# All Questions

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### $L^2$ method sovle the PDE $\bar{\partial}}u=f$, where $f\in L^{ 2 }_{ (0,p) }(\Omega )$

$\Omega$ is qseudoconvex domain，for $f\in L^{ 2 }_{ (0,p) }(\Omega )$,and ${ \bar { \partial } f }=0$. Use $L^2$ method show that there exists solution $u\in L^{ 2 }_{ (0,p-1) }(\Omega )$, such ...
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### Additive and Mutlipicative iverse problem

Find the additive and multiplicative inverse of -36. I don't remember how to do these problems. If you can help that would be great. thanks!
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### Show that the imbedding $C^{m+1}(\overline{\Omega}) \to C^{m,1}(\overline{\Omega})$ is not compact

Let $\Omega \subseteq \mathbb{R}^n$ be open. Let $C^m (\overline{\Omega})$ be the Banach space of functions such that each partial derivative $D^{\alpha}f$, $|\alpha| \le m,$ exists and is uniformly ...
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### Given a distribution to generate a set of numbers, what is probability of generating two consecutive numbers whose difference is greater than k?

Suppose I am generating a set of numbers {$x_1$, $x_2$, $x_3$ ... $x_n$} from a given probability distribution $f(x)$. Is it possible to calculate the probability of finding $x_{i+1}-x_i \geq k$, ...
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### Integration of the product of probability densities

Does a probability density $f(x|\alpha)$ multiplied by another probability density $g(\alpha)$ , where of course both integrate to one, also integrate to one if we integrate with respect to $\alpha$? ...
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### Why is $\mathbb{R}$ Not a Complete Lattice?

Why is the set of all real numbers not a complete lattice? Please give a detailed explanation!
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### A use of Hahn-Banach and Riesz Representation

Let $X$ be a compact Hausdorff topological space. Suppose $X$ is not a singleton set and $C(X)$ denotes the space of continuous functions on $X$. Do we have that for all $L \subset C(X)$ a nondense ...
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### Trouble Understanding a Problem

http://imgur.com/37FG64q I don't understand how the normal random variable with parameters 0,1 equals (x-mu)/sigma=Z ~ N(0,1). If I plug that into the pdf of the normal distribution I get ...
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### discuss on how to use SRS, one stage and two stage cluster for surver sampling

a home owner with a large library needs to estimate the purchase cost and replacement value of the book collection for insurance purposes. she has 44 shelves containing books. how you would use simple ...
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### squares which are not the sum of a square and twice a triangular number

I'm trying to determine conditions on integer squares which cannot be written as a square and twice a triangular [all numbers positive], i.e. integers $n \ge 1$ where there are no integers $a,b \ge 1$ ...
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### For any $n$, there are at most two simple groups of order $n$? [duplicate]

How do you prove that for any $n$ there are at most two simple groups of order $n$?
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### What is the big picture behind AKS algorithm?

Despite a number of question on AKS algorithm here, there does not seems to anything related to the idea behind it (for those who don't know, AKS primality testing is found in PRIMES is in P). I read ...
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### Constructing a rotation matrix from complex eigenvalues

I am trying to construct a rotation matrix $\mathbf{R}\in\mathbb{R}^{3\times3}$ rotating around an axis $\hat{n}$ in a basis $\{\hat{n},\hat{u}_{1},\hat{u}_{2}\}$. Formally: Given a basis ...
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### AIME number theory problem (unique factorization domains)

I'd greatly appreciate some help with the following problem, from a mock AIME I took. Compute the largest squarefree positive integer $n$ such that $\mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}}$ ...
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### Showing that every subgroup of a metacyclic group is metacyclic

I have tried to prove that every subgroup of a metacyclic group is metacyclic (where a group $G$ is metacyclic if it has a normal cyclic subgroup $C$ such that $G/C$ is cyclic). I believe I have a ...
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### For set of positive measure $E$, $\alpha \in (0, 1)$, there is interval $I$ such that $m(E \cap I) > \alpha \, m(I)$

I am a graduate student at Iowa State University attempting to return after a five-year hiatus and take the Real/Complex Analysis qualifier on January 8 for potential reinstatement. Since the ...
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### number of solutions of a system of linear equations

Consider a system $\Sigma (y)$ of $m$ linear equations in $n$ variables $x_1,\cdots,x_n$: $\sum_{j=1}^n a_{i,j}(y)\cdot x_j=b_i(y)$, $i=1,\cdots,m$, whose coefficients $a_{i,j}(y)$ and $b_i(y)$ are ...
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### Find $\mathsf{E}\left[(X_1+X_2)^4\mid X_1-X_2\right]$ where $X_1$ and $X_2$ are iid standard normal

Find $$\mathsf{E}\left[(X_1+X_2)^4\mid X_1-X_2\right]$$ where $X_1$ and $X_2$ are i.i.d. standard normal. I know that both $X_1+X_2$ and $X_1-X_2$ are both distributed $N(0,2)$. I'm having trouble ...
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### If $|Hom(H,G_1)| = |Hom(H,G_2)|$ for any $H$ then $G_1 \cong G_2$

Let $G_1$ and $G_2$ be two finite groups such that for any finite group $H$, $|Hom(H,G_1)| = |Hom(H,G_2)|$. How can I show that $G_1 \cong G_2$ ?
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### Find a function $f$ such that $f$ is harmonic on $D$ and $f|_{\partial D}$.

I understand its solutions in general. But my question is how to decide whether I sould take $Im z^4$ or $Re z^4$? I have two similar examples. And in one example, the real part is taken, but in ...
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### how to compute the determinant of a linear map

Let $V$ be the vector space of $m\times n$ matrices over a field $F$. Fix an $m\times m$ matrix $A$ and an $n\times n$ matrix $C$, and consider the map $\phi: V\longrightarrow V$ defined by ...
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### Reducing ${n\choose k} - {n\choose k-1}$

I'm writing computer program which on some point has to compute following formula: $${n\choose k} - {n\choose k-1}$$ Because I have following limits: $$n \le 4000, \space k \le\frac{n}{2}$$ computing ...
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### Compute the class number of $R=\mathcal{O}\cap\mathbb{Q}[\sqrt{51}]$

What is the class number of $\mathcal{O}_K$ where $K=\mathbb{Q}(\sqrt{51})$. Could you please explain.
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### Fixed Point Proof

*I would only like a hint! Not a full proof. Prove: If a function $f(x)$ is differentiable on $\mathbb{R}$ and $f'(x) < 1$ for all $x\in\mathbb{R}$, then $f(x)$ has at most one fixed point. So ...
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### A non-r.e. language whose complement is not r.e.

What's an example of a language that is not recursively enumerable and whose complement is also not recursively enumerable?
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### The translations of “unless” and “except” into symbolic logic.

The following two exercises come from Logic for Mathematicians by J.B. Rosser, chapter 2 section one page 17. I am not so sure how to interpret the words "unless" and "except". Notation: $\sim P$ ...
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### Find all possible Jordan Canonical forms

I am working through the problem below What are the possible Jordan Canonical forms for a matrix $A \in M_n$ with characteristic polynomial $p_A(t)=(t+3)^4(t-4)^2$? Give reason for your answer. ...
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### Reputation probabilities

Is there a probabilistic model for what one's reputation can be on MSE? I can of course obtain increments of points in +2, +5, + 10, + 15, +50, +100. Are there models for what my reputation will ...
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### What does this music video teach us about 863?

This delightful animation by Stefan Nadelman depicts "the additive evolution of prime numbers", set to Lost Lander's song "Wonderful World": http://www.youtube.com/watch?v=TZkQ65WAa2Q. (If you haven't ...
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### near linear distance geodesics in nonpositively curved riemannian manifolds

Let $c_v, c_w$ be two geodesics starting at a point $p\in M$, where M is a nonpositively curved, complete, smooth Riemannian manifold. Say $c_v(\varepsilon) = \exp_p(\varepsilon v)$ and ...
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### An extrasensory perception strategy :-)

Inspired by classical Joseph Banks Rhine experiments demonstrating an extrasensory perception (see, for instance, the respective chapter of Jeffrey Mishlove book “The Roots of Consciousness”), I ...
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### spherical mean of solution of the helmholtz equation

I'm stuck with this problem. Given a domain $\Omega \subset \mathbb{R}^3$ where the function $u$ satisfies: $u_{xx} +u_{yy}+u_{zz} + k^2 u = 0$, I am asked to find the spherical mean over the sphere ...
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### Prove that commutant of the group $G'$ is Abelian group.

Let the group $G$ has a faithful reducible two-dimensional representation. Prove that commutant of the group $G'$ is Abelian group. I think to so. Commutant $G'\triangleleft G$. Let $\rho$ is the ...
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### Statistics question: appropriate test of significance?

The following question came to mind while I was playing tennis. Even when I'm taking a break, I can't turn the math brain off. I'm sure this is a simple question for the statisticians here. Say a ...
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### Examples of Baire class 2 functions

Do you know of examples of Baire class 2 functions which are not Baire class 1 functions, besides the the indicator function of the rationals and the indicator function of the Cantor set?
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### Can subscripts be used like this?

I have a variable named $P$, and another three variables named $P_c$, $P_d$ and $P_u$. Now, if I define this function: $$f(x) = x_c + x_d + x_u$$ Is it correct to say that: $$f(P) = P_c + P_d + P_u$$ ...
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### A messy, messy integral

This problem's taking me a lot longer than I like (probably doing things the hard way...). I have 9 different terms to integrate, some of which are messier than others. This one, though, is messier ...
A functional $$J(y)=\int_a^b F\left(x,y(x)\right)dx, \tag{1}$$ subject to an isoperimetric constraint $$\int_a^b K(x,y)dx=l, \tag{2}$$ and a holonomic constraint $$g(x,y)=0. \tag{3}$$ Most ...