My hobbies include doing research mathematics, doing elementary mathematics, writing things in latex, cooking, correcting grammar, and being skeptical. Fortunately there is a stackexchange website for each of these compulsions.
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Jul
14 |
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measure of Haar @LSpice Yes. (Also, the phrase "Gelfand pair" was added after I made my comment.) |
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Jul
14 |
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measure of Haar Replacing $x$ with $xy^{-1}$, this is equivalent to saying that $K$-biinvariance implies right invariance with respect to conjugates of $K$. There is a counterexample with $|G| = 6$. |
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Jun
21 |
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The connection between $\pi$, $e$ and $20$ Frankly I also think $e^\pi \approx 20 + \pi$ is also at least as much coincidence as spiritual. The difference $e^\pi - 8 \pi + 2$ is only slightly less than $10^{-2}$, but it just happens to be close to the difference $22 - 7\pi$, and by subtracting the two we get something smaller than $10^{-3}$. |
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Jun
21 |
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The connection between $\pi$, $e$ and $20$ Here $\pi! = \Gamma(\pi+1)$ by definition, right? I vote coincidence. |
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Jun
10 |
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Counting equal covering sets Synonyms: regular uniform designs, configurations, tactical configurations, 1-designs. See en.wikipedia.org/wiki/…. These are also more-or-less equivalent to biregular bipartite graphs. |
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May
29 |
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Centroid of $\Omega$ and $\partial\Omega$ concides then $\Omega$ must be a ball Won't this hold for any set with (nice boundary and) a point symmetry? |
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May
26 |
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Algebraic structure on conjugacy classes spell out the "I believe it is clear" part |
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May
25 |
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Algebraic structure on conjugacy classes Not so fast. $w$ could be $x^k$ or $y^k$. |
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May
17 |
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Evaluating binomial sum with $k^2$ Here is a simple expression: $\approx e^{cN^2}$. Depending on your application this may be a good enough way to think about it. |
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May
17 |
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Proving that a group is infinite and nonabelian Try showing that $S_3$ is a quotient. |
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May
16 |
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revised |
Let $G$ be a non-abelian finite group whose all non-linear irreducible characters are faithful. Is there a classification of these groups? added 94 characters in body |
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May
16 |
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Let $G$ be a non-abelian finite group whose all non-linear irreducible characters are faithful. Is there a classification of these groups? added 94 characters in body |
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May
16 |
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answered | Let $G$ be a non-abelian finite group whose all non-linear irreducible characters are faithful. Is there a classification of these groups? |
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May
2 |
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Finite inversive planes and $PSL_2(q)$ I have voted to reopen this question. I don't think we should close questions just because they contain more than one sentence ending with a question mark. Here the questions are so closely related that it would be unnatural to answer one without answering the other. Moreover, the question was closed well after I posted an answer that I feel is complete. |
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May
1 |
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answered | Finite inversive planes and $PSL_2(q)$ |
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Apr
26 |
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Expected size of determinant of $AA^T$ for random circulant and Toeplitz matrices Any chance of getting further details about the lower bound? It seems surprising. Is it some sort of saddle-point analysis? |
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Apr
26 |
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Expected size of determinant of $AA^T$ for random circulant and Toeplitz matrices Hi, Padraig Ó Catháin and I were discussing this question and answer this week. This argument is very interesting. However, for any $(0,1)$-matrix one actually has the following variant of the Hadamard bound: $\det A \le (n+1)^{(n+1)/2} / 2^n$. This follows from $\det A = \det \begin{pmatrix} 1 & 1 \\ 0 & A\end{pmatrix} = 2^{-n} \det \begin{pmatrix} 1 & 1 \\ -1 & 2A - 1 \end{pmatrix}$ followed by the Hadamard bound for an $(n+1) \times (n+1)$ matrix. Therefore your upper bound $\lim f_{\mathrm{cir+}}$ can be improved to $0$. |
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Apr
23 |
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What is the finite group $(\operatorname {PCO}^{\circ}_{2n})^{+}(q)$ Concretely, if $M$ is the Gram matrix of the symmetric bilinear form, the conformal orthogonal group is the group of matrices $g$ such that $g^T M g = \lambda M$ for some nonzero $\lambda \in K = \mathbf{F}_q$, and this defines a natural homomorphism $c : \mathrm{CO}(M) \to K^\times$, $g \mapsto \lambda$. Note that $\det(g)^2 = c(g)^{2n}$, so $\det(g) = \pm c(g)^n$, and $\mathrm{CSO}(M) = \{g \in \mathrm{CO}(M) : \det(g) = c(g)^n\}$. |
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Apr
23 |
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What is the finite group $(\operatorname {PCO}^{\circ}_{2n})^{+}(q)$ Updated guess: CO is the group of similarities of the orthogonal form, i.e., elements that map the orthogonal form to a scalar multiple of itself. When $k$ is algebraically closed that is the same thing as scalars times O, but once you take $F$-fixed points it's not quite, but scalars times O is an index-two subgroup (for $q$ odd). Probably $\mathrm{CO}^\circ$ for Malle--Testerman corresponds to CSO for Magma. In which case, yes, $(\mathrm{PCO}^\circ_{2n})^+(q) = \mathtt{PCSOPLus}(2n, q)$. |
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Apr
23 |
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What is the finite group $(\operatorname {PCO}^{\circ}_{2n})^{+}(q)$ @DaveBenson I must be wrong then. As far as I understood CSO means scalars x SO, so PCSO would be the same thing as PSO. |