# All Questions

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### The Real solution of the equation $x^\frac13= -1$

Surely, $x=-1$ is the only solution. But, wolframalpha says there is no solution in the real field. I think this is because they transformed $x^\frac13$ to $e^{\frac13(lnx)}$. But why did they ...
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### An integration question to be solved without using differentiation under the integral sign.

$$I(\alpha)=\int_0^1 \frac{x^\alpha-1}{\ln x}dx.$$ As the title says, if someone could solve this without using the differentiation under the integral sign technique, I would be very grateful.
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### Highschool inequality 2

Given a real number $a>0$ find a $b>0$ such that $\sqrt{(x-2)^2+(y-1)^2+(z-1)^2+(ω-3)^2}<b\Longrightarrow |xyzω-6|<a$ I tried the procedure followed in highschool inequality 1 but it ...
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### Are primary ideals always contained in unique maximal ideal?

Just wondering, is this a standard fact? I notice a couple Banach algebra texts define primary ideals in this way. Another question: does this property, i.e. being contained in a unique maximal ideal, ...
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### Computing efficiently a small base to the power a large number

Is there a fast algorithm to compute an exponential with a small base, (namely , close to 1) For example, computing 1.01 to the power 100?
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### Is the flux through $A$ the same as the flux through $B$?

In the figure below, the amount of field lines through $A$ is the same as the amount of field lines through $B$, but can you say the flux through $A$ is the same as the flux through $B$ as well?
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### How to prove a function is a matrix exponential?

If $F(x) = \exp(x A) = \sum_{i = 0}^\infty \frac{1}{i!} x^i A^i$ where $F(x), A \in \mathbb{R}^{n \times n}$, then $$F(x + y) = F(x) F(y) = F(y) F(x)$$ holds. When is the ...
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### Prove if $E_1$ and $E_2$ are measurable then$m(E_1\cup E_2)+(E_2\cap E_2)=m(E_1)+m(E_2)$

by additivity $m(E_1\cup E_2)=m(E_1)+m(E_2)$ (because $E_1,E_2$ are measurable) but i don't know what to do with $E_1\cap E_2$. I tried to use demorgan's identity to solve this part but this is not ...
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### Self complementary graph with a pendant vertex

Show that if a self-complementary graph contains a pendant vertex, then it must have at least another pendant vertex. Let $G$ be a graph of order $n$, so it has $n(n-1)/4$ edges, just like its ...
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### What is the pattern of this sequence of numbers? [on hold]

$5,\,7,\,14,\,16,\,33,\,55,\,82,\,114$ I was given this sequence of numbers, but I can't find the pattern to it. I have put them into different online sequence calculators, but they did not work. ...
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### Variance of not quite the product of two independent, normally distributed random variables

Let's say I have two independent variables, $X\sim N(10,9)$ and $Y\sim N(5,4)$. $X$ represents the number of orders received in a month, and $Y$ represents the size of each order. For this example, a ...
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### Poisson process Probabilities

If I assume that {$N(t)=: t >= 0$} is a Poisson process with intensity $\lambda$. For $0<s<t$, how would I find the Pr{$N(t)>N(s)$}?
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### point of intersection of a line l and the plane p, i get 0 somehow

$L: x=\frac {y-1}{2}=\frac {z+1}{3}$, $P= x − 2y + z = 1$. Find the point of intersection of the line L and the plane P.
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### Arrangement of the word MATHEMATICS if last spot must have the letter 'T'

How many ways can the word MATHEMATICS be arranged if the last letter must be a T? My solution: There are $2$ possible choices for the last letter (There are $2$ different T's), which leaves $10$ ...
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### why does $\frac{d}{dx} log_b(x)$ not = $\frac{lnb}{x}$?

I know that $log_b(x) = \frac{lnx}{lnb}$, and that differentiating $$\frac{d}{dx}(\frac{lnx}{lnb}) = \frac{1}{lnb}\frac{d}{dx}(lnx)=\frac{1}{xlnb}$$, so where is my mistake when I do it this way: ...
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### Show that A*B and B*A have the same order [duplicate]

How can I show that the elements A * B and B * A have the same order? where A, B belong to a finite group G How can I prove that 2 elements have the same order? I was thinking of showing that ...
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### Problem in deducing the number of onto functions

Let $A, B$ have $m, n$ elements ($m > n$). Therefore, the number of onto functions from $A$ to $B$ is: $$\sum_{k = 0}^n (-1)^k \binom{n}{k} (n - k)^m.$$ How can one use the IE (Inclusion/Exclusion) ...
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### Find some n such that $|s-s_n|< 10^{-3}$

Consider the series $\sum_{n=1}^\infty \frac{1}{n^2}$. Let $s_n$ be the $n$th of the series and $s$ be the sum of the series. Find some $n$ such that$$|s-s_n|< 10^{-3}$$ Can someone please ...
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### How to measure reaction time when given a right or wrong answer test? [on hold]

Here is an example of some data. test 1 test 2 test 3 correct 20 10 13 wrong 4 8 12 Lets ...
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### If a quadratic form $f$ takes the minimum on a triangle in a vertex, what can I say about min of $f$ on edges of a subdivision?

Let $f(x)=x^2+y^2$ be the Euclidean square-norm and $A,B,C\in\mathbb{R}^2$ be vertices of a triangle $\Delta$ such that $f$ takes the maximum on $\Delta$ in $C$, the minimum in $A$ and takes the ...
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### Find preimage of a function given that the image of the function [on hold]

I have a following question that I need help on. Let $F(s,t)=\left( s^2 \cos(t), s^2 \sin(t) \right).$ Find the $F$-preimage of $[0,1] \times [0,1].$
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### Does a simplex with equal altitudes have to be equilateral?

Consider a simplex in $\mathbb{R}^d$. Assume that all its altitudes have the same length. Does it necessarily mean that the simplex is equilateral, i. e. all distances between its vertices are equal ...
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### Group Theory: Finding Homomorphisms From a Cyclic Group to an Automorphism Group.

I have to find all the homomorphisms, $$h:C_{5}\to Aut(C_{31})$$ Given that there are thirty elements in $Aut(C_{31})$, do I have to find the order of each of the elements and then see which of them ...
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### finding the probability to get a diploma

For getting a diploma a person needs to go to $3$ interviews at $3$ teachers: $A,B,C$. In each interview a teacher can give a positive opinion or negative opinion. The person will go to interview at ...
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### Is the discriminant of a polynomial surjective onto $\mathbb Z$?

Consider polynomials of degree two over $\mathbb Z$: $f = ax^2+bx+c$ The discriminant is $D = b^2-4ac$ And we can show that $D=2$ is not a possible value for $D$. I wonder if the value $D=2$ ...
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### write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$

write $\iiint_E \hspace{1mm}dV$ in 6 forms. where $E = \left\{ (x, y, z)\hspace{1mm}|0\leq z\leq x+y, x^2\leq y\leq \sqrt{x},0\leq x\leq 1\right\}$ As you can see two forms are easy. \iiint_E ...
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### Riesz Projection as a Cauchy type integral

Let \begin{equation*} f(\zeta)=\sum_{k\in\mathbb{Z}}\widehat{f}(k)\zeta^k \end{equation*} be a complex-valued function on unit circle $\mathbb{T}=\{ \zeta\in\mathbb{C}:|\zeta|=1\},$ where ...
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### Tower of Hanoi variation from Concrete Mathematics - possible arrangements

From Concrete Mathematics, there is a problem that describes a variation of the Towers of Hanoi, where the disks can not move directly from peg $A$ to peg $B$, but must go through a middle peg. ...
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### Modelling the Möbius strip using implicit functions

While researching on Möbius strips I found its parametric representation on a lot of websites claiming it is easier. Can someone please explain what problems appear when modelling the Möbius strip ...
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### Dense sets and Empty Interior

if $A$ is dense in $X$, is there a relation which shows in which cases $A$ has empty interior ? $\mathbb{Q}$ has an empty interior as a dense set in $\mathbb{R}$, so does its complementary in ...
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### Probability function and distribution - taking out fish from a pool

In a pool of fish there are 4 fish of type A, 3 fish of type B, 2 fish of type C, 1 fish of type D. We take out fish without returning them until we get fish of type C for the first time. ...
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### Does a bounded countably infinite union of sets with volume have volume?

If $A_1, A_2,...$ are sets with volume and $A= \cup_{i=1}^\infty A_i$ is a bounded set, must $A$ have volume? This was a homework problem that we went over in class, and if I remember correctly the ...
Find in scalar parametric form an equation for the line of intersection of the plane $P$ and the plane with Cartesian equation $2x + y − z = 0$. $P= x − 2y + z = 1$.
Given a unit vector $A\in \Bbb{R}^N$ and an angle $\theta$, the unit vector $P$ needs to satisfy $\left<A,P\right>=\cos\theta$. How to sample $P$ uniformly? For example: If \$A = ...