# All Questions

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### Question regarding divisibility test of 13

In order to develop a divisible test for 13, we use $1000 \equiv -1 \pmod{1001}$. I understand the idea; however, why do we use $1001$, can we use any smaller number? For example, to test for ...
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### “Discrete logarithm problem in many groups of cryptographic interest”?

In many articles and papers I find the phrase ...discrete logarithm problem in many groups of cryptographic interest... commonly used. I would like what groups are they exactly referring to when ...
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### find largest cube in lot of cuboids

sorry for my bad English... I've a system consists of many cuboids, many are adjacent to the others. My problem is... I want to find the largest cube in this system (which may consists of many of some ...
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### Show that this linear map has nontrivial kernel

$f$ is a linear map on a $k$-vector space $M$ with $f^n=0$ for some $n>0$. $k$ is a field. $k^d$ is $k$-algebra generated by $x_1,\ldots,x_d$ satisfy the relation $x_i x_j + x_j x_i =0$ for all ...
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### Quotient Space $\mathbb{R} / \mathbb{Q}$

I've just learned about topological quotient spaces and was wondering if anyone can help me with this example I thought of. Let $(\mathbb{Q}, +)$ be the usual group of rational numbers for addition, ...
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### $h+k=p-1$, $p$ prime. Prove $h!k! + (-1)^h \equiv 0 \pmod{p}$?

Suppose that $p$ is a prime. Suppose further that $h$ and $k$ are non-negative integers such that $h + k = p − 1$. I want to prove that $h!k! + (−1)^h \equiv 0 \pmod{p}$ My first thought is that by ...
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### Difference between pairwise distinct and unique?

I've come across the term "pairwise distinct" in many research papers. But, I don't understand how it differs from just saying that the elements of a set are unique instead of saying that they are ...
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### The natural inclusion of an infinite abelian group $G$ into $\widehat{\widehat{G}}$

I was recently trying to think of a simple example that demonstrates that the natural inclusion of an abelian group $G$ into ...
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### Trig identity to zeta function identity?

The inequality $$\zeta(s)^3 | \zeta(s + it)^4 \zeta(s + 2it)| \ge 1$$ follows from $$3 + 4 \cos(\theta) + \cos(2 \theta) \ge 0$$ How is that done? What is the relationship between zeta and the ...
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### Graph coloring problem (possibly related to partitions)

Given an undirected graph I'd like to color each node either black or red such that at most half of every node's neighbors have the same color as the node itself. As a first step, I'd like to show ...
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### Decidable? Turing machine runs X steps for certain input?

Question is the following language decidable: {(M)|given input "aaaaa" Turing machine M will perform at least 1295 steps} I would say, yes it is. Just let the Universal Turing Machine count each ...
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### Is language L context-free?

Is following language context-free? Alphabet: {a,b,c,d} L = {w | w is not in {aabbc,abc,add}} I think it is: {aabbc},{abc},{add} are all regular. Because of closure properties(Union) R = {w | w ...
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### Sub-discrete valuation ring?

Let $A$ be a DVR in its field of fractions $F$, and $F'\subset F$ a subfield. Then is it true that $A\cap F'$ is a DVR in $F'$? I can see that it is a valuation ring of $F'$, but how ...
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### Eigenfunction expansion solution to a PDE with a constant non homogeneous term

I'm wondering if the method of finding a solution to a nonhomogeneous PDE by the method of eigenfunction expansion works if the nonhomogeneous term is a constant, rather than a function of the ...
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### L* regular -> L regular?

If language L* (Kleene Star) is regular, does it imply that L is also regular?
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### Why can we think of the second fundamental form as a Hessian matrix?

Let $f: U \rightarrow \mathbb{R}^3$ be an immersion that parametrizes a piece of a surface, and let $(h_{ij})$ be the matrix for the second fundamental form of that surface. According to pg. 70 of ...
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### Is this matrix diagonalisable?

Let $A$ be the following matrix: $$A = \left(\begin{array}{rrr} -1 & \hphantom{-}3 & \hphantom{-}0\\ 0 & 2 & 0\\ -3 & 3 & 2 \end{array}\right).$$ I've found that the ...
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### Uniform continuity

I need to prove that if $f: (0,1) \rightarrow \mathbb{R}$ is Uniformly continuous then it is bounded. Thank you.
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### Splitting field of $x^{n}-1$ over $\mathbb{Q}$

From I.N.Herstein's Topics in Algebra. Chap 5 Sec 5.3 Page 227 Problem 8 Problem 8: If $n>1$ prove that the splitting field of $x^{n}-1$ over the field of rational numbers is of degree $\Phi(n)$ ...
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### Converting polar equation to cartesian coordinate polar equation and back again?

OK, so I have the following polar equation: $r = Θ/20$ And I would like to translate this a little to the right, and down from the polar origin. Now, I figure since I know cartesian coordinate ...
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### projective module

Let $(R,m)$ be commutative noetherian local ring with unity. Suppose $P$ is a finitely generated projective module over $R[X]$ of rank $n$ . Is $P$ free? If not,what is the counter example?
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### Finding the regular points of a rational map

Let $X$ and $Y$ be (irreducible, quasi-projective) varieties (over an algebraically closed field $k$), and $\phi: X \to Y$ be a rational map. I think I understand what it means for $\phi$ to be ...
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### Find the length of the curve: $y=\frac{x^{5}}{6}+\frac{1}{10x^{3}}\qquad 1\leq x\leq 2$

$$y=\frac{x^{5}}{6}+\frac{1}{10x^{3}}\qquad 1\leq x\leq 2$$ $$\frac{dy}{dx}=\frac{5}{6}x^{4}-\frac{3}{10x^{4}}$$ squaring this $$=\frac{25}{36}x^{8}+\frac{9}{100x^{8}}$$ Plugging into the formula ...
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### Is there a set-theoretic definition of Projective Space?

I posted this on mathOverflow previously which was the wrong place to post it and I was asked to try this forum instead. Can anyone explain this in simple terms: I met projective space via a recent ...
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### Computational Geometry journals

What are good computational geometry journals? I'm also counting geometric modeling as part of computational geometry. Are there any good journals that are considered to be non-mainstream in the US?
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### Can we make $\tan(x)$ arbitrarily close to an integer when $x\in \mathbb{Z}$?

My 7-year-old son was staring at the graph of tan() and its endlessly-repeating serpentine strokes on the number line between multiples of $\pi$ and he asked me the question in the title. More ...