# All Questions

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### Stochastic methods for solving very high-dimensional PDE

I am looking for stochastic methods for solving a very high-dimensional PDE (with one time dimension and very large number of spatial dimensions), which would reduce it to a lower-dimensional problem, ...
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### bounds on a series with binomial coefficients

I have the following series $\sum\limits_{l=1}^n {n\choose l} \alpha^{\beta^l}$ where $\alpha > 0$and $0 \leq \beta \leq 1$. Can anybody guide me how I can evaluate it or find some tight upper ...
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### What is the joint distribution of sample mean and sample variance of normal distribution?

${X_i} \sim N\left( {\mu ,{\sigma ^2}} \right)$, define $\bar X = \sum\limits_{i = 1}^n {{X_i}}$, ${S^2} = \frac{1}{{n - 1}}\sum\limits_{n = 1}^n {{{\left( {{X_i} - \bar X} \right)}^2}}$. What is the ...
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### Special fiber of $X(p)$ in characteritic $p$

Let $p \geq 5$ be a prime. Let $Y(p)$ be the fine moduli space representing elliptic curves + basis of the $p$-torsion over $\mathbb{Q}_p$ and let $Y_1(p)$ be the fine moduli space representing ...
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### Is it possible to define the density of the logistic map for $x<0$?

Probability density functions (PDF's) have inherent connections to the field of Dynamical Systems. The motivation for this question can be found in: ...
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### Estimate for Levy metric

In the Encyclopedia of Mathematics there is an inequality for Levy metric ($d_L$): $$d_L(E,F) \leq \{\beta_r(F)\}^{r/(r+1)},$$ where $E$ is a a distribution that is degenerate at zero, $\beta_r(F)$, ...
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### Weyl group actions on 0-weight spaces

For a complex simple Lie group G with a maximal torus T, we can take a highest-weight representation V of G and look at the 0-weight space, i.e. the subspace of V of elements invariant under T. This ...
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### Fixing (non)-independency of a the subfamilies of finitely many events.

I'm would be interesting in any construction of a probability space with n events (n is given), where for every subset of these events, it is also given whether or not, the family is mutually ...
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### Lower regularity version of Moser's theorem on volume elements

A theorem of Moser, published in "On the Volume Elements of a Manifold" (Transactions of the Americal Mathematical Society 120, 1965), shows that if a $C^\infty$ compact manifold $M$ has two ...
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### Cramer's theorem in Hilbert spaces

I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space: Let $X_1,X_2,\cdots$, be ...
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### How to compute $\pi_{3}$ of $L(p,q)\# L(p',q')$?

Let $L(p,q)$ be a 3-dimensional lens space, and let $L(p',q')$ be another. Is there any known result concerning the 3rd homotopy group of the connected sum $L(p,q)\# L(p',q')$? If not, I am interested ...
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### Mathematically compute - species diversity [on hold]

Using a simple model, and having these as paramters numberofspecies <- 100 meaninitialpopulationsize <- 50 sdloginitialpopulationsize <- 1 #determines variation in initial population ...
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### Estimate on sum of $J_n^4$

If $J_n(x)$ is the Bessel function of order $n$, we know that for all $x$, $$\sum_{n=-\infty}^{\infty} J_n^2(x)=J_0^2(x)+2\sum_{n=1}^{\infty} J_n^2(x)=1.$$ What is known about  ...
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### Mathematics Research and The Internet [on hold]

I reformulate here a question about Mathematics and The Internet. My questions are: What was the vital role of Mathematics research in the foundation of the Intranet ($\rightarrow{Internet}$) and, do ...

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