# All Questions

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### Why if $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic? [duplicate]

In the notes I'm studying from ( again =) ) I read: If $(\mathbf B, \cdot)$ is a finite order group with prime order then $(\mathbf B, \cdot)$ is cyclic Could someone give me a justification for ...
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### Confused about notation “:=” versus plain old “=” [duplicate]

Relating to sets, I find the following in a text book: "...the set S := {1, 2, 3}". The book has an extensive notation appendix, but the":=" notation is not included. What exactly does ":=" mean, and ...
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### How mathematics would be different if the first derivations, conjectures and theorems would be others?

I've realised that mathematics is nothing else that an implication of some assumptions (plus the assumptions themselves, of course). We have axioms and we derive new "things", new rules, ideas, ...
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### Variance computed using Taylor series does not agree with numerical experiment [migrated]

I would like to estimate an angle $\theta\in\left(-\frac{\pi}{2},\frac{\pi}{2}\right)$ given the noisy observations of its sine and cosine (this is related to my earlier question). My estimator is ...
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### A question about the definition of adjoint functors

Aluffi's "Chapter 0", on pg. 492, says the following: Let $C$ and $D$ be categories, and let $\mathcal{F}:C\to D, \mathcal{G}:D\to C$ be functors. We say that $\mathcal{F}$ and $\mathcal{G}$ are ...
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### Is it possible to divide an equilateral triangle into N equal parts?

It’s easy to divide an equilateral triangle into $n^2$, $2n^2$, $3n^2$ or $6n^2$ equal triangles. But can you divide an equilateral triangle into 5 congruent parts? Recently M. Patrakeev found an ...
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### Arrange black and white balls so that each pair of white balls is separated by at least two black balls

I am trying to solve the following question: How many linear arrangements of $m$ white balls and $(n-m)$ black balls are possible such that each pair of white balls is separated by at least two ...
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### Is incorrect the Landau's demonstration?

Leon Henkin says, at the end of his On Mathematical Induction text, Edmund Landau (Foundations of Analysis) failed to demonstrate the existence and uniqueness of adding natural numbers, because ...
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### There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them. [on hold]

There are 267 not isomorphic groups of order 64. How many of these are Abelian? Describe them.
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### (Terminology_Taylor Series) “expand at $x_0$, evaluate at x, affine approximation”

I am reading one-variable calculus book where it explains Taylor series and little confused with the following terms: (1) Expand $f(x)$ at $x_0$ (2) Evaluate $f(x)$ at x (3) Best Affine, ...
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### Expressing the orthogonal projections on a linear operator $T$'s eigenspaces as polynomials in $T$
In the inner product space $\mathbb{C}^{2}$ with its standard inner product, let $$T\begin{pmatrix} x\\y \end{pmatrix} = \begin{pmatrix} 3x+4y\\-4x+3y \end{pmatrix}$$ a linear operator. Express the ...