All Questions

Determine the closure of the set $A =\{(x,y) \in B \mid x+y > 1 \}$ in $B$, where $B$ is the open unit disk. Is the set open or closed? For the closure I got that $\operatorname{cl_{B}}A = B \cap \...

enter image description here Hi how do I proof those 2 triangles congruent to each other? Please help

any help would be nice its part ALG-1 (𝑛2+1)(𝑛2+3)

Is there any way to attain the fundamental solution of 2D Poisson's equation $\nabla^2u(\mathbf{x})=\delta(\mathbf{x})$ using the theorem of complex functions? I think a candidate solution could be to ...

In the triangle $ABC$ it holds that $|AB| = 10$, for the midpoint $C'$ of the side $AB$ it holds that $|CC'| = 8$ and the interior angle at $C$ at the triangle $BCC'$ is $\frac{\pi}{6}$. (a) What ...

my friends... Sometimes when I try to figure out the solutions of a characteristic polynomial, I realize that it's so difficult to do it by hand especially when I have a Characteristic polynomial of ...

Recently, I proved this statement: Proposition: Let $R$ be a PID. The prime ideals of $R[y]$ are precisely the ideals of the following form: $(0)$, $(f(y))$ where $f$ is an irreducible polynomial in $...

Consider the following initial value problem: $$e^{-x}y'' + ln(1 + x)y' - x^2y = 0$$ $$y(0) = 1$$ $$y'(0) = 2$$ Show that $x = 0$ is an ordinary point in the differential equation. Find the first $4$ ...

A set of numbers has an average of 100. And the largest element is 5 greater than 3 times the smallest element. Which element cannot be in the set? i) 30 ii) 80 iii) 120 iv) 154 v) 50

Below you will find the position profile in time for a servo-device I am building. Currently the motion follows the orange profile moving away from zero to a maximum value and then quickly returning ...

Let (X, τ) be a topological space where τ = {A ⊆ X : p ∈ A} ∪ {∅} (i.e. is p inclusion topology). Then show that (X, τ) is not T1 -space . I know that the definition Aspace X is called T1 space if for ...

the question is: $$2^m-7 > 10n, m=n, n\ge7$$ This has completely lost me and I can not use the calculators that work with less than or equal to. Thanks in advance!

I am having some trouble with this problem. I believe that the language that B recognizes must included a substring "bab" in-order to ever hit an acceptance state. Below is a screenshot of ...

In Cantor's Attic, it is stated that the theories 1 and 2 are equiconsistent. $\mathsf{ZFC}$ + $\delta$ is inaccessible and $V_\delta\prec V$ (the latter expressed as a scheme in the language $\{\...

I apologise if this has been asked before, its a really long one to google! Suppose S is a countable subset of R. I want to show that there is a sequence in R, such that if I choose any element from S,...

I have to prove that the solution $u$ of the following Poisson equation $$-\Delta u(x,y)=\sin(\pi x)\sin(\pi y)$$ for $(x,y)\in[0,1]^2$ with boundary conditions $$u(0,y)=u(1,y)=u(x,0)=u(x,1)=0$$ can ...

I'm flipping an unfair coin 5 times. The probability of getting heads is $2/3$ and the probability of getting tails is $1/3$. What's the probability that at least 3 of the coins end up being heads? My ...

I'm evaluating a line integral of the function $T= x^2 + 4xy + 2yz^3$ from $a = (0,0,0)$ to $b=(1,1,1)$ on the path $z = x^2$, and $y = x$ without using the fundamental theorem. My question is how to ...

When we are converting from polar to rectangular form often times in the problems while we are simplifying we divide r from all sides; however, my question is, like how we can't divide x in equations ...

Find a holomorphic injections $f$ such that: a) $\{ z\in \mathbb C: |\text{Re}z|<1\} \mapsto \{z\in \mathbb C: \text{Re}z>0, \text{Im}z>0\}$ and $\{z=0\}\mapsto \{w=1+i\}$ b) $\{z\in \mathbb ...

Let $\mathcal{H}$ be an infinite-dimensional Hilbert space, with $U$ a densely defined, unitary operator. I was wondering if such an operator is in fact a closed operator, which is equivalent to \...

I was reading a recent paper and was trying to understand the novel factor analysis method that they introduce. I am not terrific at linear algebra so I was hoping to get some intuition behind what ...

Consider two integrable functions $f(x)$ and $g(x)$, whose derivatives (and higher order derivatives) are also integrable. From the properties of the Fourier transform (and convolutions), the Fourier ...

Below are two versions of Kleene's recursion theorem. How are they related? Are they equivalent? If not, does one of them (which one?) imply the other? Note that both $U(n,x)$ and $\phi_n(x)$ is the ...

I am trying to solve this problem. What I have done is the following: Let $x_0,x_1,\dots,x_n$ to be the roots of $(n+1)$-st degree Legendre polynomial. The interpolating projection $P$ onto $\mathcal{...

The restriction conjecture says that for $1 \leq p < 2d/(d+1)$ and $(d+1)(1/p) - 2 \geq (d-1)(1 - 1/q)$, and any $f \in C_c(\mathbf{R}^d)$, $$ \| \widehat{f} \|_{L^q(S)} \lesssim \| f \|_{L^p(\...

Suppose there is an auction website. The rules are that the winner pays second-highest bid, and if a bid is made with less than 3 minutes left on the timer, the timer is reset to 3 minutes. A seller ...

I have two definitions of the Riemann mapping theorem 1.Suppose that $G$ is a simply connected domain in the complex plane, $G\neq \mathbb{C}$, and that $z_0$ is a point of $G$. There exists a unique ...

I'm studying linear and geometric algebra and I've asked myself always what is the form! I had a lot of "form" expressions such as (linear-bilinear-quadratic..), but I didn't get what is the ...

Let $E$ be an affine space attached to a $K$-vector space $T$ and $a \in V\subset E$. Consider the bijection $f:E\rightarrow T$ send $x$ to $x-a$, where $x=(x-a)+a$. Clearly $f(E)$ is a $K$-subspace ...

I am currently trying to work out the probability of winning a game. This is proportional to another value however the I dont have the true value of this other quantity. Instead it is the sum of ...

$\sin P = \frac{8}{17}$ and $\tan Q = \frac43$. If $P$ is obtuse and $Q$ is reflex, show clearly that $\cos(P − Q) = \frac{33}{85}$. My working: $$\cos P \times \cos Q + \sin P \times \sin Q$$ $$\...

Recall that the tangent bundle $TM$ of a manifold $M$ consists of all pairs $(x, \overrightarrow{v})$ where $x \in M$ and $\overrightarrow{v}$ is the tangent space $T_xM$ of $M$ at $x$. Show that $TM$ ...

What are the properties of dense sequences? Is there any relation between dense sequences and equidistributed sequences?

I'm reading through a baseball box score, and it dawned on me that this seems to be essentially what's going on in describing pitching appearances. For those who are unfamiliar with this, it's common ...

Find the radius and centre of the circle $$z\bar z + (2-3i)z + (2+3i)\bar z+ 4 = 0$$

Let $V$ be a vector space of dimension $n$ over an algebraically closed field $F$. Let $0<d<n$. Consider the map $f:G(d, V^*)\to G(n-d,V)$ between Grassmannians, given by $f(W)=\{v\in V\mid\...

Let $\overline{B}= \{(x,y) \in \Bbb{R}^2 \mid x^2+y^2 \leqslant 1 \}.$ Determine if the set $A=\{(x,y) \in \overline{B}\mid x \geqslant0 \}$ is open or closed. If I let $(x,y)\in A$ and $r >0$ ...

Having done some research on bijective numeration - that is, a number system in which every non-negative integer can be represented in exactly one way using a finite string using a finite set of ...

After thinking about how $$\left(\sum_{k=0}^\infty \frac{1}{k!}\right)^{z} = \sum_{k=0}^\infty \frac{z^k}{k!}$$ I wondered about what kind of sequence $(a_n)_{n=0}^\infty$ satisfies $$\left(\sum_{k=0}^...

Suppose $E \subset \Bbb R^n$, Lebesgue mesure $m(E) < +\infty$. $f(x), f_1(x), \cdots, f_n(x), \cdots$ are measurable functions on $E$. Prove that $f_n$ convergence to $f$ in measure $\iff$ $\lim_\...

I have letters ABCDEFG (7 letters). I want to find the probability that if I shuffle the letters and arrange them randomly that B is first and A is last (but I can pick any arbitrary two letters). I ...

The definition my teacher gave in his course are: -Let $G$ and $H$ be groups, let $f:G \longrightarrow H$ an homomorphism of groups. We call $f$ a group monomorphism if for all group $K$ and any pair ...

I feel this is more of a question in the logarithms, but I am confused why this tree reoccurrence $$T(n) = 3T(2n/3)+1$$ solution is derived from $log_{3/2}$ rather than $log_{2/3}$? My thinking ...

I am given the following graph: Question: Given the following graph of the cumulative distribution function of random variable $Y$. Calculate $P(Y=103)$ This type of question is usually easy, ...

Don't know how to derive this equation... If $f(x) = ax + by + c = 0$ is a bisector of $g(x) = a_{1}x+b_{1}y\ +c_{1}$ and $AB$, then the equation is $$ \left(a^{2}+\ b^{2}\right)g\left(x\right)\ -\ 2\...

Let $\mathcal{H}$ be a complex, separable, infinite-dimensional Hilbert space. Given an operator $T\in\mathcal{B(H)}$ and $\pi:\mathcal{B(H)}\rightarrow\mathcal{B(H)}/\mathcal{K(H)}$ the canonical ...

There are three ($A_1,A_2 and A_3)$ people sitting around the table, each with $a_1=2,a_2=3,a_3=4 $ apples. To ensure each one has equal amount of apples, A3 just gives A1 one apple. In this case, ...

What I did was considering the collection of linear bounded functionals $\left \{ Jx ; x\in E \right \}$ where $J$ is the canonical injection from $E$ to its bidual . ($\left \langle Jx,f \right \...

In a geometric sequence $u_1=125$ and $u_6 =1/25$. a) Find the value of $r$ (the common ratio). b) Find the largest value of $n$ for which $S_n < 156.22$. c) Explain why there is no value of $n$ ...