# All Questions

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### What is the probability of getting exactly one two and one three in a 5 card draw?

In a 52 cards deck, what is the probability of getting exactly one 2 and one 3 if 5 cards are drawn. I'm wondering what is the difference between doing it the following two ways. Intuitively I would ...
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### Constructing a family of sets

I am completely stuck at the following question. Suppose $X$ is an infinite set. Show that there is a family $\mathcal{F}$ of subsets of $X$ satisfying the following: (a) If $A \subseteq X$ is ...
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### Text introducing $T^{i,j}$-tensor algebra

I'm reading a lecture note here : http://www.cis.upenn.edu/~cis610/diffgeom7.pdf It introduces $T^{•,•}(M)$ the tensor algebra and says that this is a necessary tool in differential geometry. Well, ...
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### Approximating Lipschitz Functions by $C^1$ functions

According to Evans-Gariepy as a corollary of the Whitney's Extension Theorem we have the following Theorem (Approximating Lipschitz Functions) Suppose $f: \mathbb R^n \to \mathbb R$ is Lipschitz ...
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### Proving that something equals the commutator subgroup and conjugacy classes/normal subgroups

I've learned that the commutator subgroup is generated by the commutators. Now this says little about its elements (to me) because I don't see how they need to be commutators themselves. I'm ...
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### Problem solving a PDE using separation of variables and Fourier expansion

I am trying to solve the heat equation: $$\frac{\partial \theta}{\partial t}=\frac{\partial^2 \theta}{\partial x^2}$$ The boundary conditions are: $$\frac{\partial \theta}{\partial x}(x=0)=0$$ ...
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### Is there a nice proof of this zeta function integral

Could some one link me to a good proof of this integral written on the first line here. All the sites I've seen so far just introduce with the definition without explaining it. If not, could anyone at ...
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### Row replacement operation not changing the determinant

Can someone prove why a row replacement operation does not change the determinant of a matrix? **row replacement operation being adding one row to another or something of that sort
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### Statistics basics

Given that $X$ has mean $a$ and variance $b$. Then $E(X^2) = a^2 + b^2$. Why is this true? Please provide a proof alongside any other relevant information. Thanks in advance.
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### When a set of functions becomes complete?

I know that a set of functions are said to form a complete basis on an inteval if any function on that interval can be expressed as a linear combination of the functions in the set. I also know that ...
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### the sum operation in a normed space is continuous

If $(E, \parallel \parallel)$ is a normed space , then the function $+ : E \times E \rightarrow E$ , $( x , y) \mapsto x + y$ is continuous. Question: to solve this exercise ... Which it is ...
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### Methods for detecting kinks along a set of points (for interpolating)

Can anyone point to some literature on methods for detecting kinks/sharp turns with given a set of points. To give some background, I'm trying to get estimates of some function based on randomly ...
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### What is the order of

What is the order of the following: $$\frac{(33x^{7}+6)(x^{2}+3)}{\sqrt{x^3+7x^2-x+5}}$$ Would it be $$\Theta (x^{\frac{17}{2}})$$
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### Combinatorics [sandwich toppings]

I'm trying to figure out how to solve this excercise; You can order a sandwich from 5 different types of bread, you can have butter, lettuce or neither. You get to choose from 3 types of meat, and ...
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### Sum of two normal operators.

We have euclidean space, $A$ and $B$ are normal operators (definition: $A$ is normal when $AA^*=A^*A$) and we know that $Im(A)$ and $Im(B)$ are orthogonal. How can I prove that $A+B$ is normal too? I ...
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### Isometry and Extreme points

If $X$ is a Hilbert space and either $T$ or $T^*$ is an isometry, show that T is an extreme point of the closed unit ball of $B(X)$ where $B(X)$ is bounded linear functionals on $X$. Can I get some ...
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### Volume and Average Height

How do you calculate the average number of floors of buildings across a city block when all the buildings are varying heights. Followings in a listing of what the area looks like and the number of ...
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### Congruent number 23

The set of Congruent numbers are all the integer areas of rational sided right triangles. This means that if g is a Congruent number there exists some integer n such that g*n^2 is the integer area of ...
Let $f$ be absolutely integrable over all $\mathbb{R}$ and continuous and $\hat{f}(s)=0$, where $s\in \mathbb{R}$ and $|s|>a$, where $a>0$. Prove that following formula holds ...