# All Questions

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### Counterfeit money, guaranteeing a profit

Bob and I found two 50 dollar bills out of nowhere. We know they're either both legitimate or both counterfeit. If they're legitimate, they're worth 50 dollars each, otherwise 0. I get one 50 dollar ...
• 293
1 vote
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• 2,706
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### For which set $A$, Alice has a winning strategy?

Alice and Bob are playing a game. They take an integer $n>1$, and partition the set $\{1,2,...n\}$ into two non-empty subsets $A,B$. Alice takes the set $A$ and Bob takes the set $B$. They take a ...
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### Fourier multipliers on $L^2(\mu)$

On $L^2(\mathbb{R}^d)$, we have $T_m$ defined $\widehat{T_m f} = m \widehat{f}$ is a bounded operator on $L^2$ if and only if $m \in L^\infty$. What can be said about the same problem for more general ...
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### How to evaluate employees based on the NPS of their sales?

Suppose that, as a manager, I am creating a campign to evaluate employees based on the NPS (Net Promoter Score) of their sales - the objective is to reward high perfoming employees. However, due to ...
• 570
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### Uniform lower bound in the Landau Prime Ideal Theorem

Are there positive constants $\alpha,\beta,C$ such that for every number field $K$, the number of prime ideals of $\mathcal{O}_K$ of norm at most $x$ is at least $\alpha x^{\beta}$ for all $x\geq C$? ...
• 3,215
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### Paraphrasing the definition of a complete measure

I am trying to paraphrase what it means for a measure to be complete, so that I can get a better grasp of the concept. Here are the relevant definitions from my textbook, Real Analysis by Folland: ...
• 1,003
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### Sum of even binomial coefficients modulo $p$, without complex numbers

Let $p$ be a prime where $-1$ is not a quadratic residue, (no solutions to $m^2 = -1$ in $p$). I want to find an easily computable expression for $$\sum_{k=0}^n {n \choose 2k} (-x)^k$$ modulo $p$. ...
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### Is the divergence of positive definite matrix function locally Lipschitz?

Let $a:\mathbb{R}^d\to S_d^{++}$ where $S_d^{++}$ is the set of $d\times d$ positive definite matrices. Suppose that $a$ is $C^1$. Then by a theorem of Phillips and Sarason (Rogers and Williams ...
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### Is there a function whose autoconvolution is its square? $g^2(x) = g*g (x)$

I am looking for a function over the real line, $g$, with $g*g = g^2$ (or a proof that such a function doesn't exist on some space like $L_1 \cap L_2$ or $L_1 \cap L_\infty$). This relation can't hold ...
• 111
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### Inequality constraints in calculus of variations

$\def\d{\mathrm{d}}$It turns out that Yuri's answer to my earlier question, whilst correct (and I thank him for his effort), was not quite what I desired. I had not posed the question properly, so I ...
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### How can you make a decimal point for a number display on Desmos?

I have this Desmos graph which is a number display. You type a button and that number gets displayed. I'm thinking about turning it into a calculator, but first I need a decimal point. (Also the ...
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### What is the connection between algebraic groups and topoi?

I have a longstanding interest in topos theory. (See this MSE search of my questions about topos theory.) I am studying for a postgraduate research degree in linear algebraic group theory. Naturally, ...
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Let $k_1=k_2=1$ be the principal curvatures of a regular surface $S$ at point $p\in S$ and assume that there is a circle $c$ of radius $1/2$ passing through $p.$ Prove that the geodesic curvature of $... • 5,235 6 votes 0 answers 117 views +50 ### Is$10^n+n^{10}$prime for some integer$n \ge 2$? Is $$f(n):=10^n+n^{10}$$ a prime number for some integer$n\ge 2$? Two necessary conditions :$\gcd(n,10)=1$Since$n$must be odd , we must also have$11\mid n$, otherwise$11\mid f(n)$In ... • 82.3k 5 votes 0 answers 141 views +50 ### Can$!1+!2+!3+\cdots+!n$be a perfect power? Can$!1+!2+!3+\cdots+!n$be a perfect power if$n\geq3$? Note that$!n$is a subfactorial. I do know that$1!+2!+3+\cdots+n!$is only a perfect power if$n=1, 3$, since when$n\geq9, 1!+2!+3!+\cdots+9!...
Let $R=\mathbb{Z}[\sqrt{−n}]$ where $n$ is a squarefree integer greater than 3. Prove that $R$ is not a UFD. Conclude that the quadratic integer ring O is not a UFD for $D\equiv 2, 3$ mod $4$, \$D < ...