All Questions

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

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 =...

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.

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\...

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 $$\{...

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 ...

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 ...

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?

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

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

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 ...

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 $, ...

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 ...

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 $\varepsilon >0$ ...

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)!}{...

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(...

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 ...

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)...

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&...

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

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......

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 ...

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 ...

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} + \...

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 ...

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.

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 ...

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 \...

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) $\...

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 $\Delta t \in \mathbb Q$ ...

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 \ ...

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 ...

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

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{...

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)= $$\begin{pmatrix}f(u)cos(v)\\f(u)sin(v)\\g(u)\\\end{pmatrix}$$ ...

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-...

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

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 $B_s^3\in \Lambda^2_{loc}$

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 ...

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 $\frac{\pi}{a} e^{-a}$

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 $\gcd(q,n)=1$. ...

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 ...

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 ...

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", ...

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 \...

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^{\...

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: ...

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: ...

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 ...

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 ...