# All Questions

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### How do I know if a given polynomial is a quasi polynomial?

How do I know if a given polynomial is a quasi polynomial? For example, if I'm given the polynomial: $e^x\tan(x)$ or the polynomial $e^{(i-t)}t^3$, my gut feeling is that they're both not quasi ...
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### Finding median of union of two sorted (ordered) lists

We are given two sorted list of numbers $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$. Question is, how to find a median for list $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$. Algorithm should ...
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### Topology; Definition of the open ball and open sets confuses me

I just picked up T.W Gamelin’s book on topology. I started reading and got confused when I came to the definition of an open ball on the second page. It says $B(x;r) =$ All $y$ in the set $X$ such ...
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### Is this proof clear, complete and readable?

I am trying to prove this statement: $C^2/U(1)$ can be identified with $R^3$ so that the image of the $U(1)$ fixed point is $(0,0,0)$. And I was wondering if someone could tell me if the following ...
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### How to this Simplify Boolean Expression

Im very weak in math and logic, and currently try doing K-map, and got this as result: $(C'B')+(CB)$ my question is, can this be more simple? I tried it myself, but I got $0$. $(C'C)+(B'B)$ I just ...
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### formula for summation notation involving variable powers

I need help finding the formula for this summation notation: $$\sum_{k=1}^n{k^{2k} }$$ or $$1^2 + 2^4 +3^6 +.....+n^{2n}$$ And preferably not involving calculus.
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### Trying to prove concurrence of altitudes of a triangle.

I know that this question had been asked before, but I am not exactly following what the answers say. Doing my own way here: I am puuzzled how to continue? I named the points A,B,C, and the foot of ...
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### Irreducible and prime elements

In my commutative algebra lecture notes it says: A non-zero element $p$ of a ring $R$ which is not a unit of $R$ is called a prime element if $p=ab$ implies $a$ is a unit or $b$ is a unit. Is this ...
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### Permutation of positive real numbers

Consider a set of positive real numbers $\{P_1,P_2,\dots,P_n\}$ and a permutation of this set $\{Q_1,Q_2,\dots,Q_n\}$. Is it possible to find a permutation such that ...
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### Deriving angle from sin or cos

How can I derive the value in degrees of an angle starting from either the cos or sin value? $$\cos(t) = c_{1} \quad \text{or} \quad \sin(t) = c_{2}$$
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### Proving that if the semigroup (A, *) is a group, then the relation is an equivalence relation.

I'm aware that posting exam questions is probably frowned upon, but this isn't homework, I think I'm genuinely misunderstanding some part of the algebra. The question is this: Throughout this ...
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### Intersections of two parabolas given focii

As part of Voronoi's algorithm, I need to calculate the intersection of two parabolas to compute a breakpoint at run time. I've spent literally 8 hours on this, and I've only gotten my equations to ...
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### Show that $\left| \mathbb{P}(X=m)-\mathbb{P}(Y=m) \right| \le \mathbb{P}(Y\ne X)$

Let $X,Y$ two random variables of the same probability space. Show that $$\left| \mathbb{P}(X=m)-\mathbb{P}(Y=m) \right| \le \mathbb{P}(Y\ne X)$$ I think I need to start from LHS and split it ...
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### Number of teams and matches

This question has two parts. Given n players, how many different teams can be created with at least one and at most n-1 players? For example, given the four players A, B, C, and D, the following ...
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### Minimum weight vertex cover test set

I want to test the efficiency of an approximate algorithm for finding the minimum weight vertex cover. Is there an online test suite that contains weighted graphs with known vertex covers? If not, ...
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### Explanation solution partial-fraction of $\frac{x^2 + 2}{x^2 - 1}$

The partial fraction of $\dfrac{x^2+2}{x^2-1}$ is $1 + \dfrac{3}{2}\cdot(\dfrac{1}{x-1}-\dfrac{1}{x+1})$. I understand how you get $\dfrac{3}{2}\cdot(\dfrac{1}{x-1}-\dfrac{1}{x+1})$ but from where ...
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### Convergence of summable sequences

If $(a_n)$ is a sequence such that $$\lim_{n\to\infty}\frac{a_1^4+a_2^4+\dots+a_n^4}{n}=0.$$ How do I show that $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?
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### Is there a logic to formalize the concept of “understanding”

The question may seem little bit weird given that philosophers have been struggling to have a full grasp on the concept of "understanding". But I'm wondering if there are any logics (modal-based or ...
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### Find the inverse function 3

Find the inverse function for the following function: $$f(x) = \log_{\sqrt{4-x^2} } \left(x^3+5x^2-x \right)$$ Thanks.
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### Homomorphism between finite groups

I have to prove or disprove the following statement: If $\phi:G \rightarrow H$ is a homomorphism between finite groups, with non-trivial image (i.e. $\phi(G)\neq\{e_H\}$), then $\#G$ and $\#H$ ...
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### Proof, that every nonempty set of integers, not all zero, has a greatest common divisor

I'm searching for a proof or (better) a way to understand the proof from the book "Elementary methods in number theory", that every nonempty set of integers, not all zero, has a greatest common ...
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### Find sum of quartic function coefficients by its plot

the plot of quartic function $y=ax^4-x^2+bx+c$ is given: I need to find a sum of $$\frac{a}{|a|} + 2\frac{b}{|b|} + 4\frac{c}{|c|}$$ How to do it?
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### Laplace transform of $f(t)=te^{-t}\sin(2t)$

I was asked to find the laplace transform of the function $f(t)=te^{-t}\sin(2t)$ using only the properties of laplace transform, meaning, use clever tricks and the table shown at ...
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### Finding ellips equation by focuses and tangent line

The Ellips which has focuses in $(±3,0)$ and a tangent line $x+y-5=0$. I need to find ellips equation. I've founded these equations $\frac{x_{0}}{a^2} = \frac{1}{5}, \frac{y_{0}}{b^2} = \frac{1}{5}$ ...
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### How do I solve $(1+x^2)y'=\frac{1}{y}, z\geq 0, y(0)=0$?

I have this assignment: $$(1+x^2)y'=\frac{1}{y}, z\geq 0, y(0)=0$$ There was a long time ago that I solved one of those, but if I remember it right, I would want to rewrite the equation on the ...
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### Sylow subgroup of $S_{11}$

I want to construct some Sylow $3$-subgroup of $S_{11}$.This subgroup has $3^4$ elements. I know any Sylow $3$-subgroup is isomorphic to $(\mathbb{Z}/3\mathbb{Z})^3\rtimes P$ where $P$ is a Sylow ...
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### Injective homomorphism between a finite group $G$ and $GL_n(\mathbb{F}_p)$ where $p$ is prime

I'm looking for a solution to the following problem: Given a natural number $n$, a prime number $p$ and a finite group $G$, I need to find an injective homomorphism between $G$ and the group ...
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### Equation of heat conduction for spherical solid

What will be the correct Equation of heat conduction for a homogeneous spherical solid with constant thermal diffusivity K and no heat source?
### Every sequence in $\mathbb{R}$ has a monotonic subsequence
I have trouble with this kind of infinite construction in topology. Can someone check my proof is sound? Let $s$ be a sequence in $\mathbb{R}$. Then $s$ has a monotonic subsequence. There are two ...