# All Questions

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### Are Sylow p-subgroups in conjugate?

Let $G$ be an infinite group and $p$ be a prime number. Let $\mathscr{C}$ be a chain of p-subgroups of $G$ ordered by inclusion. Then, for every element of the union of $\mathscr{C}$ has an order ...
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### Problems in formalizing these sentences

This is the first sentence that I have to formalize: "Every student likes at least one type of cake" Let: $S(x)$ stands for 'x is a student' $C(x)$ stands for 'x is a type of cake' $L(x,y)$ ...
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### Prove that $|A\cap B| \le \frac {1}{2} |A|$ where $A,B$ are two subgroups of $G$

Suppose $G$ is a finite group, $A,B$ are subgroups of $G$ and $A$ isn't a subgroup of $B$. Prove (by using Lagrange's theorem) that $|A\cap B| \le \frac {1}{2} |A|$.  This is what I have so far: ...
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### Infinite group with finite order elements [closed]

Can you give me an example of an infinite group in which every element has order $3$ (except identity) ?
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### Is $c$ (as in $y=mx+c$) the $x$ or $y$ intercept?

In $y=5x+6$, is the $6$ the $y$-intercept or the $x$-intercept? I can't remember and need to know for revision.
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Deciphering the definition of the upper limit, we see that $\limsup x_n=L$ is and only if the following two conditions are fulfilled: (a) $\forall\epsilon>0\;\;\exists N\in\mathbb{N}$ such that ...
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### Relationship between asymptotic distribution and logarithmic sums of elements of subset of the natural numbers

Consider a subset $A$ of the natural numbers analogous to the primes (but rarer). Let $a_n$ denote the $n$th element of $A$, and $a(n)$ denote the number of elements of $A$ less than or equal to $n$ ...
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### Finding Modules with Given Length of Composition Series

I study a course about commutative algebra, and I saw many questions as the following: Find an example of a $\mathbb{Q}\left[\lambda_1,\lambda_2\right]$-module with $\ell\left(M\right)=3$ (i.e., ...
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### Sigma-fields and probability

I'm unsure what this question asks of me. For (i) I have given a power set with 16 elements in terms of a,b,c and d. I don't understand what I need to do for (ii). I believe (iii) is fairly ...
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### Trig substitution fails for evaluating $\int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx$?

Evaluate the integral $$\int \frac{\cos x \sin x}{\sin^2{x} + \sin x + 1} dx$$ Basically I could substitute: $t = \sin x$ and get: $$\int \frac{t}{t^2 +t + 1} dt$$ But, ...
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### zero raised to power zero in Church encoding

In Church encoding of the natural numbers in lambda calculus raising zero to the power zero gives the answer zero. Does anybody know of an encoding where the answer is 1?
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### What bound does the Hamming bound give you for the largest possible size of a $t$-error-correcting code of length $2t + 1$?

Let $\mathbb{A}$ = $\{0, 1\}$ and suppose $t$ is a positive integer. What bound does the Hamming bound give you for the largest possible size of a $t$-error correcting code of length $2t+1$? I have ...
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### Blocks in a layer? $X$ layers of blocks in a triangle, which $Y$ being the total number of blocks… (w/o using triangular numbers)

I have $32$ layers of blocks in a triangle. However, for the sake of variation, let's call that $X$. I have $1000$ square blocks total. We'll call those $Y$. The first layer of the triangle has $1$ ...
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### Proof the $\mathrm{rank(rows)=rank(columns)}$

Assume we have matrices $A=BC$.It is obvious that the $i$th row of $A$ is a linear combination of the rows of $C$ with coefficients from the $i$th row of $B$ or $b_{i1}C_1........b_{in}C_n$. ...
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### Projective Geometry with Clifford Algebra - lost Inner Product

Projective geometry may be studied with the tools of Clifford Algebras by adding a new direction (see for example this article). But as far as I understand it, only blades and null spaces are used for ...
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### Bass' paper on Gorenstein rings

I am currently reading the paper On the ubiquity of Gorenstein rings by Hyman Bass. I found difficulty to understand the proof of Proposition (7.2). Under the the following setting: $A$: commutative ...
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### Exponents for Hölder functions on metric spaces

Sometimes people talk about Hölder functions on metric spaces without mentioning the allowed range for the exponent. On manifolds, it's traditional to assume an exponent in $[0, 1]$, since all ...
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### A digraph is a graph where every edge is directed. How many digraphs on $n$ vertices are there?

So far I have that between any two vertices (say $j$ and $k$) there are 3 options. there is no edge between $j$ and $k$ there is an edge directed from $j$ to $k$ there is an edge directed from $k$ ...
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### Metric spaces - Limit points and Isolated points

I apologise for this, but I have numerous potential misunderstandings. $\Bbb R^2$ is a metric space, since it is a subspace of the Euclidean space, $\Bbb R^n$ I can look at a set $E$ with elements ...
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### Possible integer roots of polynomial with real coefficents

If $p\in\mathbb{Q}[X]$, then the rational root theorem gives us possible integer roots of $p$. If $p\in\mathbb{R}[X]$, the theorem cannot be applied. Nevertheless, triangular inequality gives us lower ...
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### Ideal of $TV$ which trivially intersects $V$

Let $V$ be a vector space over a fied $\mathbb{K}$ and let $TV = \bigoplus_{n \in \mathbb{N}} V^{\otimes n}$ be its tensor algebra. Let $b \colon V \times V \rightarrow \mathbb{K}$ be a bilinear form ...
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### Showing $\{x\in \mathbb{R}:\mu(\{x\})>0\}$ is a countable set under certain conditions.

Let $\mu$ be a measure such that $(\mathbb{R}, \mathcal{B}_{\mathbb{R}}, \mu)$ is a $\sigma$-finite measure space. I have to prove that $D=\{x\in \mathbb{R}:\mu(\{x\})>0\}$ is a countable set. Let ...
Definition: Let $F$ a figure in the plane, its symmetry group is defined by $\Sigma(F):=\{\sigma \in O(2,\Bbb R)\mid \sigma(F)=F\}$. Here $O(2,\Bbb R)$ denotes the real orthogonal group. Exercise 1: ...