# All Questions

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### Help with proof about functions and subsets

Problem: let $f: A \rightarrow B$. Prove that $f$ is injective if and only if for all $D \subset A$ we have that $f^{-1}(f(D)) = D$. Proof: => Suppose $f$ is injective. Let $x \in f^{-1}(f(D))$. ...
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### Does strictly positive density function on the real line with infinite expected value exist?

The problem is as stated in the title. I am looking for an example or a disproof, whether there exists a continuous density function on the whole real line with infinite expected value. Once again: ...
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### Reversed Cayley transformation for any unitary matrix

It is well known that if $Q$ is a unitary matrix such that $I+Q$ is invertible (where $I$ is the identity matrix), then $$A:=(I-Q)(I+Q)^{-1}$$ is skew-Hermitian ($A=-A^*=-\bar{A}^T$). This fact is ...
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### Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$.

Prove that $a+ib$ is prime in $\Bbb Z[i]$, of $a^2+b^2$ is prime in $\Bbb Z$. My Try: We can easily show that $\Bbb Z[i]$ is a FD but how can we show that $\Bbb Z[i]$ is a UFD. Because if we can show ...
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### Proving that a matrix product is singular

I just played around in mathematica and found out that it seems like if $A$ is an $m\times n$ matrix and B is an $n\times m$ matrix, with $m>n$, then $AB$ is singular. How does one go about proving ...
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### Is there a way to define the concept of manifolds so it looks more like “generalised affine spaces”?

What I have in mind is along the lines of this: Let $M$ a topological space, $V$ a normed vector space, and $$\boxminus \colon M\times M \to V,$$ $$\boxplus \colon M\times V \to M.$$ Then ...
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### Stupid question on linear subspaces formed by functions

The set of differentiable real-valued functions on (0,3) such that f'(2)=b is a subspace of $\mathbb{R}\leftarrow(0,3)$ if and only if b=0 ($\mathbb{R}\leftarrow(0,3)$ denotes the set of functions ...
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### Complex projective manifolds and smooth projective varieties

Look at the following theorem: The following two categories are equivalent: The category of smooth projective varieties over $\mathbb C$. (Where a variety is understood as in Hartshorne's ...
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### Math Combination

I have 3 bags with 3 balls each Bag 1 - white, blue , green Bag 2 - yellow, orange, purple Bag 3 - black, red, brown How do we calculate all possible 3 balls combinations(1 ball from each bag) ?
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### How does one prove that if $\cos(t) = \cos(t')$ and $\sin(t) = \sin(t')$ then $t = t' + 2k\pi$?

How does one prove that if $\cos(t) = \cos(t')$ and $\sin(t) = \sin(t')$ then $t = t' + 2k\pi$ ? I've tried proving the above statement, which I think is valid. I know $\sin(t)$ is injective on ...
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### Principle of explosion: Other arguments?

I've come across a proof-theoretic argument for explosion on Wikipedia, which is as follows: $A \ \ \wedge\sim A$ $A$ $\sim A$ $A \lor B$ $B$ $(A \ \ \lor \sim A) \implies B$ I've thought of ...
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### Simple composition factor

We know that if $e$ is a primitive idempotent in a semiperfect ring then $Re/Je$ is a simple left $R$-module, where $J$ is the Jacobson radical of $R$. Now, suppose that $e$ is an idempotent in an ...
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### Proof via strong induction of a string output

I'm still new to the whole proof thing (first class of discrete mathematics and analysis right now). I could do general induction problems, but the fact that 'n' is the output here along with the ...
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### Inclusion of Sobolev spaces with fractional order

Let $W^{k,p}\mathbb(R^n)$ the usual Sobolev space. We know that if $k>l$ and $1\leq p<q<\infty$, $(k-l)p<n$ and $$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}$$ then ...
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### Updated conditional probability - Elo rating

Consider a tournament where all teams play against each other, and suppose we have the probability for win-draw-loss for every match (e.g. with the prediction from Elo rating). Is there a feasible ...
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### probability of two successive random numbers has the same starting number

Question/problem(subtask b): What is the probability of two successive random numbers has the same starting number? What we do know is that a random number generator randomizes numbers of 6-digits ...
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### what is the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE?

I want to know the difference between renormalized solution and entropy solutions for nonlinear elliptic PDE when right hand side of the elliptic operator is $L^1.$ Also how does right hand side with ...
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### Saturation of the Babenko–Beckner inequality

The Babenko-Beckner inequality $|| \mathcal F f ||_q \geq C(q,p)||f||_p$ is a well-established theorem. It relates the $q$-Norm of a Fourier transform $\mathcal F f$ of a function $f$ to its ...
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### Understanding Tensor product of modules

This is follow up question Understanding the Details of the Construction of the Tensor Product If $M$ and $N$ are 2 A-modules, we define $Z = A^{MxN}$ as module generated by $MxN$. Then $D$ is ...
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### Boundedness and Schrödinger operators

I'm working on Schrödinger operators, currently struggling with the proof of $\nabla(\Delta + i)^{-1/2}$ bounded. The domain is $H^2(\mathbb R^3)$. So, of course $(\Delta + i)^{-1}$ is bounded, and ...
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### Find diff. function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) \neq 0, f(0,0) = 0$ and $|R(0,x)|/|x|^n \to 0 \forall n$. [on hold]

Give an example of a differentiable function $f:\mathbb{R}^2 \to \mathbb{R}$ such that $f(x,y) \neq 0, f(0,0) = 0$ and $|R(0,x)|/|x|^n \to 0 \ \forall n$. Here $R(x,y)$ denotes the error term while ...
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### Pre-dual of distributions with support in a closed subset

Usually, in order to define the space of distributions $\mathcal{D}'(\Omega)$ on an open subset $\Omega \subseteq \mathbb{R}^n$, one considers the space $\mathcal{D}(\Omega)$ of $C^\infty$-test ...