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| [[Image:Klein bottle.svg|thumb|240px|right|A two-dimensional representation of the Klein bottle [[Immersion (mathematics)|immersed]] in three-dimensional space]]
| | Seven Noxious Weeds Of Southern California<br><br>Tonight is the Miss out on The united states 2014 pageant on ABC. On Sept. fourteen, news was shared that Miss out on Oklahoma Kelsey Griswold is picked as the favored to get [http://en.wikipedia.org/wiki/Barry_Schwartz Link Schwartz] . Oklahoma has gained the title 6 times in the past.<br><br>It was throughout this early morning that Pierce woke up and found the negative news. He crawled out of mattress and noticed his elbow was incredibly swollen. On best of that, he was even now encountering the higher fever that was bothering him from the day before. Soon after the same physician reevaluated the elbow, he imagined there may possibly be a severe an infection. It was critical adequate that they despatched him back again to Boston from Link Schwartz, exactly where he was with the crew.<br><br>1) The swiftest increasing age group in the region: Individuals ages eighty five and more than- Federal Interagency Discussion board on Getting older-Related Data. More mature People in america 2004: Important indicators of well-becoming, Federal Interagency Discussion board on Growing older-Connected Statistics. Washington, DC: U.S. Government Printing Business office, 2004.<br><br>When it comes to choosing a duration it is basically just 'different strokes for exclusive people'. Individually I am a admirer of the Dash triathlons merely since I like to go genuinely truly quickly!! This length permits me to place in a overall, all-out challenging perform for a fast complete of time. The Olympic is also suited for me owing to the reality in spite of the fact that you are not able to go as tough as the Dash, it is far more swiftly paced than the Iron or fifty % Iron. I didn't knowledge like I experienced the leading pace. I imagined I could protect my lead from the swim but Lewis was riding truly solid. I am [http://Www.Sharkbayte.com/keyword/delighted delighted] with finishing next.<br><br>The economic climate took a nosedive. I discovered myself proper in there with absolutely everyone else. I had considered that my guaranteed retirement was secure. Eighty per cent of it was. Nonetheless, Link Schwartz shedding the twenty%25 is what truly harm. Just a little anything that I never ever planned on, but it transpired even so. Liberty was in risk.<br><br>Treasury Secretary Tim Geithner is in the sizzling seat as a bailout oversight committee tries to figure out exactly where $700 billion bucks went to. I smell nuts roasting.<br><br>In this economy, however, there may be other motives Link Schwartz , such as needing far more earnings, not having ample several hours in the working day to operate an additional job, insecurity at operate, and more. It is not enjoyable to know the organization you expended several years doing work for is laying off hundreds or 1000's of individuals. The dread that will come with that knowledge is palpable in the organization hallways, till the final particular person is gone, and then a short-lived sigh of relief provides way to fret about the next possible layoffs. Many individuals are trying to commence an online enterprise to give them anything to tumble again on if/when layoffs come all around yet again.<br><br>I guess the LA Lakers are using on the Utah Jazz in the fifteenth NBA championship to begin off the 2009 NBA Playoffs. The Cleveland Cavaliers take on the Detroit Pistons. The Portland Path Blazers are getting on the Houston Rockets. The Orlando Magic get on the Philadelphia 76ers. The San Antonio Spurs consider on the Dallas Mavericks in the playoffs. Atlanta will take on Miami. And last but not least, I guess my house crew, the Denver Nuggets are having on the New Orleans Hornets for their likelihood to truly become a respectable team. Any person bear in mind when the Colorado Rockies got into the World Sequence? That was fairly incredible huh? Forgive me if I manufactured any mistakes, basketball fans. I'm much more of a football supporter.<br><br>And don't forget all that money that Washington despatched to GM and Chrysler so as to hold voters in Michigan utilized? Now that point out warns that its welfare rolls are slated to increase by an further a hundred,000 in the following handful of months.<br><br>The post above is data of a common mother nature and the data supplied might not apply to your personal predicament. Please consult your financial planner or licensed professional for investment decision advice. |
| [[Image:Surface_of_Klein_bottle_with_traced_line.svg|thumb|150px|right|Structure of a three-dimensional Klein bottle]]
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| In [[mathematics]], the '''Klein bottle''' {{IPAc-en|ˈ|k|l|aɪ|n}} is an example of a [[orientability|non-orientable]] [[surface]]; informally, it is a surface (a two-dimensional [[manifold]]) in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the [[Möbius strip]] and the [[real projective plane]]. Whereas a Möbius strip is a [[surface with boundary]], a Klein bottle has no boundary (for comparison, a [[sphere]] is an orientable surface with no boundary).
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| The Klein bottle was first described in 1882 by the [[Germany|German]] mathematician [[Felix Klein]]. It may have been originally named the ''Kleinsche Fläche'' ("Klein surface") and that this was incorrectly interpreted as ''Kleinsche Flasche'' ("Klein bottle"), which ultimately led to the adoption of this term in the German language as well.<ref>{{Cite book | publisher = AMS Bookstore | isbn = 978-0-8218-4816-6 | last = Bonahon | first = Francis | title = Low-dimensional geometry: from Euclidean surfaces to hyperbolic knots | date = 2009-08-05 | page=95 | url=http://books.google.com/books?id=YZ1L8S4osKsC}}, [http://books.google.com/books?id=YZ1L8S4osKsC&pg=PA95 Extract of page 95]</ref>
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| ==Construction==
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| Start with a square, and then glue together corresponding coloured edges, in the following diagram, so that the arrows match. More formally, the Klein bottle is the [[quotient space]] described as the [[square (geometry)|square]] [0,1] × [0,1] with sides identified by the relations {{nowrap|(0, ''y'') ~ (1, ''y'')}} for {{nowrap|0 ≤ ''y'' ≤ 1}} and {{nowrap|(''x'', 0) ~ (1 − ''x'', 1)}} for {{nowrap|0 ≤ ''x'' ≤ 1}}:
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| :[[Image:Klein Bottle Folding 1.svg]]
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| This square is a [[fundamental polygon]] of the Klein bottle.
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| Note that this is an "abstract" gluing in the sense that trying to realize this in three dimensions results in a self-intersecting Klein bottle. The Klein bottle, proper, does not self-intersect. Nonetheless, there is a way to visualize the Klein bottle as being contained in four dimensions.
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| Glue the red arrows of the square together (left and right sides), resulting in a cylinder. To glue the ends together so that the arrows on the circles match, pass one end through the side of the cylinder. Note that this creates a circle of self-intersection. This is an [[immersion (mathematics)|immersion]] of the Klein bottle in three dimensions.
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| <gallery | align = center> | |
| Image:Klein Bottle Folding 1.svg
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| Image:Klein Bottle Folding 2.svg
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| Image:Klein Bottle Folding 3.svg
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| Image:Klein Bottle Folding 4.svg
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| Image:Klein Bottle Folding 5.svg
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| Image:Klein Bottle Folding 6.svg
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| </gallery> | |
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| By adding a fourth dimension to the three dimensional space, the self-intersection can be eliminated. Gently push a piece of the tube containing the intersection along the fourth dimension, out of the original three dimensional space. A useful analogy is to consider a self-intersecting curve on the plane; self-intersections can be eliminated by lifting one strand off the plane.
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| This immersion is useful for visualizing many properties of the Klein bottle. For example, the Klein bottle has no ''boundary'', where the surface stops abruptly, and it is [[orientability|non-orientable]], as reflected in the one-sidedness of the immersion.
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| [[File:Science Museum London 1110529 nevit.jpg|thumb|right|150px|Embedded Klein bottles in the [[Science Museum (London)|Science Museum in London]]]]
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| [[Image:Acme klein bottle.jpg|thumb|150px|right|A hand-blown Klein Bottle]]
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| [[Image:KleinBottle-02.png|thumb|right|150px|Dissecting the Klein bottle results in Möbius strips.]]
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| The common physical model of a Klein bottle is a similar construction. The [[Science Museum (London)|Science Museum in London]] has on display a collection of hand-blown glass Klein bottles, exhibiting many variations on this topological theme. The bottles date from 1995 and were made for the museum by [[Alan Bennett (glass blower)|Alan Bennett]].<ref>[http://web.archive.org/web/20061128155852/http://www.sciencemuseum.org.uk/on-line/surfaces/new.asp Strange Surfaces: New Ideas<!-- Bot generated title -->]</ref>
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| ==Properties==
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| Like the [[Möbius strip]], the Klein bottle is a two-dimensional differentiable [[manifold]] which is not [[orientability|orientable]]. Unlike the Möbius strip, the Klein bottle is a ''closed'' manifold, meaning it is a [[Compact space|compact]] manifold without boundary. While the Möbius strip can be embedded in three-dimensional [[Euclidean space]] '''R'''<sup>3</sup>, the Klein bottle cannot. It can be embedded in '''R'''<sup>4</sup>, however.
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| The Klein bottle can be seen as a [[fiber bundle]] over the [[circle]] ''S''<sup>1</sup>, with fibre ''S''<sup>1</sup>, as follows: one takes the square (modulo the edge identifying equivalence relation) from above to be ''E'', the total space, while the base space ''B'' is given by the unit interval in ''y'', modulo ''1~0''. The projection π:''E''→''B'' is then given by π([''x'', ''y'']) = [''y''].
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| The Klein bottle can be constructed (in a mathematical sense, because it cannot be done without allowing the surface to intersect itself) by joining the edges of two Möbius strips together, as described in the following [[limerick (poetry)|limerick]] by [[Leo Moser]]:<ref>{{cite book |author=David Darling|title=The Universal Book of Mathematics|publisher=John Wiley & Sons |year=2004 |pages= |isbn=|page=176}}</ref>
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| : A mathematician named Klein
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| : Thought the Möbius band was divine.
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| : Said he: "If you glue
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| : The edges of two,
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| : You'll get a weird bottle like mine."
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| The initial construction of the Klein bottle by identifying opposite edges of a square shows that the Klein bottle is a [[CW complex]] with one 0-cell ''P'', two 1-cells ''C''<sub>1</sub>, ''C''<sub>2</sub> and one 2-cell ''D''. Its [[Euler characteristic]] is therefore 1-2+1 = 0. The boundary homomorphism is given by ∂''D'' = 2''C''<sub>1</sub> and ∂''C''<sub>1</sub>=∂''C''<sub>1</sub>=0, yielding the [[cellular homology|homology groups]] of the Klein bottle ''K'' to be H<sub>0</sub>(''K'','''Z''')='''Z''', H<sub>1</sub>(''K'','''Z''')='''Z'''×('''Z'''/2'''Z''') and H<sub>''n''</sub>(''K'','''Z''') = 0 for ''n''>1.
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| There is a 2-1 [[covering map]] from the [[torus]] to the Klein bottle, because two copies of the [[fundamental region]] of the Klein bottle, one being placed next to the mirror image of the other, yield a fundamental region of the torus. The [[universal cover]] of both the torus and the Klein bottle is the plane '''R'''<sup>2</sup>.
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| The [[fundamental group]] of the Klein bottle can be determined as the [[Deck transformation#Deck transformation group, regular covers|group of deck transformations]] of the universal cover and has the [[presentation of a group|presentation]] <''a'',''b'' | ''ab'' = ''b''<sup>−1</sup>''a''>.
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| Six colors suffice to color any map on the surface of a Klein bottle; this is the only exception to
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| the [[Heawood conjecture]], a generalization of the [[four color theorem]], which would require seven.
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| A Klein bottle is homeomorphic to the [[connected sum]] of two [[projective plane]]s. It is also homeomorphic to a sphere plus two [[cross cap]]s.
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| When embedded in Euclidean space the Klein bottle is one-sided. However there are other topological 3-spaces, and in some of the non-orientable examples a Klein bottle can be embedded such that it is two-sided, though due to the nature of the space it remains non-orientable.<ref>{{Cite book | publisher = CRC Press | isbn = 978-0-8247-0709-5 | last = Weeks | first = Jeffrey | title = The shape of space, 2nd Edn. | year = 2002 }}</ref>
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| ==Dissection==
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| Dissecting a Klein bottle into halves along its [[plane of symmetry]] results in two mirror image [[Möbius strip]]s, i.e. one with a left-handed half-twist and the other with a right-handed half-twist (one of these is pictured on the right). Remember that the intersection pictured isn't really there.
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| ==Simple-closed curves==
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| One description of the types of simple-closed curves that may appear on the surface of the Klein bottle is given by the use of the first homology group of the Klein bottle calculated with integer coefficients. This group is isomorphic to '''''Z'''×'''Z<sub>2</sub>'''''. Up to reversal of orientation, the only homology classes which contain simple-closed curves are as follows: (0,0), (1,0), (1,1), (2,0), (0,1). Up to reversal of the orientation of a simple closed curve, if it lies within one of the two crosscaps that make up the Klein bottle, then it is in homology class (1,0) or (1,1); if it cuts the Klein bottle into two Möbius bands, then it is in homology class (2,0); if it cuts the Klein bottle into an annulus, then it is in homology class (0,1); and if bounds a disk, then it is in homology class (0,0).
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| ==Parameterization==
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| [[Image:KleinBottle-Figure8-01.png|thumb|left|The "figure 8" immersion of the Klein bottle.]]
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| [[Image:Kleinbagel cross section.png|thumb|left|Klein bagel cross section employing a figure eight curve (the [[lemniscate of Gerono]]).]]
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| === The figure 8 immersion ===
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| The "figure 8" [[Immersion (mathematics)|immersion]] (Klein bagel) of the Klein bottle has a particularly simple parameterization. It is that of a "figure-8" torus with a 180 degree "Möbius" twist inserted:
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| :<math>\begin{align}
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| x & = \left(r + \cos\frac{\theta}{2}\sin v - \sin\frac{\theta}{2}\sin 2v\right) \cos \theta\\
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| y & = \left(r + \cos\frac{\theta}{2}\sin v - \sin\frac{\theta}{2}\sin 2v\right) \sin \theta\\
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| z & = \sin\frac{\theta}{2}\sin v + \cos\frac{\theta}{2}\sin 2v
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| \end{align}</math>
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| for 0 ≤ ''θ'' < 2π, 0 ≤ ''v'' < 2π and ''r'' > 2.
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| In this immersion, the self-intersection circle (when ''v'' = 0, π) is a geometric [[circle]] in the ''xy''-plane. The positive constant ''r'' is the radius of this circle. The parameter ''θ'' gives the angle in the ''xy''-plane, and ''v'' specifies the position around the 8-shaped cross section. With the above parameterization the cross section is a 2:1 [[Lissajous curve]].
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| In four dimensions this surface can be made non-intersecting by adding a little ''v'' dependent "bump" to the fourth ''w'' axis at the intersection point. E.g.
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| :<math>\begin{align}
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| w & = \cos v
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| \end{align}</math>
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| === 4-D non-intersecting ===
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| Another non-intersecting 4-D parameterization is modeled after that of the [[Flat_torus#Flat_torus|flat torus]]:
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| :<math>\ \begin{align}
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| x & = R\left(\cos\frac{\theta}{2}\cos v-\sin\frac{\theta}{2}\sin 2v\right) \\ y & = R\left(\sin\frac{\theta}{2}\cos v+\cos\frac{\theta}{2}\sin 2v\right) \\ z & = P\cos\theta\left(1+e\sin v\right) \\ w &= P\sin\theta\left(1+e\sin v\right)
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| \end{align}</math>
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| where ''R'' and ''P'' are constants that determine aspect ratio, ''θ'' and ''v'' are similar to as defined above. ''v'' determines the position around the figure-8 as well as the position in the x-y plane. ''θ'' determines the rotational angle of the figure-8 as well and the position around the z-w plane. ''e'' is any small constant and ''e''sin''v'' is a small ''v'' depended bump in ''z-w'' space to avoid self intersection. The ''v'' bump causes the self intersecting 2-D/planar figure-8 to spread out into a 3-D stylized "potato chip" or saddle shape in the x-y-w and x-y-z space viewed edge on. When ''e=0'' the self intersection is a circle in the z-w plane <0, 0, cos''θ'', sin''θ''>.
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| === Bottle shape ===
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| The parameterization of the 3-dimensional immersion of the bottle itself is much more complicated. Here is a version found by Robert Israel:
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| [[File:Klein bottle translucent.png|thumb|right|Klein Bottle with slight transparency]]
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| :<math>\begin{align}
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| x(u,v) &= -\frac{2}{15} \cos u (3 \cos{v}-30 \sin{u}+90 \cos^4{u} \sin{u} \\
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| &\quad -60 \cos^6{u} \sin{u}+5 \cos{u} \cos{v} \sin{u}) \\
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| y(u,v) &= -\frac{1}{15} \sin u (3 \cos{v}-3 \cos^2{u} \cos{v}-48 \cos^4{u} \cos{v}+ 48 \cos^6{u} \\
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| &\quad \cos{v}-60 \sin{u}+5 \cos{u} \cos{v} \sin{u}-5 \cos^3{u} \cos{v} \sin{u}-80 \\
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| &\quad \cos^5{u} \cos{v} \sin{u}+80 \cos^7{u} \cos{v} \sin{u}) \\
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| z(u,v) &= \frac{2}{15} (3+5 \cos{u} \sin{u}) \sin{v}
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| \end{align}</math>
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| for 0 ≤ ''u'' < π and 0 ≤ ''v'' < 2π.
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| ==Generalizations==
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| The generalization of the Klein bottle to higher [[genus (mathematics)|genus]] is given in the article on the [[fundamental polygon]].
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| In another order of ideas, constructing [[3-manifold]]s, it is known that a '''solid Klein bottle''' is [[homeomorphic|topologically equivalent]] with the [[Cartesian product]]: <math>\scriptstyle M\ddot{o}\times I</math>, the [[Mobius band]] times an interval. The [[solid Klein bottle]] is the non-orientable version of the '''solid torus''', equivalent to <math>\scriptstyle D^2\times S^1</math>.
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| ==Klein surface==
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| A '''Klein surface''' is, as for [[Riemann surface]]s, a surface with an atlas allowing the [[transition function]]s to be composed using [[complex conjugation]]. One can obtain the so-called [[dianalytic structure]] of the space.
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| ==See also==
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| * [[Algebraic topology]]
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| * [[Alice universe]]
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| * [[Systoles of surfaces#Klein bottle|Bavard's Klein bottle systolic inequality]]
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| * [[Boy's surface]]
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| * [[Ouroboros]]
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| * [[Sphericon]]
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| * [[Topology]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| * {{MathWorld|urlname=KleinBottle|title=Klein Bottle}}
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| * A classical on the theory of '''Klein surfaces''' is [http://www.zentralblatt-math.org/zmath/en/search/scans.html?volume_=225&count_=158] of Alling-Greenleaf
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| ==External links==
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| {{Commons category|Klein bottle}}
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| * [http://plus.maths.org/issue26/features/mathart/index-gifd.html Imaging Maths - The Klein Bottle]
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| * [http://www.kleinbottle.com/meter_tall_klein_bottle.html The biggest Klein bottle in all the world]
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| * [http://www.youtube.com/watch?v=E8rifKlq5hc Klein Bottle animation: produced for a topology seminar at the Leibniz University Hannover.] [http://www-ifm.math.uni-hannover.de/~fugru/video/top/05_top_Kleins_Bottle.mov]
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| * [http://www.youtube.com/watch?v=sRTKSzAOBr4&fmt=22 Klein Bottle animation from 2010 including a car ride through the bottle and the original description by Felix Klein: produced at the Free University Berlin.]
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| * [http://www.geometrygames.org/TorusGames/index.html.en Torus Games] Free downloadable games for Windows and Mac OS X that highlight the topologies of the Torus and Klein Bottle.
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| * [https://github.com/danfuzz/xscreensaver/blob/master/hacks/glx/klein.man Klein Bottle], [[XScreenSaver]] "hack". A screensaver for [[X Window System|X 11]] and [[OS X]] featuring an animated Klein Bottle.
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| {{PlanetMath attribution|id=4249|title=Klein bottle}}
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| [[Category:Surfaces]]
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| [[Category:Geometric topology]]
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| [[Category:Impossible objects]]
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Seven Noxious Weeds Of Southern California
Tonight is the Miss out on The united states 2014 pageant on ABC. On Sept. fourteen, news was shared that Miss out on Oklahoma Kelsey Griswold is picked as the favored to get Link Schwartz . Oklahoma has gained the title 6 times in the past.
It was throughout this early morning that Pierce woke up and found the negative news. He crawled out of mattress and noticed his elbow was incredibly swollen. On best of that, he was even now encountering the higher fever that was bothering him from the day before. Soon after the same physician reevaluated the elbow, he imagined there may possibly be a severe an infection. It was critical adequate that they despatched him back again to Boston from Link Schwartz, exactly where he was with the crew.
1) The swiftest increasing age group in the region: Individuals ages eighty five and more than- Federal Interagency Discussion board on Getting older-Related Data. More mature People in america 2004: Important indicators of well-becoming, Federal Interagency Discussion board on Growing older-Connected Statistics. Washington, DC: U.S. Government Printing Business office, 2004.
When it comes to choosing a duration it is basically just 'different strokes for exclusive people'. Individually I am a admirer of the Dash triathlons merely since I like to go genuinely truly quickly!! This length permits me to place in a overall, all-out challenging perform for a fast complete of time. The Olympic is also suited for me owing to the reality in spite of the fact that you are not able to go as tough as the Dash, it is far more swiftly paced than the Iron or fifty % Iron. I didn't knowledge like I experienced the leading pace. I imagined I could protect my lead from the swim but Lewis was riding truly solid. I am delighted with finishing next.
The economic climate took a nosedive. I discovered myself proper in there with absolutely everyone else. I had considered that my guaranteed retirement was secure. Eighty per cent of it was. Nonetheless, Link Schwartz shedding the twenty%25 is what truly harm. Just a little anything that I never ever planned on, but it transpired even so. Liberty was in risk.
Treasury Secretary Tim Geithner is in the sizzling seat as a bailout oversight committee tries to figure out exactly where $700 billion bucks went to. I smell nuts roasting.
In this economy, however, there may be other motives Link Schwartz , such as needing far more earnings, not having ample several hours in the working day to operate an additional job, insecurity at operate, and more. It is not enjoyable to know the organization you expended several years doing work for is laying off hundreds or 1000's of individuals. The dread that will come with that knowledge is palpable in the organization hallways, till the final particular person is gone, and then a short-lived sigh of relief provides way to fret about the next possible layoffs. Many individuals are trying to commence an online enterprise to give them anything to tumble again on if/when layoffs come all around yet again.
I guess the LA Lakers are using on the Utah Jazz in the fifteenth NBA championship to begin off the 2009 NBA Playoffs. The Cleveland Cavaliers take on the Detroit Pistons. The Portland Path Blazers are getting on the Houston Rockets. The Orlando Magic get on the Philadelphia 76ers. The San Antonio Spurs consider on the Dallas Mavericks in the playoffs. Atlanta will take on Miami. And last but not least, I guess my house crew, the Denver Nuggets are having on the New Orleans Hornets for their likelihood to truly become a respectable team. Any person bear in mind when the Colorado Rockies got into the World Sequence? That was fairly incredible huh? Forgive me if I manufactured any mistakes, basketball fans. I'm much more of a football supporter.
And don't forget all that money that Washington despatched to GM and Chrysler so as to hold voters in Michigan utilized? Now that point out warns that its welfare rolls are slated to increase by an further a hundred,000 in the following handful of months.
The post above is data of a common mother nature and the data supplied might not apply to your personal predicament. Please consult your financial planner or licensed professional for investment decision advice.