Redshift conjecture: Difference between revisions

Latest revision as of 16:06, 5 January 2015

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-[[File:Steiner-erz-def.png|250px|thumb|1. Definition of the Steiner generation of a conic section]]
+Cheap Hostgator Hosting - [http://sabinegardiner.wordpress.com/ http://sabinegardiner.wordpress.com/]. Selecting the right webhosting service is important. The last thing you wish to do is get a website published with a hosting service (after finding out to use that hosting service), and afterwards be disappointed with the service and have to mess around with moving your website to another hosting service.<br><br>I'm a happy HostGator customer. I use both HostGator and Bluehost. I recommend both. This article is an extensive HostGator review.<br><br>I'll begin with one of the most important factors to consider: Cost<br><br>HostGator has several pricing strategies. They are as follows (since the date this article was published):.<br><br>Hatchling Plan: as reduced as $4.95 per month with a 3 year commitment. On this plan you can host one site.<br>Baby Plan: as low as $7.95 per month with a 3 year commitment. On this plan you can host an unlimited variety of internet sites.<br>Business Plan: as low as $12.95 per month with 3 year dedication.<br><br>You don't need to dedicate for 3 years. With shorter term commitments, you'll pay slightly even more per month.<br><br>You can also get your very own devoted server if this is something you such as to have.<br><br>Unlimited variety of internet sites.<br><br>I would not think about a hosting service that didn't let me host an endless number of sites for one month-to-month price under $10. I have a number of sites and I such as the adaptability of having the ability to develop even more websites at no extra cost (except for registering the domain).<br><br>If you choose the Infant Strategy (this is the plan I have) or Business Plan, you can host as lots of websites on as lots of domain names as you such as. This is where using a hosting service like HostGator can conserve you a terrific deal of money in the long run versus making use of a website home builder or having a [http://Www.europeana.eu/portal/search.html?query=website+designer website designer] host your websites. When you make use of a website home builder or have a website designer host your site, you'll generally pay additional for each added site (or each additional batch of sites).<br><br>The downside of course, is you must manage your very own hosting. The good news is, this isn't hard with the user friendly CPanel and technical support.<br><br>Domain registration.<br><br>You can easily sign up domain names with HostGator. You have to pay for each domain. REMEMBER to set your domain on auto-renew (and that your invoicing info is established to auto-pay) so your domain registration doesn't lapse. Absolutely nothing could be even worse than developing an excellent internet site then to lose it all because you forgot to renew your domain. It's possible then that another person registers your domain name and you can't get it once again. That might be disastrous.<br><br>All HostGator prepares offer unlimited bandwidth.<br><br>This is excellent and you need to need this with any website hosting service. I would rule out utilizing a hosting service that didn't offer unrestricted bandwidth.<br><br>Easy-to-use CPanel Dash panel.<br><br>You manage your internet sites with HostGator in a dashboard described as a CPanel. The CPanel is a user friendly interface to manage your internet sites and domains.<br><br>Easy website installment.<br><br>I specifically require with any hosting service that I use can install WordPress with virtually a single click or a series of basic clicks. HostGator provides Fantastico De Luxe and QuickInstall choices for quickly installing WordPress and lots of various other scripts to build your site (i.e. Joomla and others).<br><br>Access to 4,500 website templates.<br><br>For any internet designer, this is huge. This is a wonderful method to build sites inexpensively for customers. You can inspect out these design templates for yourself on the HostGator website without having to sign up.<br><br>Free website home builders.<br><br>With an account, you can easily build an internet site with one of two website home builders you get access to. The 2 website builders are:.<br><br>Trendy Site Home builder, and.<br>Site Studio website home builder.<br><br>Note, you can only use the website home builders for one site on your account. What this implies is if you get an account where you can host endless domains, you can only construct one website with a website builder.<br><br>Email accounts.<br><br>You get endless POP3 e-mail accounts with SMTP. Having e-mail accounts on your customized domain names is more expert than a gmail or hotmail e-mail account.<br><br>45 day refund assure.<br><br>You can get your refund if you cancel your account within 45 days if HostGator isn't really for you.<br><br>Video tutorials.<br><br>HostGator gives you access to numerous video tutorials that step you through lots of processes.<br><br>Consumer support.<br><br>The drivers for technical support know a lot about working in HostGator. If you have a concern about personalizing a WordPress theme, HostGator will not help you (I discovered this to be the case with Bluehost.<br><br>1 Criticism of HostGator.<br><br>I had to call HostGator to verify my account upon opening it. This didn't take long, but it was an additional step. I would have chosen merely to register and start without needing to call them for confirmation.
-[[File:Persp-geradenb.png|250px|thumb|2. Perspective mapping between lines]]
-[[File:Steiner-erz-beisp.png|250px|thumb|Example of a Steiner generation: generation of a point]]
-The '''Steiner theorem''', or '''Steiner generation of a conic ''', named after the Swiss mathematician [[Jakob Steiner]], is an alternative method to define a non-degenerate [[Quadric (projective geometry)|projective conic section]] in a [[projective plane]] over  a [[Field (mathematics)|field]]:
-*Given two pencils <math>B(U),B(V)</math> of lines at two points <math>U,V</math> (all lines containing <math>U</math> and  <math>V</math> resp.) and a projective but not perspective mapping <math>\pi</math> of <math>B(U)</math> onto <math>B(V)</math>. Then the intersection points of corresponding lines form a non-degenerate projective conic section <ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891&nbsp;kB), p.&nbsp;38.</ref><ref>''Jacob Steiner’s Vorlesungen über synthetische Geometrie'', B. G. Teubner, Leipzig 1867 (bei Google Books: [http://books.google.de/books?id=jCgPAAAAQAAJ]), 2. Teil, p.&nbsp;96</ref> (1. picture)
-A ''perspective'' mapping <math>\pi</math> of a pencil <math>B(U)</math> onto a pencil <math>B(V)</math>  is a [[bijection]] (1-1 correspondence) such that corresponding lines intersect on a fixed line <math>a</math>, which is called the ''axis'' of the perspectivity <math>\pi</math> (2. picture).
-A ''projective '' mapping is a finite sequence of perspective mappings.
-Examples of commonly used fields are the real numbers <math>\R</math>, the rational numbers <math>\Q</math> or the complex numbers <math>\C</math>. Even finite fields are allowed.
-''Remark:''
-The fundamental theorem for projective planes <ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891&nbsp;kB), p.&nbsp;19.</ref> states, that a projective mapping in a projective plane over a field (pappian plane) is uniquely determined by prescribing the images of three lines. That means for the Steiner generation of a conic section: besides two points <math>U,V</math> only the images of 3 lines have to be given. From these 5 items (2 points, 3 lines) the conic section is uniquely determined.
-''Remark:''
-The notation "perspective" is due to the dual statement: The projection of the points on a line <math> a</math> from a center <math>Z</math> onto a line <math>b</math> is called perspective.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891&nbsp;kB), p.&nbsp;19.</ref>
-== Example ==
-For the following example the images of the lines <math> a,u,w</math> (see picture) are given: <math>\pi(a)=b, \pi(u)=w, \pi(w)=v</math>. The projective mapping <math>\pi</math> is the product of the following perspective  mappings <math>\pi_b,\pi_a</math>: 1) <math>\pi_b</math> is the perspective mapping of the pencil at point <math>U</math> onto the pencil at point <math>O</math> with axis <math>b</math>. 2) <math>\pi_a</math> is the perspective mapping of the pencil at point <math>O</math> onto the pencil at point <math>V</math> with axis <math>a</math>.
-First one should check that <math>\pi=\pi_a\pi_b</math> has the properties: <math>\pi(a)=b, \pi(u)=w, \pi(w)=v</math>. Hence for any line <math>g</math> the image <math>\pi(g)=\pi_a\pi_b(g)</math> can be constructed and therefore the images of an arbitrary set of points. The lines <math>u</math> and <math>v</math> contain only the conic points <math>U</math> and <math>V</math> resp.. Hence <math>u</math> and <math>v</math> are tangent lines of the generated conic section.
-The '''proof''', that this method generates a conic section can be done by switching to the affine restriction with line <math>w</math> as line at infinity, point <math>O</math> as the origin of a coordinate system with points <math>U,V</math> as points at infinity of the x- and y-axis resp. and point <math>E=(1,1)</math>. The affine part of the generated curve  appears to be the hyperbola <math>y=1/x</math>.<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891&nbsp;kB), p.&nbsp;38.</ref>
-''Remark:''
-#The Steiner generation of a conic section provides simple methods for the construction of [[ellipse]]s, [[parabola]]s and [[hyperbola]]s which are commonly called the ''parallelogram methods''.
-#The figur, which appears while constructing a point (3. picture) is the 4-point-degeneration of [[Pascal's theorem]].<ref>[http://www.mathematik.tu-darmstadt.de/~ehartmann/circlegeom.pdf ''Planar Circle Geometries, an Introduction to Moebius-, Laguerre- and Minkowski Planes.''](PDF; 891&nbsp;kB), p.&nbsp;32.</ref>
-==Notes==
-{{reflist}}
-[[Category:Conic sections]]
-[[Category:Theorems in projective geometry]]

Redshift conjecture: Difference between revisions

Latest revision as of 16:06, 5 January 2015

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