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8 views

Closure and the openness of a set

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 \...
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5 views

Proof congruent

enter image description here Hi how do I proof those 2 triangles congruent to each other? Please help
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0answers
20 views

i need help with this question its for my ALG-1 quiz [closed]

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

The fundamental solution to Poisson equation using complex function analysis

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 ...
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10 views

How can we calculate the side $AC$?

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 ...
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7 views

Characteristic polynomial's solutions

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 ...
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0answers
10 views

What is the Krull dimension of a polynomial ring over a PID?

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 $...
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10 views

Initial Value Problem to find non-zero terms and convergence

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 $4$ ...
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14 views

A set of numbers has an average of 100

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
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9 views

Replace linear return with cosine return

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 ...
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14 views

T1 space in general topology

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 ...
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21 views

Proof by Induction, any help would be greatly appreciated [closed]

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!
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3answers
19 views

B = {w | w = xbaby, where x, y ∈ Σ ∗}, Σ = {a, b} - DFA

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 ...
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1answer
9 views

The consistency of “Ord is Mahlo” and inaccessible correct cardinal

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 $\{\...
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0answers
16 views

If S is countable then there exists a sequence which has a subsequence converging to a given element in S

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,...
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0answers
13 views

Poisson equation, exact solution

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 ...
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1answer
22 views

Flipping an unfair coin 5 times

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 ...
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1answer
12 views

Parametrizing a curve between surfaces with boundary values

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 ...
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0answers
12 views

How can we cancel out r when we are converting from polar to rectangular form or vice versa?

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 ...
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0answers
10 views

Find a holomorphic injections

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 ...
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8 views

A unitary operator is a closed operator

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 \...
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4 views

Performing principal components analysis on the linear approximation of a time-series from another time-series

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 ...
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0answers
8 views

Fourier Transform Commutes Derivatives

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 ...
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9 views

Two versions of Kleene's recursion theorem - what's the relationship between them?

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 ...
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5 views

The norm of the interpolating projection $P$ onto $\mathcal{P}_n$ at the roots of the (n + 1)-st Legrendre polynomial

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{...
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4 views

Restriction-Extension Duality when $q = 1$

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(\...
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1answer
12 views

Strategy for second-price auction with multiples

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 ...
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1answer
13 views

Are these two definitions of the Riemann mapping theorem equivalent?

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 ...
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22 views

What is the form?

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 ...
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6 views

Condition for a subset of an affine space to be affine

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 ...
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12 views

How to calculate the error due to expected value?

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 ...
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8 views

Showing $\cos(P − Q) = \frac{33}{85}$ for obtuse $P$ and reflex $Q$ such that $\sin P = \frac{8}{17}$ and $\tan Q = \frac43$,

$\sin P = \frac{8}{17}$ and $\tan Q = \frac43$. If $P$ is obtuse and $Q$ is reflex, show clearly that $\cos(P − Q) = \frac{33}{85}$. My working: $$\cos P \times \cos Q + \sin P \times \sin Q$$ $$\...
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0answers
9 views

Tangent bundle is an oriented manifold

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 $TM$ ...
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0answers
15 views

Relation between dense and equidistributed sequence

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

Is there a term for using different numerical bases on either side of a decimal point?

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 ...
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1answer
16 views

In this question we have to find the radius and centre of circle in complex Number

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

Map from the Grassmannian of the dual to the Grassmannian of the space - is it regular?

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\...
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2answers
16 views

Determine if the set $A=\{(x,y) \in \overline{B}\mid x \geqslant0 \}$ is open or closed.

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 $r >0$ ...
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0answers
7 views

How to deal with irrational values with a bijective number system?

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 ...
1
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1answer
30 views

Infinite series raised to a power being a power series

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}^...
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0answers
6 views

Equivalent condition of convergence in measure

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_\...
1
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1answer
10 views

Probability of arrangement of letters following specific rules

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 ...
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0answers
29 views

Why some people define separately the concepts of an injective homomorphism and a group monomorphism if they are clearly the same?

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 ...
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0answers
7 views

Recurrence Tree Method: T(n) = 3(2n/3) + 1 Why is the solution in log base 3/2 and not base log 2/3?

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 ...
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0answers
11 views

Getting $P(Y=y)$ while having the cumulative distribution function graph without info about the $y$ axis

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, ...
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0answers
9 views

If $f(x)= 0$ is a bisector of $ g(x) = 0$ and $AB$, then derive the equation for $AB$

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\...
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0answers
7 views

Find essential spectrum of an operator

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 ...
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2answers
14 views

How to pass minimum count of apples so that every one around the table has same amount of apples?

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, ...
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0answers
8 views

$K$ is weakly*-compact $\Rightarrow $ $K$ is norm-bounded

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 \...
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0answers
18 views

In a geometric sequence $u_1=125$ and $u_6 =1/25$

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 $n$ ...

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