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Spaces with unique endomorphism monoids

If $(X,\tau)$ is a topological space, let $\text{End}(X)$ denote the collection of all continuous maps $f: X\to X$. With composition, this becomes the endomorphism monoid $(\text{End}(X), \circ)$. We say that the space $X$ has a unique endomorphism monoid if $\text{End}(X) \cong \text{End}(Y)$ as monoids, for some space $Y$, then the spaces $X$ and $Y$ are […]

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Spaces with unique endomorphism monoids

If $(X,\tau)$ is a topological space, let $\text{End}(X)$ denote the collection of all continuous maps $f: X\to X$. With composition, this becomes the endomorphism monoid $(\text{End}(X), \circ)$. We say that the space $X$ has a unique endomorphism monoid if $\text{End}(X) \cong \text{End}(Y)$ as monoids, for some space $Y$, then the spaces $X$ and $Y$ are […]

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Sum of Bessel functions to the fourth power, $\sum_{k\in\mathbb{Z}} J_k(x)^4$

Let $J_k$ denote the $k$-th order Bessel function of the first kind. I know that $$\sum_{k\in\mathbb{Z}} J_{\mu-k}(x) J_{\nu-k}(y) = J_{\mu-\nu}(x-y) \quad \forall x,y\in\mathbb{R},\mu,\nu\in\mathbb{Z},$$ so in particular, $\sum_{k\in\mathbb{Z}} J_k(x)^2 = 1 \ \forall x\in\mathbb{R}$. Now I was wondering whether there is a closed form expression for $$\sum_{k\in\mathbb{Z}} J_k(x)^4,$$ if $x\in\mathbb{R}$. Or, more generally, for $$\sum_{k\in\mathbb{Z}} J_{\mu-k}(x)^2 […]

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Noether’s Normalization Lemma of $k[x,y] /(xy)$

Let $R = k[x,y] / (xy)$. I want to find $x_{1} \in R$ such that $R$ is an integral extension of $k[x_{1}]$, where $k$ is some algebraically closed field. I believe $d = 1$ as stated in noether’s normalization lemma, but I could be wrong. I am having trouble finding such an element. Namely, if […]

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Dense set of compact-open topology

If $X,Y$ be topological spaces with $X$ locally-compact and Hausdorff and $Y$ non-trivial, then $C(X,Y)$ is closed in $Y^X$ when the latter is equipped with the compact-open topology. So it’s clearly not dense. However, I’m curious, what are some “nicely behaved” dense subsets of $Y^X$ (besides $Y^X-F$ where $F$ is a finite set of functions […]

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if two random variables are jointly Gaussian, does that mean they are Gaussian individually? or vice-versa?

I am new into the term of jointly Gaussian. I was wondering, suppose X and Y are two random variables having a Gaussian distribution. My question is whether they will be jointly Gaussian? Or in the opposite sense, if (X,Y) is jointly Gaussian, does that mean X and Y both have Gaussian distribution? What is […]

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$ [x,y]=y, [y,z]=z, [z,x]=x$ implies $x=y=z=e$ in a group

This is a question about commutators. I’m reading Lang’s Algebra, and in the section about free groups, Lang asserts that in a group $G$, if $x,y,z\in G$ satisfies $$ [x,y]=y, [y,z]=z, [z,x]=x$$ where $[x,y]=xyx^{-1}y^{1}$ is the commutator of $x$ and $y$, then we must have $x=y=z=e$. How does this hold? From the above relations, we […]

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Folland Exercise 4 chapter 5

If $X$, $Y$ are normed vector spaces the map $(T,x)\to Tx$ is continuous from $L(X,Y)\times x$ to $Y$. (That is, if $T_n\to T$ and $x_n\to x$ then $T_n x_n \to Tx$) My professor told me to use $\epsilon$ proof. So here is the proof: $$\forall\epsilon>0 \ \exists N_1\in\Bbb N \text{ s.t } \forall n\ge N_1: […]

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Prove that √det(ATA)=॥xxy॥

If x,y ∈ R3 and A is a 2 by 3 matrix whose columns are x and y. Is there a way to solve this without solving the square root of the det?

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Folland Exercise 4 chapter 5

*If $X$, $Y$ are normed vector spaces the map (T,x)$\rightarrow$Tx is continuous from $L(X,Y)\times x$ to $Y$. ( That is, if $T_n\rightarrow T$ and $x_n\rightarrow x$ then $T_nx_n\rightarrow Tx$ My professor told me to use $\epsilon$ proof. So here is the proof: $\forall\epsilon>0$ $\exists N_1\in\Bbb N$ s.t $\forall n\geq N_1$ $||T_n-T||<\epsilon/(2(||x||+1))$. $\forall\epsilon>0$ $\exists N_2\in\Bbb N$ […]