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

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0answers
15 views

Prove equivalence to the Euclidean Parallel Postulate

Show that this statement (P): The opposite sides of a parallelogram are congruent is equivalent to the H.E.P.P (Q): For every line $l$ and every point $p$ not lying on $l$ there is at most ...
1
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1answer
29 views

Question about package of cookies that is a random variable

The weight in grams of package of cookies is a random variable with expected value of $300$ grams $\color{blue}{A)}$ assume that X is normally distributed with standard deviation of $15 $ grams ...
4
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5answers
102 views

$p,q,r$ primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational.

I want to prove that for $p,q,r$ different primes, $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is irrational. Is the following proof correct? If $\sqrt{p}+\sqrt{q}+\sqrt{r}$ is rational, then ...
1
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2answers
292 views

Is it mathematically wrong to prove the Intermediate Value Theorem informally?

I have been looking at various proofs of the IVT, and, perhaps the simplest I have encountered makes use of the Completeness Axiom for real numbers and Bolzano's Theorem, which, honestly, I find a bit ...
2
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2answers
36 views

$rk(A^2)=rk(B^2) \implies rk(A)=rk(B)$ is it true?

The original statement is this: given A,B matrices $n \times n$, if $A^2$ is "Left-Right equivalent" to $B^2$ then A is LR equivalent to B (is it true or false?) I know that A is LR equivalent to B ...
2
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0answers
86 views

Is this elementary proof of fundamental theorem of algebra correct?

There is an elementary proof which considers the polynomial $ p(z) $, $z \in \mathbb{C}$ as a function of $ (r,\theta) $ where $ z= r e^{-i \theta} $, $r\ge 0$ . There are two assumptions- Assumption ...
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0answers
17 views

Isomorphism Between Splitting Fields

Let $f \in F[x]$ be irreducible and split in $E/F$. Suppose that $F \cong F'$ via $\phi$, and let $f' \in F'[x]$ be polynomial obtained by applying $\phi$ to the coefficients of $f$. It can be shown ...
2
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2answers
28 views

Statistics question on basil bush random variable

The height, $H$, in meters of a basil bush is a random variable with the probability density function $f_{_H}(t)=e^t,\;0\leq t\leq H_0$ such that $H_0$ is the maximal height. $\color{blue}{(1)}$ I ...
2
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1answer
37 views

Show that $\{w_1,\dots,w_p,v_1,\dots,v_q\}$ is an orthogonal set and spans $\mathbb{R}^n$

These series of questions build up on each other i'm stucked on the last one, i'm also not sure if all of these work but I am pretty convinced they do. Let $W$ be a subspace of $\mathbb{R}^n$ with an ...
4
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0answers
50 views

Finding the expected value and variance of ${X^3}$

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the expected value and variance of $\color{blue}{X^3}$ and I know that: cumulative ...
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1answer
40 views

Finding the expected value and variance of $X$

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the expected value and variance of $X$ and I know that: cumulative distribution function: ...
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1answer
21 views

Finding the probability density function and the distribute accumulate function

For a random variable $X$, $(X^3-1)$ is uniformly distributed in the interval $[0,7]$ I need to find the probability density function and the cumulative distribution function of $X$ My attempt: ...
2
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0answers
24 views

Upper bound for $|\zeta'(s)|$ near the line $\sigma=1$, a detailed proof

In page 285 Apostol leaves as a reader's asigment the proof that $|\zeta'(s)|=O(\log^{2}t)$, this is for every $T>0$ there exists a positive constant $K$ (depending on T) such that ...
2
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3answers
61 views

Prove that series $ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $ and $ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $ have the same radius of convergence.

I want to prove that these two power series $$ \sum^{+\infty}_{n=0}a_n(x-x_0)^n $$ and $$ \sum^{+\infty}_{n=0}(n+1)a_{n+1}(x-x_0)^n $$ have the same radius of convergence. What I've done so far is: ...
1
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0answers
20 views

Integral of ratio of complex polynomials

Let $p(z),q(z) \in \mathbb{C}[z]$ two polynomials with coefficients in $\mathbb{C}$ s.t. $deg(p) = m$, $deg(q) = n$ and $n \ge m +2$. I need to show that $$ \lim_{R \to \infty} \int_{|z| = R} ...
10
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2answers
120 views

$\lim x_n^{x_n}=4$ prove that $\lim x_n=2$ [duplicate]

Let $(x_n)$ be a sequence of real numbers, such that: $\lim x_n^{x_n}=4$, prove that $\lim x_n=2$ I'm not sure if my proof is right. I assumed that $\lim x_n $ isn't 2 and using Cauchy's criterion: ...
1
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1answer
27 views

Proof of the ultraparallel theorem in the Beltrami Klein model

I was reading (and editing) the proof mentioned at https://en.wikipedia.org/wiki/Ultraparallel_theorem#Proof_in_the_Beltrami-Klein_model and noticed it is not correct. (the ultra parallel theorem is ...
0
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0answers
16 views

Proof verification regarding asymptotics

Prove $f(z)=o(\phi (z))$ implies $f(z)=O(\phi (z))$ as $ z \to 0$. My proof is: The assumed statement is $$\lim_{z \to 0} \frac{ f(z)}{\phi (z)}=0$$ That statement implies the function $\frac{ ...
2
votes
1answer
33 views

Sufficient condition for $E(wu\mid v)=0$ given that $E(u\mid v)=0$?

I'm trying to figure out what condition concerning $w$ and $v$ would be enough for me to infer that $E(wu\mid v)=0$ given that I already know $E(u\mid v)=0$. Clearly, $w$ is a constant works: ...
2
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2answers
44 views

Finding $P$ such that $P^TAP$ is a diagonal matrix

Let $$A = \left(\begin{array}{cc} 2&3 \\ 3&4 \end{array}\right) \in M_n(\mathbb{C})$$ Find $P$ such that $P^TAP = D$ where $D$ is a diagonal matrix. So here's the solution: $$A = ...
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2answers
85 views

Complex Analysis ( Limits at a point ).

We need to prove that $ \lim_{z \to z_{0}}(z^{2}+c)$ = $z_{0}^{2}+c$ , where c is a complex constant , using $\epsilon - \delta$ definition , where $z , z_{0}$ are complex variables. What I tried : ...
3
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1answer
45 views

How to use the $b\cdot\nabla$ operator?

While trying to prove $$[c\cdot (b\cdot\nabla) - b\cdot(c\cdot\nabla)]a = (\nabla\times a) \cdot (b\times c)$$ I had some difficulties on how to treat the term $(b\cdot\nabla)$. It seems that ...
1
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1answer
26 views

Counter example: $X$ and $Y$ normal imply $(X,Y)$ bivariate normal

I vaguely remember this construction from one of my courses: Suppose that $X\sim N(0,1)$ and $Z$ is $\pm 1$ with probability $\frac{1}{2}$ each. If $X$ and $Z$ are independent, then $Y\equiv XZ$ is ...
1
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0answers
72 views

Prove that “No one likes Reggae music” is the same as “Everyone does not like Reggae music”.

I interpreted this as a case of the extension of De Morgan's Law to quantifiers. https://en.wikipedia.org/wiki/De_Morgan%27s_laws#Extensions I know that similar questions have been asked before about ...
2
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2answers
43 views

Set theory: $A-(B-C)=(A-B)\cup C$.

I'm working through the set theory exercises in Apostol's Calculus Volume 1. I'm down to the last problem: Show that one of the following results is always correct and the other one is sometimes ...
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3answers
24 views

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$

Prove that if $U$ is an orthogonal $n\times n$ matrix, then the rows of $U$ form an orthonormal basis for $\mathbb{R}^n$ I'm unsure how to proceed with proving this. Basically my idea is as ...
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1answer
18 views

Finding potential function of $\vec F =xy^2 \hat i +y x^2 \hat j$

$$\vec F =xy^2 \hat i +y x^2 \hat j$$ My attempt: $$P=U_{x}=xy^2$$ $$Q=U_{y}=x^2y$$ $$\Longrightarrow U=\int P dx=\frac{x^2}{2}y+C(y)$$ $$ U_{y}=\frac{x^2}{2}+C'(y)=Q=x^2y$$ ...
1
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1answer
22 views

Prove that if $\forall A \in \mathcal F (B\subseteq A)$ then $B \subseteq \bigcap \mathcal F $

This is Velleman's exercise 3.3.10. Suppose that $\mathcal F$ is a nonempty family of sets, B is a set, and $\forall A \in \mathcal F (B\subseteq A)$. Prove that $B \subseteq \bigcap \mathcal F $. My ...
2
votes
2answers
47 views

Evaluate $\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$

I need to evaluate the following integral using Green's theorem $$\oint_{C} e^{-x} \sin y \;dx+e^{-x} \cos y\;dy$$ $C$: from point $E \to F\to G\to H$ ...
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0answers
21 views

Proof of x = 0 modulo 3 only if the sum of its digits 0 modulo 3 [duplicate]

Okey, lets beggin from a helpfull proposition I've already proved: $$$$ if $a_i\equiv b_i\:\forall 0\le i\le m$ then to any $m$ numbers: $p_1,p_2,...,p_m\in \mathbb{Z}$ $$\sum ...
2
votes
2answers
43 views

Curl of a vector field cross itself

How we can use the property that $$A×(B×C) = B(A.C)- C(A.B)$$ to prove the relation: $$a×(∇×a) = ∇ (a^2/2) -(a.∇)a.$$ When I use it, the result directly appear to be $$∇(|a|^2 )-(a.∇)a$$ instead of ...
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0answers
11 views

2HP - Edge Disjoint Hamilton Paths

G is a directed graph and s and t are 2 vertices in G. $2HC = \{(G, s, t): \; G $ has at least 2 edge-disjoint Hamilton paths $\}$ Prove that $2HC$ is NP-Complete. I'm trying to reduce UHAMPATH to ...
2
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1answer
22 views

Using addition and subtraction in algebraic proving in set theory

I am trying to prove (using algebraic way) the following statement: A∆B=A iff B=∅ So it goes like this in one direction: A∆B=A A∆B∆A=A∆A (I added ∆A to both sides) B∆A∆A=A∆A (commutativity) B∆∅=∅ ...
3
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0answers
62 views

Is my proof rigorous? (Archimedes area of parabola)

I am currently reading Apostol's Calculus volume 1 and was revising the part where the area of a parabolic segment is found. I decided to write my own proof similar to the one in the book, which I ...
2
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0answers
27 views

Generating all coprime pairs

The Wikipedia article on coprime integers has a brief section on generating all coprime pairs. All pairs of positive coprime numbers $(m,n)$ (with $m>n$) can be arranged in two disjoint ...
3
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2answers
528 views

A curious proof of L'Hospital's rule

I quote P. Nahin When Least is Best (2004), pp. 173-174 "Since $g(x)=R(x)h(x)$, then differentiation of both sides gives $$g'(x)=R(x)h'(x)+R'(x)h(x).$$ Since $\lim_{x \rightarrow 0} h(x)=0$, and we ...
2
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3answers
42 views

Proving $|x+y|=|x|+|y| \iff x\cdot y \geq 0$

Prove: $|x+y|=|x|+|y| \iff x\cdot y \geq 0$. $|x+y|=|x|+|y| \iff x+y=x+y$ and $-(x+y)=-x-y \iff \{x,y\}\geq 0$ and $\{x,y\}\leq 0 \iff x\cdot y\geq 0$ in both cases.
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1answer
31 views

Let $W$ be a subspace of $\mathbb{R}^n$ spanned by $n$ non-zero orthogonal vectors. Show that $W=\mathbb{R}^n$.

Let $W$ be a subspace of $\mathbb{R}^n$ spanned by $n$ non-zero orthogonal vectors. Show that $W=\mathbb{R}^n$. My approach: Suppose span$\{u_1,\dots,u_n\}=W$, where$\{u_1,\dots,u_n\}$ is a set ...
0
votes
0answers
9 views

No nontrivial invariant subspaces iff characteristic polynomial is irreducible

Say $V$ is a nonzero, finite dimensional vector space over $F$, and $T\in \mathcal{L}(V)$. I want to show that the only $T$-invariant subspaces of $V$ are trivial iff $f_T$, the characteristic ...
5
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1answer
31 views

Prove that the given condition implies analytic continuation

Here is an old qual problem I'm working on, I have some idea, but I'm not sure if I'm correct or not. I would be happy if anyone could possibly confirm or correct me: Let $U=\{z\in \mathbb{C} : ...
1
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1answer
53 views

A consequence of Cesàro's theorem

Here is the statement : "Let $(a_n)_{n\ge 1}$ a real or complex sequence and $l \in \bar{\mathbb{R}}$. If $\lim \limits_{n\to +\infty} a_{n+1} - a_{n}=l$, then $\lim \limits_{n\to ...
2
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1answer
31 views

Bounded sets equivalent definition

Let $X$ be a metric space, and $E\subset X$. I have two definitions of a bounded set, I want to prove they are equivalent. Definition 1: $\exists M:\exists q\in X:\forall p\in E:d(p,q)<M$ ...
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2answers
64 views

Intuition behind constrution of the Hyperreals

Just want to attempt to check if my understanding/intuition for the construction of the Hyperreal numbers via an ultraproduct is correct. Appreciate any corrections or help. So Hyperreals are ...
1
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0answers
23 views

Proof check - $L(\mathbb{F})\to L_{V}(\mathbb{F})$ for $L=\mathfrak{sl}_{2}$ is an isomorphism.

Let $L=\mathfrak{sl}_{2}$ with basis $(x,y,h)$ and maximal toral subalgebra (or CSA) $H$, $char\mathbb{F}>2$ and $V=V(m)$ an irreducible $L$-module with highest weight $m\in\mathbb{Z}^{+}$. Let ...
2
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2answers
88 views

Digit sum of $n^2$ is 44

Is there a whole number $x$ such that the sum of the digits of $x^2$ equals 44? I would like someone to tell me if my thoughts are correct. The remainder of a number a divided with 9 is the same as ...
0
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1answer
42 views

Proof about the difference between right and left ideals in a ring

I have tried get a version of the proof stating that a left ideals of a ring is not, in general, a right ideal, and viceversa. Is my formulation right? Comments and corrections are welcome. I have ...
3
votes
2answers
58 views

Kernel of ring homomorphism

Let $\phi: R \to R'$ be a ring isomorphism and $I$ an ideal of $R$. Define $\phi(I)=\{\phi(i): i \in I\}$. Show that $\frac RI \cong \frac {R'}{\phi(I)}$. To use the first isomorphism theorem, ...
2
votes
1answer
27 views

Uniform Continuity implies Continuity

Let $f$ be a function from a metric space $X$ to a metric space $Y$. Show that if $f$ is uniformly continuous on $X$ then $f$ is continuous on $X$. Show that the converse is not true. Uniform ...
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2answers
38 views

Technical question in Vandermonde determinat proof

I can follow the proof given in (2nd proof, or the induction proof), until the sentence: "From the Expansion Theorem for Determinants‎, we can see that the coefficient of $x_k$ is:". I don't ...
0
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0answers
39 views

Help fix this proof.

What is wrong with this proof? I followed the example of the answer to another one of my questions, here Define a general recurrence relation as $$f(x)^2=A(x)+B(x)f(x+n).$$ Substitute the root ...