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 Apr20 revised Fast method or direct generation of random upper triangular matrix using integer-restricted gauss elimination fixed constraint Apr19 asked Fast method or direct generation of random upper triangular matrix using integer-restricted gauss elimination Apr15 awarded Informed Apr12 accepted maximizing a coordinate of $x^T A^T A x \leq r^2$ Apr12 comment maximizing a coordinate of $x^T A^T A x \leq r^2$ I'm assuming $\mathbf{A}^{-T} = \left(\mathbf{A}^T\right)^{-1} = \left(\mathbf{A}^{-1}\right)^T$? - I have never seen this notation before but it makes sense. Very nice solution, thanks! Apr12 awarded Curious Apr11 revised maximizing a coordinate of $x^T A^T A x \leq r^2$ added 12 characters in body Apr11 comment maximizing a coordinate of $x^T A^T A x \leq r^2$ For where I need it, my $\mathbf{A}$ matrix in this case is a collection of vectors describing a regular lattice in $\mathbb{R}^n$. That should rule out singular matrices. I guess I should add that in my question. Ok, so it WAS that division and my actual solution simply is $x=\left(A^T A\right)^{-1} e_\alpha$. Thanks! Apr11 revised maximizing a coordinate of $x^T A^T A x \leq r^2$ added 1 character in body Apr11 revised maximizing a coordinate of $x^T A^T A x \leq r^2$ correcting the derivatives to use $\partial$ Apr11 asked maximizing a coordinate of $x^T A^T A x \leq r^2$ Mar21 revised How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$? I believe that's the correct formula. In general there are ${{n}\choose{k}} 2^{n-k}$ $k$-dimensional sub-cells for the $n$-hypercube. Mar21 suggested approved edit on How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$? Mar21 accepted How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$? Mar21 comment How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$? @DanielRust ... if there is a highest finite dimension to get to all valid quasi-crystal structures in 3D (at least those of the cut-and-project variety) or if I just have to arbitrarily stop at some number of dimensions. Would there still be a reasonable parametrization for the more complex case of Meyer or Delone sets? Mar21 comment How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$? @DanielRust what I really want is an as general as possible without going overboard representation of (quasi-)crystal structures (in $\mathbb{R}^3$) which I can use in a genetic algorithm. I am supposed to find minimal energy configurations of soft particles using this method. I assume most such states will constitute classical non-quasi crystal structures but I want to avoid assumptions. The computer is supposed to answer that question after all. Cutting through rotated fixed-dimensional spaces can easily be represented by a bunch of angles. That's why I want to know ... Mar21 comment How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$? @DanielRust so what you are saying is that my first remark of being able to find "any" quasi-crystal by cutting through high-dimensional spaces at an angle isn't entirely correct? Based on what I quickly found just now on Delone set, I guess they (or, perhaps more restrictively, Meyer Sets?) are indeed also a definition of quasi crystals. So there are some of those that can't be achieved by the above procedure? I guess I'll be content with the smaller set of the cut-and-project method for now. Mar21 comment How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$? @DanielRust One thing to surely exclude is hyperbolic or spherical lattices. I only need euclidean ones. And as said, in case the general case is too complex, I'm majorly interested in the flat 3D-case. So if there are subtle differences between definitions as they approach $\infty$ Dimensions or something, that shouldn't matter at all. Mar21 comment How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$? @DanielRust I am looking for crystals and quasi-crystals that can actually happen in chemicals. Not sure what is the right definition for that but defining them as being uniformly dense structures where any local patch can be brought to locally coincide with any other such local patch or something seems fine. Approaching it from a material physics angle, choose what ever works. I have no idea how to make it more precise - I don't know how various definitions in the literature differ. Mar21 asked How to generally describe all possible quasi-crystal structures in $\mathbb{R}^3$?