# All Questions

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### Set theoretic identities involving the Cartesian product

In "Naive Set Theory" by Halmos (amazon link) there is a chapter involving ordered pairs which eventually mentions the Cartesian product and many of it's properties. For the sake of completeness I ...
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### Submartingale example: proof

I am trying to prove if the process $M_t = e^{W_t^2-t}$ is a submartingale ($W_t$ is the Wiener Process). The proof becomes a bit difficult, to the point where I am unsure how to move forward. Let ...
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### What is the number of x-intercepts in this graph of sine?

The function : $y=3-4\sin(2\pi x-3\pi)$ .. how many $x$-intercepts over the interval $[0,2]$? I am confused if they're 3 or 5 because there are 3 $x$-intercepts that are really intercepting ...
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### 4 points not separable by SVM

We know in a support vector machine: Considering we have a linear feature mapping $\phi(x_n)=x_n$ and the XOR problem. We have 2 classes in $R^2$, class 1 $t_+=+1$ and class 2 $t_-=-1$ and 4 ...
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### Cubic factoring question

I'm trying to figure out how a colleague factored an expression. I don't get how: $$a^3+a^2b-(b+1)=(a-1)[a^2+a(b+1)+(b+1)]$$ Multiplying the result I see it's true, but not sure how he got there..is ...
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### Normally distributed data or not

Can I say that the datas are normally distributed? I would say yes, but I am not entirely sure.
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### Unique solution?

If I have the function $f \colon \mathbb{R} \to \mathbb{R}$ with $f'(t) = k \cdot f(t)$ how can I argue that this solution has to be of the form $f(t) = Ce^{kt}$ and can't look any different? ...
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### Matrix calculus: rules for partial traces

I'm trying to understand a paper and have trouble seeing why the following can be written: $Tr_E\{[ \rho,V] \} = \sigma Tr_E\{\rho_E V\} - Tr_E\{ V \rho_E \} \sigma$, when we know the following ...
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### a trig question

Because my book doesn't have solutions to these problems, I'm checking here if I solved them correctly (I know it's all probably wrong): 1) $$\tan(\pi+\frac{x}{3})>0$$ What I noticed first is that ...
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### Find the primitive of a particular complex function

Let $D_1=\{|z-a|<r_1\}$,$D_2=\{|z-b|<r_2\}$,$D_3=\{|z-c|<r_3\}$ such that $D_1,D_2,D_3$ are disjoint.Let $f$ be an analytic function in $\mathbb{C}\diagdown(D_1\cup D_2\cup D_3)$.Prove that ...
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### invariants of a Lie algebra

What does it mean by "constructing invariants" in algebraic topology or algebra in general? How to define a "invariant" in algebra? What does it mean by the "invariant of a Lie algebra"?
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### if $g: \mathbb{R}^k \rightarrow \mathbb{R}$ continuous, $f_i=X \rightarrow \mathbb{R}$ measurable prove $h(x) = g(f_1(x),…,f_k(x))$ is measurable.

if $g: \mathbb{R}^k \rightarrow \mathbb{R}$ is continuous and $F_i=X \rightarrow \mathbb{R}, i = 1,2,...,k$ is measurable Prove that $h(x) = g(f_1(x),f_2(x),...,f_k(x))$ is measurable. So far I ...
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### What is $\lim\limits_{x\to 0}\left(\dfrac{x}{e^{-x}+x-1}\right)^x$

What is $$\lim_{x\to 0}\left(\frac{x}{e^{-x}+x-1}\right)^x$$ Using the expansion of $e^x$, I get that the function $$y=\left(\frac{x}{e^{-x}+x-1}\right)^x$$ is not defined for negative ...
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### Integrate $\int_a^b e^{- \cos(t)} dt$

I am looking for an explicit representation of $\int_a^b e^{- \cos(t)} dt$. The only way I could imagine to find the antiderivative is to expand this function in spherical harmonics or use the taylor ...
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### Is it possible that $N_G(H)=H$ and $N_G(K)=K$ where $K\subsetneq H$?

Is it possible that $N_G(H)=H$ and $N_G(K)=K$ where $K \subsetneq H$ and $H,K$ are proper subgroups of $G$ ?
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### A question about uniform convergence of $g_n=f\left(\frac xn\right)$

Could you give me some hint how to prove this statement: Suppose $f(x)$ is some function on R. Prove: If $g_n=f\left(\frac xn\right)$ converges uniformly to zero on R than $f(x)=0$ for all x. I ...
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### 52-card trick for a larger deck?

Long ago someone demonstrated the following card trick with a standard 52-card deck: (1) A volunteer selects 5 cards from a shuffled deck, which the performer does not see. (2) The assistant puts ...
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There is a pentagon inscribed in a circle with a diameter of 10. What is the area and perimeter? Is the answer 20 and 25? I tried using examples and applying them to the problem.
### Can a continuous function on a compact group $G$ be interpreted as the sum in $C(G)$ of its Fourier series?
For a given function $f\in C(G)$ on a compact group $G$ its Fourier transform is defined as the family of operators  \widehat{f}_\sigma=\int_Gf(t)\cdot\sigma(t^{-1}) \ \text{d}\ t,\quad ...
Define $(m,n)$ to be a special pair if $n=m \cdot Pd(n)$. Where $Pd(n)$ is the product of digits $n$. Then I have the following conjecture - For every $m$ with no digit of $m$ being $0$ , there ...