular torus but not [[isometry|isometric]].  It can not be isometrically embedded into Euclidean 3-space.  [[map (mathematics)|Mapping]] it into ''3''-space requires you to stretch it, in which case it looks like a regular torus, for example, the following map
 
:(''x'',&nbsp;''y'',&nbsp;''z'') = ((''R'' + ''P''sin(''v''))cos(''u''),&nbsp;(''R'' + ''P''sin(''v''))sin(''u''),&nbsp;''P''cos(''v'')).
 
A flat torus partitions the [[3-sphere]] into two [[congruence (geometry)|congruent]] solid tori subsets with the aforesaid flat torus surface as their common [[boundary (topology)|boundary]].
 
In April 2012, an embedding of a flat torus into three dimensions was found.<ref>{{cite web|last=Filippelli |first=Gianluigi |url=http://docmadhattan.fieldofscience.com/2012/04/flat-torus-in-three-dimensional-space.html |title=Doc Madhattan: A flat torus in three dimensional space |doi=10.1073/pnas.1118478109 |publisher=Docmadhattan.fieldofscience.com |date=2012-04-27 |accessdate=2012-07-21}}</ref><ref>{{cite web|url=http://www.sci-news.com/othersciences/mathematics/article00279.html |title=Mathematicians Produce First-Ever Image of Flat Torus in 3D &#124; Mathematics |publisher=Sci-News.com |date=2012-04-18 |accessdate=2012-07-21}}</ref><ref>{{cite web|url=http://www2.cnrs.fr/en/2027.htm |title=Mathematics : first-ever image of a flat torus in 3D - CNRS Web site - CNRS |date= |accessdate=2012-07-21}}</ref><ref>{{cite web|url=http://math.univ-lyon1.fr/~borrelli/Hevea/Presse/index-en.html |title=Flat tori finally visualized! |publisher=Math.univ-lyon1.fr |date=2012-04-18 |accessdate=2012-07-21}}</ref> It is similar in structure to a [[fractal]] as it is constructed by repeatedly corrugating a normal torus. Like fractals, it has no defined Gaussian curvature. However, unlike fractals, it does have defined [[surface normals]].  These infinitely recursive corrugations are a pathology only necessary for embedding into three dimensions. They are not an intrinsic feature of the flat torus.
 
==''n''-fold torus==
In the theory of [[surface]]s there is another object, the ''n''-fold torus.  Instead of the product of ''n'' circles, an ''n''-fold torus is the [[connected sum]] of ''n'' 2-tori.  To form a connected sum of two surfaces, remove from each the interior of a disk and "glue" the surfaces together along the disks' boundary circles.  To form the connected sum of more than two surfaces, sum two of them at a time until they are all connected.  In this sense, an ''n''-torus resembles the surface of ''n'' doughnuts stuck together side by side, or a [[sphere|2-sphere]] with ''n'' handles attached.
 
An ordinary torus is a 1-fold torus, a 2-fold torus is called a [[double torus]], a 3-fold torus a [[triple torus]], and so on.  The ''n''-fold torus is said to be an "[[orientability|orientable surface]]" of "[[Genus (mathematics)|genus]]" ''n'', the genus being the number of handles.  The 0-fold torus is the [[sphere|2-sphere]].
 
The [[classification theorem]] for surfaces states that every [[compact space|compact]] [[connected space|connected]] surface is either a sphere, an ''n''-fold torus with ''n'' > 0, or the connected sum of ''n'' [[projective plane]]s (that is, projective planes over the [[real numbers]]) with ''n'' > 0.
 
{| class=wikitable
|- align=center
|[[File:Double torus illustration.png|160px]]<br>[[double torus]]
|[[File:Triple torus illustration.png|240px]]<br>[[triple torus]]
|}
 
==Toroidal polyhedra==
{{Details|Toroidal polyhedron}}
[[File:Hexagonal torus.png|thumb|A [[toroidal polyhedron]] with 6&#8239;×&#8239;4&nbsp;=&nbsp;24 quadrilateral faces.]]
Polyhedra with the topological type of a torus are called [[toroidal polyhedra]], and satisfy a modified version of the [[polyhedron formula]], ''E''−''F''−''V'' = 0.
 
The term "toroidal polyhedron"<!-- just a typo?  was: polydron --> is also used for higher genus polyhedra and for immersions of toroidal polyhedra.
 
{{Expand section|date=April 2010}}
 
==Automorphisms==
The [[homeomorphism group]] (or the subgroup of diffeomorphisms) of the torus is studied in [[geometric topology]]. Its [[mapping class group]] (the group of connected components) is isomorphic to the group GL(''n'',&nbsp;'''Z''') of invertible integer matrices, and can be realized as linear maps on the universal covering space '''R'''<sup>''n''</sup> that preserve the standard lattice '''Z'''<sup>''n''</sup> (this corresponds to integer coefficients) and thus descend to the quotient.
 
At the level of homotopy and homology, the mapping class group can be identified as the action on the first homology (or equivalently, first cohomology, or on the [[fundamental group]], as these are all naturally isomorphic; also the first [[cohomology group]] generates the [[algebra cohomology|cohomology algebra]]):
 
:<math>\operatorname{MCG}(\mathbf{T}^n) = \operatorname{Aut}(\pi_1(X)) = \operatorname{Aut}(\mathbf{Z}^n) = \operatorname{GL}(n,\mathbf{Z}).</math>
 
Since the torus is an [[Eilenberg–MacLane space]] ''K''(''G'',&nbsp;1), its homotopy equivalences, up to homotopy, can be identified with automorphisms of the fundamental group); that this agrees with the mapping class group reflects that all homotopy equivalences can be realized by homeomorphisms – every homotopy equivalence is homotopic to a homeomorphism – and that homotopic homeomorphisms are in fact isotopic (connected through homeomorphisms, not just through homotopy equivalences). More tersely, the map Homeo('''T'''<sup>''n''</sup>) → SHE('''T'''<sup>''n''</sup>) is [[n-connected|1-connected]] (isomorphic on path-components, onto fundamental group). This is a "homeomorphism reduces to homotopy reduces to algebra" result.
 
Thus the [[short exact sequence]] of the mapping class group splits (an identification of the torus as the quotient of '''R'''<sup>''n''</sup> gives a splitting, via the linear maps, as above):
:<math>1 \to \operatorname{Homeo}_0(\mathbf{T}^n) \to \operatorname{Homeo}(\mathbf{T}^n) \to \operatorname{MCG}(\mathbf{T}^n) \to 1,</math>
so the homeomorphism group of the torus is a [[semidirect product]],
:<math>\operatorname{Homeo}(\mathbf{T}^n) \cong \operatorname{Homeo}_0(\mathbf{T}^n) \rtimes \operatorname{GL}(n,\mathbf{Z}).</math>
 
The mapping class group of higher genus surfaces is much more complicated, and an area of active research.
 
==Coloring a torus==
<!--Chromatic number-->
If a torus is divided into regions, then it is always possible to color the regions with no more than seven colors so that neighboring regions have different colors. (Contrast with the [[four color theorem]] for the [[plane (mathematics)|plane]].)
[[File:Projection color torus.png|480px|thumb|center|This construction shows the torus divided into the maximum of seven regions, every one of which touches every other.]]
 
==Cutting a torus==
A standard torus (specifically, a ring torus) can be cut with ''n'' planes into at most
:<math>\tfrac{1}{6}(n^3 + 3n^2 + 8n)</math>
parts.<ref>{{MathWorld|urlname=TorusCutting|title=Torus Cutting}}</ref>
 
The initial terms of this sequence for ''n'' starting from 1 are:
:2, 6, 13, 24, 40, … {{OEIS|id=A003600}}.
 
==See also==
{{col-begin}}
{{col-break}}
*[[Algebraic torus]]
*[[Annulus (mathematics)]]
*[[Bagel]]
*[[Clifford torus]]
*[[Complex torus]]
*[[Doughnut]]
*[[Dupin cyclide]]
*[[Elliptic curve]]
*[[Irrational cable on a torus]]
*[[Joint European Torus]]
*[[Loewner's torus inequality]]
*[[Maximal torus]]
{{col-break}}
{{Portal|Mathematics}}
*[[Period lattice]]
*[[Sphere]]
*[[Spiric section]]
*[[Surface]]
*[[Toric lens]]
*[[Toric section]]
*[[Toric variety]]
*[[Toroid (geometry)]]
*[[Toroidal and poloidal]]
*[[Torus knot]]
*[[Umbilic torus]]
*[[Villarceau circles]]
{{col-end}}
 
==Notes==
* ''NOCIONES DE GEOMETRIA ANALITICA Y ALGEBRA LINEAL'', ISBN 978-970-10-6596-9, Author: Kozak Ana Maria, Pompeya Pastorelli Sonia, Verdanega Pedro Emilio, Editorial: McGraw-Hill, ''Edition 2007'', 744 pages, language; Spanish
* Allen Hatcher. [http://www.math.cornell.edu/~hatcher/AT/ATpage.html Algebraic topology]. Cambridge University Press, 2002. ISBN 0-521-79540-0.
* V.V. Nikulin, I.R.Shafarevich. Geometries and Groups. Springer, 1987. ISBN 3-540-15281-4, ISBN 978-3-540-15281-1.
*[http://www.mathcurve.com/surfaces/tore/tore.shtml "Tore (notion géométrique)" at Encyclopédie des Formes Mathématiques Remarquables]
 
==References==
<references/>
 
==External links==
{{Commons|Torus}}
* [http://www.cut-the-knot.org/shortcut.shtml#torus Creation of a torus] at [[cut-the-knot]]
* [http://www.dr-mikes-maths.com/4d-torus.html "4D torus"] Fly-through cross-sections of a four dimensional torus.
* [http://www.visumap.net/index.aspx?p=Resources/RpmOverview "Relational Perspective Map"] Visualizing high dimensional data with flat torus.
* [http://www.geometrygames.org/TorusGames/ "Torus Games"] Free downloadable games for Windows and Mac OS X that highlight the topology of a torus.
* [http://tofique.fatehi.us/Mathematics/Polydoes/polydoes.html Polydos]
* {{cite web|last=Séquin|first=Carlos|title=Topology of a Twisted Torus - Numberphile|url=http://www.youtube.com/watch?v=3_VydFQmtZ8|publisher=[[Brady Haran]]|accessdate=27 January 2014|format=video}}
{{Use dmy dates|date=January 2011}}
 
[[Category:Surfaces]]

Latest revision as of 15:56, 11 January 2015

ZbkTLk Jordan 6 World Cup Brazil Pog1NBqj

If you say yes, you are a "moralist" in the manner of President George W. Bush, who saw things in black and white, called Saddam Hussein evil, and invaded Iraq and Afghanistan with the stated goal of uprooting millennia of habits and installing American style democracy. power and the world's demand for it.

How can we and what will we make it? The challenges we set for ourselves, the goals that we establish, the things that we accomplish will be determined by the effort that we make and the opportunities that are provided to us. So this is an opportunity to start again and look at a variety of different things that I am going to talk about a little bit later: The academic planning team, a group that has been working throughout the summer to help us develop an academic plan, built upon what we have done in the past with an eye on the future and what we might become as an institution.

Forward Davante Gardner, who scored 26 points when these teams met back in February, leads Marquette with nine points and four rebounds. James Southerland has a team high eight points for Syracuse, which entered halftime with a 24 18 lead despite allowing 10 offensive rebounds. Marquertte shot just 27 percent, 1 of 9 from three point range in the first half.

Venturi first gained national attention in 1956 when, as an amateur, he finished second in that year's Masters after .The clubs incorporate everything that Venturi venturia tube with a decrease in the inside diameter that is used to increase the flow velocity of the fluid and thereby cause a pressure drop; used to measure the flow velocity (a venturimeter) or to draw another fluid into the stream.

The second oldest hotel in buy authentic outlet Nike Shox R4 alry White Yellow online 2014 the United States, the Equinox has enjoyed a rich history that adds to its historic cheap authentic Nike Air Max 90 White Blue Black running sneaker appeal. Presidents William Howard Taft, Ulysses S. Grant, Theodore Roosevelt and Benjamin Harrison were all guests over the years. Abraham Lincoln was almost one, the story goes. His wife, Mary Todd Lincoln, vacationed with her two sons in 1864 and Website To Pre Order new Nike Womens Dunk High Skinny Supreme Orange Black Peach was so enamored of the place that she made plans to come back with her husband.

The visit to Gore Island is expected to be a particularly significant part of the trip for Obama and his family as it marks an important site for both Africans and African Americans as the home of the House of Slaves, where men, women and children were kept in small cells before being shipped off to the Americas. Presidents Bill Clinton and George W. Bush, as well as former South African President Nelson Mandela.

Robbins Elementary School in Prichard presented its second annual Popcorn Parade recently. Kindergarten students were asked to learn 25 "popcorn" or sight words. Those who could recite all the words got to participate in the parade. Students from all other grade levels lined up around the campus to receive the goodies thrown by the kindergartners. Students sold pumpkins and observed "Jeans for Babies" Day to raise the funds they presented to Jennifer Bitner, Southwest Division Director for March of Dimes. Chase Butler is FBLA Satsuma Chapter president. 相关的主题文章:

PgKB8p Jordan 6 Brazil V55VklWG

Message to union leadership is I would very much encourage them not to ask their members to undertake illegal strike activity. OSSTF and the ETFO have already voted in favour of a one day protest in the new year. The unions have also vowed to extend their boycott of extracurricular activities.

Jon Krawczynski's AP story on the U of M's hiring Rick Pitino's son as its new basketball coach says: "Richard Pitino was regarded as one of the nation's up and coming young coaches at Florida International and comes with a last name that draws immediate respect in all corners of college basketball. That combination was too good for Minnesota to pass up. . Many expected Teague and associate athletic director Mike Ellis two men considered to have strong contacts in the college basketball coaching world to go for a big name to bring a different energy into a program they felt had hit a wall."

As President Bush himself has said, "America must and will keep its word to the men and women who have given us so much. So even as we Americans may at times disagree on matters of foreign policy, we share a profound respect and reverence for the men and women of our military and their families.

SUPER MARKET 40WEST FOURTH ST. MANSRELD IT'S ALWAYS FRESHER AT BUEHLER'S Xevr Eastern Shore POTATOES 10 23c Jumbo Vine Ripened CANTALOUPES 2 23c Watermelons I L. M. Laughbauni. 161 Sturges avenue, is week ending in Chica go. III. Adario BIUEFS Mrs. Ralph Kyle ?nd Mrs. Mat tic Hoffman of Mansfield were callers at the R.

You ask any guy growing up what the worst insult was and it's always, "You throw like a girl."I know I've heard that plenty of times, but I'd never really thought about it like that.You look at every horrific act Nike Air Max 95 Black Black mens shoes on sale and they are never perpetrated by a woman. The Boston Marathon bombings? You didn't even consider that it was a woman.

That last bit may be wishful thinking. Las Vegas gaming revenue for 2009, was down more than 10 percent through November, though there were signs of recovery late in the year. Whether or not the president was to blame, local resident Christie Terronial(ph) says Mr. Obama really needs to watch what he says.

I really love the fact that issues like these are hot button issues that ignorant racists can't ignore and it forces them to expose their Nike Free Run 2 2012 Royal Blue Navy White size 6 bias in public. People who believe that white people have gotten beyond the issue of race only need to read what you people are writing. sad.

But Obama has also argued that Washington's 51 year economic embargo on Cuba should remain in force, and his administration has imposed tens of millions of dollars in fines on international companies for buy cheap New Nike Air Max 90 White Black Grey Yellow Release Date 2014 Online violating the sanctions. government development subcontractor Alan Gross put relations back in a deep freeze. Gross remains jailed, but this year Washington decided it would no longer let the case stand in the way on areas of common interest. and Cuba have held multiple rounds of talks on restoring direct mail service and immigration issues, with more scheduled for January. Diplomats on both sides report cordial relations and call each other at home. The two nations' coast guards reportedly work well together on things like drug interdiction. 相关的主题文章:

k5ZeQI World Cup Brazil 6s o7Nze6nP

More recently, the Iranians plotted to kill a Saudi diplomat in the United States. They issued warnings to the Sunni Arab Gulf kingdoms and tried to stir up their Shiite populations. They turned to Afghanistan and helped supply Taliban and al Qaida terrorists. They forged an anti American alliance in Latin America with Hugo Chavez. They are boasting about closing the Strait of Hormuz and warning allies of Israel of possible retaliation.

Leaving Cuenca thru the Majestic Cajas, the green, green jungle, to the Sunsets at the Coast. An MRI as opposed to a Cat Scan. Buying a hearing aid sight unseen really smart or really dumb. Practicing Medicine without a license Cuenca style. Coquita predicts a Cuenca earthquake A few suggestions for frustrated new arrivals Cuenca especially.

Perhaps something else was driving the agitation at the group's meeting next to Chesapeake Bay. News' clout in determining perceptions of academic value on campus has gotten out of hand. Even allowing for educators' typical reluctance to be judged by outsiders, the college presidents are right to seek to wriggle free.

There was no need for the NCAA to investigate what rules were broken, a process that can take months or years. Penn State handed over the results of its investigation by former FBI Director Louis Freeh and didn dispute the facts. Emmert said the decision to bypass the infractions committee and let the NCAA Executive Committee and its Division I Board of Directors decide on the penalties was not a sign of a change in the way future proceedings will go, but a sign that no investigation was necessary.

This is a trend rather than a rule: Rudy Giuliani and Michael Bloomberg, both mayors of New York, have at different times looked like plausible presidents in waiting. But the point stands City Hall is more than an apprenticeship for No 10 and should be a cause for celebration. In the Boris versus Ken bouts, we have witnessed the emergence of a red blooded, gloves off civic politics that does not depend upon the national political landscape for its energy or for its buy authentic Sale New 2014 Nike Air Max Ltd 2 White Black importance. This contest is about the future of the world's greatest city, not the respective standing of Ed Miliband and David Cameron.

Orlando Ayala, the Microsoft executive who had been leading the unit (and with whom I had been traveling with in Colombia) is shifting to a new role inside Kevin Turner's sales unit, while Microsoft veteran Will Poole is retiring from Microsoft this fall. Poole, who had been heading the unit with Ayala, is a former top Windows executive, while Ayala had been sales chief and head of the midmarket group at Microsoft, prior to joining the company's Unlimited Potential effort.

But what happens when someone who is supposed to be enforcing that rule ends up breaking it himself? That the White Chocolate Pink Womens Dunk High For Real question being asked at Sky Harbor Airport. FOX 10 Marc Martinez reports.Dramatic weight loss: 2 people lose a combined 240 poundsDramatic weight loss: 2 people shop Nike Air Max Wavy White Black 2014 For Sale lose a combined 240 poundsFOX 10 Kari Lake introduces us to two people who spent the last year doing what so many others only dream about and it was family that made all the difference.FOX 10 Kari Lake introduces us to two people who spent the last year doing what so many others only dream about and it was family that made all the difference. 相关的主题文章:

A1ipXj Buy Cheap Jordans 2014 RxzPOmuY

C. 106, Jordan 10 Lady Liberty 2 401 (2).[11] Section 2 503 (1) describes acceptable methods of a seller tender of delivery and states jordan 4 oreo the following: of delivery requires that the seller put and hold conforming goods Legend Blue 11s at the buyer Jordan 11 Legend Blue disposition and give the buyer any notification reasonably necessary to enable him to take delivery. The manner, time and place for tender are determined by the agreement and this Article, and in particular . Jordan 10 Bulls Over Broadway

If you haven heard of Anvil yet, it because the hit documentary about this middle aged Canadian Metal band Anvil! The Story of Anvil is just reaching Cincinnati (it opens Friday at the Esquire Theatre). jordan 4 oreo It the story of a marginally successful (at best) Rock act that has been trying to make it big jordan 7 french blue for some 30 Jordan 10 Bulls Over Broadway years. (The band, supported by VH1 Classics in the wake of the film strong reception elsewhere, will perform and answer questions after a Lady Liberty 10s midnight Saturday screening jordan 7 french blue at the Esquire)..

Lower level compression stockings are sold over the counter and are 10 15mmHg and 15 20mmHg. You don't need a Jordan 11 Legend Blue doctor's order for them and they are considered moderate compression. This is the level that many people begin cheap jordans with to see if they Jordan 11 Legend Blue will actually help.. Jordan 10 Lady Liberty

Year we definitely had some struggles on the field, especially early on, stated head coach Chad Webb. I thought we were able to make some positive adjustments down the stretch towards the end jordan 4 oreo of conference play. I think advancing through the tournament was a great experience for those young girls and there are several things we can build on going into this season. 相关的主题文章：

AchYks buy jordans sYL5Ma4p

We don't handle Legend Blue 11s anything as a company for labor or products other than customer service. We consult with Retailers and Restaurants to help them grow their business, but dont' charge them for consulting services. Jordan 11 Legend Blue We guide them to use our niche birthday mailer program to help them grow their sales, Legend Blue 11s along with Thank You follow up mailers thereafter.

Mar 2, 2002 Eleven people were killed and over 50 were injured, 4 critically, in a suicide bombing at 19:15 on Saturday evening near a yeshiva in the Jordan 11 Legend Blue ultra Orthodox Beit Yisrael neighborhood in the center of Jerusalem Jordan 11 Legend Blue where people had gathered for a bar jordan 4 oreo mitzva celebration. The terrorist detonated the bomb next to a group of jordan 4 oreo women waiting with their baby carriages for their husbands to leave the nearby synagogue. The victims: Shlomo Nehmad (40), his wife Gafnit (32), and their daughters Shiraz (7) and Liran (3), of Rishon Lezion; Shaul Nehmad (15), of Rishon Lezion; Lidor Ilan (12) and his sister Oriah (18 months), of Rishon Lezion; Jordan 10 Lady Liberty Tzofia Ya'arit Eliyahu (23) and her son Ya'akov Avraham (7 months), of Jerusalem.

JON EWALL Jordan 10 Lady Liberty 10 months ago from usa Level 2 CommenterA very interesting article since in the Jordan 10 Lady Liberty past french blue 7s few cheap jordans weeks many of the articles FAIL to report cheap jordans what the Republican House has proposed in an jordan 4 oreo attempt to move a immigration bill. The country must not allow the Tea Party Caucus in the Republican Congress to lay traps in a legislation that is mean to move the country forward as a whole. Congress (House Senate) either of them should be referred to as Republican Congress ,the President in his speeches will refer to the same. Check these links for jordan 7 french blue the another view of your hub.. 相关的主题文章：

pIk4cx Buy Cheap Jordans 2014 ci04gQ3L

The Boilermakers did a make a late rally and had a chance to tie or win, but the Badgers made the plays on this jordan 4 oreo night. They prevailed 68 66, tying a school record by hitting 11 3 pointers and winning in Mackey Arena for the first cheap jordans time since 1984. Don focus too much on the final 30 seconds because the Badgers won this game in the jordan 7 french blue first half, hitting 8 of 12 from 3 point range..

CHICAGO (FOX 32 News) If you are looking for a good film to check out this weekend, look no further! FOX 32's in house movie expert Jake jordan 7 french blue Hamilton has released his list of "Top 10 Movies of 2014."INTERSTELLAR is, for a cheap jordans lot of reasons, an important study of space and time not just for the adventurers on the screen, but for myself. Unmeetable) Jordan 10 Bulls Over Broadway expectations sugarplum fairy dreams of a cinematic masterpiece and walking out mildly disappointed.So, I stepped away from the film and decided, once the hype both internal and external had died down, Legend Blue 11s that I'd revisit it once jordan 7 french blue more several weeks later.Space and time.And that's when I discovered not necessarily Jordan 11 Legend Blue what I was hoping for, but what it was INTERSTELLAR is a small study of love performed on a grand stage of discovery. An epic study of people told against a backdrop of their desperate attempt to survive, this is a Nolan we've never seen jordan 4 oreo before, jordan 7 french blue and I Jordan 10 Lady Liberty hope to one day see again.As gorgeous as the eye melting visuals are, this is a movie about the relationship Lady Liberty 10s between a Legend Blue 11s father and his child and the tragedy, hope and loss it must endure in order to survive.Whether it's the most important relationships in our lives or just a movie, sometimes the things we care about don't turn out the way we want them to cheap jordans and that's a good thing.Sometimes they just need space and time.9. 相关的主题文章：

9YoTxE jordans dN7gr3uh

On jordan 7 french blue December 26, cheap jordans 1997, MPD officer Don Williams was shot and killed outside Crumpy's Comedy Club where he was working security.In May 2013, after a conviction, a stint on death row, a successful appeal, two subsequent mistrials, and french blue 7s a plea deal. Timothy McKinney walked out of Jordan 10 Lady Liberty jail on 15.5 years time served.Police say McKinney pulled jordan 7 french blue the trigger in an argument another man. Now, he Jordan 10 Lady Liberty is in jail on an attempted murder charge.More >>An oil tycoon's ex wife refused a $975 million check because she says she's entitled to more.More >>Goose Creek man sentenced to prison, probation in 'horrific' 2013 animal jordan 7 french blue cruelty caseGoose Creek man jordan 7 french blue sentenced to prison, probation in 'horrific' 2013 animal cruelty caseUpdated: Wednesday, January 7 2015 4:06 PM EST2015 01 07 21:06:02 GMTA judge jordan 4 oreo sentenced a Goose Creek man charged with 10 counts of animal cruelty to jordan 4 oreo a combination of prison time and probation in connection with a 2013 incident.More >>A judge sentenced a Goose Creek man charged with 10 counts of animal cruelty to a combination of prison time and probation in connection with a 2013 incident.

Ancient Greece and RomeBread takes center stageBarley flatbreads were the norm cheap jordans in Greece. Plato (400 BCE) encouraged cheap jordans the ideal cheap jordans man to eat wholemeal barley bread.[5413] Jordan 11 Legend Blue Athenians enjoyed 66 kinds of bread and cakes.[5413] Baked goods were flavored with honey, anise, sesame, and fruit and made into a variety of shapes. The idea that jordan 4 oreo white bread is a sign of purity and status comes from the Greeks. 相关的主题文章：

Retrieved from "https://en.formulasearchengine.com/index.php?title=Andromeda_Galaxy&oldid=285579"