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

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2
votes
1answer
21 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
votes
2answers
37 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 = ...
1
vote
2answers
61 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
votes
1answer
41 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
vote
1answer
25 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
vote
0answers
64 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
votes
2answers
42 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 ...
0
votes
3answers
22 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 ...
1
vote
1answer
17 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
vote
1answer
19 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
46 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 $E \to F\to G\to H$ ...
1
vote
0answers
18 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 ...
1
vote
0answers
9 views

2HC - 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
votes
1answer
20 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
votes
0answers
60 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
votes
0answers
25 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
votes
2answers
515 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
votes
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.
0
votes
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
votes
1answer
28 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
vote
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
votes
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: $\exists M:\exists q\in X:\forall p\in E:d(p,q)<M$ ...
4
votes
2answers
63 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
vote
0answers
21 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 $(x,y,h)$, $char\mathbb{F}>2$ and $V=V(m)$ an irreducible $L$-module with highest weight $m\in\mathbb{Z}^{+}$. Let also ...
2
votes
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
votes
1answer
41 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
57 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 ...
1
vote
2answers
36 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
votes
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 ...
0
votes
1answer
32 views

My problem in the definition of Dirichlet generating function?

In the definition of Dirichlet generating function "for the square-free numbers " is: $$ \frac{\zeta(s)}{\zeta(2s)}=\sum_{n=1}^{\infty} \frac {|\mu(n)|}{n^s} $$ where $\mu$ is Moebius ...
1
vote
1answer
30 views

If $A\subseteq\Bbb R$ is nonempty with $|A|\ge 2$, then $A$ totally disconnected $\iff A^\circ=\emptyset$.

In the course of working on an exercise, I came up with the claim given in the title. Just looking for verification. $\underline{\text{Claim: } A\text{ is totally disconnected}\iff ...
2
votes
1answer
35 views

Showing a nonabelian group of order 21 has an automorphism that is not inner.

I've seen at least 3 proofs of this on here, but most involved techniques I don't think I'm comfortable with, so I wanted to see if this one works: Since $21=3\cdot 7$, up to isomorphism there's only ...
3
votes
3answers
81 views

Evaluate $\iint_{R}(x^2+y^2)dxdy$

$$\iint_{R}(x^2+y^2)dxdy$$ $$0\leq r\leq 2 \,\, ,\frac{\pi}{4}\leq \theta\leq\frac{3\pi}{4}$$ My attempt : Jacobian=r $$=\iint_{R}(x^2+y^2)dxdy$$ $$x:=r\cos \theta \,\,\,,y:=r\cos \theta$$ ...
1
vote
1answer
23 views

Under the Borel measure associated to the Cantor function each of the intervals remaining in the construction of the Cantor set has measure $2 ^{-n}$

Let $f$ be a function such that agrees with the cantor function on $[0,1]$, vanishes on $(-\infty,0)$, and is identically $1$ on $(1,+\infty)$ and let $\mu_f$ the Borel measure associated to $f$. Show ...
1
vote
0answers
27 views

Uniformly bounded family of harmonic functions

I am pretty sure other questions on this site can answer this problem, but I'm really interested in knowing if this particular solution is valid. Thanks. Question: Let $U$ consists of the set of ...
-1
votes
1answer
104 views

Injections, Surjections, Bijections [on hold]

So i was given a question that asks me to determine whether the function is injective, bijective, or surjective. If you answer bijective than determine the functions inverse, domain, and target space. ...
2
votes
2answers
48 views

Proving by induction $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + nq^{n+1}}{(1-q)^{2}}$

The context is as follows: I am asking this question because I would like feedback; I am a beginner to mathematical proofs. We wish to show $\sum\limits_{k=1}^{n}kq^{k-1} = \frac{1-(n+1)q^{n} + ...
4
votes
1answer
97 views

Possible proof of Fermat's Last Theorem for prime exponents greater than 2

I would appreciate if someone could check my attempt in proving the Fermat's Last Theorem for prime exponents greater than $2$. Firstly, let's prove a couple of lemmas which state that sum or ...
0
votes
2answers
31 views

Simple matrix derivative identity

Is the following correct, and is there some kind of similar identity when $x$ and $y$ are matrices? For $A \in \mathbb{R}^{n \times n}$, $\nabla_A x^T A y = x y^T$. And my proof: ...
0
votes
2answers
22 views

Spot the error in experimenting with contradiction on 5's rationality.

Let $5=\frac ab$ $\forall\ a,b\ \epsilon\ N$. And $(a,b)=1$ Squaring both sides, $25b^2=a^2$ Thus, $25|a^2$; $25|a$ So $a=25m$ Substituting, $25b^2=25^2m^2$ So $b^2=25m^2$ So $25|b$ (By the ...
-1
votes
0answers
70 views

Determine whether it is injective, surjective, bijective or neither injective nor surjective [closed]

The question i was given asked (a) Determine whether it is injective, surjective, bijective or neither injective nor surjective. (b) If you answered "bijective" in part (a) determine the ...
0
votes
1answer
25 views

defining a sequence of numbers L n≥1, and prove something about it

So i was given this question just to try for practice to test our knowledge and i'm really confused on how to go about this question. How should i go about this question, i'm confused with the greek ...
-4
votes
0answers
44 views

prove that for any 2n≥2 and any \a ​1 ​​ ,…,a ​n ​​ ∈N, we have the following: [closed]

So the question I was given goes like this we will introduce a mystery function,P:N→N. We don't know a formula for P (and we won't be able to determine one!) but we do know that P satisfies the ...
1
vote
3answers
72 views

Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$

This is Velleman's exercise 3.3.4. Suppose $A\subseteq \mathscr P (A)$. Prove that $ \mathscr P (A)\subseteq \mathscr P ( \mathscr P (A))$. I started reexpressing the terms in their equivalent forms ...
2
votes
3answers
74 views

Proving $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$ using strong induction

The question says $10\cdot n=0$ for all $n\in\mathbb{Z}$ with $n\geq 0$. Here is my proof by strong induction: Base case: $10\cdot0=0$. Let $k\geq 0$, and suppose that for any $m\leq k$ we have ...
0
votes
3answers
59 views

Help with proof: $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$

The question is: If $A,B$ are any $m\times n$ matrices, prove that $\mathrm{rank}(A+B) \le \mathrm{rank}(A) + \mathrm{rank}(B)$. ($\mathrm{rank}(A)$ is the dimension of the column space of $A$, ...
2
votes
2answers
24 views

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle$.

Let $H$ be normal subgroup of $G$. If $G/H$ is cyclic group generated by $aH$, prove that $G=KH$ where $K=\langle a\rangle $. I would like someone to check my solution. First of i will prove that $G$ ...