# All Questions

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### What are coquantales?

A quantale can be defined as a monoid in the monoidal category $\mathbf{Sup}$ of complete join semilattices, equipped with tensor product. What are examples of non-diagonal comonoids in ...
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### best book for practicing unsolved problems in differential equation and linear algebra

I started reading differential equation and linear algebra. Can anyone provide the link/book name where I may get many questions to practice. Generally, in the end of book only few problems are there. ...
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### How to express the equations as the Square root Like?

$$2\sin \frac{\pi}{16}= \sqrt{2-\sqrt{2+\sqrt{2}}}$$ What law is need to be applied here? Do I have to make the $\frac{\pi}{16}$ in a form that will be give us $\sqrt{2}$ like sin 45 degree?
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### Endomorphisms of a ring

Let $R$ be a ring with identity and let $R^n=P⊕P'$ be a direct sum decomposition with right $R$-modules as its components. We take $e\in\operatorname{End}(R^n_R)$ as the projection of $R^n$ onto $P$, ...
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### Evaluating a surface integral using symmetry

Let $S$ denote the surface of the cylinder $x^2+y^2=4,-2\le z\le2$ and consider the surface integral $$\int_{S}(z-x^2-y^2)\,dS$$ How can one use geometry and symmetry to evaluate the integral without ...
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### How to construct this modular Galois representation?

I want to know how to construct this modular Galois representation: The $\bmod p$ representation is semi-simple. For any lattice $T$ , $\dfrac{T}{p^{3}T}$ does not have a $G$-stable cyclic subgroup ...
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### How to find $\max|f(z)|$ in complex analysis?

The $M-L$ estimation lemma inequality states: $$\left |\int_\Gamma f(z) dz\right| < ML(\Gamma)$$ Where $M = \max|f(z)|$ and $L(\Gamma)$ is the arc length of $\Gamma$. Here: Wikipedia: ...
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### Problem Solving Question With Polynomials

For any polynomial $p$ with real coefficients, let $$S(p):= \{x\in \mathbb{R} \mid p(x) \in \mathbb{Z}\}$$ Prove that if $p$, $q$ are two polynomials such that $S(p) = S(q)$, then either ...
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### Sufficient condition for $M$ to have constant curvature

I decided to keep my original question. However, I'm having trouble only in a part of it (check NOTE) Let's consider a Riemannian manifold $(M,g)$, with the Levi-Civita connection $\nabla$. I would ...
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### Subgroups isomorphic to $\mathbb{Z}_{5}\oplus\mathbb{Z}_{5}$

Let $A=\mathbb{Z}_{360}\oplus\mathbb{Z}_{150}\oplus\mathbb{Z}_{75}\oplus\mathbb{Z}_{3}$. I need to calculate the number of subgroups of $A$ which is isomorphic to $\mathbb{Z}_{5}\oplus\mathbb{Z}_{5}$ ...
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### Proof of the Curtis-Hedlund Theorem: Why is there a function $\mu\colon A^S\to A$ such that $\tau(x)(1_G)=\mu(x_{|S})$ for all $x\in A^G$?

Here is the Curtis-Hedlund Theorem and its proof [the sets $V(\cdot,\cdot)$ used in this proof are explained below.]: My problem is I am not sure that I have understand that correctly. So I ...
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### How to derive the expected value of even powers of a standard normal random variable?

I am trying to prove that, for a standard normal random variable $Z \sim N(0,1)$, ${\mathbb E}[z^n]=n!!$ for even values of $n$. What I'm doing is integrating the p.d.f. of $Z$ which is ...
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### Finding a function in a differential equation

$$\left [ (m+1)f(x)+\frac{\mathrm{d}(\xi(x) f(x))}{\mathrm{d}x} \right ]y^{m}+\left [ (n+1)g(x)+\frac{\mathrm{d}(\xi(x) g(x))}{\mathrm{d}x} \right ]y^{n}=0$$ Find $\xi(x)$ I tried to do the ...
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### What is the connection with quadratic map

While reading Prof. Tao's Wordpress blog. I noticed he mentioned a different function $\displaystyle\Lambda_2(n):= \sum_{d|n}\mu(d)\log^2(n/d)\ldots(\ast)$ and said that this function vanishes ...
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### Number of Solutions of $y^2-6y+2x^2+8x=367$? [closed]

Find the number of solutions in integers to the equation $$y^2-6y+2x^2+8x=367$$ How should I go about solving this? Thanks!
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### Infinite number of poles in a domain of a meromorphic function?

Let $U\in \mathbb{C}$ be an open set and $f$ be meromorphic. Let $\gamma$ be a simple closed path in $U$ and let $D$ be the interior of $\gamma$ in $U$. Also suppose there are no zeros on the trace of ...
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### Need help with proof of $SO(3)$ is path connected

I am working on an exercise in Tapp's matrix groups for undergraduates. It is a proof that $SO(3)$ is path-connected. $SO(3)$ is the group $$SO(3) = \{A \in O(n)\mid \det A = 1 \}$$ where $O(n)$ is ...
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### Maximum and minimum average question

During a cricket match, India playing against NZ scored in the following manner: Partnership | Runs scored $1$st wicket$~~$ | $112$ $2$nd wicket | $58$ $3$rd wicket$~$ | $72$ $4$th wicket$~$ | $9$ ...
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### Strange Sigma Notation

How do I interpret this form of sigma notation? Do e1 and e2 take on all combinations of 1 and -1? If they do, what's the point? They just get multiplied inside the sum! FYI, this comes from equation ...
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### Identifying the types of singularities, poles and branch points on different functions

Someone please confirm that what i'm doing below is correct, -Thanks. $$sin(1-z^{-1}) \tag{1}$$ $sin(1-z^{-1})=\frac{z-1}{z}-\frac{(z-1)^3}{3!z^3}+\frac{(z-1)^5}{5!z^5}$ -has only an essential ...
### Why do we have $u_n=\bigl(\frac{n}{n+1}\bigr)^{n^2}=\exp(-n+o(n) )$
Why do we have $u_n=\left(\dfrac{n}{n+1}\right)^{n^2}=\exp\left(-n+o\left(n\right) \right)$ My attempts : $\ln\left(1+\dfrac 1n\right)=\dfrac 1n-\dfrac1{2n^2}+o\left(\dfrac{1}{n^2}\right)$ but I ...