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I need to find the cardinality of this set : {x+y\sqrt2}+z\sqrt3} | x,y,z \in \mathbbQ}

I would really appreciate it if someone can help me start this problem. not asking for solution of course.
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3 views

Fubini's Theorem on Ito's integral?

It is well known that Fubini's Theorem allows us to swap two interated integrals. In stochastic calculus, Fubini's Theorem (under certain conditions) ensures that $$\mathbb{E}\int_0^t g(W_s) ds =...
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4 views

How can a log function become a gamma distribution?

This is the question I want to know how Wi is a gamma(1,1) distribution and sigma(Wi) a gamma(n,1) distribution. To me I dont get the intuition of log fundtion becoming a function of exp. Thanx.
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6 views

Are the functions injective and surjective?

I want to check if the following functions are injective and surjective. $f:\mathbb{Z}\rightarrow \mathbb{N}$, $x\mapsto \begin{cases}2x-1 & \text{ falls } x>0 \\ -2x & \text{ falls } x\...
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0answers
6 views

Does $\mathcal B([0,1])=\{U\cap [0,1]\mid U\in \mathcal B(\mathbb R)\}?

Denote $\mathcal B(X)$ the Borel set of $X\subset \mathbb R$. Does $$\mathcal B([0,1])\underset{(1)}{=}\{U\cap [0,1]\mid U\in \mathcal B(\mathbb R)\}\ ?$$ For (1) I tried as follow. We have that $$\{...
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0answers
10 views

Sample the vectors in the plane

Let n be a non zero natural number and n vectors in plane $\vec{v_{1}},\vec{v_{2}},...,\vec{v_{n-1}},\vec{v_{n}}$.Prove that there exist $a_{1},a_{2},a_{3},...,a_{n}\in\left \{ 1,-1 \right \}$ for ...
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3 views

Collecting the most presents possible with a limited distance that you can travel.

The following map shows towns (in purple) and the number of presents at each town (also in purple). The towns are connecting by roads (in orange) and the length of those roads is shown in orange (the ...
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0answers
12 views

How to prove that R is a binary relation?

R={(x,y)|x,y are 5 bit strings with equal zeros number} Prove that R is Equivalence relation. How much classes does it have?
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1answer
11 views

evaluation of ODE

I'm currently going through analysis. While learning ODE's I found some step I can't understand. $\dfrac{dp(x)}{p(x)}=\dfrac{d x}{1+x} \Rightarrow \ln p(x)=\ln(x+1)+\ln C$
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17 views

Find the smallest natural number N for which N!> 10^N

Find the smallest natural number N for which N!> 10^N Please help me with this problem.
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0answers
5 views

First hitting time inequality

Let $X_t$ be a continuous time stochastic process. Define the first hitting time of $X$ as $\tau = \inf~\{ t \leq t': \lvert X_t \rvert \geq \lambda \}$ for some $\lambda$, $t'$ It is a known fact ...
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1answer
9 views

Is the endomorphism of M (where M is an A-module) a non-commutative ring or not?

I was reading the "Undergraduate Commutative Algebra". It formalises the definition of Module. Consider M, an A-module where A is a ring. It defines $\mu_f : M \to M$ for the map $m \mapsto fm $, ...
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2answers
19 views

A system of linear equations with solution $x=20$ and $y=20$

I am trying to come up with a system with solution $x=20$ and $y=20$. This is what I have now: $$\begin{array}{|l} 3(x-3y)-(2x-3y)=-100 \\ 4(x-3y)+2(2x-3y)=-200\end{array}$$ The expected solution is ...
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7 views

Do $f=o(g)$ and $\lvert h\rvert\leq f$ imply that $h=o(g)$?

Suppose (1) $f(x)=\mathcal{o}(g(x))$ as $x\to\infty$ (2) $\lvert h(x)\rvert\leq f(x)$. Does this imply that $h(x)=\mathcal{o}(g(x))$ as $x\to\infty$? My would say: Yes! Let $\varepsilon >0$ ...
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0answers
19 views

Calculate the sum: $C_{2n}^n+2C_{2n-1}^n+4C_{2n-2}^n+…+2^nC_n^n$

Calculate this sum:$$C_{2n}^n+2C_{2n-1}^n+4C_{2n-2}^n+...+2^nC_n^n.$$ What I tried: $$ C^n_{2n}=\frac{(2n)!}{(n!)^2}$$ $$ 2C^n_{2n-1}=\frac{2(2n-1)!}{n!(n-1)!}=\frac{2n(2n)!}{n!n!(2n)}=\frac{(2n)!}{...
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1answer
12 views

Fubini's theorem versus two different values on double integrals - question about infinite series

I am stuck trying to understand answer given on this Stack Exchange question. Where do both $$ \sum_{x=1}^\infty\sum_{y=1}^\infty f(x,y) = f(1,1) = 1.5 $$ and $$\sum_{y=1}^\infty\sum_{x=1}^\infty f(...
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1answer
18 views

Which is the function that the zero-set of which is Möbius strip?

Many surfaces can be expressed by implicit functions. As described later, the spherical surface, the side surface of the cylinder, and the torus can be expressed by implicit functions. However, I have ...
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2answers
22 views

The real roots of an equation $f(x) = 0, f'(x) = 0$

Prove that, if $f(x) = x^2(1-x)^2$, then the roots of the equation $f''(x) = 0$ are distinct, and lie between 0 and 1. Prove that the corresponding result for the equation $g'''(x) = 0$, where $g(x)...
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1answer
15 views

Inverse using Gauss-Jordan elimination

Assuming $x,y,z \neq 0$, find the inverse of the following matrix using Gauss-Jordan elimination. $$\begin{bmatrix} 1&1&1&1\\ 1&1+x&1&1\\ 1&1&1+y&1\\ 1&1&...
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3answers
32 views

$\lim_{x \to 0}\Big(\frac{\sin{x}}{x}\Big)^{\frac{\sin{x}}{x-\sin{x}}}$

$$\lim_{x \to 0}\Bigg(\frac{\sin{x}}{x}\Bigg)^{\frac{\sin{x}}{x-\sin{x}}}$$ I endeavored to use my head and analyze this task I had received a few answers to already: $\lim \limits_{x \to 0^+} (1+\...
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0answers
8 views

Can we define an interpretation in which a=a is false for some term a? [duplicate]

Given an interpretation is a structure $\langle D,\nu\rangle$ with a non-empty domain $D$ and a function $\nu$ that assigns referents to terms, extensions to predicates and truth values to sentences......
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2answers
10 views

Expected value $\mathbb{E}(Y_n)$ in conditional case

Let $\lbrace Y_n \rbrace_{n \in \mathbb{N_0}}$ be a family of random variables and $\mathbb{P}(Y_0=0)=1$. $Y_n$ is conditional on $\lbrace Y_1=y_1,...,Y_{n-1}=y_{n-1} \rbrace$ unifmorly distributed ...
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0answers
3 views

How do I use Prefix sum and cumulative sum technique in this problem?

Problem: Some person has been given N jobs which are to be done only in intervals given for eg: [T1 T2] if given [2,4] he can do his job only in between these two time states of a day.This man wants ...
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1answer
12 views

How to express $num \cdot 1.06 ^{x - 1} + num \cdot 1.06 ^{x - 2} + \dots + num \cdot 1.06 ^{x - x}$ mathematically?

How to express $$ num \cdot 1.06 ^{x - 1} + num \cdot 1.06 ^{x - 2} + \dots + num \cdot 1.06 ^{x - x} $$ mathematically? I think it can be: $$ x \cdot num \cdot (1.06 ^{x - 1} + 1.06 ^{x - 2} + \...
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0answers
18 views

How to show that $f(x,y)$ is continuously differentiable on $\mathbb{R}^2$?

I have been given the function $f(x,y)=\begin{cases} \frac{x^3y-xy^3}{x^2+y^2}, \quad \quad (x,y)\neq 0 ; \\ 0 \quad \quad \quad \quad \quad (x,y)=0.\end{cases}$. Is it enough to compute all ...
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3answers
23 views

Show that $(a+b)^n/(a^n+b^n)$ diverges

Show that $(a+b)^n/(a^n+b^n)$ diverges I want to show that this sequence diverges for a>0 and b>0. I can show that the sequence diverges for a=b but I am stuck after that.
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0answers
11 views

Group $G$ isomorphic to direct product $G \times G$ [duplicate]

I am looking for a group $G$ with $\vert G \vert > 1$ which is isomorphic to the direct product $G \times G$. I have been looking for such a group among non-finite groups, such as $\mathbb{Z}$ or ...
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0answers
6 views

Confusion About Area Distortion and Orientation of Ellipsoid

Let $K = \{ (x,y,z) \mathbb{R}^3 | (\frac xa)^2 +(\frac yb)^2 +(\frac zc)^2 =1\}$ oriented by outward-pointing unit normal field. Consider $f(x,y,z) = ( \frac {ay}b, \frac{bz}c, \frac{cx}a)$, $f: K \...
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0answers
9 views

vol$(\Omega)=\sum$vol$Q_j$ where $Q_j$ are open boxes

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with $C^\infty$ boundary. For every $\epsilon >0$ there exist $l<m$ and open disjoint boxes $Q_1,...,Q_m\subset\mathbb{R}^n$ such that (a) $\...
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1answer
11 views

Finding largest viable step size, simple

I have a set of rational numbers $t_0, t_1, t_2, \dots ,t_n$ in strictly ascending order $t_i < t_j$ where $i < j$. My goal is to find the largest possible step size $\Delta t \in \mathbb Q$ ...
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2answers
15 views

Equivalent definition of convexity with $\theta \in (0,1)$ instead of $[0,1]$.

We defined a functional $\Phi: V \to \mathbb{R}$ ($V$ is a Banach space) to be convex if $$ \tag{1} \Phi((1 - \theta) v + \theta w) \le (1 - \theta) \Phi(v) + \theta \Phi(w) \quad \forall v,w \in V \ ...
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1answer
26 views

Why is $\left(\frac{\partial}{\partial x_i}\right)f = \left(\frac{\partial f}{\partial x_i}\right)$

i'm currently reading An Introduction to Morse Theory by Yukio Matsumoto and on p.62 it says A vector field itself is sort of a differential operator, since it assigns to each point a "tangent ...
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2answers
21 views

How to simplify this summation $\sum_{r=0}^{20}(-1)^r(r+2)(r+1)\\$

$\sum_{r=0}^{20}(-1)^r(r+2)(r+1)\\$ What is the total sum?
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1answer
39 views

Show that $\lim_{x\to 0} \frac{\tan x\tan^{-1}x-x^2}{x^6}=\frac{1}{18}$

Show that $$\lim_{x\to 0} \frac{\tan x\tan^{-1}x-x^2}{x^6}=\frac{1}{18}.$$ Proceed: $$\tan x\tan^{-1}x=(x+\dfrac{1}{3}x^3+\dfrac{2}{15}x^5+O(x^7)) (x-\frac{1}{3}x^3+\frac{1}{5}x^5+O(x^7))=x^2+\frac{...
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0answers
10 views

Gaussian and Mean Curvature of a Sphere

I need to calculate the Gaussian and Mean curvatures of a sphere of radius a. Writing the equation of the sphere in the form p(u,v)= $$\begin{pmatrix}f(u)cos(v)\\f(u)sin(v)\\g(u)\\\end{pmatrix}$$ ...
1
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1answer
13 views

Does $ \int _1^{\infty }\frac{\sinh (a \log (x))}{\sqrt{x}} $ converge or diverge?

How do I find out if $ \int_1^{\infty } \frac{\sinh (a \log (x))}{\sqrt{x}} \, dx $ diverges or converges? Wolfram says that: $ \int_1^{\infty } \frac{\sinh (a \log (x))}{\sqrt{x}} \, dx=\frac{4 a}{1-...
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0answers
13 views

Possible problems with definied log(z).

I am trying understand why I consider some of special contours from: $$ (z)^{s-1}=e^{(s-1) \log (s)} $$ and in this contour. Why I could define $z^{s-1} ?$ reference
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1answer
18 views

Expectation of stochastic integral

I'm not able to understand why $$E\bigg[\int_0^tB_s^3dBs\bigg]=0$$ it would be true if $B_s^3 \in \Lambda^2$ but it seems to me that $B_s^3\in \Lambda^2_{loc}$
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1answer
17 views

What is the angle between the asymptotes of the hyperbola $5x^2-2\sqrt 7 xy-y^2-2x+1=0$?

What is the angle between the asymptotes of this hyperbola? $$5x^2-2\sqrt 7 xy-y^2-2x+1=0$$ I used $S+\lambda=0$ and used straight line condition to find combined equation to asymptotes. Then how ...
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0answers
19 views

How to calculate this integral with residues $\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2+a^2} dx$ [duplicate]

Anyone can help me with this integral? $\int_{-\infty}^{\infty} \frac{\cos(x)}{x^2+a^2} dx$ My professor tell me the answer is $\frac{\pi}{a} e^{-a}$
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0answers
8 views

On the biconditional $I(n^2) = 2 - \frac{5}{3q} \iff (k = 1 \land q = 5)$, where $q^k n^2$ is an odd perfect number

MOTIVATION Let $N$ be an odd perfect number given in the so-called Eulerian form $$N = q^k n^2,$$ i.e., $q$ is the special / Euler prime satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$. ...
1
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2answers
14 views

How to find the rotation matrix to rotate 3-d vectors such that the z-component of the rotated vectors is approximately constant?

I have an N-body simulation. Each body in the simulation has an array of positions as a function of time. For example, the body Earth has the following positional ...
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0answers
9 views

Numerically stable algorithm to find most likely

I have $N$ Normal random variables $x_i$, each one with mean and variance $\mu_i$, $\sigma^2_i$. Given a value $L$, is there a numerically stable way to compute which of those has the higher ...
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5answers
26 views

Having trouble proving that a series is divergent

I am trying to prove that $$\sum_{n=1}^\infty \frac{n}{\sqrt{n+1}}$$ diverges without checking the limit, bounds or doing any other lengthy steps, as it should be seen as divergent "immediately", ...
0
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1answer
15 views

normal Distribution of the Sum of independent Variables \ Using Z table

We throw Fake Cube(Containing $1,2,3,4,5,6$) $~300$ times . Probability of geting $6$ is $\frac{1}{2}$ Probability of geting $1,2,3,4,5~$ is $~0.1$ What is the probability that that the Sum : $~~S \...
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0answers
14 views

Can we solve this integral equation?

Let $(E,\mathcal E,\lambda)$ be a measure space, $p,q_i$ be positive probability densities on $(E,\mathcal E,\lambda)$ for $i=1,2$, $\mu:=p\lambda$, $\sigma_{ij}:E^2\to[0,\infty)$ be $\mathcal E^{\...
1
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2answers
20 views

Compute $\int_Rx^2+y^2$, where $R=\{(x,y)\in\mathbb{R}^2:|x|\leq|y|\leq2\}$.

Compute the integral of $f(x,y)=x^2+y^2$ over $R$, where $R=\{(x,y)\in\mathbb{R}^2:|x|\leq|y|\leq2\}$. Writing the region as $R=\{(x,y)\in\mathbb{R}^2:-2\leq y\leq2, -y\leq x\leq y\}$, we have that: ...
0
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0answers
6 views

Numerical Simulation Streamfunction

So I'm given the task of calculating the streamfunction with given angles of the flow on different carthesian coordinates (as shown in the picture)TaskAerodynamics. The translation of exercise one: ...
-1
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2answers
21 views

how do I change the basis of orthogonal vector? I'm not getting the correct answer

Given vector $\mathbf{v} = \begin{bmatrix} 5 \\ -1 \end{bmatrix}$, $\mathbf{b_1} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$ and $\mathbf{b_2} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$ all written in the ...
0
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1answer
13 views

In a convex 1897-sided polygon distances from chosen point to all sides independent of the choice of the point

Prove that in a convex 1897-sided polygon, in which all internal angles are equal, for any point inside this polygon, the sum of the distances from this point to all sides of the polygon is ...

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