# All Questions

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### 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 ...
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### 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 ...
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### 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?.
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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}} ...
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### 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 ...
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### 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 ...
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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 $|a_{i,j}| ... 2answers 33 views ### Proof with subspaces Prove: If$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) ... 0answers 13 views ### Solving integral with spherical bessel functions I would like to find if possible a solution (closed form) for the following integral: $$\frac{1}{2 \pi}\cdot\int\limits_0^{2\pi}\exp\bigg[-ia(\cos x+\sin x)\bigg]\,J_{0.5}(b\cos x)\,J_{0.5}(b\sin ... 0answers 17 views ### Newton method for p-adic fields I want to understand where the last line comes from. I.e. why there is the p^{2n-2ka} term. I tried to use the estimate formula for the reminder but it doesn't work for me... 0answers 14 views ### Does a K_n with n pendants have a name? Consider the graph we get by taking the complete graph on n vertices, and then attaching a pendant vertex to each of the n vertices by an edge. Does such a graph have a name, i.e. do such graphs ... 0answers 13 views ### What is a purely inseparable extension? There are many different definitions of purely inseparable extension, and below is what I have chosen for my definition. (Since I don't know what is a standard one, if you know please tell me what ... 0answers 24 views ### A Combinational identity using permutations For a distribution {p_1,p_2, …,p_m}, with p_i>0 and\sum_1^m{p_i}=1 , let J be a subset of size j, and m>j\geq1. It holds that:$$\int_0^1\prod_{i \in J} (x^{-p_i}-1) dx = ... 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 ... 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 ... 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 ... 1answer 18 views ### Arithmetic modulo$n$when$n>ar=a \pmod n$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 ... 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? 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 ... 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 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 ... 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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