# All Questions

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### Solutions of $2^x 7^{1/x}\le 14$

The solution is supposed to be $(-\infty,0)$ and $[1,\log_2 7]$. What I get when solving the problem is $(-\infty, \log_2 7]$. Where did I get it wrong? I start by dividing both sides by 14, then ...
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### Seminar concearning Spectral Theory of Differential Operators?

I must prepare a seminar about spectral theory of linear partial differential operators. However, I'm at a loss as to a nice reference. I'm looking for something that fits in a graduate spectral ...
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### Expressing problems in canonical form for solving with simplex

The Picnic Hamper Company has a store containing 10,000kg of nuts, 4000 packs of smoked salmon, 2000 bottles of wine and 1500 Victoria sponges. It intends to use these goods to make up three different ...
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### 2 Definitions of Holomorphic functions on Riemann surfaces

In a lecture that I currently attend we defined Riemann surfaces and holomorphic mappings on it somewhat different than in another lecture that I attended a year ago. My question is: Are these ...
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### Sequences formed by integer evaluations of polynomials modulo $p^{k}$, where $p$ is a prime number and $k \in \Bbb{N}$.

I have the following question. Let $p$ be a prime number and $k$ a positive integer. Let $(a_{n})_{n \in \Bbb{Z}}$ be a two-way sequence in $\Bbb{Z} / p^{k} \Bbb{Z}$. Then is it true that ...
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### Higher dimensional SDEs cannot be compressed into a one dimensional SDE.

This question comes from quantitative finance but I think it's true in general outside that setting. I'm trying to make sense of the idea that if a process depends on at least two noises there ...
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### Prove the surjectivity of this injective linear map

I am working on the following problem. Let $g : V\to V$ be linear and injective, where $V$ is a vector space over the field K. Prove that, if $V$ is finite-dimensional, then $g$ is surjective. In an ...
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### An awkward Functional Equation

Find all $f:\mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(f(y)+x^2+1)+2x=y+(f(x+1))^2$$ for all $x,y \in \mathbb{R}.$ I proved that $f$ is bijective, but I am stuck there. Any help please?
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### Rootspace decomposition of a Lie algebra

$\DeclareMathOperator{\ad}{ad}$ Let $L$ be a non-zero Lie algebra which is semi simple. Then $L$ contains a toral element and hence a non-trivial toral subalgebra. Let $H$ denote a maximal toral ...
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### Elementary question in Group Theory with less prerequisite

Here I am posing a problem, which my beginning students of algebra were discussing for long time. Question: Without using theorem of Cauchy or Sylow, can we show that a group of order $15$ contains ...
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### formula to establish correlation between multiple library functions

I am trying to predict the change in timeliness of holds delivery relative to number of owned Bestsellers, number of holds and the checkout window. (yes, it really is a library question). To do this ...
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### logarithmic Series

I'm aware that by properties of logarithm $$\sum_{k=1}^n \ln (k) = \ln (n!)$$ My question is if $$\sum_{k=1}^n \ln^2 (k) = \ln^2 (n!)?$$ Because when I am verifying the value where $n = 5$, I get ...
### How many three digit numbers with increasing digits can be formed from the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$?
Suppose we pick 3 numbers $x,y,z \in \{1,2,3,4,5,6,7,8\}$ and form a 3 digit number $xyz$ how many possible combinations numbers can we create such that $x < y < z$. For example $357$ ...