Archive for algebra
data science ≠ algebra ???
Posted in Books, Kids, Statistics, University life with tags algebra, calculus, California, data science, high school mathematics, NYT, public universities, The New York Times, USA on September 7, 2023 by xi'an
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Posted in Books, Kids, Statistics with tags algebra, CNRS, conjugate normal model, cross validated, LaTeX, MathJax, Overleaf, PLMlatex on March 17, 2021 by xi'an
Le Monde puzzle [#960]
Posted in Kids, R with tags algebra, congruence, Le Monde, mathematical puzzle, number theory, R on April 28, 2016 by xi'anAn arithmetic Le Monde mathematical puzzle:
Given an integer k>1, consider the sequence defined by F(1)=1+1 mod k, F²(1)=F(1)+2 mod k, F³(1)=F²(1)+3 mod k, &tc. [With this notation, F is not necessarily a function.] For which value of k is the sequence the entire {0,1,…,k-1} set?
This leads to an easy brute force resolution, for instance writing the R function
[sourcecode language=”r” gutter=”false”]
crkl<-function(k) return(unique(cumsum(1:(2*k))%%k))
[/sourcecode]
where 2k is a sufficient substitute for ∞. Then the cases where the successive images of 1 visit the entire set {0,1,…,k-1} are given by
[sourcecode language=”r” gutter=”false”]
> for (i in 2:550) if (length(crkl(i))==i) print(i)
[1] 2
[1] 4
[1] 8
[1] 16
[1] 32
[1] 64
[1] 128
[1] 256
[1] 512
[/sourcecode]
which suspiciously looks like the result that only the powers of 2 k=2,2²,2³,… lead to a complete exploration of the set {0,1,…,k-1}. Checking a few series in the plane back from Warwick, I quickly found that when k is odd, (1) the sequence is of period k and (2) there is symmetry in the sequence, which means it only takes (k-1)/2 values. For k even, there is a more complicated symmetry, with the sequence being of period 2k, symmetric around its two middle values, and taking the values 1,2,..,1+k(2k+1)/4,..,1+k(k+1)/2. Those values cannot cover the set {0,1,…,k-1} if two are equal, which means an i(i+1)/2 congruent to zero modulo k, hence equal to k. This is clearly impossible when k is a power of 2 because i and i+1 cannot both divide a power of 2. I waited for the published solution as of yesterday’s and the complete argument was to show that when N=2p, the corresponding sequence [for N] is made (modulo p) of the sequence for p plus the same sequence translated by p. The one for N is complete only if the one for p is complete, which by recursion eliminates all cases but the powers of 2…
statistics in Le Monde
Posted in Books, Kids, Statistics, University life with tags algebra, capture-recapture, Cornell, Le Monde, macroeconomics, Reinhart, roadkill, Rogoff on April 28, 2013 by xi'anIn the current weekend edition of Le Monde, science leaflet (soon to disappear from the weekend edition alas!, Pierre Barthélémy wrote his tribune on a (not that recent) PLoS paper on roadkills that seems to use capture-recapture (or should) to evaluate the real number of roadkills from their disappearance rate. And Cédric Villani muses in his carte blanche on the relevance of mathematical models in social sciences, using the recent blunder by Reinhart and Rogoff as an argument: this sounds for the least extreme as there are many counter-examples in political sciences, sociology, psychology, &tc. I think he is missing the point that, while all models are wrong (in the sense physical models can be “right”), there are some models that can prove useful. A last item of interest (?) was the announcement of the new volume in the maths popularisation series, which is dedicated to the fourth dimension. I hope they also deal with higher dimensions, otherwise it could get quickly boring! It reminded me of the textbook I had to teach from the semester I taught basic vector space algebra in Cornell: the chapter on dimension 2 got followed by one on dimension 3, then one on dimension 4…
five years in Edinburgh
Posted in Kids, Mountains, Travel, University life with tags algebra, applied mathematics, Edinburgh, finance, Holyrood Park, postdoctoral position, Statistics, University of Edinburgh on April 12, 2013 by xi'an
Got an email with this tantalizing offer of a five year postdoctoral position in mathematics at the University of Edinburgh:
Chancellor's Fellowship (five positions) [tenure-track posts at Lecturer or Reader level] Applications are invited for up to five Chancellor's Fellowship posts in Mathematics. Each Fellowship provides a research-focused reduced-teaching position for up to 5 years, followed immediately by a standard open-ended (ie "tenured") lectureship or readership post. Applicants should have research interests in any area of: Applied and Computational Mathematics Financial Mathematics Mathematical Physics Operational Research Pure Mathematics Statistics One of the positions will be specifically dedicated to algebra (Representation Theory, very broadly conceived). Applicants will have a research record of the highest calibre, exhibiting the potential to become an international leader. We welcome candidates whose interests may also reach out to other disciplines. Appointment will normally be made on the Lecturer scale, £37,382 - £44,607. Dependent on experience, and in exceptional circumstances, appointment may be to Senior Lecturer/Reader level for which the salary scale is £47,314 - £53,233. Interviews will be held during May 2013. Applications containing a detailed CV and an outline of a proposed research programme should be made online
