# All Questions

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### Some integrals with respect to a Gaussian measure on a Hilbert space

Assume that $(H,\langle\cdot,\cdot\rangle)$ is a real separable Hilbert space equipped with a Gaussian measure $\mu$ with a mean $m$ and a covariance operator $C$. Let $x\in H$ be a fixed vector. What ...
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### product distinct prime factors of prime(n)-1 and prime(n)+1 [on hold]

The prime 127 has 127-1=126 with distinct prime factors 2,3,7 and 127+1=128 with distinct prime factors of only 2; hence 2*3*7=42<127. Log 127/42=q=1.296. Are such primes common? Can a value of ...
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### The Linnik problem for dimension $2$

For $N$ an integer, let $$\Omega_N:=\left\{\frac{\alpha}{\|\alpha\|}|\alpha \in\textbf{Z}^n~\text{and}~\|\alpha\|^2=N\right\}.$$ For $n=3$, Linnik asked if the set $\Omega_N$ was uniformly distributed ...
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### Finding the unique Nash equilibrium [on hold]

$$m^2(1-m)[(1+m)^2 - R] x^3 + [6m^2R + 12mR - 2m(1-m^2)(6+2m) - 4tm^2(1+m)^2] x^2 + [(1-m)(6+2m)^2 + 8tm(1+m)(6+2m)] x - 4t(6+2m)^2=0$$ where: m ∈ (0, 0.5), R ∈ [0, 0.25], t ∈ [0, 1], x ∈ [0, 2]...
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### On the size of residue class

Let $n \in \mathbb{N}$ be a odd number. Let $S \subseteq \{1,3,5,7,...,n-2,n\}$ and $|S|$ is even number. Let $R_i^k=\{a \mid a \in S \text{ } \&\text{ } a\equiv i \text{ }(mod \text{ } k)\}$ ...
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### Markov Chain: Number of communicating classes of a power of the irreducible transition matrix [on hold]

Suppose $P$ is an irreducible transition matrix, with period $d$. Consider the transition matrix $P_k$. In terms of $d$ and $k$, how many communicating classes does $P_k$ have, and what is the period ...
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### Why is every object cofibrant in an excellent model category?

In Appendix A.3 of the book higher topos theory appears the notion of an excellent model category (see Definition A.3.2.16). The main feature of this notion is that when $\mathbf{S}$ is an excellent ...
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### Examples of maps with nontrivial Hopf invariant but Lusternik-Schnirelmann category of the cofiber doesn't increase?

Let $A$ be a suspension and $X$ be a space with Lusternik-Schnirelmann category $n$ and let $\alpha: A\to X$. It is easy to see that the cofiber $C_\alpha$ has $\mathrm{cat}(C_\alpha) \leq n+1$. One ...
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### Application(s) of complex dynamical system into some other areas of mathematics [on hold]

Complex dynamical system is a very active branch in mathematics. I wondered are there some nice applications of complex dynamical system into the some other areas of mathematics.
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### Considering a matrix with integrer entries over $\mathbb{Z}/p \mathbb{Z}$, does it remain full rank? [on hold]

Suppose I have an $m \times n$ matrix $M$ with integer coefficients, and suppose it has full rank. Let $p$ be a prime and now consider the matrix $\bar{M}$ over $\mathbb{Z}/p \mathbb{Z}$. Is it true ...
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### the relation between projective and quasi-projective modules

An $R$-module $M$ is called quasi-projective if $\text{Hom}_R(M,M)\to\text{Hom}_R(M,N)$ is surjective for every surjective homomorphism $M\twoheadrightarrow N$. What are the rings $R$ for which every ...
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### Approximating formal surfaces by analytic surfaces

Let $C$ be a smooth projective curve contained in a smooth projective $3$-fold (everything defined over $\mathbb C$). Let $\widehat X$ be the formal completion of $X$ along $C$ and suppose there ...
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### Renorming a Banach space to make projections contractive

Let $X$ be a Banach space and $P$ be a projection in $B(X)$. Then $X$ can be renormed so that $P$ has norm $1$. Can the same be done for a family of projections? That is, given finitely many ...
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### Combination of certain linear-programming topics new?

Consider the combination of the following topics, aimed at a future book on Linear Programming: Generalization of certain parts of the polyhedron theory and of the Simplex Algorithm to arbitrary ...