[{"id":37,"title":"Is numerical analysis statisitcal at heart?","sha":"b4b4bea3213964f132dd13f780ff870d","user_id":57,"state":"published","body":"**[Interpolation and approximation of continuous functions by Hermite polynomials](https://statisticalnumericalanalysis.wordpress.com/2011/12/08/interpolation-and-approximation-of-continuous-functions-by-hermite-polynomials/)**\r\n\r\nThe [Chebyshev points](http://en.wikipedia.org/wiki/Chebyshev_nodes) in [-1,1] are the zeros of the [Chebyshev polynomials](http://en.wikipedia.org/wiki/Chebyshev_polynomials). \r\n\r\nFejér uniformly approximated arbitrary continuous functions f:[-1,1]\u0026rarr; \u0026#8477; by use of [Hermite polynomials](http://en.wikipedia.org/wiki/Hermite_polynomials) at Chebyshev points ([Interpolation and Approximation, P. J. Davis, 1963, pp. 118-121](http://www.amazon.com/Interpolation-Approximation-Dover-Books-Mathematics/dp/0486624951/ref=sr_1_1?ie=UTF8\u0026qid=1423937109\u0026sr=8-1\u0026keywords=Interpolation+and+Approximation)). These are polynomials that agree with the function at the Chebyshev points and whose derivative there is 0.\r\n\r\nChebyshev points have a \u0026#946;(1/2, 1/2) distribution. The generic orbits of the chaotic function 4x(1-x) also have a \u0026#946;(1/2, 1/2) distribution ([1](http://en.wikipedia.org/wiki/Logistic_map#Chaos_and_the_logistic_map)), and a large enough finite portion of such an orbit lies very close to the the Chebyshev points: could we use these points in place of Chebyshev points? More generally:\r\n\r\n**Question**: If f:[-1,1] \u0026rarr; \u0026#8477; is a continuous function and for each integer n, \u0026#915;(n) is a random sample of n points from a \u0026#946; (1/2,1/2) distribution on [-1,1] do the Hermite-Fejér polynomials corresponding to the grid points \u0026#915;(n) converge uniformly to f with probability 1?\r\n\r\nThe answer is likely \"yes\": [\"Polynomial Approximation\" by Gordon K. Smyth, pp. 5-6](http://www.statsci.org/smyth/pubs/EoB/bap064-.pdf).\r\n\r\nCan we formulate other ideas of numerical analysis as statistical theorems (with convergence in probability)? ([2](https://statisticalnumericalanalysis.wordpress.com/)).\r\n","created_at":"2015-02-14T18:23:32.159Z","updated_at":"2019-04-20T18:29:35.902Z","zenodo_id":22276,"doi":"http://dx.doi.org/10.5281/zenodo.15135","subject":null,"tags":["interpolation","approximation","fejer","hermite","chebyshev"],"vote_count":6,"view_count":1,"score":6.1,"deleted":false,"muted":false,"tweeted":false,"attachment_file_name":null,"attachment_content_type":null,"attachment_file_size":null,"attachment_updated_at":null},{"id":49,"title":"Those who walk among us","sha":"013f3addfa3f5a7db2871a922349b3dd","user_id":57,"state":"published","body":"An older definition of a species is “the largest group of organisms capable of interbreeding and producing fertile offspring” ([1]( http://en.wikipedia.org/wiki/Species)). It can be difficult to decide to which species an organism belongs, and especially so in the context of “cryptic species” in which individual organisms in related, but different species, are very similar in appearance (see, e.g. [2]( http://www.sciencedirect.com/science/article/pii/S1567134812002808),[3]( http://ww.w.sandflycatalog.org/files/pdfs/MS09N01P001.PDF),[4]( http://www.josephrossano.com/InstallationCrypticSpecies.php)). \r\n\r\nHuman evolution is a commonly accepted aspect of evolution ([5]( http://humanorigins.si.edu/resources/intro-human-evolution)), and several people have speculated whether there is a new human species that we haven’t noticed. See, e.g. ([6]( http://www.popsci.com/article/science/human-species-still-evolving)) in which Nye discusses the possibility and likelihood of a “Homo superius” walking among us. Nye assumes that such a new species would have to “outcompete” Homo sapiens, rather than live side by side as in a cryptic species.\r\n\r\nIs it possible that such a cryptic species of humans already exists? I like to imagine there is, and my bet is on “Homo systemicus”, whose most characteristic trait is to be able to think fluently and flexibly about issues and problems at a systemic level, rather than simply through the levels of examples. I suspect Alexander Grothendieck ([7](http://en.wikipedia.org/wiki/Alexander_Grothendieck)), for example, might have belonged to a postulated species Homo systemicus. \r\n","created_at":"2015-02-17T12:43:35.788Z","updated_at":"2015-02-22T01:10:57.051Z","zenodo_id":22943,"doi":"http://dx.doi.org/10.5281/zenodo.15452","subject":null,"tags":["evolution","human_species"],"vote_count":0,"view_count":1,"score":0.1,"deleted":false,"muted":true,"tweeted":false,"attachment_file_name":null,"attachment_content_type":null,"attachment_file_size":null,"attachment_updated_at":null},{"id":28,"title":"Is teaching about instruction or selection?","sha":"b0b3c84a223e16c8f066b8770831a962","user_id":57,"state":"published","body":"Teaching is commonly associated with instruction, yet in evolution, [immunology] (http://post.queensu.ca/~forsdyke/theorimm0.htm#Introduction), and [neuroscience] (http://www.mindcreators.com/NeuralDarwinism.htm), instructional theories are largely defunct. \r\n\r\nWe propose a co-immunity theory of teaching, where attempts by a teacher to alter student neuronal structure to accommodate cultural ideas and practices is sort of a reverse to the function of the immune system, which exists to preserve the physical self, while teaching episodes are designed to alter the mental self.\r\n\r\nThis is a theory of teaching that is based on the [inter-subjective relationship](http://en.wikipedia.org/wiki/Intersubjectivity) between teacher and learner. This theory posits that teaching does not, as is commonly assumed, take place via instruction from teacher to students, but rather through a process of selection in the learner’s brain, stimulated by materials and activities utilized by the teacher. In this theory, the mechanism that drives the selection process in learners’ brains is [co-regulated emotional signaling](http://www.apadivisions.org/division-39/publications/reviews/first-idea.aspx) between teacher and learner. From this perspective, the power of [formative assessment](http://weaeducation.typepad.co.uk/files/blackbox-1.pdf) is that it intrinsically carries with it emotional aspects for both learner and teacher, in that it provides a feedback relationship between them both, and so, according to the [Greenspan \u0026 Shanker](http://www.amazon.com/First-Idea-Language-Intelligence-Ancestors/dp/0306814498/ref=sr_1_1?ie=UTF8\u0026qid=1423770340\u0026sr=8-1\u0026keywords=Greenspan++Shanker) theory of cognitive symbolic development, promotes cognitive development.","created_at":"2015-02-12T19:53:14.014Z","updated_at":"2022-09-01T10:08:35.625Z","zenodo_id":22127,"doi":"http://dx.doi.org/10.5281/zenodo.15082","subject":null,"tags":["teaching","instruction","selection"],"vote_count":18,"view_count":1,"score":18.1,"deleted":false,"muted":false,"tweeted":false,"attachment_file_name":null,"attachment_content_type":null,"attachment_file_size":null,"attachment_updated_at":null}]