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A former participant of math olypmiads. I started to think about mathematics lately.

Most recent interests:

  • Gauge integrals: did you know that there exists an integral, whose definition is a generalization of Riemann's definition, yet its strength exceeds that of Lebesgue integral? Instead of limiting your partitions by a positive constant $\delta$: $t_{k+1}-t_k <\delta$, you allow restrictions by a positive function (the "guage") $\delta$: $t_{k+1} - t_{k} < \delta(c_k)$, where $c_k \in [t_k,t_{k+1}]$.
  • Primitive parts of sequences: Take a integer linear recurrence sequence $a_n$ arising from a monic quadratic equation in $\mathbb{Z}[x]$: $x^2 + ax +b = 0$. Examples: Fibonacci, Lucas, $\frac{a^n-b^n}{a-b}$ (where either both $a,b$ are integers, or are conjugate algebraic integers). Then: 1) Under some conditions, $a_n$ has a primitive divisor - a prime divisiors not dividing the previous elements (a statement which translates into statement about order of elements) - that's Zsigmondy's Theorem. 2) $\prod_{d|n} a_d^{\mu(\frac{n}{d})}$ is an integer sequence! For $\frac{a^n-b^n}{a-b}$ it yields the Cyclotomic homogeneous polynomials, and for Fibonacci it yields a sequence called its primitive part.

Dec
20
revised Odd-Order Groups with Cyclic p-Sylow Subgroups (for smallest p | G)
added 144 characters in body
Dec
20
comment Odd-Order Groups with Cyclic p-Sylow Subgroups (for smallest p | G)
@Timbuc I changed my wording. (BTW, $C_p \times C_p$ indeed gives a new example, yet $C_{p^2}$ is still a particular case of my example, by taking the field $\mathbb{F}_{q}$ to be a quadratic extension of a prime field.)
Dec
20
comment Odd-Order Groups with Cyclic p-Sylow Subgroups (for smallest p | G)
@Timbuc I meant "The subgroup generated by raising elements of $\mathbb{F}_{q}^{\times}$ to the $c$-power". As you said, your example can be expanded and is a generalization of my example. If no other example will be found in the next days I'll accept it as the answer, but for the mean time I'm leaving it as is.
Dec
20
comment Odd-Order Groups with Cyclic p-Sylow Subgroups (for smallest p | G)
@Myself Yes, those examples are the same. This is $G_{p,\frac{p-1}{q}}$. Can be seen by writing the group operation explicitly. In particular, this means we can't get more examples by taking non-abelian groups whose order is a product of 2 odd primes.
Dec
20
revised Odd-Order Groups with Cyclic p-Sylow Subgroups (for smallest p | G)
deleted 10 characters in body
Dec
20
asked Odd-Order Groups with Cyclic p-Sylow Subgroups (for smallest p | G)
Oct
8
awarded  Popular Question
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30
awarded  Explainer
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awarded  Autobiographer
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awarded  Popular Question
Sep
7
comment Problem with orientation in Stokes' Theorem
@lmsteffan I point my right hand's thumb in the direction of the normal (at a point near the curve). My other four fingers naturally curl in some direction which dictates the orientation of the curve.
Sep
7
asked Problem with orientation in Stokes' Theorem
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2
comment Stokes' Theorem - Stuck with a non-elementary integral
No mistake - this clever parameterization works (leads to $-8$). Thank you.
Sep
2
accepted Stokes' Theorem - Stuck with a non-elementary integral
Sep
2
comment Stokes' Theorem - Stuck with a non-elementary integral
@Semiclassical - Thanks, that's a useful and clever remark. We can replace $\vec{F}$ with $(4z,0,x)$ and use the parametrization suggested by Jyrki - $(x,y,z)=(\sin(2\theta), 2\sin^2\theta, 2\sin \theta), \theta \in [0,\pi]$.
Sep
2
asked Stokes' Theorem - Stuck with a non-elementary integral
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2
awarded  Curious
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awarded  Yearling
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22
awarded  Necromancer