# All Questions

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### Calculation of the integral related to the gravitational shock wave

The following integral $$\int\limits_0^\infty \frac{\cos{\left(\frac{1}{2}\sqrt{3}s\right)}}{\sqrt{\cosh{s}-\cos{\theta}}}\,ds$$ can be found in the paper Tevian Dray and Gerard 't Hooft, The ...
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### Generic Smoothness Type of Results in Positive Characteristic

Let $f:X\to Y$ be a surjective morphism between two projective varieties over a field of characteristic $p>0$. Also assume that $f_*\mathcal{O}_X=\mathcal{O}_Y$, and $X$ is smooth. We know that ...
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### The line graph of a complete graph

There exist a $\left\{P_{5},C_{4}\right\}$- decomposition of the graph $L(K_{9})$
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### Difference in the Four Color Theorem

How is proving that any planar graph with maximum degree of four has a four coloring, different from proving the four color theorem? If they are different, then how would one prove it?
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### Topological Derivation of Leray Spectral Sequence

I'm interested in computing - to the extent possible - the Leray spectral sequence for a particular map which is almost, but not quite, a fiber bundle (e.g. a Seifert fiber space). The hardest step ...
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### Adherent value of sin(sqrt(n)) [on hold]

I have been struggling against this question for several hours and I really need some help. It is from my undergraduate real analysis course. Your time and help is greatly appreciated. I got struck ...
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### Isotopy class of closed 2-ball embedded in R^3

My intuition tells me that any two topological embeddings of the closed 2-ball (aka unit disk) into $R^3$ are isotopic in $R^3$. Is this correct? Maybe easy to prove? It seems like it should be easy ...
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### Use of infinitude of primes in the Green-Tao theorem

In a video I watched last night on nuking mathematical mosquitos, Matt Parker gave the following proof of the infinitude of primes: suppose there are finitely many primes. The Green-Tao theorem says ...
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### Factorisation of twisted polynomials

Let $K=\mathbb{C}((t))$ and let $K_m=\mathbb{C}((t^{1/m}))$. let $K\{x\}$ denote the ring of twisted polynomials. The addition in this ring is defined as usual, but the multiplication is adjusted by ...
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### List of counting proofs instead of linear algebra method in combinatorics

I've just come across this proof of the Graham-Pollak Theorem by Sundar Vishwanathan (thanks to Konrad Swanepoel's sporadic comments about it on this site), that must be called beautiful after its ...
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### Is there a generalization for the discrete fourier transform whereby eigenvalues are other roots of unity?

The eigenvalues of the discrete fourier transform are $\{1, -1, i, -i\}$ in approximately equal proportions. https://en.wikipedia.org/wiki/Discrete_Fourier_transform#Eigenvalues_and_eigenvectors Is ...
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### Loxodrome loop on a surface of revolution

Prove that there can be a closed loxodrommic curve on a surface of revolution only when it is doubly connected.
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### Finding topological properties under a metric on set of composition operators of L2 [on hold]

We define a new metric on all composition operators in L2: ‎‎‎‎‎$‎$‎‎‎‎d‎‎{R}(‎A‎,B)=‎\sqrt{‎{‎‎‎‎\paralle‎l ‎P‎{R(A)}‎‎- ‎‎‎‎‎‎ ...
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### Square wave in the limit of infinite frequency [on hold]

What is the new function obtained when one takes the limit of the square wave function when the frequency is taken to infinity? Does it depend on how the function is written down (e.g. defined as ...
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### A non locally compact group of finite topological dimension?

Is there a topological group which is Hausdorff, first countable, locally connected and has finite topological dimension, yet fails to be locally compact?
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### Opposite of an E2-algebra

Suppose $C$ is the monoidal $\infty$-category of modules over an $\mathcal{E}_2$-ring spectrum $A$. Let $C' = C$ as a category, but with opposite monoidal structure to $C$. Is $C'$ the category of ...
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### CLT for sums of an infinite sequence of rv with an asymptotic distribution

Excuse me if the question is ill-posed. I'll do my best to explain the problem.I have a vector $(x^{(n)}_1, x^{(n)}_2, \ldots x^{(n)}_n),$ whose individual components can be shown to be asymptotically ...
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### Has the attempt of proof of the Frankl conjecture by Vladimir Blinovsky been checked? [on hold]

I found his article in arxiv: http://arxiv.org/pdf/1507.01270.pdf. But i didn't find any response to the article and as I'm an undergraduate I have no knowledge to judge if this approach is promising. ...
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### Hahn-Banach theorem for arbitrary locally compact fields?

Does anyone know if the Hahn-Banach theorem is true for every locally compact field? Specifically, let $F$ be a finite algebraic extension of either $Q_p$, the $p$-adic completion of $Q$, or of ...
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### smoothness of boundary under Riemann mapping

Suppose there is a smooth Jordan curve separating the complex plane. For complicity, assume the curve is given by a graph $(x, \phi(x))$, where $\phi(x)$ is smooth, bounded, and derivatives are ...
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### an example for fundamental group of graph of groups [on hold]

suppose we have a graph $X$ with the vertex set $\left\lbrace v_1,v_2,v_3 \right\rbrace$ and the edge set $\left\lbrace e_1,e_2,e_3 \right\rbrace$ like a triangle. let $(\Gamma,X)$ be a graph of ...
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### Chern-Simons forms, characteristic numbers, and boundary terms?

For any principal $G$-bundle $P \to M$ with principal connection $\omega$, given a $G$-invariant polynomial $p: \mathfrak{g} \to \mathbb{R}$ we can construct a form $p(F_\omega)$ on $P$ which descends ...
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### Integral of cos(x) wrt t when integral of sin(x) wrt t is known [on hold]

If $\int_0^T sin(\theta) dt = A$, where $\theta$ is a variable, A is constant. Then can we find out $\int_0^T cos(\theta) dt$ = ?
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### $V(A)$ semi group of equivalent projections in $M_∞(A)$ cancelative?

I found in the book of Murphy, C*- Algebras and Operator Theory, the Theorem 7.1.2 : Let A be an unital C* algebra, the semi group $V(A)$ of equivalent projections (under Murray Von Neumann ...
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### Is support function of a convex curve in $\mathbb{R}^2$ absolutely continuous? [on hold]

There is an example of a convex function of two variables that is convex but not absolutely continuous. The level set of this function is a convex curve. To construct one example of such a function ...
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### HNN extension group with finitely generated base

Let $B$ be a group and let $A_1$ and $A_2$ subgroups of $B$ with $\phi :A_1\rightarrow A_2$ an isomorphism. Let $\left<t\right>$ be the infinite cyclic group, generated by a new element $t$. The ...
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### Solvable Lie algebra whose nilradical is not characteristic

It is well known that the nilradical of a finite-dimensional Lie algebra over a field of characteristic p > 0 need not be characteristic (that is, invariant under all derivations of the algebra), but ...
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### Sufficient conditions for $\sum_{n \ge 1} a_n e^{-(a_1+\cdots+a_n) s} \sim \frac{1}{s}$ as $s \to 0^+$

Let $(a_n)_{n \ge 1}$ be a sequence of non-negative real numbers such that $\sum_{n \ge 1} a_n = \infty$, and set $\lambda_n := a_1 + \cdots + a_n$ for each $n$. Then the (generalized Dirichlet) ...
Let $G$ be a locally compact group and $H\subset G$ a closed and cocompact subgroup. I wish to consider bounded continuous functions from $G$ to $\mathbb{C}$ that are periodic in the following strong ...