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 Mar 23 awarded Necromancer May 5 awarded Good Answer Sep 18 awarded Yearling Apr 17 awarded Nice Answer Dec 10 answered Abstract Thought vs Calculation Dec 10 comment Can we alter the axioms of Euclidean space to have $\mathbb{Q}^3$ as a unique model? For me a more natural candidate would be the $A^3$, where $A$ is the set of real rational integers. Otherwise, you would end up with certain standard curves (which physisist are interested) only having finitely many points. The question is then: what is arclenght or area and how do you calculate it? Dec 9 comment Direct proof of irrationality? Is using the fact that any non-integer algebraic integer is irrational considered a proof by contradiction? What about using that $x^2-2$ is irreducible (by Eisenstein) thus has only irrational real roots? Nov 23 comment Are Penrose tilings universal? Do aperiodic universal tilings exist? A 1x1 square and a 2x1 ddomino tile is universal. Also any set containing a 1x1 tile and a set of "supertiles" build from 1x1 tiles is universal. You can do the same on any lattice: pick a fundamental cell and supertiles build by copies from the fundamental cell (for example, hexagons and triangles would work; or a pararlelogram and half of it). I suspect that this is actually the only possibe case, but not sure. I am pretty sure that no aperiodic tile can be universal. Nov 7 comment fibonacci series mod a number Also note that the order of $GL_2(\Z_k)$ is much smaller than $n$, and the order of the matrix divides this order. If $l$ is the reminder of $n$ divided by this order, then $A^n=A^l \mod p$. Last but not least it is enough to consider the subgroup of matrices of $det =\pm1$. Oct 30 awarded Commentator Oct 30 comment How should one think about non-Hausdorff topologies? An elementary often used example of this type is the trig circle. When we look to the trig circle we view it as $\R/ 2\pi \Z$, yet often we work with real numbers not classes. Oct 28 comment Does $\frac{\mbox{lcm}(1,2,\dots,n+1)}{\mbox{lcm}(1,2,\dots,n)}\to\infty$? There exists a "simple" formula for the seuence: $$\frac{{\rm lcm} (1,2,3,.., n+1)}{\rm lcm} (1,2,3,.., n)}$$ is $p$ if (n+1)= p^ \alpha$for some prime$p$and$1$otherwise. Oct 28 revised When are infinitely many points in the orbit of a polynomial integers? deleted 27 characters in body Oct 28 revised When are infinitely many points in the orbit of a polynomial integers? added 79 characters in body; added 12 characters in body; added 221 characters in body Oct 28 answered When are infinitely many points in the orbit of a polynomial integers? Oct 27 comment describe subsets of the integers closed under the binary operation Ax+By Actually if$A=2, B=-1$or more generarily$A+B=1$the minimal set$S$is$S=\{ 1 \}\$. Oct 27 revised describe subsets of the integers closed under the binary operation Ax+By added 384 characters in body Oct 27 answered describe subsets of the integers closed under the binary operation Ax+By Oct 24 comment Finding the determinant of a matrix with LU composition Is there any reason why you want to calculate the determinant this way? This looks like an overkill, row reduction is much faster and as far as I know the easy LU decomposition algorithms include row-reducing the matrix. If you are really interested in the LU decomposition, there are couple algorithms listed on Wikipedia. A good description of the process (and for many other basic math results) is also given at this page: tutorial.math.lamar.edu/Classes/LinAlg/LUDecomposition.aspx Oct 21 answered Locating a submatrix within a matrix