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Thu, 24 Jul 2014 20:16:13 +0000hourly1http://wordpress.com/Comment on Resultants by shaffaf
http://rigtriv.wordpress.com/2008/07/29/resultants/#comment-7796
Thu, 24 Jul 2014 20:16:13 +0000http://rigtriv.wordpress.com/?p=279#comment-7796How can you decide if two quartic polynomial have a common root by the concept of Galois group?we know that all the roots of these two polynomials are complex roots.
]]>Comment on Parallel Parking by Parallel Parking a Car | Wet Savanna Animals
http://rigtriv.wordpress.com/2007/10/01/parallel-parking/#comment-7737
Fri, 18 Jul 2014 05:58:16 +0000http://rigtriv.wordpress.com/2007/10/01/parallel-parking/#comment-7737[…] Charles Siegel, Parallel Parking page 1st of October 2007 on his blog “Rigorous Trivialities&#… […]
]]>Comment on Parallel Parking by Rod Vance
http://rigtriv.wordpress.com/2007/10/01/parallel-parking/#comment-7255
Tue, 17 Jun 2014 11:38:28 +0000http://rigtriv.wordpress.com/2007/10/01/parallel-parking/#comment-7255I get exactly the same outcome as jwwash: you get a Lie algebra of countably infinite dimension: you can keep getting arbitrary high powers of $\cos\theta$, $\sin\theta$ times forward slide and side slide. So I agree with you: we cannot find the Lie group that acts on the car configuration space by the methods of Nelson and Burke. Rossmann makes a direct copy BTW. Anyhow, you can solve the problem by leaving the steering angle out of the configuration space and instead saying that a general transformation available to the driver is of the form $\exp(s(D + \kappa\,R))$, where $D$ drives the car forwards, $R$ rotates it about the midpoint of its hinder axle and now you say that $\kappa$, the curvature of the car’s path is now a “control force”, a parameter that parameterises the family of basic transformations. So our configuration space is now $\mathbb{R}^2\times\mathbb{T}^1$ comprising the car’s $x,\,y$ position and its heading $\phi$. You can clearly get from $x,y,\phi$ to any point in Nelson’s / Burke’s configuration space $\mathbb{R}^2\times\mathbb{T}^2$ by imparting a drive angle. It is then a fairly simple matter to prove that the smallest Lie group containing the basic transformations of the form $\exp(s(D + \kappa\,R))$ is indeed $\mathbb{E}(2)$, the Euclidean group of all orientation preserving isometries of the plane: rotations and translations. See my calculations at http://www.wetsavannaanimals.net/wordpress/lie-theoretical-systems-theory
]]>Comment on Parallel Parking by Chapter 13: Lie Theoretical Systems Theory | Wet Savanna Animals
http://rigtriv.wordpress.com/2007/10/01/parallel-parking/#comment-6021
Fri, 30 May 2014 01:08:45 +0000http://rigtriv.wordpress.com/2007/10/01/parallel-parking/#comment-6021[…] Charles Siegel, Parallel Parking page 1st of October 2007 on his blog “Rigorous Trivialities&#… […]
]]>Comment on Review: LiveScribe Echo Digital Pen by Check out the Latest Digital Product Reviews on my blog.
http://rigtriv.wordpress.com/2011/04/04/review-livescribe-echo-digital-pen/#comment-5803
Sun, 25 May 2014 15:57:37 +0000http://rigtriv.wordpress.com/?p=1885#comment-5803Excellent post but I was wanting to know if you could write a litte more on this subject?
I’d be very thankful if you could elaborate a little bit further.

Many thanks!

]]>Comment on Tangent Spaces and Singular Points by Charles Siegel
http://rigtriv.wordpress.com/2008/01/23/tangent-spaces-and-singular-points/#comment-4972
Fri, 18 Apr 2014 06:54:57 +0000http://rigtriv.wordpress.com/2008/01/23/tangent-spaces-and-singular-points/#comment-4972Yes, at a singular point, you can have an arbitrarily large tangent space. To see this, look at the union of the coordinate axes in . Then the tangent space is dimensional.
]]>Comment on Tangent Spaces and Singular Points by Anonymous
http://rigtriv.wordpress.com/2008/01/23/tangent-spaces-and-singular-points/#comment-4962
Thu, 17 Apr 2014 20:06:32 +0000http://rigtriv.wordpress.com/2008/01/23/tangent-spaces-and-singular-points/#comment-4962For a curve, can the dimension of the tangent space be arbitrary big ?
]]>Comment on Suggestions and Requests by Charles Siegel
http://rigtriv.wordpress.com/suggestions-and-requests/#comment-4837
Sun, 30 Mar 2014 01:06:14 +0000http://rigtriv.wordpress.com/?page_id=144#comment-4837Removed, I haven’t been maintaining the blog for awhile, so the comment hygiene has declined. Apologies.
]]>Comment on Suggestions and Requests by Toby
http://rigtriv.wordpress.com/suggestions-and-requests/#comment-4834
Sat, 29 Mar 2014 10:16:35 +0000http://rigtriv.wordpress.com/?page_id=144#comment-4834comment 3 on this page :http://rigtriv.wordpress.com/2009/11/03/chern-classes-part-1/ is spam and not something authorised by the linked company. appreciate removal help.
]]>Comment on Algebraic Geometry Belongs to Sheaves by Andrew Macfarlane
http://rigtriv.wordpress.com/2010/02/05/algebraic-geometry-belongs-to-sheaves/#comment-4720
Sun, 09 Mar 2014 07:09:58 +0000http://rigtriv.wordpress.com/?p=1509#comment-4720I believe that within a some decades that sheaves will be seen as major tool in the creation of functors (“comparisons”, “measures of structure”) on large systems. For example “Structures of systems 1. Cohomology of manufacturing and supply network-like systems” http://www.tandfonline.com/doi/full/10.1080/03081079.2014.888551.(Int.Jnl.Gen.Sys.)
The analysis of large information and engineering systems (as opposed to building them) has few good analytical tools. Sheaves and their various cohomologies are a way to “integrate” the local and global. This might require the development of (co)homology-like tools that are defined not on groups and rings but perhaps lattices or graphs – the “(co)homology” looking for special objects such as modular lattices or expander graphs. The concept of a Topos might be drafted for logics of such systems with Grothendieck topologies coming from special classes of subsystems. With society becoming more and more dependent on large-scale systems these sprawling logics are ripe for some serious analytical tools.
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