# All Questions

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### linear independence of $\sin(k \pi / m)$

I have tried searching the literature for a result like the following, but have not found anything. For a positive integer $m$, is it known that $$\{ \sin (k \pi / m): 1 \leq k \leq m/2, (k,m)=1 \}$$ ...
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### What is the shatter coefficient / VC - dimension of some hypothesis set?

Let $H:=\{h:\mathbb{N}_0^n \rightarrow \{0,1\}| h(x_1,\cdots,x_n) = \mathbb{1}_0(\sum_{i\in I}{x_i}-\sum_{j \notin I}{x_j}) \text{ for some } I \subset \{1,\cdots,n\}\}$ where $\mathbb{1}$ is the ...
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### What mathematical background is preliminary for reading and understanding books/papers on wavelets? [migrated]

Please excuse my english. I have had the following math courses for mechatronics engineering education: Calculus (single and multivariable) Linear algebra (introductory) Differential equations (ode'...
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### Representation Theorem for levy process

The answer given by The Bridge, Martingale representation theorem for Levy processes was useful for me. Thank you. But I have a question, can this theorem be given for $X_t$ being not just $R^n$ ...
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### Constructible sheaves on general stratified spaces

I am not an expert in the field, so my question might be rather standard. Let $X$ be a compact metric space. Assume that $X=\cup_{i=1}^NS_i$ is a finite disjoint union of locally closed topological ...
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### Interpolation inequality for fractional Sobolev spaces

In Theorem 5.2 of the book Robert A. Adams and John J. F. Fournier, MR 2424078 Sobolev spaces, ISBN: 0-12-044143-8. is stated that, for any integers $0 \leq j \leq m$ and $u \in W^{m,p}(\Omega)$ (...
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### Reference - Generalized Hodge conjecture for triangulated motives

GHC for triangulated motives: The Hodge conjecture holds and an object $\rm M \in Dmg$ is effective if and only if its Hodge realization is effective. I would like to know some references on GHC ...
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### Splitting the Tits algebras of a anisotropic group

Assume we are given an anisotropic algebraic group $G$ over a field $k$, having non trivial Tits algebras (i am interested in the $E_7$ adjoint cases). Question: Is it possible that there exists a ...
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### approaches to Apollonius circle problems

I've been looking for solutions to finding the set of circles tangent to two other circles. one circle can be inverted to a line, but two circles can be mapped to a line and a circle or equivalently ...
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### An upper bound for the number of prime numbers in non-linear progressions

Let $f(x)$ be a non-linear polynomial over $\mathbb{Z}$. Consider the following sum $\pi_f(x):= \#\{y: y< x \text{ and } f(y) \text{ is prime} \}$. Can we get an upper bound for $\pi_f(x)$?
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### Is a C*-algebra with an isomorphic predual a von Neumann algebra?

It is well-known that a C*-algebra $A$ is a von Neumann algebra if and only if it has an isometric predual, that is, if and only if there exists a Banach space $X$ such that $A$ is isometrically ...
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### Is there a schema category for hyperstructures?

I am completely fascinated by Niels Baas' notion of hyperstructures, chiefly because I can see how such gadgets could be used in modeling both biological and social systems, or other evolutionary ...
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### Does every smooth manifold admit a metric with bounded geometry and uniform growth?

Let $M$ be a smooth manifold, $g_M$ a Riemannian metric, and consider for $x\in M$ the volume growth function, $gr_x$ that maps $r>0$ to the volume $vol_{g_M}(B(x,r))$. My interest is to see ...
### Reasons for $\alpha>-\frac{1}{2}$ constraint in texts regarding Gegenbauer polynomials $C^{(\alpha)}_k(x)$
In texts regarding the Gegenbauer polynomials $C^{(\alpha)}_k(x)$, I often see the constraint $\alpha>-\frac{1}{2}$ alongside definitions and identities. I understand that the orthogonality ...