# All Questions

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### marginal distribution of Ornstein Uhlenbeck process

I am learning the OU process. For now, what I can understand is that the OU process is the strong solution of a SDE $d\sigma²(t)=-\lambda \sigma²(t)dt+dz(\lambda t)$ where z is the compound possion ...
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### What is the point of triangulating topological spaces?

In a general sense, what is the purpose to triangulating, for example, a 3-dimensional topological space? What advantages does it give if we can triangulate a Seifert-Weber space into 23 tetrahedra? ...
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### Domain of the Function Square Root of 12th Degree Polynomial

Find the Domain of $$f(x)=\frac{1}{\sqrt{x^{12}-x^9+x^4-x+1}}$$ My Try: The Domain is given by $$x^{12}-x^9+x^4-x+1 \gt 0$$ $\implies$ $$x(x-1)(x^2+x+1)(x^8+1)+1 \gt 0$$ Please help me how to ...
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### Am I wrong ? (2)

Let $X=C[0,1]$ be the space of real continous functions on $[0,1]$. $X$ is a Banach space with the two norms $$|f|_\infty=\sup_{s\in[0,1]}|f(s)|$$ and ...
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### Show that $\forall x \in \mathbb R$ with $x>0$, the improper cosine integral exists and is riemann integrable

This is the cosine integral: $$\operatorname{Ci}(x):=-\int_x^\infty \dfrac {\cos t}t dt.$$ I need to show that the improper Riemann-Integral exists. I've searched on the web for two hours now and no ...
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### is retract of a hausdorff space closed in that space?

If $Z$ is a topological space, we call $Y\subset Z$ a retract of $Z$ if there is a continuous map $r:Z \rightarrow Y$ such that $r(y)=y$ for all $y\in Y$. If $Z$ is Hausdorff and $Y$ a retract of ...
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### matrix inverse in tensor notation

Suppose there is a matrix $A$ that transforms vectors, $$Y = A x$$ Now express this in some other coordinate system, with $x = B z, \,\, y = B w$, so \begin{align*} & Bw = A B z \\ ...
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### find center of sphere, of sphere inscribed into cone at deepest position

How to inscribe a ball into a cone? If I position a ball into a cone at the deepest position possible, cut a plane centric through that 3D object and just look at that plane, then I assumed that: a) ...
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Let $R$ be a principal ideal domain $a,b\in R$ with $a$ not equal to $0$. We know $(a)+(b)$ is an ideal of $R$. Suppose that $\gcd(a,b)=1$, show that $(a)+(b)=R$.
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### Finding ideals in the ring $\mathbf{Z}_{12}$

I have problem with this task. Anybody can show how designate all ideals in the ring $(\mathbf{Z}_{12},+_{12},\cdot_{12})$?
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### Given two concentric circles with radiuses r < R, can we estimate the number of chords in between the circles?

Given two concentric circles with radiuses $r < R$, can we estimate the number of chords in between the circles? With more details, fix a point $P$ on the inner circle. Trace a tangent to the ...
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### Average value calculation

A company has 2 mines: A and B. The production per person in mine A was 22,6 tons last year, and 27,9 tons in mine B. Last year 60% of the total production came from mine A. The question is the total ...
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### What statistical test should i use for the given scenario?

I have an hypothesis that in people who have used both ios and android, a majority of them say 70% prefer android. I have a sample size of 50-60 users who have used both ios and android. What ...
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### Joint probability generating functions, help please!

With a sequence of $N$ independent Bernoulli trials performed, where $N \in \mathbb{Z}^+$ and the probability of success on any trial is $p$, and $S$ and $F$ being total number of success and fails ...
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### Euclidean algorithm of two polynomials

I got stuck on this question: Find the monic gcd of $f(x)=x^5-6x^4+13x^3-11x^2+x+5$ and $g(x)=x^2-3x+2$. I worked through the Euclidean algorithm, first multiplying $g(x)$ with $x^3$ but then the ...
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### hard inequalities

I have to find real $x$ that satisfy the equation: $\dfrac{x^7}{7} = 1+10^{1/7}x(x^2-10^{1/7})^2$ I saw that the way is to look for solution of the form: $x = a^{1/7}+b^{1/7}$. my question is: how ...
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### Is there a term for an endomorphism defined up to conjugation by an automorphism?

Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that ...
127 views

### If $X\sim exp(\lambda)$ what is the PDF of $X^2$?

If $X\sim exp(\lambda)$ what is the Probability density function of $X^2$? I'd like to know how to calculate it, and what is the way... Thank you!
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### How prove $a_{n}=[\sqrt{2}n]+[\sqrt{5}n]$ Contains infinitely even numbers.

let sequence $$a_{n}=[\sqrt{2}n]+[\sqrt{5}n]$$ where $[x]$ is the largest integer not greater than $x$ show that $\{a_{n}\}$ Contains infinitely even numbers. also I guess ...
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### 2 circle questions

1) I need a hint on this one that I know how to solve using trigonometry but not geometry: Find the equation of a circle that touches $x$-axis in the $(0, 0)$, and touches the circle of a known ...
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### Polynom Space, check if U a base

$R_5[x]$ is a polynom space which is lower than 5 over R (Including the zero polynom). Given: $U = \{p(x) \in R_5[x] | p(0) = p(1) = p(2)\}$ Prove that U is a sub-space of $R_5[x]$. Find a base to ...
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### Fractions from least to greatest

What is the fastest way to find the least common denominator of all the fractions without losing too much time? 7/9 , 1/4, 14/15, 2/3, 1/2 Thanks.
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### When is a number in $\mathbb{Q}(\sqrt{a_1},\ldots,\sqrt{a_n})$?

Given an algebraic number $\alpha$ with minimal polynomial $P(x)$ of degree $2^n$, how can I decide if there are integers $a_1,\ldots,a_n$ such that ...
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### Give an example of a faithful representation of $D_8$ of degree 3.

Give an example of a faithful representation of $D_8$ of degree 3. So $D_8$=<$a,b : a^4=b^2=1, ab=ba^{-1}$>. A representation is faithful if ker(p)=e. The solution to this question i am given is ...
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### Hypothesis testing Mu question

We are told that we are testing for age and that the alternative hypothesis is: I claim that the true mean age of English students is greater than $19$ years. Then presumably the null hypothesis is: ...
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### Find $5$ digit number equal to sum of all $3$ digit numbers with distinct digits that can be formed from it

Find a number $N$ with five digits, all different and none zero, which equals the sum of all distinct three digit numbers whose digits are all different and are all digits of $N$. What could be done ...
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### Probability generating function question

The probability generating function of a non-negative, integer valued random variable $A$ is given by: $G(b) = \cfrac{e^{2(b-1)}}{2-b}, (|b| \lt 2)$ To determine ...
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### Proof of Markov Property

I'm trying to understand a simple proof for the markov property which states that: "$A_1$, $A_3$ are conditionally independent given $A_2$ iff $P(A_3 | A_1 \cap A_2)=P(A_3|A_2)$" The Proof begins as ...
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### Can we solve this equation $\frac{\cos\theta}{\cos{\theta}^2}=k$

I was in doubt that we can solve these type of Equation or not: $\frac{\cos\theta}{\cos{\theta}^2}=k$ where $k$ is a given constant.
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### Exercise 5 .39 page 46 Real and Abstract Analysis [Hewitt and Stromberg]

How to show that every non-Archimedian field contains a subfield algebraically and order isomorphic to $\mathbb{Q}(t)$, where $\mathbb{Q}(t)$ is the field of rational functions with coefficient in ...
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### numerical update rule for discretized hawkes excitation process

So I think I am just misunderstanding some simple notation or something and would appreciate some help. I am trying to replicate this model in an agent based model, but I cannot seem to figure out the ...
From Hatcher's SSAT, If the coefficient group $G$ is a field, then $H_n(X;G)$ is the direct sum $\oplus_p E^\infty_{p,n-p}$ of the terms along the $n^\text{th}$ diagonal of the $E^\infty$ page. ...
### About definition of $L^\infty(0,T;L^\infty(\Omega))$ and null sets
The norm in $L^\infty(0,T;L^\infty(\Omega))$ is $$\text{esssup}_{t \in [0,T]}\text{esssup}_{x \in \Omega}|u(t,x)|$$ In the inner essential supremum, can the null set (on which the function fails to ...