# All Questions

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### function mapping odd numbers to counting numbers

Mapping even numbers to counting numbers is straight forward. Without introducing any other variable: i = 0, 2, 4, 6, . . . if i > 0: count = i/2 what about ...
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### About the constants between the Ky-Fan norm and the Trace norm

Are there known constants $\alpha$ and $\beta$ such that we can write, $\alpha \text{Tr} (M_n) \leq \text{Ky-Fan-k}(M_n) \leq \beta \text{Tr} (M_n)$? where $M$ is a $n \times n$ real symmetric ...
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### similarity metric for geometries [on hold]

I'm searching for a method to calculate the degree of similarity of two given geometries. These geometries can be of any type and can have an arbitrary shape. For the sake of simplicity, I primarily ...
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### Can some exotic sphere be diffeomorphically embedded into some $R^n$?

Can some (or perhaps every) exotic sphere be diffeomorphically embedded into some $R^n$? How does such an embedding (if it exists) look like? Note that the answer to the related question Can one give ...
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### Measures on a unit sphere of a Hilbert space

Consider a real separable infinite-dimensional Hilbert space $H$. Let $S=\{h\in H\mid \|h\|=1\}$ be a unit sphere in $H$. What are the most natural measures on $S$? Is there a (Borel) measure $\mu$ on ...
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### Reference quest: variety of lines and variety of planes

Let $X\subset \mathbb P_{\mathbb C}^n$ be a smooth projective variety, $F(X)\subset G(2,n+1)$ its Fano variety of lines and $$I_F=\left\{([l],[l'])\in F(X)\times F(X), l\cap l'\neq \emptyset\right\}$$ ...
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### optimize a Quadratic Matrix Programming with multi-spherical constraints

I have got the following quadratic problem restricted on the Cartesian product of Euclidean spheres. $\underset{X \in \mathbb{R}^{n\times 3}}{\text{min}}$ $Q(X) = \frac{1}{2} Tr(X^TA X) + Tr(B^T X)$ ...
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### On spatial tensor products of von Neumann algebras

Let $H$ be a Hilbert space, and let $A_1,A_2,A_3\subset B(H)$ be three commuting von Neumann algebras. We write $\odot$ for the algebraic tensor product, and $\bar\otimes$ for the spatial tensor ...
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### Relative Leopoldt defect

Let F be a totally real field such that the Leopoldt conjecture holds at a prime number $p$ and $M$ be a quadratic totally real extension of $F$. Is there a bound of the Leopoldt defect of $M$ ?
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### Are the theorems of Ergodic Theory valid for non-probability spaces?

The theorems in Ergodic Theory have assumed a probability measure, always. I am interested to know if they hold even when the space is not equipped with a probability measure. In other words, if my ...
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### How can you prove that f(x) DNE when f(x)+f(2-x)=3x^2+4x+2? [on hold]

How can you prove that f(x) DNE when f(x)+f(2-x)=3x^2+4x+2?
Suppose that $p$ is prime and $q$ is an even number divides $p-1$, such that $q<\frac{p-1}{q}$ and $u$ has order $q$ modulo $p$. Let $S$ be the subgroup of $Z^*_p$ consisting of the powers of $u$. ...
Let us consider the Klein-Gordon equation $$(\Box +m^2)u=0,$$ where $u$ is a scalar valued function, $m\geq 0$, $\Box=\frac{\partial^2}{\partial x_0^2}-\sum_{i=1}^d\frac{\partial^2}{\partial x_i^2}$. ...