# All Questions

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### Prove for any integer $N$ that there exists $n > N$ where $n!-1$ is not a prime

I was thinking about Euclid's proof of the infinitude of primes and the fact that we could make the argument about $n!-1$ instead of $n!+1$ when I wondered if it would be easy to prove that for any ...
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### If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$.

If $\mu(E)=0$, show that $\mu(E\cup A)=\mu(A\setminus E)=\mu(A)$. I just started learning about measure this week, so I don't know any theory about measure except the definition of outer measure ...
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### …is the closed form for sequence A_n. Find c using the Fibonacci and Lucas number sequences.

Let \begin{align*} A_0 &= 6 <br/> A_1 &= 5 <br /> A_n &= A_{n - 1} + A_{n - 2} \; \textrm{for} \; n \geq 2. \end{align*} There is a unique ordered pair $(c,d)$ such that ...
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### Is there a divergence theorem for differential forms?

For a vector field $X\in\Gamma(TM)$ on a closed Riemannian manifold $(M,g)$ we have \begin{align*} \int_M\text{div}X\;\mu=0, \end{align*} where \begin{align*} ...
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### Boundary conditions

I am kinda confuse with the second part of my homework. I did the first part (3/a and 4/a) without any problem, but part b for both problems I don't get it at all. I try to plug the boundaries in the ...
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### What does the spectrum of the Grothendieck ring of varieties look like?

Let $k$ be a field (if you want, $k=\mathbb C$). The Grothendieck group of varieties is the Abelian group generated by isomorphism classes of $k$-varieties, subject to the relation ...
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### Why is it ok to factor an equation with no limit so it has a limit?

I'm just starting out in calculus, so please bear with me if this is not a sensible question. In the book I'm reading, the author gives the example of the problem of finding the limit of ...
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### Is there a formalism for a universal mathematical representation of algorithms?

I don't know if my question is correct so excuse me if I'm not 100% clear about what I would want to know. Is there a formalism which can capture all possible algorithms (mathematically speaking) ? ...
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### How important is the own talent for research of your PhD supervisor?

Currently I am in the process of finding a PhD. Some potential supervisors are more didactical than others, some are nicer and warmer than others, and some are more famous mathematicians than others. ...
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### Primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$

I am looking for the minimal primary decomposition of $(0)$ in $k[X,Y,Z]/(ZY,ZX^2,Z-XY)$. I realize that this is a similar question to some of the previous ones, but the ring is different than in ...
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### Angle quadrisection in a triangle

In triangle ABC, AB=84, BC=112, and AC=98. Angle B is bisected by line segment BE, with point E on AC. Angles ABE and CBE are similarly bisected by line segments BD and BF, respectively. What is ...
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### Finding a point on a curve where the tangent is parallel to another line noted

Find the point on the curve $y=x^2+2$ where the tangent is parallel to the line $2x+y-1=0$ I understand the answer is $(-1,3)$ but I can't find a way to get there... Thanks
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### What is a conventional name for a set of values having no properties except that values are distinct?

I know essentially nothing about math but I'm interested in very low-level concepts. I'm thinking of something like a finite or infinite set (although I'm not married to consider sets per se, maybe ...
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### Inverse of a product in a group can be wriiten as the product of the inverses of each element in reverse order

Let $(G,\circ)$ be a group and let $g_1,...,g_n\in G, n\in\aleph$. Prove that $(g_1\circ ...\circ g_n)^{-1}=g_n^{-1}\circ ...\circ g_1^{-1}$ I tried this by induction but was unsure how to take out ...
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### Interchange rows in a matrix without using interchange operation

I'm sure that it's already out there somewhere in the abyss that is page 37 on google, so I apologize. I haven't been able to find it. Given some arbitrary matrix, how can two rows be interchanged ...
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### Find basis of subspaces

I don't know how to create basis of V1 and V2. If I want to prove M1^2=M1, do I need to find matrix representation of M1 first? Thanks!!!!!!
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### How to find the orthogonal of a vector space

Let $V$ be a vector space over a field $F$ equipped with a symmetric bilinear form $B$. Let $W$ be a vector subspace of $V$. I know that we define the orthogonal complement $W^\bot$ to be ...
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### How to get from the statement $(AB'+C'A'+C'B')$ to equivalent statement $(AB'+C'A')$?

I've been working a Boolean algebra problem for probably 2 hours at this point, and while I arrive at a much simplified equivalent expression, there's a simpler one yet. Basically, I start out with a ...
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### Finding the probability density function of $U=Y_1+Y_2$

Let $(Y_1,Y_2)$ denote a random sample of $n=2$ from the uniform distribution function on (0,1). (1) Find the probability density function $U=Y_1+Y_2$ (2) Find $E(U)$ I am unsure of how to bound ...
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### Is every arrangement reachable by shuffling this way?

Suppose we have a vertical stack of $n$ distinguishable coins, each of which is either heads-up or tails-up. Let a shuffle be the following procedure: divide the stack at will into a top- and ...
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### Universal Cover of $\mathbb{R}P^{2}$ minus a point

I've already calculated that the fundamental group of $\mathbb{R}P^{2}$ minus a point as $\mathbb{Z}$ since we can think of real projected space as an oriented unit square, and puncturing it we can ...
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### The Purpose of Master Thesis

I am posting this question in the aftermath of the earlier posting in this link. Here are what I would like to know more about master and PhD thesis: (1) I understand that schools have different ...
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### $\mathrm{lcm}(b,c)$ from $\mathrm{lcm}(a,b)$ and $\mathrm{lcm}(a,c)$

Given that lcm$(a,b)=60$ and lcm$(a,c)=270$, find lcm$(b,c)$ I believe you're supposed to use the rule lcm$(a,b)=p_1^{\text{max}(r_1,s_1)}\cdots p_m^{\text{max}(r_m,s_m)}$ Here's my work so far: ...
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### Finding an order of a coset in $A/B$ where $A$ is a free abelian group and $B$ is a subgroup.

Let $A$ be a free abelian group with basis $x_1,x_2,x_3$ and let $B$ be a subgroup of A generated by $x_1+x_2+4x_3, 2x_1-x_1+2x_3$. In the group $A/B$ find the order of the coset $(x_1+2x_3)+B$. How ...
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### Divergence structure equation

Consider Laplace's equation with potential function $c$: $$-\Delta u + cu = 0, \tag{*}$$ and the divergence structure equation $$-\operatorname{div}(aDv)=0, \tag{**}$$ where the function $a$ is ...
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### Difference between necessary and necessary but not sufficient?

This is from Discrete Mathematics and its applications I read up on necessary and sufficient from here What is the difference between necessary and sufficient conditions? If p->q (p implies ...
Suppose $H$ is a Hilbert space (much of this still works if it's just a Banach space), $x\in H$, and $(x_n)$ a sequence in $H$. Does $x_n\to x$ strongly in H iff $x_n\to x$ as operators in the strong ...