For questions concerning a specific proof, asking for verification, identifying errors, suggestions for improvement, etc.

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29 views

Understanding functions in $\mathbb{P}^1$

I just need solution check: I am given function $f\in k(\mathbb{P}^1)$ that sends $(x:y) $ in $\frac{x}{y}$. If I understood this right, this is just coordinate function $x$ since it send and point ...
0
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0answers
62 views

Did the book make a mistake in the identity$\frac{1}{\cos 2x+\tan x} = \sin 2x$?

EDIT: OP says that they misread the text from which this question was drawn. It actually said $$\frac{1}{\cot(2x) + \tan(x)} = \sin(2x)$$ where $\cos$ has been replaced by $\cot$; that identity is ...
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3answers
34 views

Verification of a proof in Measure Theory

Let $m$ be the Lebesgue measure on $\Bbb R$ and $f:\Bbb R\to [0,\infty)$ be a Lebesgue integrable function. Show that $\exists $ a measurable set $E\subset [0,\infty)$ such that $m(E)\neq m(f^{-1}(E)$ ...
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2answers
37 views

Poles of $\frac{\sin z}{z}$

I need to find the poles of the following function $$f(z) = \frac{\sin z}{z}$$ However, as ${z \rightarrow 0 }$, $f(z) \rightarrow 1$. So, I think I should rule out $z = 0 $ case. But I am not able ...
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0answers
35 views

What is the probability of getting STRAIGHT FLUSH in a $13$-card poker game?

What is the probability of getting STRAIGHT FLUSH in a $13$-card poker game? Here is my attempt: A straight flush is five cards in sequence and of the same suit, but not ace king queen jack ten. ...
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1answer
39 views

What is the probability of getting FOUR OF A KIND in a $13$-card poker game?

What is the probability of getting FOUR OF A KIND in a $13$-card poker game? Here is my attempt: The setup for the required poker hand would either be: $$AAAABCDEFGHIJ,$$ $$AAAABBBBCDEFG,$$ or ...
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1answer
44 views

Determine if the function is continuous and differentiable on the closed interval $[0,\frac{1}{\pi}]$

Define $G(x)=\int_0^xg(x),$ where g is given by the following: $$g(x) = \begin{cases} \sin\frac{2}{x} & \textrm{ if $x\ne 0$} \\ 0 & \textrm{ if $x =0$} \\ \end{cases} $$ Is ...
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1answer
26 views

Prove that $m^*(B) = \inf \{ m^*(A) : A \text{ open}, A\supset B\}$

I would like to prove that for $B \subset \mathbb{R}$, $m^*(B) = \inf \{ m^*(A) : A \text{ open}, A\supset B\}$. $m^*$ refers to the outer measure. Could you please see if the following is correct? ...
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2answers
23 views

Proving that a common divisor of two variables is also a divisor of the sum of the two variables

if $k~|~a$ and $k~|~b$ then $k~|~as+bt$ for all $s,t \in \mathbb{Z}$ is what I'm trying to prove so I thought I should start by proving that $k~|~a+b$ if $k~|~a$ and $k~|~b$. since $a = \prod ...
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1answer
24 views

$\lVert f \rVert_{\Phi} < \infty$ for every $f$ measurable and satisfying a certain condition

Let $\Phi : [0, \infty) \rightarrow [0, \infty)$ a convex, strictly incrasing function, with $\Phi(0)=0$. Let $L^{\Phi}(0,1)=\{f:(0,1)\rightarrow\mathbb{R} \text{ measurable}:\int_0^1\Phi\left( ...
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2answers
40 views

Answer not matching for calculating the value of an integral

Consider the following integral $$\int_ {-\infty}^{\infty} \frac{x^2 - x + 2 }{x^4 + 10x^2 + 9} dx \!$$ I need to find its value using residue calculus. So, I considered the complex function $$\int_ ...
0
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2answers
36 views

What is the probability of getting NO PAIRS in a $13$-card poker game?

What is the probability of getting NO PAIRS in a $13$-card poker game? Here is my attempt: The setup for the required poker hand would be: $$ABCDEFGHIJKLM$$ where $A, B, \ldots, M$ are distinct ...
0
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1answer
21 views

Polynomials and Lipschitz function

Let $f(x) = x^4 + 11x^2 + 9x -5$ and let $M > 0$. Show that f is a Lipschitz function on the interval $[-M, M]$ I honestly cannot figure out how to start this proof. Nothing similiar is in the ...
2
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1answer
66 views

$f$ is continuous, is $1/f$ continuous

Let $f: A \rightarrow \Bbb{R}$ be uniformly continuous. Suppose there exist $k>0$ s.t. $|f(x)| \ge k$ for all $x \in A$. Show that the function $1/f$ is also uniformly continuous on $A$. My proof ...
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0answers
9 views

Nonuniform Continuity Criteria with cos(1/x)

Use the nonuniform continuity criteron to show that $f(x) = cos\left(\frac{1}{x}\right)$ is not uniformly continuous on $(0, \infty)$ My proof: Let $(x_n)$ be defined by $x_n = \frac{1}{2n\pi}$ ...
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0answers
41 views

Verification: $S^{\perp} = ({span(S)})^\perp = (\overline{S})^\perp = (\overline{span(S)})^\perp$?

Let V be an inner product space (not necessarily Hilbert) and S a subset. This question Orthogonal complement of a Hilbert Space deals with $S^{\perp} = (\overline{span(S)})^\perp$. I'm looking for ...
0
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2answers
21 views

Continuous Functions and neighborhood

Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be continuous and let $\beta \in \Bbb{R}$. Show that if $x_0 \in \Bbb{R}$ is such that $f(x_0) < \beta$, then there exist a $\delta$ - neighborhood $V$ of $x_0$ ...
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1answer
26 views

Let $G$ be a graph with $n$ vertices. Prove that $\chi(G) \ge \frac{n}{\alpha(G)}$

$\chi$ is the chromatic number of $G$, and $\alpha$ is the independence number of $G$. I know that if $G$ has a proper coloring, then the set of vertices with a particular color is an independent ...
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2answers
33 views

Check: Radius of Convergence of the Sum of these Complex Taylor Series

I just found the following Taylor series expansions around $z=0$ for the following functions: $\displaystyle \frac{1}{z^{2}-5z+6} = \frac{1}{(z-2)(z-3)} = \frac{-1}{(z-2)} + \frac{1}{(z-3)} = ...
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0answers
63 views

If $K\cap\Bbb Q^{\text{cycl}}=\Bbb Q(\zeta_m)$ and $K/\Bbb Q$ Galois, then $\text{Gal}(K(\zeta_n)/K)\cong\text{Gal}(\Bbb Q(\zeta_n)/\Bbb Q(\zeta_m))$

$\DeclareMathOperator{\Gal}{Gal}$ Here is my argument: Induction on the number of primes dividing $n/m$. If there are two primes (i.e., $K(\zeta_n) = K(\zeta_{q_1},\zeta_{q_2})$, where ...
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0answers
12 views

Proof verification: $\frac{d(\nu_1\times \nu_2)}{d(\nu_1 \times \nu_2)}(x_1,x_2)=\frac{d\nu_1}{d\mu_1)}(x_1)\frac{d\nu_2}{d\mu_2}(x_2).$

This is exercise 3.12 from Folland's Real Analysis. It took me a long times to come up with a solution to this problem, and I'd appreciate it if anyone could verify if my answer is correct. For ...
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2answers
410 views

Forcing Bijectivity

I'm working out of the Nakahara text in mathematical physics, and I'm presented with a map $f: \mathbb{R} \rightarrow \mathbb{R}$ defined by $ f:x \mapsto \sin(x) $, and told that it is neither ...
0
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2answers
24 views

Prove that DE || BC

Let M be the midpoint of side BC in triangle ABC. The angle bisector of BMA intersects AB in D, while the angle bisector of CMA intersects AC in E. How can i prove that DE||BC? I drew out the ...
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2answers
42 views

Probability density function of the negative of a random variable (Exercise 4.1.2 from Grimmett and Stirzaker)

Find the density function of $Y = a X$, where $a > 0$, in terms of the density function of $X$. Show that the continuous random variables $X$ and $-X$ have the same distribution function if and ...
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2answers
40 views

If $a \equiv b$ mod n, then $ac \equiv b(c+n)$ mod n

Show that $a\equiv b$ mod n implies that $ac \equiv b(c+n)$ mod n. My proof attempt: If $a \equiv b$ mod n, then $n|(b-a)$ which implies that $(b-a) = nx$ for some $x \in \mathbb{Z}$. Which ...
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2answers
57 views

Inequality from IMO 2000 problem 4 question $\Pi_{cyc}\left(a-1+\frac{1}{b}\right)\leq 1$ $abc=1$

I know the problem is repeated but my question is somehow different. I want to know whether my proof is correct because I have troubles with the last part. Since $abc=1$ we can homogenize the ...
0
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0answers
18 views

implication of continuity of function in proving convergence of sequence

I am reading a paper named "Asynchronous Broadcast-Based Convex Optimization Over a Network" by Nedic and I am confused about a part I attach as follows. I don't know the rational deducting from the ...
0
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1answer
16 views

Convergence of functionals in the dual space

Let $M\subseteq X$ be a subset of a normed space. I have been asked to show that the annihilator of $M$,$M^a$ is closed. To do this I assume that it isn't closed. I.e there exists some functional ...
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0answers
23 views

Calculating the gradient of a scalar field. Am I missing something?

Question Find the gradient of $f$ at each point where it exists. $f:\Bbb{R^3} \to \Bbb{R} ~,~f(x,y,z)=xy^3z-\sin(x).$ Attempted solution $\text{grad } f=\nabla ...
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0answers
22 views

Proof-sketch of the Uniqueness Theorem for first order differential equations

I intuitively convinced myself that solutions for first order differential equations are unique but the argument I thought of seems very iffy. I suspect it might not be formalisable, or even worse, be ...
0
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1answer
18 views

Proof of equivalency in disjoint sets.

Prove, If A, B, C, and D are sets with |A|=|B| and |C|=|D| and if A and C are disjoint and B and D are disjoint, then |A ∪ C|= |B ∪ D|. Would I start this proof using the definition of disjoint ...
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0answers
16 views

Geometric series of a $V_{2} = \sum_{t=0}^{\infty } \beta ^{t} \frac{\left ( \gamma^{t} \right )^{1-\rho}}{1-\rho}$

What is the geometric series of this $$V_{2} = \sum_{t=0}^{\infty } \beta ^{t} \frac{\left ( \gamma^{t} \right )^{1-\rho}}{1-\rho}?$$ I am getting this answer: $$V_{2} = \frac{1}{1-\rho} ...
0
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0answers
24 views

The Sobolev Space $W^{m,p}$ is a closed subspace of $W^{1,p}$.

(This is a proof verification, I need to know if the procedure and steps are correct and if the proof is complete) I want to show that the Sobolev Space $W^{m,p}(I)$ is a closed subspace of ...
2
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1answer
60 views

Proove $3^n$ divides $a(n)$ for all integers $n\ge 1$

Q. Define a sequence of integers $a_1$, $a_2$, $a_3$... $a_1=3$, $a_2=18$ and $a_n=6a_{n-1} - 9a_{n-2}$ for each integer $n\ge 3$. Prove by strong induction that $3^n$ divides $a_n$ for all integers ...
2
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1answer
30 views

Is this proof of divergence of an alternating series correct?

Determine whether the series $\sum_{r=1}^\infty (-1)^{r-1}(\sqrt{r+1}-\sqrt{r})$ is convergent, absolutely convergent, or divergent. The way the textbook did it is that they let $b_r = ...
5
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1answer
64 views

Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers.

Prove that the determinant of an invertible matrix $A$ is equal to $±1$ when all of the entries of $A$ and $A^{−1}$ are integers. I can explain the answer but would like help translating it into ...
2
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0answers
38 views

Prove $x^n$ is Uniformly Continuous on a Bounded Subset of $\mathbb{R}$

Hello I want to prove that $f(x)=x^n$ is uniformly continuous on any bounded subset of $\mathbb{R}$. I'm wondering if my proof is correct, and I'm primarily wondering if my choice of $\delta$ is ...
5
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1answer
88 views

Proving $(A\times B)^- = A^-\times B^-$ (closure of cartesian product)

My proof, for: $$(A\times B)^- = A^-\times B^-$$ using the metric $$d''((a_1,a_2),(b_1,b_2)) = max\{d_1(a_1,b_1),d_2(a_2,b_2)\}$$ $\rightarrow$ Well, if $a = (a_1,a_2)\in (A\times B)^-$ then: ...
0
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1answer
76 views

Prove that $f : \mathbb R \smallsetminus \{−1\} \to\mathbb R \smallsetminus \{1\}$ given by $f(x) = \frac{x − 3}{x + 1}$ is bijective

I know for a function to be bijective it must be one to one and onto. Here's what I have Take by cases Case 1 (one to one) $$ \begin{align*} \frac{x-3}{x+1} &= \frac{y-3}{y+1} \\[1ex] ...
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1answer
47 views

Modulo and Congruence Class Proof.

(A) Determine all the equivalence classes for the relation of congruence modulo $5$. (B) Give the partition $P$ of $ℤ$ associated with the relation of congruence modulo $5$. In order to ...
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3answers
42 views

Does the proportion pass through 1/2?

Michael Jordan is shooting free throws. He misses his first one. At the end of the day, he has made $99$% of his free throws. At some point during the day, did he necessarily have a $50$% success ...
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1answer
24 views

Proof Verification: A differentiable aplication $\psi : M\to N$ is differentiable if and only if: $\psi^{*}f\in C^{\infty}(M)$

Show that a differentiable aplication $\psi$ over $M$ to a differentiable variety $N$ is differentiable if and only if: $$\psi^{*}f\in C^{\infty}(M)$$ For: $f\in C^{\infty}(N)$ Where $\psi^{*}f$ is ...
2
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1answer
81 views

$\lim_{x \rightarrow 0} \frac{\sin(x)}{x}$ using $\epsilon - \delta$ definition

So I'm trying to evaluate this limit without using squeeze theorem and doing it by $ \epsilon - \delta$ definition. Here's my attempt: $$ \left| \frac{\sin(x)}{x} - 1 \right| \leq \left| ...
0
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1answer
44 views

Proof of $\partial\partial\partial S=\partial\partial S$

How can I prove in metric space that $$\partial\partial\partial S=\partial\partial S$$ using the proposition of the boundary below? $$\partial S=\text{cl}S \cap \text{cl}(S^c)$$ I found that if ...
3
votes
1answer
335 views

Determining the Collatz Series as a Tree of $\forall\mathbb{N}$

I'm proposing a proof for the Collatz Conjecture; and should like to take answers in terms of validation or contradiction to the arguments proposed. The conjecture states, where; $$ T(n) = ...
-5
votes
1answer
57 views

Proof that $0^0 \neq 1$ [closed]

Suppose that $t = \sqrt{t}^{\sqrt{t}}$, then, it follows that; $$ t^{\sqrt{t}} = \sqrt{t}^{t} \\ \frac{1}{2}t\ln{\left(t\right)} = \sqrt{t}\ln{\left(t\right)} \\ \ln{\left(t\right)}\left[\frac{1}{2}t ...
0
votes
1answer
18 views

Congruent powers implies numbers are congruent

Let $N\in\mathbb{N}$, and let $m,n$ be coprime. Also, suppose $a,b$ are relatively prime to $N$, and that $$ a^n\equiv b^n\mod{N},\ a^m\equiv b^m\mod{N} $$ I need to show that $a\equiv b\mod{N}$. I ...
0
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0answers
24 views

Curl by dual: need help with this example

I came across this old thread here: "... Actually, there is a generalization of curl to any dimension. If you have a vector field, you can take its dual. So, the dual of $4 i + 2 j + 3 k$ would just ...
0
votes
2answers
22 views

Sequence Convergence proof check

I am trying to solve the following sequence question: Show that $\{a_n\}_{n=1}^\infty$ converges to $A$ if and only if $\{a_n - A\}_{n=1}^\infty$ converges to $0$. Proof: Assume ...
0
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0answers
17 views

Is $\int f d|\nu|=\int f d|\nu^{+}|+\int f d|\nu^{-}|$? Where $\nu$ is a signed measure.

Let $\nu$ be a signed measure on $(X, \mathcal{M})$. If $f\in L^1(\nu)$, then $\int f d|\nu|=\int f d|\nu^{+}|+\int f d|\nu^{-}|$. Is this statement true? I know that it holds for nonnegative ...