Comments for The Lumber Room http://shreevatsa.wordpress.com "Consign them to dust and damp by way of preserving them" Wed, 10 Apr 2013 13:58:36 +0000 hourly 1 http://wordpress.com/ Comment on The functional equation f(x+y) = f(x)f(y) by KVM http://shreevatsa.wordpress.com/2013/04/08/the-functional-equation-fxy-fxfy/#comment-11239 Wed, 10 Apr 2013 13:58:36 +0000 http://shreevatsa.wordpress.com/?p=2695#comment-11239 Very nice!

]]> Comment on Trajectory of a point moving with acceleration perpendicular to velocity by Karthik http://shreevatsa.wordpress.com/2013/04/07/trajectory-of-a-point-moving-with-acceleration-perpendicular-to-velocity/#comment-11234 Mon, 08 Apr 2013 10:24:35 +0000 http://shreevatsa.wordpress.com/?p=2676#comment-11234 The ODE: If it’s linear, homogeneous and has constant coefficients, then the solution can be expressed using \exp(\alpha t). I don’t remember how I know this either! Wikipedia’s explanation is about the best I’ve found in the past.

I do believe there’s a theorem that says an n^{th} order linear homogeneous ODE (variable coefficients) has an n dimensional solution space, so the n independent exponentials do form a basis.

]]> Comment on Trajectory of a point moving with acceleration perpendicular to velocity by S http://shreevatsa.wordpress.com/2013/04/07/trajectory-of-a-point-moving-with-acceleration-perpendicular-to-velocity/#comment-11233 Mon, 08 Apr 2013 10:21:24 +0000 http://shreevatsa.wordpress.com/?p=2676#comment-11233 I don’t know if it’s built-in with WordPress proper, but it does come with the default WordPress.com setup. Documented here (it’s somewhat limited, but will do).
(Also one can’t install one’s own plugins on blogs hosted on WordPress.com, at least not with a free account.)
(Also Trevisan has written LaTeX2WP to make things easier.)

]]> Comment on Trajectory of a point moving with acceleration perpendicular to velocity by Karthik http://shreevatsa.wordpress.com/2013/04/07/trajectory-of-a-point-moving-with-acceleration-perpendicular-to-velocity/#comment-11232 Mon, 08 Apr 2013 10:10:48 +0000 http://shreevatsa.wordpress.com/?p=2676#comment-11232 Thanks, that’s useful to know. Does WP have Latex support (in this form) built in or have you installed a plugin of some sort?

I see now some Emacs gunk also made its way into the comment. The perils of composing blog comments in the scratch buffer, it seems. :-)

]]> Comment on Trajectory of a point moving with acceleration perpendicular to velocity by S http://shreevatsa.wordpress.com/2013/04/07/trajectory-of-a-point-moving-with-acceleration-perpendicular-to-velocity/#comment-11231 Mon, 08 Apr 2013 09:55:03 +0000 http://shreevatsa.wordpress.com/?p=2676#comment-11231 Read it — thanks, that’s clean. Yes, this makes sense, and roughly this sort of reasoning went into the previous post on sin and cos, which is ultimately unsurprising (both are about circles).

About the ODE: are you saying that because it’s a linear/homogeneous differential equation, the solution must be of the form \exp(\alpha t)? I don’t know (or if I did, I don’t remember — I don’t even remember if I did know!) why this should be so. This is the k=1 case in this, but the argument in that article that “In order for the sum of multiple derivatives of a function to sum up to zero, the derivatives must cancel each other out and the only way for them to do so is for the derivatives to have the same form as the initial function” seems unsatisfying to me.

]]> Comment on Trajectory of a point moving with acceleration perpendicular to velocity by S http://shreevatsa.wordpress.com/2013/04/07/trajectory-of-a-point-moving-with-acceleration-perpendicular-to-velocity/#comment-11230 Mon, 08 Apr 2013 09:36:10 +0000 http://shreevatsa.wordpress.com/?p=2676#comment-11230 WordPress.com uses $ latex ... $ syntax (without the space between $ and latex) instead of $ ... $ syntax… I’ve edited your comment; now to read it. :-)

(The two expressions with \underbar{r_0} gave “Formula does not parse” errors, so I’ve changed them to \underbar{r0} (ugly-looking, but still better than raw))

]]> Comment on Trajectory of a point moving with acceleration perpendicular to velocity by Karthik http://shreevatsa.wordpress.com/2013/04/07/trajectory-of-a-point-moving-with-acceleration-perpendicular-to-velocity/#comment-11229 Mon, 08 Apr 2013 08:53:39 +0000 http://shreevatsa.wordpress.com/?p=2676#comment-11229 Ah, jeez. I don’t know why I thought the comments would handle TeX input.

]]> Comment on Trajectory of a point moving with acceleration perpendicular to velocity by Karthik http://shreevatsa.wordpress.com/2013/04/07/trajectory-of-a-point-moving-with-acceleration-perpendicular-to-velocity/#comment-11228 Mon, 08 Apr 2013 08:52:44 +0000 http://shreevatsa.wordpress.com/?p=2676#comment-11228 I tried to solve this problem independent of a coordinate system. I used a tensor, but it turns out you don’t really need one:

Let \underbar{Q} be an orthogonal tensor that rotates its argument(operand?) by 90 degrees, so that det(\underbar{Q}) = 1 and \underbar{a} \cdot \underbar{Q} \underbar{a} = \underbar{0} \forall \underbar{a}. Then \dot{\underbar{v}} = \underbar{Q} \underbar{v} by your definition.

Also, \underbar{v} \cdot \underbar{Q} \underbar{v} = 0, so \underbar{v} \cdot \dot{\underbar{v}} = 0. This can be written as \frac{1}{2} \dot{v^2} = 0, so \underbar{v} has constant magnitude.

Finally, \dot{\underbar{v}} = \underbar{Q} \underbar{v} can be integrated to give \underbar{v} = \underbar{Q} (\underbar{r} - \underbar{r0}) because \underbar{Q} is not time dependent. Choosing our origin carefully so that \underbar{r0} = 0, We get \underbar{v} = \underbar{Q} \underbar{r} , the same as the relation between \dot{\underbar{v}} and \underbar{v}. The same step as above gives \frac{1}{2}\dot{r^2} = 0, so \underbar{r} has constant magnitude as well.

In two dimensions, r=const tells us that it moves in a circle about the origin, v=const tells us that the circular motion is uniform.

Incidentally, I think setting \underbar{a} \cdot \underbar{Q} \underbar{a} = k a^2 where unlike above, k is not zero, describes a (linear) spiral instead. (Not sure, just spitballing.) An interesting question is what the spirals look like when k(t) is not a constant.

About the solution of the ODE, the best I could do was note that it’s linear/homogeneous, so exp(\alpha t) must be a solution for some \alpha. Solving the resulting polynomial gives \alpha = \pm i, which are sines/cosines. This simply shifts the problem one link up the chain of reasoning, but there’s my engineering education for you.

(global-set-key (kbd “C-`”) ‘underbar)

]]> Comment on Lattice points visible from the origin by josephpzambrano http://shreevatsa.wordpress.com/2008/11/07/lattice-points-visible-from-the-origin/#comment-11133 Sun, 07 Apr 2013 19:03:13 +0000 http://shreevatsa.wordpress.com/?p=628#comment-11133 No problem at all, and yes that is what I was asking. I suppose a linear transformation taking the standard lattice into one generated by a different basis set must be an isomorphism to preserve the mentioned proportion. In which case, I wonder what happens when the mapping isn’t quite as nice.

]]> Comment on Lattice points visible from the origin by S http://shreevatsa.wordpress.com/2008/11/07/lattice-points-visible-from-the-origin/#comment-11109 Sun, 07 Apr 2013 15:59:20 +0000 http://shreevatsa.wordpress.com/?p=628#comment-11109 Hi, sorry for the delay in replying… commenting on old posts is perfectly fine, but I may not know the answer. :-) In this case, I’m not even sure I understand the question: are you asking whether, if we take the lattice generated by (say) (2,3) and (-1,5), and count the number of visible lattice points, the proportion will be the same? I don’t think that’s going to be the case, though I haven’t given enough thought as to why, or what the right answer will be.

]]> Comment on A better keyboard layout for typing IAST on Mac OS X (based on EasyUnicode) by Typing Kannada on Mac OS X | The Lumber Room http://shreevatsa.wordpress.com/2013/01/22/a-better-keyboard-layout-for-typing-iast-on-mac-os-x-based-on-easyunicode/#comment-11108 Sun, 07 Apr 2013 15:50:03 +0000 http://shreevatsa.wordpress.com/?p=2602#comment-11108 [...] now have three input methods: US, EasyIAST (see earlier post), and MacUIM [...]

]]> Comment on Lattice points visible from the origin by josephpzambrano http://shreevatsa.wordpress.com/2008/11/07/lattice-points-visible-from-the-origin/#comment-10023 Sun, 31 Mar 2013 15:41:25 +0000 http://shreevatsa.wordpress.com/?p=628#comment-10023 I realize this is an old post and I apologize if commenting on it at this point is frowned upon but does this analysis carry over to lattices generated by any basis set?

]]> Comment on A better keyboard layout for typing IAST on Mac OS X (based on EasyUnicode) by S http://shreevatsa.wordpress.com/2013/01/22/a-better-keyboard-layout-for-typing-iast-on-mac-os-x-based-on-easyunicode/#comment-7650 Thu, 07 Mar 2013 17:22:56 +0000 http://shreevatsa.wordpress.com/?p=2602#comment-7650 Glad to have helped. If you try out either EasyUnicode or my EasyIAST above, and have any reactions, I’d be interested in knowing them. Regards,

]]> Comment on A better keyboard layout for typing IAST on Mac OS X (based on EasyUnicode) by govindkanshi http://shreevatsa.wordpress.com/2013/01/22/a-better-keyboard-layout-for-typing-iast-on-mac-os-x-based-on-easyunicode/#comment-7649 Thu, 07 Mar 2013 09:55:57 +0000 http://shreevatsa.wordpress.com/?p=2602#comment-7649 Thanks for sharing this information. I was looking for layout information which is easier to get used to for long time.

]]> Comment on Examples of bad design by Design | Interaction Design http://shreevatsa.wordpress.com/2010/06/29/examples-of-bad-design-2/#comment-7647 Mon, 04 Mar 2013 02:14:10 +0000 http://shreevatsa.wordpress.com/?p=1937#comment-7647 [...] Souce: http://shreevatsa.wordpress.com/2010/06/29/examples-of-bad-design-2/ [...]

]]> Comment on Moving rows in OO.o Calc (OpenOffice.org Spreadsheet) by Also Anonymous http://shreevatsa.wordpress.com/2006/06/11/moving-rows-in-ooo-calc-openofficeorg-spreadsheet/#comment-7645 Mon, 25 Feb 2013 14:08:31 +0000 http://shreevatsa.wordpress.com/2006/06/11/moving-rows-in-ooo-calc-openofficeorg-spreadsheet/#comment-7645 holding alt while moving is nice.. thanks Anonymous.

]]> Comment on dpkg error: trying to overwrite X, which is also in package Y by fiskfisk http://shreevatsa.wordpress.com/2006/04/16/dpkg-error-trying-to-overwrite-x-which-is-also-in-package-y/#comment-7644 Thu, 21 Feb 2013 19:17:28 +0000 http://shreevatsa.wordpress.com/2006/04/16/dpkg-error-trying-to-overwrite-x-which-is-also-in-package-y/#comment-7644 Thank you, that saved quite some time! \o/

]]> Comment on Using Gmail with mutt, the minimal way (IMAP update) by Baz http://shreevatsa.wordpress.com/2007/07/31/using-gmail-with-mutt-the-minimal-way/#comment-7515 Thu, 24 Jan 2013 20:18:13 +0000 http://shreevatsa.wordpress.com/2007/07/31/using-gmail-with-mutt-the-minimal-way/#comment-7515 Which is more likely – that someone will break into your server and look at this file, or that someone will watch you typing your password over your shoulder?

]]> Comment on The potions puzzle by Snape in Harry Potter and the Philosopher’s Stone by shvedsky http://shreevatsa.wordpress.com/2012/10/22/the-potions-puzzle-by-snape-in-harry-potter-and-the-philosophers-stone/#comment-7487 Mon, 14 Jan 2013 07:21:11 +0000 http://shreevatsa.wordpress.com/?p=2589#comment-7487 Just in case you haven’t seen it before, google for “Harry Potter and the Methods of Rationality”. It’s a fanfic work (not finished yet, but ongoing), where everybody acts rationally, in full awareness of all the magical world uncovered through the series from the very beginning. And Harry is also very clever, tries to perform scientific experiments on magic, maybe because his stepfather is a university professor. :) It mocks stupid rules of quiddich, introduces much more “clever plotting”, contains many geek references, and is fairly well-written.

]]> Comment on Premature optimization is the root of all evil by Software optimization – a good excuse to go gaga ? | tamasrev http://shreevatsa.wordpress.com/2008/05/16/premature-optimization-is-the-root-of-all-evil/#comment-7478 Sat, 15 Dec 2012 23:01:49 +0000 http://shreevatsa.wordpress.com/?p=450#comment-7478 [...] well-known since the seventies that premature optimization is root of all evil. This saying is so old that it was part of my CS education in the early 2000′s. (The best [...]

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