# All Questions

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### Height function of a hypersurface

I was reading an article by do Carmo and Warner, which says: "By the height function for an oriented hypersurface at a point $p$ we shall mean the function defined on a neighborhood of the origin in ...
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### divergence theorem with singularity at r = 0

I am trying to evaluate the volume integral given by \begin{align} \int_V [\nabla(\vec{x} \cdot \vec{u}) - \nabla \cdot (\vec{x}\vec{a})] dV \end{align} where $\vec{x}$ is the position vector and ...
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### Folland 8.20 (Fourier Analysis)

I'm stuck a bit on this problem from Folland: The first part I can't figure out at all. The second part, I know: $\|Pf(x)\|_1 = |Pf(x)| = |\int f(x,y)dy| \leq \int |f(x,y)|dy$. If the last term is ...
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### $X,Y$ are iid from distribution $F$, which is a continuous function, then is $P(X=Y)>0$?

Suppose $X,Y$ are iid random variables from a distribution function $F$, which is a continuous function. Then is it true that $P(X=Y)>0$? For me, the answer is trivially YES. Well, why not? We ...
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### Linear Regression without X? : Using procedure to construct the GLS, derive the Best Linear Unbiased Estimator of α and compute its variance

(Have been working in matrix algebra) Given model: yi = a + ei ( y_i= α+ϵ_i ) That is y subset i and error term subset i Where the expected value of each error term for each entry is = 0 ...
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### Can we exploit FFT for evaluating quadratic on regular, gridded data with stationary covariance?

I would like to evaluate the quadratic $\mathbf{y}^{T}K^{-1}\mathbf{y}$ with the following assumptions: $\mathbf{y}$ is on a regular grid, same spacing in all directions $K$ is separable and ...
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### Is T an infinity spectrum whenever T is a spectrum?

Definitions: For a given first order sentence $\phi$ define $\text{spectrum}(\phi)$ to be the set of all cardinalities of the finite models of $\phi$. A set $S\subseteq\mathbb N_+$ is said to be a ...
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### Should The domain in Simple approximation theorem be measurable?

Here's Royden's version of The Simple Approximation Theorem, My question is do we really need domain $E$ be measurable? Or in other way, do we always define measurable function on a measurable ...
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### Existence of Type I Self Dual Codes

How would I prove Type I self-dual codes (binary self-dual codes that are not necessarily doubly even) exist for every even length $n$?
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### Laurent Series about $z=0$ of $f(z) = \frac{1}{z^3 - iz}$

So far: $$\frac{1}{z^3 - iz} = \frac{1}{z(z^2 - i)} = \frac{i}{z} - \frac{iz}{z^2 - i}$$ Now I see that: $$\frac{-iz}{z^2 - i} = z\left(\frac{i}{i - z^2}\right),$$ and this is where I get stuck. ...
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### Does $\displaystyle \lim_{x \rightarrow 0^{+}}f(\ln x+ x)=-\infty$ imply $\displaystyle \lim_{x \rightarrow -\infty}f(x)=-\infty$?

Let $f$ be a function defined on $\mathbb{R}$ such that $\displaystyle \lim_{x \rightarrow 0^{+}}f(\ln x+ x)=-\infty$. True or false? $\displaystyle \lim_{x \rightarrow -\infty}f(x)=-\infty$. ...
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### Isomorphism of a ring of matrices

Is it possible for a ring of matrices to isomorphic a ring of numbers? Suppose $$R = \begin{pmatrix} a & b \\ -3b & a \\ \end{pmatrix} a,b \in \mathbb Z$$ Can $R$ ...
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### Show that the operator induced by $T$ on the quotient space $V/\operatorname{ker} (T-5I)$

A linear operator $T$ on a complex vector space $V$ has characteristic polynomial $x^3(x-5)^2$ and minimal polynomial $x^2(x-5)$. Show that the operator induced by $T$ on the quotient space ...
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### Determine the maximum and the minimum of an expression

Let $x,y,z \in \Bbb R, x,y,z \gt 0$ such that $x^2+y^2+z^2=1$. Determine tha maximum and the minimum possible values of the expression $$\frac {x^3+y^3+z^3} {x+y+z}.$$
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### Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale?

Let $(x_n,\mathcal{F}_n, n\ge 1)$ be a martingale diference. Is $(x_n\mathbf{1}_{\{ |x_n|\le a_n \}},\mathcal{F}_n, n\ge 1)$ a martingale and why?? $a_n$ is a constant.
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### Financial mathematics- finding yield rates for bonds

I'm not sure if its appropriate to post here but oh well QF put me on hold. Joe must pay liabilities of $1,000$ due $6$ months from now and another $1,000$ due one year from now. There are two ...
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### Regularity connected union of smooth sets

Consider a countable family of bounded, open connected sets with smooth boundary $S_i \subset\mathbb{R}^n$. What can be said about the regularity of the boundary of $S = \bigcup_{i = 1}^{\infty} S_i$ ...
### what is the CDF of $f(x)=\frac{3x^2}{2}$?
This is probably a dumb question but I just want to make sure. The pdf is $f(x)=\frac{3x^2}{2}$ if $-1 \leq 0 \leq 1$. The CDF is $F(x)=\frac{x^3}{2}$ but with what bounds? sorry if this is an easy ...