0
votes
0answers
5 views

How to get this very simplified demographic forecast?

I'm working on the simulation of a population growth. The variables and hypothesizes are the following: Lifetime: X years (X constant for everybody, yeah !) Initial population: Y people (with always ...
0
votes
0answers
3 views

Movement of birds - Acceleration, Velocity, Time and Displacement. Needed for an assignment

Hi so there are a quandary of birds sitting on a tree.There are 3 teams observing the movement of the birds. Team 1 observes that on their first flight the birds move a short distance across a branch ...
0
votes
0answers
8 views

a question on Frechet derivative

Suppose the derivative of a functional is given by $\int_{\Omega}(\vec{v}.\nabla u)|\nabla u|^{p-2} \phi$ for $\phi\in W_0^{1,p}(\Omega)$, then what is the functional?.
0
votes
0answers
6 views

limsup facts - which imply which?

According to this answer on my previous question, $\limsup X_n = \bigcap_{n \geq 1} \bigcup_{m \geq n} X_n$ $= \bigcap_{n \geq 1} \bigcup_{m \geq n} X_n$ $= \bigcap_{n \geq k} \bigcup_{m \geq n} ...
0
votes
1answer
17 views

If $X$ has a Poisson distribution with $E[X]=\lambda$, does $Var[X^2]=4\lambda^3+6\lambda^2+\lambda$?

Suppose $X$ has a Poisson distribution with mean (and therefore variance) $\lambda$. Using Excel to explore properties of the distribution of $X^2$ with some small integer values of $\lambda$ I ...
0
votes
2answers
10 views

Proper use of indicator function

Given a set $X$ and a subset $A \subseteq X$ the indicator function $\boldsymbol{1}_{A} : X \rightarrow \{0,1\}$ of $A$ is defined as $$\boldsymbol{1}_{A}(x) = \begin{cases} 1 & \text{if } x \in A ...
1
vote
0answers
13 views

How many possible words of this type can be formed?

We are making $10$ letter words using the letters $A,C,G,T$. How many possible words are there of the form $A...AC...CG...GT...T$ This is where all of the $A's$ go before the all of the $C's$ and ...
0
votes
0answers
8 views

Uncountable “relatively independent” subset of finite dimensional vector spaces over an uncountable field

Let $V$ be a $n$ dimensional vector space over an uncountable field ; then does there always exist an uncountable subset $S$ of $V$ such that any $n$ vectors of $S$ are linearly independent ? ( I can ...
1
vote
0answers
9 views

Example 2.3.4 in Hartshorne algebraic geometry

I am reading Hartshorne algebraic geometry. Example 2.3.4. Let $k$ be an algebraically closed field, and consider the affine plane over $k$, defined as $A^2_k = \text {Spec} k[x,y]$ . The closed ...
1
vote
1answer
21 views

Weird contradiction between equations

A guy that I tutor came to me with the following question: The time it takes for body $A$ to pass 160 km is 5 hours longer than the time it takes for body B to pass 90 km. The speed of body A is ...
0
votes
0answers
35 views

verify that the set $\{0,1,2,3\}$ is not a group under multiplication modulo $4$

Given the set $\{0,1,2,3\}$: -Associativity holds for this set -Closure holds for this set (constructing the Cayley table, all entries in the tables are in this set). -there is an identity element ...
1
vote
0answers
6 views

min and max number of hexagons in hexagonal tiling

Is there a way to calculate the maximum and minimum number of hexagons in a hexagonal tiling of a surface with regular identical size hexagons, knowing the area of the surface and the area of the ...
0
votes
1answer
13 views

Speed at which hands of clock approaching one another.

A clock's hour hand's length is $1$, and its minute hand's length is $r$. First I had to find the distance between the tips of the hands at 4:00. I did this using the law of cosines. This gives me ...
0
votes
0answers
9 views

How do I compute the norm of a non-principal ideal of the ring of integers of a quadratic field without using ''large'' results

I am trying to compute the norm of the ideal $I=(7, 1+\sqrt{15}) \trianglelefteq \mathbb Z[\sqrt{15}],$ the ring of integers of $\mathbb Q[\sqrt{15}].$ I knew $I^2$ would be principal, as $I\bar ...
0
votes
0answers
30 views

Eyebrow calculation

Given a width of 71 and a height of 35, what are the following dimensions: left side, right side, radius, and base?
-8
votes
0answers
28 views

find the value of Sequential , it is easy question [on hold]

\begin{equation*} 1+1+3/4+1/4+5/16+3/16+7/64+5/64+..... \end{equation*}
0
votes
1answer
13 views

Is it true that the number of arbitrary constants in the solution always equal to order of the ordinary differential equation?

Is it true that the number of arbitrary constants in the solution (if solutions exist) always equal to order of an ordinary differential equation? If yes, how to "prove" such a statement, if it can be ...
1
vote
1answer
18 views

How to prove that the dependent variable could not be expressed explicitly in terms of the independent variable(s)?

Consider the equation that $$xy=\log{y}+1\text{.}$$ How does one prove that $y$ cannot be expressed explicitly in terms of $x$? By the way, I do not know how the adverb "explicitly" is strictly ...
-1
votes
2answers
51 views

Why is this true? $(\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C=1 \implies A+B+C=\pi)$

Why is this True? $$\cos^2A+\cos^2B+\cos^2C+2\cos A\cos B\cos C = 1 \Rightarrow A+B+C = \pi$$ with this assumption that $$0\leq A,B,C<\frac{\pi}{2}$$
0
votes
1answer
13 views

Show $\lim X_k < \infty$ is in tail sigma-algebra

Show $\lim X_k < \infty$ is in tail sigma-algebra Given random variables $X_1, X_2, X_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(X_{n+1}, X_{n+2}, ...)$ be their tail sigma-algebra. For ...
0
votes
0answers
13 views

Show $\sum_k X_k < \infty$ is in tail sigma-algebra

Show $\sum_k X_k < \infty$ is in tail sigma-algebra Given random variables $X_1, X_2, X_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(X_{n+1}, X_{n+2}, ...)$ be their tail sigma-algebra. For ...
1
vote
0answers
9 views

What is the pdf of $X$, where $dX_t = -aX_t + d N_t, N_t$ is a compound Poisson process?

I would like to find the probability density function (at stationarity) of the random variable $X_t$, where (I'm not sure this notation makes sense, I'm not very familiar with the stochastic calculus ...
-1
votes
1answer
9 views

Inversion of a matrix in a system of linear inequalities

I would like to know if someone knows sufficient conditions on $A\in\mathbb{R}^{n\times n}$ and $b\in\mathbb{R}^{n}$ such that for all $x\in\mathbb{R}^{n}$: $$Ax\leq b \Rightarrow x\leq A^{-1}b \text{ ...
0
votes
0answers
17 views

Consider extension $[\mathbb{Q}(\alpha\ ):\mathbb{Q}]$ where $\alpha\ $ is zero of $P(x) = x^4 + 9x^{2} + 15 $.

Consider extension $[\mathbb{Q}(\alpha\ ):\mathbb{Q}]$ where $\alpha\ $ is zero of $p(x) = x^4 + 9x^{2} + 15 $. Find $[\mathbb{Q}(\alpha\ ):\mathbb{Q}(\alpha^2 + 3)]$. My attempt: By Eisenstein's ...
-1
votes
0answers
6 views

Integration of a tensor product

I would like to compute the integral of a diadic tensor on the sphere, but I can't see how to do this : $$\int_{\Bbb S^2}\mathbf{\vec n} \otimes\mathbf{\vec n}\mathrm{d}^2\mathbf{\vec n} $$ Where ...
0
votes
1answer
13 views

Largest number for which a laurent series converges

For part $(a)$ I got summation from $\sum^{\infty}_{n=0}(-1)^n\frac{z^{2n}}{(2n+1)!}$ Is this correct? Could someone explain how to do part (b) because I have no idea where to start Thanks
0
votes
0answers
2 views

An MCQ for finding the extremal of the functional $J = \int_{a}^{b} F(x, y, y^{'})$

Consider a functional $$J = \int_{a}^{b} F(x, y, y^{'}),$$ where $F(x, y, y^{'}) = \frac{1 + y^{2}}{(y^{'})^2}$ for admissible function $y(x).$ Which of the following are extremals for $J$? $y(x) = ...
1
vote
0answers
17 views

Is it 3-D Catalan numbers?

I am studying Catalan numbers recently but I think that how about 3-D Catalan? So that I imagine following situation ; A man travel through the path-way parallel to $ x, y, z $ axis from O ...
0
votes
0answers
20 views

How to find kth smallest value of a linear equation

Here's a question that was asked in IOITC 2009 India. Even though it should have a solution related to algorithms, yet I post it here as it is pretty "number-theoretic". Indraneel loves posing ...
-4
votes
1answer
51 views

IMO 2015 Problem 3 [on hold]

Let $n$ and $k$ be positive integers. Prove that if $n$ is relatively prime with $30$, then there exist integers $a$ and $b$, each relatively prime with $n$, such that $\frac{a^2-b^2+k}{n}$ is an ...
0
votes
1answer
12 views

Finding surface of revolution isometric to helicoid

I'm trying to find a function $f(x)$ such that the two surfaces given below are isometric: $$f_1(x,y) = (ax \cos(y), ax \sin(y), y)$$ $$f_2(x,y) = (f(x)\cos(y), f(x)\sin(y), x)$$ Now I understand ...
0
votes
0answers
15 views

An⇀̸A in L1[−π;π] ( An is partial fourier sum )

Let \begin{equation*} (A_n x)(t) = \frac{a_0}{2} + \sum\limits_{k=1}^n a_k cos(kt) + b_k sin(kt), \\ a_k = \frac{1}{\sqrt{\pi}} \int_{-\pi}^{\pi} x(t) cos(kt) dt, \\ b_k = \frac{1}{\sqrt{\pi}} ...
1
vote
1answer
34 views

How many $10$ letter anagrams of KOLMOGOROV don't contain the subword GROOV?

How many $10$ letter anagrams of KOLMOGOROV don't contain the subword GROOV? Not sure how to do this one. Obviously there are $\frac{10!}{4!}$ anagrams of KOLMOGOROV but I'm not sure how to account ...
0
votes
0answers
7 views

Show that a given sigma field is the smallest one containing the given class of sets

I've been trying to solve the following question from Leo Breiman, Probability but getting stuck in how to proceed and have few doubts as well. Define $\mathcal{B}^{(\infty)}$ as the smallest ...
1
vote
0answers
27 views

Assume that the sum of absolute values of all entries of $A$ equals to $1$. What is the maximal possible value of $\det(A)$?

Let $A$ be an $n \times n$ matrix and assume that the sum of absolute values of all its entries equals to $1$. What is the maximal possible value of $\det(A)$? My attempt: We know that $|a_{i,j}| ...
0
votes
2answers
33 views

Proof with subspaces

Prove: If $V$ and $W$ are three-dimensional subspaces of $\Bbb R^5$, then $V$ and $W$ must have a non-zero vector in common. (Hint: start with bases for the two sub-spaces, making six vectors in all) ...
1
vote
0answers
13 views

Solving integral with spherical bessel functions

I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,J_{0.5}(b\cos x)\,J_{0.5}(b\sin ...
0
votes
0answers
17 views

Newton method for $p$-adic fields

I want to understand where the last line comes from. I.e. why there is the $p^{2n-2ka}$ term. I tried to use the estimate formula for the reminder but it doesn't work for me...
0
votes
0answers
14 views

Does a $K_n$ with $n$ pendants have a name?

Consider the graph we get by taking the complete graph on $n$ vertices, and then attaching a pendant vertex to each of the $n$ vertices by an edge. Does such a graph have a name, i.e. do such graphs ...
0
votes
0answers
13 views

What is a purely inseparable extension?

There are many different definitions of purely inseparable extension, and below is what I have chosen for my definition. (Since I don't know what is a standard one, if you know please tell me what ...
1
vote
0answers
24 views

A Combinational identity using permutations

For a distribution {$p_1,p_2, …,p_m$}, with $p_i>0$ and$\sum_1^m{p_i}=1$ , let $J$ be a subset of size $j$, and $m>j\geq1$. It holds that: $$\int_0^1\prod_{i \in J} (x^{-p_i}-1) dx = ...
1
vote
1answer
16 views

Fatou's Lemma conditions for strict inequality

Under what conditions do we have equality (resp. strict inequality) in Fatou's Lemma? If the sequence $f_n$ is convergent, then it is obvious that equality holds. Is it the only case? There are some ...
3
votes
1answer
17 views

Calculate moment of inertia of Koch snowflake

That's just a fun question. Please, be creative. Suppose having a Koch snowflake. The area inside this curve is having the total mass $M$ and the length of the first iteration is $L$ (a simple ...
0
votes
0answers
6 views

Separability of the Wasserstein space with respect to $W_2(\cdot,.) +|\phi(\cdot) - \phi(.)|$

I would be thankful, if someone could give me some short proof or reference for the following problem. Given a lower semi-continuous and geodesically convex functional on the Wasserstein space ...
0
votes
1answer
18 views

Arithmetic modulo $n$ when $n>a$

$r=a \pmod n$ can be rewritten as $a = qn + r$ where $a$ and $n$ are positive and non-zero integers and $q$ is a unique integer. When solving for $a \pmod n$ such that $a$ is greater than $n$, it is ...
1
vote
1answer
28 views

Roots of the complex equations

Find all the roots for the following equation. $2x^4-x^3-x^2+3x+1=0$ My attempt, I factorised it to $(x+1)(2x^3-3x^2+2x+1)=0$ So I know one of its roots is -1. How to proceed then?
0
votes
0answers
6 views

Show $\lim \frac{\sum_{j=1}^{k} X_j}{k} < \infty$ is in tail sigma-algebra

Show $\lim \frac{\sum_{j=1}^{k} X_j}{k} < \infty$ is in tail sigma-algebra Given random variables $X_1, X_2, X_3, ...$, let $\tau = \bigcap_{n\geq1} \sigma(X_{n+1}, X_{n+2}, ...)$ be their tail ...
0
votes
1answer
12 views

Calculate laurent series in the following regions [on hold]

Can someone help me with this? Not sure where to start... I've split it up into partial fractions but not sure what to do now Thanks
4
votes
0answers
40 views

integrate $\int \frac{1}{e^{x}+e^{ax}+e^{a^{2}x}} \, dx$

I've been trying to integrate $$ \int \frac{1}{e^{x}+e^{\omega x}+e^{\omega^{2}x}} \, dx $$ where $\omega=e^{2i\pi/3}$ but to no avail. I've tried substituting in $u=e^{(1+\omega)x}$ but ended up ...
0
votes
4answers
31 views

For every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$

In $R^3$,show that for every integer vector $\overrightarrow{a}$,there is a integer vector $\overrightarrow{b}$ such that $\overrightarrow{a}\bot\overrightarrow{b}$ Generally,in $R^n$,for every ...

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