# All Questions

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### Evaluate the infinite series $\sum_{r=1}^{\infty} \frac{2^{-r}}{r^{2}}$

My teacher posed an infinite series question to me today and I'm not quite sure how to start to go about it. $$\sum_{r=1}^{\infty} \dfrac{2^{-r}}{r^{2}}$$ Any hints would be much appreciated.
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### Good books on combinatorics

I have a math Ph.D. but my knowledge of combinatorics sucks and I simply don't know how to compute anything more complicated, i.e. what happens when we put restrictions on the allowed configurations ...
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### least square solution for obtaining a line 3d by intersecting many planes

If I have been given 4 planes and I know only a point (just a point lie on the plane e.g. o1,o2, o3 & o4) and normal vector of each plane (n1, n2, n3 & n4). Then, by intersecting all 4 (or ...
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### Prove that if R and S are nonzero rings then $R\times S$ is never a field.

This question has been asked on here before but I'm looking for some additional insights. The question comes from the section of the Dummit and Foote textbook on the Chinese Remainder Theorem. All of ...
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### How to prove this set has volume 0?

Let $A \subset \mathbb R^n$ be an open set and the function $g: A\to \mathbb R^n$ be a $C^1$ function. Let $$B=\{x\in A: Jg(x)=0\}$$ ($Jg$ means the Jacobian of $g$), then $g(B)$ is of volume zero. ...
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### Colored balls puzzle

Imagine you have $n$ balls in a bag that are colored from $1$ to $n$. At each turn you take two balls at random out that have different colors and color one the color of the other. You then put them ...
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### Eigenvectors of a Matrix Homework

Find eigenvectors of: \begin{pmatrix}4 & 2 \\ 2 & 3 \end{pmatrix} $$\det(A-\lambda I) = \lambda^2-7 \lambda+8=0 \iff \lambda_1=\frac{7+\sqrt{17}}{2} \ \lor \ \lambda_2= \frac{7-\sqrt{17}}{2}$$ ...
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### Is this series of hermite polynomials $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} H_n(x)$ a known function?

Is this series of hermite polynomials $\sum_{n=0}^\infty \frac{(-1)^n}{\sqrt{2^n n!}} H_n(x)$ a known function? Thank you!
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### Conditional probability measurement with n-grams

Let's say you have an alphabet with two words, a,b and a long string of these letters. You measure the probability of finding an $a$ or $b$ to be $p(a)=3/4$ and ...
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### How can I solve this recurrence?

I have a weird recurrence relation and don't know how to solve it: $$a_n = pa_{n-1} + qa_{n+1} + cb_n$$ $$b_n = p'b_{n+1} + q'a_n$$ $$a_0 = 1$$ $p,q,c,p',q' \in [0,1]$ and $p+q+c=1,p'+q'=1$. Thanks ...
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### How to calculate this integral?

Define $$F=(x^2+y-4,3xy,2xz+z^2)$$ Compute the integral of Curl F over the surface $x^2+y^2+z^2=16, z\geq 0$
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### Matrices with eigenvalues 0 and 1

How can you describe all $2\times 2$ matrices whose eigenvalues are 0 and 1? My attempt: I know that 0 and 1 has to be solutions of the characteristic polynomial. And I've considered some examples ...
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### Taking a derivative of the same equation in different form produces different results

I have a feeling this question is going to have an obvious answer, but I'm left a bit puzzled. I have the following equation: $$\tag 1 \frac{1}{S(x)} = \frac{W}{12}\cdot(1 - U(x)^2)$$, where $W$ is a ...
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### Does $S^1 \times S^2$ nontrivially covers itself?

I am not sure how to show whether or not $S^1 \times S^2$ nontrivially covers itself. Some help would be appreciated. Thanks
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### $AB - BA = I$ in Hilbert Space [duplicate]

Let H be a Hilbert space and $A$ and $B$ be bounded operators in $H$. How can I prove that $AB - BA = I$ is not possible ? Probably this is as easy as in the matrices case, but I couldn't prove it. ...
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### Prove that the $2$ form defines a symplectic structure

Prove that the $2$ form $$\omega = -2[(1+x_2^2)dx_1 \wedge dx_2 + dx_1 \wedge dx_3 + dx_3 \wedge dx_4]$$ defines a symplectic structure on $\mathbb{R}_x^4$. My definition of as ...
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### Integrating a partial fraction with multiple quadratic denominators

When integrating a real rational fraction, you first break it into partial fractions. You then end up with fractions with linear denominators $\frac{A}{(x-b)^n}$, which are easy. You also end up with ...
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### Clues to prove average in T is minor or equal than average in a smaller inner interval.

Suppose I want to prove (or disprove) this assertion Let $f$ be a discrete function, $T,h,k$ are constants So these terms are averages over $T$ and over $h$ $\sum\limits_{i=0}^{T}\frac {f(i)}{T}$ ...
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### Breaking a contour integral into 3 separate contours?

We can try to integrate the following function around a counter-clockwise circular contour: $$\frac{x^3}{(x-1)(x-2)(x-3)}$$ Can someone show how to use the Cauchy–Goursat theorem (explained here and ...
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### Stability under supremum of sets of social choice function with single peaked preferences

Here is a question emerging from reading Moulin, H. (1980). On strategy-proofness and single peakedness. Public Choice, 35(4), 437–455. The setting is as follows: A non-empty finite set of ...
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### Is the validity of measuring area by approximation an assumption of calculus?

The assumption that if you subdivide an area into more and more sub intervals, the approximation gets better and better. Has this been formally proved, or is it just intuition? Thanks!
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### PDE: Why do they have the wrong units?

Take a look, for example, at the telegrapher's equations (let's look at the voltage one). They have the wrong units. Equation $u_{x} = Li_{t} + Ri$ *where $u$ is potential in volts $V$, $L$ is ...
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### Homeomorphism on Identification Space

Let $\sim$ be and equivalence relation on the unit line $X=[0,1]$ defined by $x\sim y$ if either $x=y$ or $\textbf{both}$ $x$ and $y$ $\in$ {${0,\frac{1}{2},1}$}. Construct a homeomorphism ...
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### Why isn't this limit equal to $0$?

$f(2)=4$, $g(2)=9$, $f'(2)=g'(2)$. $\displaystyle \lim_{x \to 2} \frac{ \sqrt{f(x)}-2} { \sqrt{g(x)}-2}$. Why isn't this limit equal to $0$? Since $f$ and $g$ are differentiable at $x=2$, that ...
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### How does the Tarski axiom relate to the Grothendieck universe?

It's claimed, e.g. in the formulation of Tarski–Grothendieck set theory, that the Tarski axiom implies that for every set, there is a Grothendieck universe containing it. However, I can't see how ...
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### Solve the following integral using substitution only?

Can you solve the following integral using only substitution? $$\int \dfrac{dx}{\left(\sqrt{x^2-4}\right)^3}$$ I saw a solution to this which began with $x=2\sec(u)$, but is there another way to solve ...
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### why is an open faithfully-flat morphism fpqc?

Why is an open faithfully-flat morphism fpqc? In other words, why must an open faithfully flat morphism $X\rightarrow Y$ have the property that around every $x\in X$, there is an open nbhd $U$ of ...
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### normal vector condition on 1 component

If you have two normal random variables $Z_1$ and $Z_2$ possibly correlated namely you have a multivariate normal distribution $Z= (Z_1, Z_2)$ What is the conditional distribution $(Z_1|Z_2 =a)$? ...
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### Finding distributional limit

How to find $\lim_{\varepsilon\rightarrow 0+}f_{\varepsilon}$ in $D'(R)$, if $f_\varepsilon$ is defined as: $f_\varepsilon(x)=\frac{1}{\varepsilon^3}$ for ...
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### Topological boundary vs geometric boundary

Let $M_1=B((0,0),1)=\{(x,y) \mid x^2+y^2<1\}$ $M_2=\{(x,y) \mid x^2+y^2\le1\}$ What are the interior of $M_1$ and $M_2$ ? And What are the boundary of $M_1$ and $M_2$ ? How to find them? ...
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### Sequential criterion for differentiability

Is there a sequential criterion for differentiability,just like there is one for continuity ? If not then,why so ? I'm studying undergraduate real analysis and haven't really come across one. ...
I want to evaluate this integral using Residue Theorem $$\int_C^\ \frac{4z} {z^4 +6z^2 +1} dz =$$ $$C : |z| = 1$$ so I substitute letting $$\ W = z ^ {2 }$$ $$dw = 2z dz$$ and the ...