0
votes
0answers
98 views

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 ...
2
votes
2answers
168 views

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? ...
3
votes
2answers
462 views

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 ...
0
votes
1answer
61 views

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 ...
0
votes
1answer
57 views

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 ...
1
vote
1answer
338 views

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 ...
1
vote
1answer
435 views

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 \\ ...
3
votes
2answers
213 views

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) ...
0
votes
1answer
137 views

Addition of Ideals

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 $(a)+(b)=R$.
0
votes
0answers
57 views

Finding ideals in the ring $\mathbf{Z}_{12}$

I have problem with this task. Anybody can show how designate all ideals in the ring $(\mathbf{Z}_{12},+_{12},\cdot_{12})$?
-1
votes
1answer
145 views

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 ...
0
votes
1answer
42 views

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 ...
1
vote
1answer
166 views

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 ...
0
votes
1answer
75 views

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 ...
1
vote
1answer
71 views

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 ...
1
vote
0answers
64 views

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 ...
0
votes
0answers
31 views

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 ...
0
votes
3answers
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!
8
votes
1answer
152 views

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 ...
1
vote
1answer
42 views

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 ...
0
votes
1answer
21 views

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 ...
0
votes
2answers
34 views

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.
12
votes
1answer
144 views

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 ...
0
votes
0answers
56 views

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 ...
0
votes
0answers
31 views

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: ...
1
vote
1answer
41 views

Normalisation of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ is UFD…

We know that the normalisation of $\mathbb{Z}$ in $\mathbb{Q}(\sqrt{d})$ where $d\in \mathbb{Z}$ is $$O=\mathbb{Z}[\beta], \text{$\beta=\sqrt{d}$ if $d\equiv2,3 \pmod 4$}; \ \frac{1+\sqrt{d}}{2} ...
1
vote
2answers
254 views

Eignvalues of Laplacian operator and Sobolev spaces

Let $\Omega$ an open bounded in $\mathbb{R}^n$ and $I$ un interval in $\mathbb{R}$, let $t_0 \in I$, $f\in H^1_0(\Omega)$, $g\in L^2(\Omega)$. Let $(\lambda_n)_{n\in \mathbb{N}}$ the eigen values od ...
1
vote
1answer
54 views

Prove that $ A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$

I'd like to get some help So I need to prove that when $A^2=0 \Leftrightarrow \mbox{ Row}A⊥\mbox{Col}A$ Linear Algebra, of course. Thanks
1
vote
1answer
35 views

How to show that a representation is supercuspidal?

Let $G$ be a reductive group. If we know a representation $\pi$ of $G$ explicitly, how could we determine that $\pi$ is supercuspidal or not? Are there some references about this? Thank you very much. ...
0
votes
0answers
33 views

Why does this theorem provide a neccesary and sufficient condition for a $2 \times 2$ linear system of congruences to have a unique solution?

Why does the following theorem both provide a neccesary and sufficient condition for a $2 \times 2$ linear system of congruences to have a unique solution ? I see that $\gcd((ad-bc),m) = 1$ is a ...
5
votes
3answers
270 views

Are there other approaches for the foundations of mathematics, other than logic and set theory?

Are there other approaches for the foundations of mathematics, other than logic and set theory? And why does set theory begin talking about objects and groups of objects. Is it proven somewhere that ...
0
votes
1answer
29 views

probability function

have adiscreet random variable and cumulative distribution function ,how do i get the probability function,mean ,variance x 1 2 3 4 5 6 7 8 F(x) 0.1 0.2 0.25 0.4 ...
0
votes
1answer
41 views

Highest Common Factor unique factorisation domain

Let $R$ be a UFD (unique factorization domain) $a, b, c \in R$ nonzero. Take $d = hcf(a,b)$ Show that $a = dx$ and $b = dy$ for some $x,y \in R$ so that $hcf(x,y) = 1$ This seems like it should be ...
1
vote
0answers
33 views

Density of $C^0([0,T]\times M)$ in $L^p([0,T]\times M)$?

Let $M$ be a compact $C^2$ hypersurface embedded in $\mathbb{R}^n$ of dimension $n-1$. Is the space $C^0([0,T]\times M)$ dense in $L^p([0,T]\times M)$? How do I prove this or what theorem can I use? ...
0
votes
0answers
58 views

Showing that triangles in $\mathbb{Z}$ are thin

If we let $\mathbb{Z}$ be generated by $\{3,5\}$ then I have a question which asks me to show that geodesic triangles are $k$-thin and to find a minimum bound on $k$. I have been thinking about this ...
2
votes
1answer
51 views

If $f$ is an anti-symmetric polynomial, then $f=g\prod_{1\leq i < j\leq n}(X_i-X_j)$ for some $g$ symmetric

So we have the situation that $f\in K[X_1,...,X_n]$ is anti-symmetric, which means that $\sigma (f)=\pm f$ where it is a plus if $\sigma$ is an even permutation on the $X_i$ and a minus if it is not ...
0
votes
1answer
35 views

The gradient in different dimensions

I study to final exam in calc 3. Question: Are my thoughts about the gradient correct? The gradient is a normal vector to a plane given a point in $xyz$-plane. With this vector you can calculate ...
1
vote
1answer
48 views

$T_3$ is quasi isometric to $T_4$

I have a question which asks me to show that $T_3$ is quasi isometric to $T_4$, that is the three and 4 valence trees. I know that this means that I have to define a map $f:T_3\rightarrow T_4$ such ...
1
vote
0answers
48 views

Counting problem (Combinatorics/Discrete math)

A restaurant has a menu containing 12 starters, 22 main courses. a) 7 friends visit. 5 have a starter, all 7 have a main course. How many ways to do this? - My answer: (12)5 x (22)7 (i.e. 12x11x...x8 ...
3
votes
2answers
94 views

How can there exist an isomorphism between this group and the cyclic group $(\mathbb Z,+)$?

I have a group over $\mathbb Z$, defined by the binary operation $*$, such that $a*b:=a+b+2$. From the previous exercise, I have deduced that the identity-element is $-2$ and that it is an abelian ...
0
votes
1answer
703 views

8 less than triple a number is equal to -5 (I'm trying to find the unknown number)

It's a hard question... for my thinking! I've tried many solutions but haven't figured it out. (Let $M$ be the unknown number) I've tried: $M^3-8=-5$ Then $M^3-8+8=5+8$ Then I think the answer ...
0
votes
2answers
279 views

How to prove this Gram determinant

Let $E$ be an Euclidian oriented vector space of dimension $3$ and $x,y,u,w \in E$. How do we prove (without coodinates) $$ \det \begin{pmatrix} \langle x,u \rangle & \langle x,w \rangle \\ ...
0
votes
1answer
80 views

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 ...
0
votes
1answer
83 views

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 ...
0
votes
2answers
49 views

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 ...
2
votes
1answer
70 views

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.
0
votes
0answers
53 views

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 ...
0
votes
1answer
124 views

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 ...
3
votes
1answer
148 views

extension problem of a spectral sequence

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. ...
1
vote
1answer
33 views

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 ...

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