Hypsicles

2014-01-16T00:32:09Z

88.192.133.216: /* Life and work */

In [[Riemann surface]] theory and [[hyperbolic geometry]], the '''Macbeath surface''', also called '''Macbeath's curve''' or the '''Fricke–Macbeath curve''', is the genus-7 [[Hurwitz surface]].

The [[automorphism group]] of the Macbeath surface is the [[simple group]] [[Projective linear group|PSL(2,8)]], consisting of 504 symmetries.<ref name="w">{{harvtxt|Wohlfahrt|1985}}.</ref>

==Triangle group construction==

The surface's [[Fuchsian group]] can be constructed as the principal congruence subgroup of the [[(2,3,7) triangle group]] in a suitable tower of principal congruence subgroups. Here the choices of quaternion algebra and [[Hurwitz quaternion order]] are described at the triangle group page. Choosing the ideal <math>\langle 2 \rangle</math> in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7. Its [[systolic geometry|systole]] is about 5.796, and the number of systolic loops is 126 according to R. Vogeler's calculations.

==Historical note==

This surface was originally discovered by {{harvs|first=Robert|last=Fricke|authorlink=Robert Fricke|year=1899|txt}}, but named after [[Alexander Macbeath|Alexander Murray Macbeath]] due to his later independent rediscovery of the same curve.<ref>{{harvtxt|Macbeath|1965}}.</ref> Elkies writes that the equivalence between the curves studied by Fricke and Macbeath "may first have been observed by [[Jean-Pierre Serre|Serre]] in a 24.vii.1990 letter to [[Shreeram Shankar Abhyankar|Abhyankar]]".<ref>{{harvtxt|Elkies|1998}}.</ref>

==See also==
* [[Klein quartic]]
* [[First Hurwitz triplet]]

==Notes==
{{reflist}}

==References==
{{refbegin}}
*{{citation|last1=Berry|first1=Kevin|last2=Tretkoff|first2=Marvin|contribution=The period matrix of Macbeath's curve of genus seven|title=Curves, Jacobians, and abelian varieties, Amherst, MA, 1990|pages=31–40|publisher=Contemp. Math., 136, Amer. Math. Soc.|location=Providence, RI|year=1992|mr=1188192 }}.
*{{citation|last1=Bujalance|first1=Emilio|last2=Costa|first2=Antonio F.|contribution=Study of the symmetries of the Macbeath surface|title=Mathematical contributions|pages=375–385|publisher=Editorial Complutense|location=Madrid|year=1994|mr=1303808 }}.
*{{citation|last=Elkies|first=N. D.|authorlink=Noam Elkies|contribution=Shimura curve computations|title=Algorithmic Number Theory: Third International Symposium, ANTS-III|publisher=Springer-Verlag, Lecture Notes in Computer Science 1423|year=1998|doi=10.1007/BFb0054849|arxiv=math.NT/0005160 |pages=1–47|volume=1423|series=Lecture Notes in Computer Science|editor1-last=Buhler|editor1-first=Joe P.|isbn=3-540-64657-4}}.
*{{citation|last=Fricke|first=R.|authorlink=Robert Fricke|title=Ueber eine einfache Gruppe von 504 Operationen|journal=[[Mathematische Annalen]]|volume=52|year=1899|pages=321–339|doi=10.1007/BF01476163|issue=2–3}}.
*{{citation|last=Gofmann|first=R.|title=Weierstrass points on Macbeath's curve|journal=Vestnik Moskov. Univ. Ser. I Mat. Mekh.|year=1989|issue=5|pages=31–33|volume=104|mr=1029778 }}. Translation in ''Moscow Univ. Math. Bull.'' '''44''' (1989), no. 5, 37–40.
*{{citation|last=Macbeath|first=A.|authorlink=Alexander Macbeath|title=On a curve of genus 7|journal=[[Proceedings of the London Mathematical Society]]|volume=15|year=1965|pages=527–542|doi=10.1112/plms/s3-15.1.527}}.
*{{citation|last=Vogeler|first=R.|title=On the geometry of Hurwitz surfaces|journal=Florida State University thesis|year=2003}}.
*{{citation|last=Wohlfahrt|first=K.|title=Macbeath's curve and the modular group|journal=Glasgow Math. J.|volume=27|year=1985|pages=239–247|mr=0819842 |doi=10.1017/S0017089500006212}}. Corrigendum, vol. 28, no. 2, 1986, p. 241, {{MR|0848433}}.
{{refend}}

[[Category:Hyperbolic geometry]]
[[Category:Riemann surfaces]]
[[Category:Riemannian geometry]]
[[Category:Differential geometry of surfaces]]
[[Category:Systolic geometry]]

Template:National Bolshevism sidebar

2013-12-27T22:53:27Z

88.192.91.207:

Hostgator Coupon - [http://dawonls.dothome.co.kr/db/?document_srl=347024 http://dawonls.dothome.co.kr/db/?document_srl=347024]. Hostgator is a big independently possessed hosting business running from multiple state of the art facilities in Dallas, Texas. Hostgator was established in 2002, since then they have actually expanded rapidly and presently host over 400,000 internet sites. [http://hostgatorcouponoffers.com/ hostgatorcouponoffers.com]Hostgator provides lots of different internet hosting bundles, and deal with a broad variety of clients. From the first time web designer who requires basic, stress free hosting for their individual site; all the way with to large companies, who need specialist committed hosting services. Hostgator's hosting plans can be split into 3 groups; standard shared hosting plans (ideal for the substantial majority people), reseller hosting plans (these are primarily for people and companies that wish to "resell" their account resources to clients of their own), and lastly devoted server strategies (these accounts offer customer their own server, so they do not have to share its resources with anyone else). Extremely few of us will every need a devoted server so this review will focus on the shared hosting strategies that Hostgator offer. Features Hostgator's three primary shared hosting strategies are named: "Hatchling" (the entry level strategy priced at $6.95 / month), "Child" (this is the most popular strategy, and it is most likely to satisfy the requirements of a very large range of clients), and "Swamp" (much the exact same as the "Infant" plan, however with increases in bandwidth and disk space, priced at $14.95 / month). For a complete list of attributes and a side by side contrast of all hosting strategies you should go to Hostgator's website here. Below is an evaluation of the most crucial features of the "Infant" plan, this is most likely the most suitable package for a lot of users, and it is our preferred plan. Disk area 100GB - This amount has been recently upgraded by Hostgator from 5GB to a massive 100GB. All individuals are most likely to discover it impossible to tire this amount of disk space. Bandwidth 1000GB/month - Likewise enhanced is the bandwidth allocation, from 75GB to a seemingly extreme 1000GB. Again virtually zero opportunity of utilizing all that, but it's good to understand that you certainly will not be facing additional fees for going over your restriction, even if you have a very busy internet site(s). Site Studio website home builder - This is a wonderful cost-free program that enables you to develop your internet site from scratch. You has more than 500 [http://Statigr.am/tag/templates templates] and color design to select from. You need no HTML experience, or code writing understanding. It provides the most basic possible method of making an expert looking website in a very brief area of time. However, there is no have to take my word for it, as Hostgator provides us with a demo version of the software application on their site! Limitless add-on domains - This truly is the stand out function of this hosting strategy (the "Hatchling" strategy just allows 1 domain), and enables you to host as lots of internet sites as you like on a single account, at no additional expense. This permits you to make full use of your huge bandwidth and disk space allowances, and host several sites at a fraction of the typical cost. 99.9 % Uptime assurance - This essentially tells us that Hostgator is a serious host offering a dependable service. If uptime drops below this figure in any given month then you do not spend for that months hosting, it's as basic as that! We would never think about making use of a host that did not provide a solid uptime assurance; this is simply due to the fact that there is only one great reason a host will not offer an uptime assurance - undependable uptime! 30 Day refund assurance - This has become a quite standard feature in the hosting neighborhood, though it's nice to have for included peace of mind. Instant setup - A lot of hosting carriers take 24-48 hours to setup your account however Hostgator promises to have you going in under 15 minutes (they do not charge a setup charge either)! Limitless MySQL data sources - This is extremely helpful since each Fantastico (see below) script needs its own MySQL database. Fantastico DeLuxe - This outstanding program enables you to instantly set up over 50 scripts through your control panel. Scripts include blogs, online forums, galleries, buying carts, and more. Limitless email accounts - Allows you to have as lots of, or as couple of, different email addresses as you like. It includes many functions and offers excellent performance. Finest of all, there is a complete working trial on the Hostgator internet site, so you can check it out yourself! Client service and technical support Hostgator offers us 24/7 phone support, and live online talk. The truth that you are offered 2 options to receive immediate technical support at any time of the day is wonderful. Our experience has always been extremely great when getting in touch with Hostgator, their operatives are really respectful and most notably they seem to understand their stuff when dealing with technical problems. Efficiency The performance from Hostgator's servers is outstanding! Hostgator place much tighter limitations on the number of internet sites sharing the same server compared with most various other shared hosting suppliers. This provides greater dependability since less stress is put on the servers; and it likewise significantly improves the speed at which your web pages run. Server performance is an additional among the crucial areas where Hostgator identify themselves from the group of other web hosts. Our verdict Overall there is a lot to like about the way Hostgator works, they truly do seem to have a good grasp on exactly what the typical client requires from a hosting company. Seldom do you come across reports of dissatisfied Hostgator customers, and after hosting with them ourselves we now understand why! Dependability, efficiency, and uptime from the servers is first rate, likewise customer care is means ahead of a lot of other host out there. However, the area where Hostgator truly stands out is value for cash. At simply $9.95 / month for the "Infant" strategy (that includes unrestricted domains); anybody looking to host more than one site has a quite simple choice to make. The other terrific aspect of this bundle is that you do not have to register to an agreement. You pay on a month-by-month basis, which leaves you free of cost to cancel your account at any time without losing cash. To sum up, we do not truly have any bad things about Hostgator; at the prices they charge, we genuinely feel that they provide an exceptional service. Keep up the excellent work Hostgator!

formulasearchengine - User contributions [en]

Hypsicles

Template:National Bolshevism sidebar