# All Questions

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### Irreducible polynomials in $\mathbb{Z}_3/\left\langle x^2+1\right\rangle$

We wish to find all the irreducible polynomials in $\mathbb{Z}_3/\left\langle x^2+1\right\rangle$. I came across this problem in my course on advanced algebra. I have little knowledge about the ...
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### Mathematical induction.

Use mathematical induction (and proof by division into cases) to show that any postage of at least 12 cents can be obtained using 3 cent and 7 cent stamps. I thought this was the simple kind of ...
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### Determining a succinct big $\Theta$ expression

Determine a succinct big-$\Theta$ expression for the growth of function $$(\log^{50}n)n^2 + n^{2.1}(\log n^4) + 1000n^2 + 100000000n$$
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### Paramaterization of paraboloid and plane.

Consider the paraboloid $z=x^2+y^2$. The plane $2x-4y+z-6=0$ cuts the paraboloid, its intersection being a curve. Find "the natural" parameterization of this curve. I have set each equation equal ...
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### Evaluating and Simplifying a Double Integral

I have an integral as follows $f(t) = \int_r^\infty \frac{(sP)^{1-\rho}t^{-\alpha/2}}{1+(sP)^{1-\rho}t^{-\alpha/2}} \;dt$ I wish to get rid of the $s$ in $f(t)$ because this is an inner integral ...
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### normal distribution under special condition

Given Gaussian random var U∼N(−1,1) and V∼N(1,1), are T=(U+V, U−2V) (T is a 2-element vector) and W={U with 50% chance, V with 50% chance) also Gaussian random var? If they are, what is the mean and ...
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### Determining the value of A given $Z_4=Z_8\oplus Z_2/A$

Let Z define the integers and $Z_a$ define the integer group modulo a. I want to determine what A is. Given $Z_4=Z_8\oplus Z_2/A$, where $A\subset Z_8\oplus Z_2$, am I able to just say that given ...
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### Determining a lower bound on the order of a group based on its presentation

I am reading Abstract Algebra book by Dummit and Foote (3-rd edition). On pages 26-27 they define a dihedral group: $D_{2n} = \langle r,s | r^n = s^2 = 1, rs = sr^{-1} \rangle$ The authors ...
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### Just a little poker question i was wondering about?

There are six players remaining in an 18-player tournament. The top four players win prizemoney. All remaining players understand and utilize both equity and pot odds to make mathematically sound ...
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### Combinatorial problem of choosing points inside an equilateral triangle without them being too close.

Determine the smallest integer $m_n$ which satisfies the following property: If $m_n$ points are chosen inside an equilateral triangle of sides 1, then at least two of them are at distance ...
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### if [a,b] is a subset of(c,d) what relationships exist between a,c and b,d?

I want to know what is the relationships I couldn't figure it out???? if $[a,b]$ is a subset of $(c,d)$ what relationships exist between $a$, $c$ and $b$, $d$?
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### Limit $n \rightarrow \infty \frac{n}{e^x-1} \sin\frac{x}{n}$

I am just working through some practice questions and cannot seem to get this one. Plugging this into wolfram alpha I know the limit should be $\frac{x}{e^x-1}$, but I am having a bit of trouble ...
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### Statistics (unsure how to do it)

A person's resting heart rate is the lowest number of heart beats per minute when fully relaxed and without distractions. Age, fitness, genetics, health status and gender affect the resting heart ...
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### All Possible Pairs of 18

I will be having 18 Students in my class this year. I'd like to have them learn in pairs rotating every day with a different student in the class. What are all the possible pairings. for example on ...
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### The convergence of a product of sequences converging in $L^2$.

Earlier today I found myself pondering the following question for which I do not have a reasonable answer. Suppose $f_m\to f$ and $g_m\to g$ in $L^2$. Moreover suppose that $f_m g_m\in L^2$ for ...
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### Regularity of elliptic PDE in terms of weighted Sobolev space

There seems to be a whole industry of weighted ver. of the Poincaré's inequality (Weighted Poincare Inequality) I wonder if there are results like, weighted $L^2$ equivalent of (interior/boundary) ...
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### Why $\Bbb{E}f$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$?

In my professor's lecture note there is a remark saying that "$\Bbb{E}[f]$ is $\cal F_0$ measurable if $f$ is independent of $\cal F_0$". I think this should be easy, but I just don't see why. Can ...
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### If $|H|=112$ then $A_7\cap H \lhd H$?

I posted this because Alex Clark asked in chat and I'm not sure how to proceed. Let $G$ be a group such that it has a fixed subgroup isomorphic to $A_7$, which we done simply by $A_7$. Let $H$ be a ...
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### Given $k$ distinct linear operators, prove such an $\alpha$ exists

I have $k$ distinct linear operators $\{\phi_i\}$ which act on $V$, a vector space on some number field $K$ (in the sense that $\Bbb Q$ is the smallest possible one). Now I have to prove that there ...
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### Random matrix theory

A random symmetric $2 \times 2$ matrix $A = \begin{pmatrix} a_{11} & a_{12} \\ a_{12} & a_{22}\end{pmatrix}$ is a member of the gaussian orthogonal ensemble (GOE), if it satisfies three ...
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### Help in solving an integral.

I am trying to evaluate this integral, but could not find a solution. I tried it, assuming it to be product of two exponential and then tried integration by parts but it does not lead to anywhere. Can ...
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### Partitioning a number as a sum of $k$ non-zero numbers, but order does not matter

I would like some confirmation regarding my logic here, which I feel is 'suspiciously straightforward'. Say I wish to express a number as the sum of $10$ non-zero numbers, where order does not ...
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### What's the necessary condition for that any three vectors are parallel to the edges of a triangle in the plane?

What's the necessary condition for that any three vectors are parallel to the edges of a triangle in the plane? I answered the following: The necessary condition is that the vectors are ...
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### On the Spivak's proof of the theorem 3-11 (calculus on manifolds)

In second paragraph of the case 1 within the proof: What is $U$ s.t $A\subset U$ and satisfies in the proof of the case 1 of theorem 3-11. $\psi_i$ is defined on $U_i$ and its support is not ...
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### Normal Matrix with Real Eigenvalues is Hermitian

Let $A$ be a normal matrix. Then I want to show that, if $A$ has real eigenvalues, $A$ is Hermitian. (Notation: * denotes the complex conjugate, T denotes the transpose, and $\dagger$ denotes the ...
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### Definition of $f \vee g$ and $f \wedge g$

In Olav Kallenberg's Foundations of Modern Probability he uses the notation $f \vee g$ and $f \wedge g$ where $f, g$ are two functions from a set $\Omega$ to $\mathbb{R}$. What does this notation ...
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### Not sure what formula to use? (what to solve for?)

The question states, "The weight of people in a certain pacific island is normally distributed with a mean of 175 lb. and a standard deviation of 33 lb. They want to design a one-person canoe that ...
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### Orthogonal Position and Velocity Vectors

Is it true that if the position and velocity vectors of a moving particle are always perpendicular the path of the particle is on a sphere? If so how do I prove it? Geometrically I believe it makes ...
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### Express the root of the equation by Lambert W function

Lambert $W$ function is know as the root to the transcendent equation $$We^{W}=x$$ nevertheless, recent I found a equation of the type $$\sqrt{F}e^{1/F}=1/x$$ where $F=F(x)$. I just wonder that, ...
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### Fourier transform of $1 - \cos(xe^{-x^2})$

Is there a closed form expression or maybe an infinite series? If not is there a "good" approximation to it? Even a "good" approximation of the fourier transform close to zero frequency would do. Can ...
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### Proving $\sin^2(x) + \cos^2(x) =1$ using calculus

Ok so the book in which I found this doesn't say mention the trigonometric functions by name but the question is: Let $s(x)$ and $c(x)$ be functions satisfying $s'(x)=c(x)$ and $c'(x)= -s(x)$ for ...
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### Quiver algebra as a wreath product?

I'm having trouble understanding a definition of a quiver Hecke algebra. Suppose $k$ is a commutative ring, and $\Omega$ a finite set. We build a quiver $Q_{\Omega,n}$ with vertex set $\Omega^n$. ...
### Proving the asymptotic relationship between $(lg\cdot n)^{0.5}$ and $lg\cdot (n^{0.5})$?
Say $f(n) = (lg\cdot n)^{0.5}$ and $g(n) = lg\cdot (n^{0.5})$ It would appear that $f(n) = O(g(n))$ for $n \gt 55$ correct? How do I go about proving the the relationship for this?