# All Questions

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### A special type of differential equations

Working in optimal control of PDEs, I came across a type of evolution problem that has instead of an initial condition a link between the initial state and the final state. Here is a simplified ...
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269 views

### Points on curves of genus 3

Let $Y$ be a smooth complex projective curve of genus two, $X$ a Galois cover of degree two of $Y$ and $K$ the canonical divisor of $X$. Let $i$ be the involution of $X$ over $Y$. Can one find a point ...
• 219
1 vote
74 views

### Proof that $\max_i(d_i^{(n)})/ \sqrt{n}$ tends to $0$ when $\sum_i (d_i^{(n)})^2 / n$ converges

Let $d_i^{(n)} \in \mathbb{N}$ be a sequence of integer valued sequences such that $$\lim_{n\rightarrow\infty} \sum_{i=1}^n \frac{(d_i^{(n)})^2}{n} = C < \infty$$ Is it ...
79 views

### Truncated circle method and partitions

Let $p(n)$ be the unrestricted partition function. Then it is known that its generating function $F(z)$ is expressible as an infinite product $$F(z)=1+\sum_{n\ge1}p(n)z^n=\prod_{k\ge1}(1-z^k)^{-1}.$$...
• 444
1 vote
68 views
+100

### Popular algorithms (stopping rules) with output - a prefix of a permutation

What are some popular settings, when we look at the elements of a randomly generated permutation one by one, and we use certain stopping rule which, as a result gives us a prefix of the observed ...
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### Find the integer solution to when $15 \cdot 2^n + 1$ is a perfect square [closed]

I tried to solve it using the $\text{mod}$ but the only reasonable observation made by me was that $n≥3$ except when $n=0$ using ($\text{mod}$ $8$), which did not help, unfortunately. Then I tried to ...
1 vote
38 views

### Lax pair and cubic nonlinear Schrödinger equation

Motivation: I'm trying to understand the Section 4 in the this section 4 in paper" Low regularity conservation laws for integrable PDE by Killip-Visan-Zhang ö It reads as follows: many completely ...
• 133
1 vote
131 views

### Concrete sheaves

On the nLab, given a local $S$-topos $E$, a concrete sheaf is defined as an object that is separated with respect to the local isomorphisms (the morphisms that are inverted by the global sections ...
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1 vote
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### Topological rings with a final topology

Given a family of ring homomorphisms $\phi_i : X \rightarrow Y_i$ where each $Y_i$ is a topological ring and consider the initial topology on $X$, i.e. the coarest topology such that each map is ...
• 171
1 vote
38 views

### Sherman-Davis type inequalities for non-negative operator in a Hilbert space with trivial kernel

Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of ...
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26 views

### Use Green's theorem to find a line integral for a open curve [migrated]

An area $R$ in the $xy$-plane is bounded by two curves $C_1$ and $C_2$. The two curves form a closed curve $C$ where the positive direction of rotation is anti-clockwise. The two curves can be ...
1 vote
40 views

### Is it true that any semisimple Jordan algebra has the unit element?

I found this theorem in Minnesota notes of Koecher [Thm. 9 p.70]: Thm. Any semisimple Jordan algebra has a unit element. During the text Koecher do not state that the Jordan algebra has to be finite ...
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29 views

### On boundedness of fractional integral with respect to the fractional Laplacian

Let $\alpha \in [0,1]$ and $p> 1$. Consider the smooth function $\varphi (x)= \sqrt{x^2+1}$, for all $x\in \mathbb{R}$, is it true that \begin{split} \int_{\mathbb{T}^2}\varphi ^{p-...
• 1
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+50

### Learning roadmap for admissible representations of $\widehat{\mathfrak{g}}$ (affine Lie algebras)

Let $\mathfrak{g}$ be a finite dimensional semisimple Lie algebra over $\mathbf{C}$. A priori one might expect the representation theory of the affine Lie algebra $\widehat{\mathfrak{g}}$ (the Lie ...
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### What are the solutions in numbers of $xyz \mid x^n + y^n + z^n$, $x,y,z$ globally coprime

What are globally coprime integers $x,y,z\in \mathbb Z^*$ such that $xyz$ divide $x^n + y^n + z^n$? I have no other motivation for that problem but its inherent beauty and interest. Note that it can ...
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21 views

### Integration and Gaussian Elimination questions using an Expression [closed]

I am learning Integration and Gaussian Elimination and have reached this question which has left me completely stumped. I don't know where to begin and am looking for any advice on how to solve it. ...
• 1
1 vote
99 views

### Proof of lower bound on variance

I'm reading through the paper Poincaré type and spectral gap inequalities with fractional Laplacians on Hamming cube. However, I'm having a difficult time understanding the following proof: Lemma 2.1 ...
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### Young tableaux — irreps correspondence for simple complex Lie algebras

$\DeclareMathOperator\GL{GL}\DeclareMathOperator\SL{SL}$I have learned that Young tableaux which were originally introduced to study the irreducible representations of finite symmetric groups $S_n$ ...
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52 views

### Equivalence of Hilbert space norm associated to the harmonic oscillator and a sum of Sobolev and weighted $L^2$ norms

I have seen an equivalence claimed in a few places, but I do not know of a reference that actually proves it with details and it has been a while since I took graduate courses on all this. Apologies ...
55 views

### Can an upper bound for $r_{0}(n)$ be reached from a duality principle about the distinct primes $n$ "defines"?

Under Goldbach's conjecture, denote by $r_{0}(n)$ the smallest non negative integer $r$ such that both $n-r$ and $n+r$ are prime and by $k_{0}(n):=\pi(n+r_{0}(n))-\pi(n-r_{0}(n))$, so that $k_{0}(n)$ ...
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### How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?
Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$. How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on \$\Bbb C^+_*=\{z\in\Bbb C:\text{...