# plm

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bio website location age member for 2 years, 6 months seen Mar 10 at 15:31 profile views 68

My name is Paul Le Meur. I live in Bordeaux, France.

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 May15 comment Proving $\cos \theta = \sin(\frac{\pi}{2} - \theta)$ for all $\theta$On the page you link to the author forgets to mention that putting the graph upside down is only replacing $t$ with $-t$ for the sine function because it is an odd function $sin(-t)=-sin(t)$. In general this does not hold, you invert the graph sending $f$ to $-f$ (not $t$), in particular $cos(-t)=cos(t)$, which allows to have the "symmetric" formulas $sin(t)=cos(\pi/2-t)$, $cos(t)=sin(\pi/2-t)$. May15 comment Proving $\cos \theta = \sin(\frac{\pi}{2} - \theta)$ for all $\theta$If you compose $f$ with $x\rightarrow x+T, T>0$, you shift its graph to the left, because lower values of x are (almost?) universally put on the left. You may check that on a graphing software -like Wolfram Alpha which is what you probably meant. May15 comment The topology on $\mathbb{R}$ with sub-basis consisting of all half open intervals $[a,b)$.If all else fails you may still answer the question by "Asking on math.stackexchange.". I am sure they would at least give partial credit. May15 comment Truth table reductionTake an undecidable problem $A$, like the decision problem for Peano Arithmetic ($A$ is a subset of the integers consisting of Gödel codes of true sentences in PA), and a decidable one $B$, like the set of all natural numbers, or the empty set. Then your truth-table reduction from $B$ to $A$ may ask $0=0$ or $0=1$ to the oracle for $A$, depending on what $B$ you picked, and the decision just be equal to what the oracle says -you could do the opposite, switch 0s and 1s appropriately. But a truth-table reduction (or Turing reduction) from $A$ to $B$ would imply that $A$ is decidable. May14 comment Maximize lateral surface of a square pyramid in a sphereI looked at the picture now, I thought a scoop was a half-sphere. Taking a full sphere changes the problem, but it must not really be harder. May14 comment Maximize lateral surface of a square pyramid in a sphereTo prove that the symmetric solution I proposed is maximal should not be too hard. And for Fourier analysis fans there must be explanations in terms of that and perhaps interpretations in terms of volumes on Lie groups with their Haar measure. I add this just to say something grandiose and surely wrong, my trademark -at least there is the Lebesgue measure involved :). May14 comment Maximize lateral surface of a square pyramid in a sphereHmm, I guess the problem asks for the maximum total area of the pyramid. And anyway you are certainly allowed to put the pyramid upside down. So you get the square face of area 200 plus 4 triangular faces of base the same length as the square and height $5\sqrt{6}=\sqrt{(5\sqrt 2)^2+10^2}$. The triangular faces have area $10\sqrt 2\cdot 5\sqrt{6}=100\sqrt 3$, so you have a total area of $200+400\sqrt 3\approx 900$. May14 comment Maximize lateral surface of a square pyramid in a sphereI am not too sure what the problem asks. If you take smaller and smaller triangular faces, you get a pyramid that is flatter and flatter, tending to a square in fact, a 2-dimensional set. Then the maximal square is contained in the same plane as the boundary of the scoop, and therefore has sides of length $10\sqrt 2$ and area $200$. May13 comment Differential equations: Connection between repeated roots of characteristic equation and generalized eigenvectors@GerryMyerson, Ok nomenclature varies. Some call a spiral "focus", some "sink/source", some "spiral sink/spiral source", and I guess there are other terms. I thought the term focus was for the simplest (un)stable node, where the matrix is a multiple of the identity, and all lines are eigenlines. Conclusion: Gerry is right again. :) May12 revised Differential equations: Connection between repeated roots of characteristic equation and generalized eigenvectorsadded 625 characters in body May12 revised Differential equations: Connection between repeated roots of characteristic equation and generalized eigenvectorsadded 625 characters in body May12 comment Differential equations: Connection between repeated roots of characteristic equation and generalized eigenvectors@GerryMyerson: True, they do not spiral. They bend once. Google has examples google.com/… For spiraling orbits you would not call the origin a focus, but rather a (spiraling) sink or source -this is the conjugate eigenvalue case, with nonzero real part. May11 revised Differential equations: Connection between repeated roots of characteristic equation and generalized eigenvectorsadded 419 characters in body May11 answered Differential equations: Connection between repeated roots of characteristic equation and generalized eigenvectors May8 comment eigenvector computation@JohnSmith, As Marvis and Robert said no: take $D=I$ the identity in $\mathbb Q^2$ and $C$ the translation to the right by one (a Jordan block with 3 $1$s). $CD=C$ is not diagonalizable. (Robert was quicker. :) May7 comment When did mathematicians think of axiomatically building or defining operations etc?@ZhenLin, At least I did it in comments, not in a response. :) I was serious (though consciously speculative), I even quoted wikipedia. I also really liked the insight that people well before Dedekind certainly had probably (or not) thought about axiomatizing the natural numbers. And the comments on Gauss too. Of course I may be wrong but if I am right the fact that it's puzzling to you would show that was real insight. :) May7 comment When did mathematicians think of axiomatically building or defining operations etc?Well, on the other hand, I remember Gauss commented about Fermat's last theorem being "unprovable" (in his lifetime) or Goldbach's conjecture, and many similarly simple statements in number theory, so he must clearly have thought very generally about the process of doing research in mathematics. Then it is natural to assume that he also understood how we create definitions, auxiliary/novel structures in mathematics, with operations and relations (which is why we call them "structures"), and look at their properties as wholes, therefore that he had a basic understanding of category theory. May7 comment When did mathematicians think of axiomatically building or defining operations etc?I would comment that Gauss, Cauchy, Riemann, and big minds around there, perhaps Euler, probably thought about defining the natural numbers and their operations via successor. Dedekind defined primitive recursive functions. I would say that Gauss probably thought about primitive recursive functions, and perhaps about groups, axiomatically. I don't think he came to the idea of sets, he seems to have been biased against such philisophical topics, and he probably did not have enough examples to feel the need for/use of category theory. May7 comment When did mathematicians think of axiomatically building or defining operations etc?I think I remember something about van der Waerden axiomatizing vector spaces, in a textbook... From wiki: "Walther von Dyck (1882) gave the first statement of the modern definition of an abstract group." And regarding vdW and his textbook "Algebra" wiki says: "This work systematized an ample body of research by Emmy Noether, David Hilbert, Richard Dedekind, and Emil Artin." May7 comment When did mathematicians think of axiomatically building or defining operations etc?The theory of sets is quite a different matter -from binary operations and arithmetic. It was invented, in the sense of developing its foundations and yoga, by Cantor, but there must clearly have been people (philosophers?) thinking about it before.