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');\n }\n }\n else {\n $(\"#searchresults\").empty();\n }\n };\n $(\"#searchbox\").keyup(doSearch);\n });})(jQuery);\n ~~~~\n\n 5. When run, this script uses the [jQuery][] library to asynchronously fetch\n `site.json`, to unhide the searchbox on success, and to wait for jQuery\n [`keyup`] events.\n \n `keyup` events are then handled by selecting keys from the previously\n fetched `site` JavaScript object whose values are matched by the `pat`\n [JavaScript regexp][] (which was, in turn, built from the value of the\n `#searchbox` text field).\n \n Finally, the matching object keys are transformed into links and added to\n the `#searchresults` element's `innerHTML`.\n\n[`pandoc`]: http://johnmacfarlane.net/pandoc/\n[`pandoc` Markdown]: http://johnmacfarlane.net/pandoc/README.html#pandocs-markdown\n[GNU Make]: http://www.gnu.org/software/make/manual/make.html\n[`find`]: http://www.gnu.org/software/findutils/manual/html_mono/find.html\n[jQuery]: http://docs.jquery.com/Main_Page\n[`keyup`]: http://api.jquery.com/keyup/\n[JavaScript regexp]: http://www.regular-expressions.info/javascript.html\n[JSON]: http://json.org\n[JavaScript]: http://en.wikipedia.org/wiki/JavaScript\n", "title": "Site Search", "updated": 1329774795.0}, {"id": "urn:uuid:751779fc-603e-4c20-a230-a9909a4bba91", "link": "http://mstone.info/posts/thoughts-20120212/", "published": 1329022800.0, "text": "% Thoughts for February 6-12\n% Michael Stone\n% February 12, 2012\n\nThis week, I was pleasantly reminded of the power of learning by Evelyn\nGlennie's talk on [how to listen][ted-eg] and of the power of parsimony by the\nfirst chapter of [Walter Rudin][rudin]'s book on [Real and Complex\nAnalysis][rca].\n\n[ted-eg]: http://www.ted.com/talks/evelyn_glennie_shows_how_to_listen.html\n[rudin]: http://www.math.wisc.edu/oldhome/news/WRudinObit.html\n[rca]: http://www.amazon.com/Complex-Analysis-International-Applied-Mathematics/dp/0070542341\n", "title": "Thoughts for February 6-12", "updated": 1329110677.0}, {"id": "urn:uuid:6afc8c1c-54de-444f-bff8-493268e8a4e4", "link": "http://mstone.info/posts/thoughts-20120205/", "published": 1328418000.0, "text": "% Thoughts for Jan. 30 -- Feb. 5.\n% Michael Stone\n% February 5, 2012\n\nAs I mentioned [last week], I've recently become interested in trying to fill\nin some gaps in my mathematical education, beginning with the theory of\ndifferential equations. To that end, I began by recalling a (very mildly\ncareful) definition of the word [\"equation\"](../equations-vs-definitions).\nNow, let's try out my definitions on [Coddington's][diffeq] exercise \u00a71.6.1(c).\n\nTo begin, let\n\n * $\\mathbb{N}$ be the non-negative integers,\n * $\\mathbb{C}$ be the complex numbers,\n * $I = [p, q] \\subset \\mathbb{R}$ be a closed real interval,\n * $(A, B)$ be the space of all functions from $A$ to $B$,\n * $C^1\\!\\!(A, B)$ be the space of continuously differentiable functions from $A$ to $B$,\n * $L(A, B)$ be the space of linear transformations from $A$ to $B$,\n * $D : C^1\\!\\!(A, B) \\to (A, L(A,B))$ be the total derivative operator.\n\nExercise \u00a71.6.1(c) asks us to find all the solutions of the equation\n\n$$y' - 2y = x^2 + x$$\n\nThis means that we're looking for a parameterization of the subset $K \\subset\nC^1\\!\\!(I, \\mathbb{C})$ satisfying the identity, for all $k \\in K$ and $x \\in I$:\n\n$$(Dk)(x) - 2k(x) - x^2 - x = 0$$\n\nWe are also given, via Theorem \u00a71.6.2 that for all $a \\in \\mathbb{C}$ and $b\n\\in C^0\\!\\!(I, \\mathbb{C})$ continuous on $I$, the sets of solutions $V$ of\ndifferential equations of the form:\n\n$$y' + ay = b(x)$$\n\nall have, for some $x_0 \\in I$ and for all $c \\in \\mathbb{C}$, the form\n\n$$V = \\left\\{\\phi : \\phi(x) = e^{-ax} \\int_{x_0}^{x} e^{at} b(t) \\mathrm{d}t + ce^{-ax}\\right\\}$$\n\nUnifying the signature of Theorem \u00a71.6.2 with the equation of Exercise\n\u00a71.6.1(c), we find that $a = -2$ and $b(x) = x^2 + x$ and that\n\n$$K = \\left\\{\\lambda x. e^{2x} \\int_{x_0}^{x} e^{-2t}(t^2 + t) \\mathrm{d}t + ce^{2x}\\right\\}_{c \\in \\mathbb{C}}$$\n\nOur remaining task is to simplify $K$ by simplifying\n\n$$\\int e^{-2t}(t^2 + t) \\mathrm{d}t$$\n\nTo that end, we start off knowing that, for all $a, b, c \\in \\mathbb{C}$ and\nfor all $n \\in N$,\n $$\\int e^{c t} = \\frac{1}{c} e^{c t}, \\int t^n = \\frac{1}{n+1} t^{n+1}, \\mbox{ and } \\int \\left(af + bg\\right) = a \\int f + b \\int g$$\nNext, via the product rule for differentiation, we know that\n $$D(e^{ct} t^n) = D(e^{ct}) t^n + e^{ct} D(t^n) = c e^{ct} t^{n} + n e^{ct} t^{n-1}$$\nNext, while we don't yet know how to compute $\\int e^{-2t}(t^2 + t)$, we might\nhope to find an appropriate antiderivative by solving\n $$e^{-2t}(t^2 + t) = D\\left(aF + bG + cH\\right)$$\nfor some $a, b, c \\in \\mathbb{C}$ and for $F = e^{-2t} t^2$, $G = e^{-2t} t^1$, and $H = e^{-2t} t^0$.\n\nFortunately indeed, this question reduces to solving the following system of\nthree linear equations in the Euclidean vector space generated by the basis\n$\\left\\{x^a e^{-2bx}\\right\\}_{a, b \\in \\mathbb{N}, a < 3}$ with coefficients in\n$\\mathbb{C}$:\n\n$$\n\\begin{align*}\n\\left[F + G\\right] &= \\left[\\begin{array}{c}DF \\\\ DG \\\\ DH\\end{array}\\right]^T \\left[\\begin{array}{c}a \\\\ b \\\\ c\\end{array}\\right] \\\\ \\\\\n \\left[F + G\\right] &= \\left[\\begin{array}{c}-2F + 2G \\\\ -2G + H \\\\ -2H \\end{array}\\right]^T \\left[\\begin{array}{c}a \\\\ b \\\\ c\\end{array}\\right] \\\\ \\\\\n \\left[\\begin{array}{c}F \\\\ G \\\\ H\\end{array}\\right]^T \\left[\\begin{array}{c}1 \\\\ 1 \\\\ 0\\end{array}\\right] &= \\left[\\begin{array}{c}F \\\\ G \\\\ H\\end{array}\\right]^T \\left[\\begin{array}{ccc}-2 & 0 & 0 \\\\ 2 & -2 & 0 \\\\ 0 & 1 & -2 \\end{array}\\right] \\left[\\begin{array}{c}a \\\\ b \\\\ c\\end{array}\\right] \\\\ \\\\\n \\left[\\begin{array}{c}1 \\\\ 1 \\\\ 0\\end{array}\\right] &= \\left[\\begin{array}{ccc}-2 & 0 & 0 \\\\ 2 & -2 & 0 \\\\ 0 & 1 & -2 \\end{array}\\right] \\left[\\begin{array}{c}a \\\\ b \\\\ c\\end{array}\\right]\n\\end{align*}\n$$\n\nFrom here, a quick trip through [Octave] yields:\n\n~~~~ { .octave }\noctave:1> A = [-2, 0, 0; 2, -2, 0; 0, 1, -2]\nA =\n\n -2 0 0\n 2 -2 0\n 0 1 -2\n\noctave:2> inv(A)\nans =\n\n -0.50000 0.00000 0.00000\n -0.50000 -0.50000 0.00000\n -0.25000 -0.25000 -0.50000\n\noctave:3> inv(A) * [1; 1; 0]\nans =\n\n -0.50000\n -1.00000\n -0.50000\n~~~~\n\nThus we have\n $$\\left[\\begin{array}{c}a \\\\ b \\\\ c\\end{array}\\right] = \\frac{-1}{2} \\left[\\begin{array}{c} 1 \\\\ 2 \\\\ 1 \\end{array}\\right]$$\nand\n $$\\int e^{-2t}(t^2 + t) \\mathrm{d}t = \\int D\\left(aF + bG + cH\\right) = aF + bG + cH$$\nand, finally,\n $$K = \\left\\{\\lambda x. \\frac{-1}{2} (x^2 + 2x + 1) + ce^{2x}\\right\\}_{c \\in \\mathbb{C}}$$\n\n[diffeq]: http://www.amazon.com/Introduction-Ordinary-Differential-Equations-Mathematics/dp/0486659429\n[last week]: http://mstone.info/posts/thoughts-20120129/\n[Octave]: http://www.gnu.org/software/octave/\n", "title": "Thoughts for Jan. 30 -- Feb. 5.", "updated": 1329767589.0}, {"id": "urn:uuid:0982d0bd-d670-4d86-beca-0df4d11dae10", "link": "http://mstone.info/posts/equations-vs-definitions/", "published": 1328418000.0, "text": "% Equations vs. Definitions\n% Michael Stone\n% February 5, 2012\n\nAbout a year ago, in the context of a course on Scheme, a friend and colleague\nasked me:\n\n > *What's the difference between an equation and a definition?*\n\nHere is my much-belated answer, resurrected as a reminder and a guide to myself\nas I [begin][last week] my own study of equations (this time of the\n[differential][diffeq] variety)...\n\n\n\nThe character \"`=`\" is commonly used to signify six different things; namely:\n\n * an equivalence relation\n * the standard (numeric) equality procedure\n * equations\n * identities\n * definitions\n * assignments\n\nEquivalence relations are reflexive, symmetric, and transitive binary\nrelations.\n\nThe standard (numeric) equality procedure is a procedure whose application to\narguments evaluates to `#t` when all arguments are numerically equal and to\n`#f` otherwise.\n\n~~~~ { .scheme }\n(= 1 1) ;; -> #t\n(= 1 2) ;; -> #f\n~~~~\n\nEquations are combinations whose head is the equals symbol, like:\n\n~~~~ { .scheme }\n(= y (* 2 x))\n~~~~\n\nLike any syntax, equations can be assigned several interpretations. One\ninterpretation that is popular in math is to treat equations as defining\nrelations; that is, as describing one or more sets of solutions of the\nequation(s).\n\n\n\nA solution is an environment in which a set of equations all evaluate to `#t`.\nFor example, the evaluation of the following \"let\" expression proves that\n`(4, 2)` is a solution of the equation `y = 2*x`:\n\n~~~~ { .scheme }\n(let ((y 4) (x 2)) (= y (* 2 x))) ;; -> #t\n~~~~\n\nIdentities[^identities] are equations that have been (perhaps implicitly) universally\nquantified over a set of satisfying assignments. Identities are mostly used to\nrelate values like `2`, `+`, and `*` as in:\n\n~~~~ { .scheme }\n(\u2200 ((n \u2115)) (= (+ n n) (* 2 n)))\n~~~~\n\n[^identities]: While there are few identities written in Scheme, there are\n several computer algebra systems written in Scheme-like languages in which\n plenty of identities have been written.\n\nDefinitions are compound terms whose evaluation has the side-effect of\ninstalling a new variable-to-value binding into the current environment:\n\n~~~~ { .Scheme }\n(define f (\u03bb (x) (* 2 x)))\nf ;; -> $