Tagged Questions

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

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1
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0answers
13 views

$f: \mathbb{R} \to \mathbb{R}$ and $f(x)>0$ $\forall x \in \mathbb{R}$. Prove $\exists \epsilon >0: f(x) > \epsilon$ for all $x \in [-10,10]$

Is my proof below valid as a solution to the problem? Problem: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function such that $f(x)>0$ for all $x \in \mathbb{R}$. Prove there exists an ...
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0answers
12 views

What is the identity in A8 [on hold]

What is the identity in A8 . I am not sure about the identity element in A8 ANSWER (12)
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0answers
16 views
1
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2answers
37 views

In metric space, $X$ is connected and $X\subset Y \subset \bar X$, prove that $Y$ is connected.

Question: $X$ is connected and $X\subset Y\subset\bar X$, prove that $Y$ is connected. This is one of my midterm questions this morning. I couldn't figure it out. But now I came up with this proof, ...
1
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0answers
22 views

Trigonometry of tetrahedron

I'm trying to develop the algebraic proofs for these two formulas that appear on the webpage below! The image below is of an unfolded non-regular tetrahedron. Triangle B represents the dihedral angle ...
1
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4answers
19 views

Limit of $\frac{2^n+2.71^n+\frac{4^n}{n^4}}{\frac{4^n}{n^4}+n^33^n}$ - what is wrong with my proof?

Here's quite a hairy sequence, the limit of which I need to find as $n\rightarrow\infty$: $$\frac{2^n+2.71^n+\frac{4^n}{n^4}}{\frac{4^n}{n^4}+n^33^n}$$ The Squeeze Theorem seemed like a good idea so ...
2
votes
1answer
35 views

Associates in a non-integral domain

I have tried to give a proof of the following theorem but I feel very unsure and would be very grateful if someone would check it for me Many thanks! Theorem. Let R be the ring $C[0,1]$ of ...
0
votes
1answer
8 views

How to prove that at Complete Binary Tree (CBT) at height $h$ we have $2^h$ leaves

I try to prove it by induction, please tell me if I'm right... The induction assumption - For every CBT at height $h$ there is $2^h$ leaves. The base of the induction is right (I'm writing this proof ...
3
votes
1answer
29 views

About generated $\sigma$-algebras (proof verification).

It would be really helpful if anyone would browse through this and tell me if my solution is ok. Here is the question: Let $C \subset 2^X$ be a collection of subsets. Show that for every $K \in ...
1
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0answers
16 views

Uniqueness of smallest element in poset

Prove that a smallest element, if it exists, is determined uniquely. This is follows directly for definition. An element $a \in X$ is called the smallest element of $(X, \preceq)$ if for every $x \in ...
3
votes
2answers
55 views

Integer solutions of the equation $7(a^2+b^2)=(c^2+d^2)$

What are all the integer solutions of the equation $7(a^2+b^2)=(c^2+d^2)$ First thing to note is that $c=7C$ and $d=7D$ and substituting it in the original equation yields an equation that is ...
3
votes
0answers
39 views

Two versions of invariance of domain theorem?

While reading about Invariance of Domain Theorem, I noted that there are two common version of it, one saying that $f(U)$ is open, while another saying that $f$ is a homeomorphism (and sometimes both ...
2
votes
0answers
27 views

James $\ell_1$-theorem: problem in the proof

I am struggling with the very last estimate in the proof of James' $\ell_1$-theorem. (Please see below an excerpt from Albiac and Kalton's fantastic book Topics in Banach space theory.) I don't see ...
1
vote
3answers
33 views

If $x>y$, then $\lfloor x\rfloor\ge \lfloor y\rfloor$, formal proof

For x ∈ ℝ, define by: ⌊x⌋ ∈ ℤ ∧ ⌊x⌋ ≤ x ∧ (∀z ∈ ℤ, z ≤ x ⇒ z ≤ ⌊x⌋). Claim 1.1: ∀x ∈ ℝ, ∀y ∈ ℝ, x > y ⇒ ⌊x⌋ ≥ ⌊y⌋. Assume, x, y ∈ ℝ # Domain assumption ...
0
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1answer
14 views

Primality Test for Safe Primes

Is this proof acceptable ? Theorem Let $N$ be of the form $N=2p +1$ with $p$ prime , then $N$ is prime iff $N \mid 2^{2p}-1$ Proof In one direction , if $2p+1$ is a prime then by Fermat ...
0
votes
0answers
21 views

Continuity of translation on $L^1$ on the reals ($\int |f(x+h)-f(x)|\,dx\to 0$)

Let $f$ be a real valued, Lebesgue integrable function on $\mathbb{R}$. prove $$\lim_{t \to 0} \int_{\mathbb R} |f(x+t)-f(x)|\, dx=0.$$ I solved it in this way. is it correct? Since $f(x)$ is ...
0
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0answers
14 views

The Answer to the problem Prove that there is a 1-1 correspondence between the set of subgroups of Z/NZ and the set of the positive divisors of N [duplicate]

I need to Prove that there is a 1-1 correspondence between the set of subgroups of Z/NZ and the set of the positive divisors of N My attempt: We first define $B=\{d>0: divisor of N\}$, ...
0
votes
1answer
28 views

Am I correct with this change of variable?

I have been solving a problem from a paper I read related to poisson point processes and for some reason I am not reaching the same result the paper has. The problem is re-expressing an expression by ...
0
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1answer
21 views

Verification of proof that NM is a normal subgroup of G if M and N are both normal subgroups of G

My proof is as follows: M is a subgroup of G means $g_1mg_1^{-1}$ is part of M and likewise N is a subgroup of G means $g_2mg_2^{-1}$. To prove our claim do the following: $$g_1mg_1^{-1} ...
1
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0answers
14 views

Verification of Proof that if N is a normal subgroup and H is any subgroup HN={hn| h in H and n in N} is a subgroup

My proof is as follows: I only have question for the closure portion. Closure: Let h1, h2 be in H and n1, n2 be in N. Since N is normal we can say $Nh_2=h_2N$. This also means $n_1h_2=h_2n_3$ So ...
3
votes
2answers
30 views

Verification of Proof that a nonabelian group G of order pq where p and q are primes has a trivial center

A nonabelian group $G$ of order $pq$ where $p$ and $q$ are primes has a trivial center My Proof is as follows: Assume we have nonabelian group $G$ of order $pq$ where both $p$ and $q$ are ...
1
vote
1answer
26 views

Verification of Proof that if G is not abelian G/Z(G) is not cyclic

I will prove this by the contrapositive: "If G/Z(G) is cyclic then G is abelian" Proof: We assume that G/Z(G) is cyclic. This means it is generated by a left coset $(aZ(G))^n$=e for some integer n. ...
0
votes
0answers
28 views

Is this derivative of $\frac{\partial x}{\partial P}$ correct?

By IFT, let $x^* + \Phi \left (\frac{f(x)+\check{G}}{a(\hat{G}+f(x))} \right ) - 1 = 0 \equiv F$. $P=f(x)$ is a convex function, where $f'<0$, and $f''>0$. I want to find $\frac{\partial ...
1
vote
1answer
20 views

Verifying a bound on the norm of an operator in $l_2$.

The problem: Define $L: l_2 \rightarrow l_2$ by $L(x_1, x_2, ...) = (y_1, y_2, ...)$, where $y_n = (x_1 + x_2 + ... + x_n)/n^2$. Show that $||L|| \leq (\sum_{n=1}^\infty 1/n^2)^{1/2}$. My proof: ...
0
votes
0answers
29 views

Why this theorem is invalid?

I know that the following theorem (and by extension the proof) are invalid: Incorrect Theorem. Suppose that $x$ and $y$ are real numbers and $x\neq 3$. If $x^2y = 9y$ then $y = 0$. Proof. ...
3
votes
1answer
29 views

A function $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ with $\varphi(x)=x,$ $\|\varphi(y)-x\|\leq K\|y-x\|^\alpha$ for $\alpha>1, K>0$

If we have a function $\varphi:\mathbb{R}^n\to\mathbb{R}^n$ with $\varphi(x)=x,$ $\|\varphi(y)-x\|\leq K\|y-x\|^\alpha$ for $K>0,$ and we define $\varphi^1:=\varphi, ...
0
votes
1answer
20 views

a proof for a probably common problem?

Can someone provide a proof for the following problem? I know that this might be a common proof to some common problem that I am yet to know, and that if someone would leave a proof it would give me ...
0
votes
0answers
22 views

Computer verification of Fermat's Last Theorem - status

My question is about the status of proof verification...and specifically about Fermat's last theorem. How close are we to having computers able to verify theorems of this complexity. What about the ...
0
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0answers
26 views

prove well-ordering of nonnull subset of positive ints using weak induction

Let $S\subseteq Z^+$. If $S$ has one element it must be the smallest element and hence it is well-ordered. Assume true for $S$ having $n$ elements. If $S$ has $n+1$ elements if the smallest is ...
1
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1answer
39 views

Suppose $x \in \mathbb{R}$. If $x^3-x>0$, then $x>-1$. Contrapositive proof

Suppose $x \in \mathbb{R}$. If $x^3-x>0$, then $x>-1$. Proof (Contrapositive). Suppose $x \leq -1$. It follows that $x^3 \leq x \leq -1$. Picking $x=-1$, the quantity $x^3-x=0$, otherwise it is ...
3
votes
1answer
27 views

Is it true that $n_p!\le |G|$?

Let $G$ be a finite group and $n_p:=|\text{Syl}_p(G)|$. Is it true that $n_p!\le |G|$ ? I've shown that it's true, but I'm not so sure, can you check my proof? Proof. Let $G$ act on ...
0
votes
1answer
13 views

Prove asymptotic bound by the substitution method

I need to prove that $T(n) = 4T(n/2) + n^2lgn = \mathcal{O}(n^2lg^2n)$ by using the substitution method. Unfortunately, I'm not able to identify the error in my train of thought. For the problem at ...
1
vote
1answer
17 views

Finding $\sup$ and $\inf$ of $\{\frac{nk}{1+2n+3k} : n,k \in \Bbb{N}\}$

I'm trying to solve the following problem: Find $\sup$ and $\inf$ of $A=\{\frac{nk}{1+2n+3k} : n,k \in \Bbb{N}\}$ and maximal and minimal element of this set. As for $\sup(A)$ and $\max(A)$ I tried ...
1
vote
1answer
28 views

How do I prove that a subset is closed in the topological space of $n \times n$-matrices.

Consider the topological space $M$ of $n \times n$ matrices over $\mathbb{R}$ equipped with the standard topology. Let $\mathcal{A} \subset M$ be the set of matrices such that $det(A) = 1$ for $ A ...
0
votes
0answers
30 views

How to prove isomorphism between these two graphs

I thought that the best way to approach this problem was to use a direct proof and say that since the graphs have the same number of vertices G1: {v1, v2, ..., vi, ..., vk} and G2 : {b1, b2, ..., ...
0
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2answers
26 views

Prove by contradiction $a \in C$, if $a \in A \land a \not\in B \setminus C$

This is the exercise I have: Suppose that A ⊆ B, a ∈ A and a $\not\in$ B\C. Prove by the method of contradiction that a ∈ C Proving by contradiction means that if I find a contradiction ...
2
votes
4answers
56 views

Prove $Q \rightarrow \neg(Q \rightarrow \neg P)$

I have an exercise about proving statements: Suppose that P is true. Prove that Q → ¬(Q → ¬P ) is true Givens: $P$ $Q \rightarrow \neg P$ Goal: $\neg Q$ which I simply prove ...
2
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0answers
23 views

How to prove this statement $x \not\in D$ then $x \in B$

I am quite a beginner writing proofs, that's why I am asking such a simple question. I have an exercise: Suppose A\B ⊆ C ∩D and x ∈ A. Prove (by using proof techniques) that if $x \notin D$ ...
2
votes
1answer
32 views

Determine whether $\phi$ is a homomorphism

Let $\phi: \mathbb{Z}_6 \rightarrow \mathbb{Z}_2$ be given by $\phi(x)=$the remainder of $x$ when divided by $2$, as in the division algorithm. Let $\phi: \mathbb{Z}_9 \rightarrow ...
3
votes
2answers
22 views

Division with remainder

I have proved the division with remainder theorem: If a $\in \mathbb{Z}$ and $d \in \mathbb{N}$ then there exists unique numbers $q,r \in \mathbb{Z}$ such that $a=dq+r$ where $0\le r<d$. I proved ...
1
vote
1answer
26 views

Prove that $C_H(K) = N_H(K)$ for $G=H \rtimes_{\phi} K$

Let $H,K$ be group where $\phi: K \rightarrow \operatorname{Aut}(H)$ is a homomorphism. Also, let $G=H \rtimes_{\phi} K$. Show that $C_H(K) = N_H(K)$ Proof: Let $h\in N_H(K)=\{h\in H: ...
1
vote
2answers
35 views

Proof by Induction that $16 \mid 5^n - 4n - 1$

Using induction, prove that $16\mid 5^n - 4n - 1$ for $n$ in $\mathbb{N}$ Here's what I have and what I'm stuck on: basis: $n = 1$, $5 - 4(1) - 1 = 0$ and $16\mid 0$. Hypothesis: Assume true for ...
2
votes
0answers
15 views

Rearrangement of absolutely convergent series

I would be very grateful if someone would verify whether my proof below is correct. Many thanks. Theorem. $\,$ Let $(b_k)$ be a rearrangement of the complex sequence $(a_k)$. If $\sum_{k\geq 0}a_k = ...
0
votes
1answer
24 views

Evaluating a limit that involves a summation

I was solving a physics problem and I got this expression: $E=\lim_{N \to \infty}\left[\dfrac{k_0Q}{2R^2}\dfrac{1}{N}\sum\limits_{i=0}^{N/2-1}\left(\sec{\dfrac{i\pi}{N}}\right)\right]$ I'm not sure ...
2
votes
0answers
28 views

Period of a pendulum

Consider the pendulum problem $\frac{d^2x}{dt^2}+\sin(x)=0$ $\frac{dx}{dt}(0)=v_0=0$ $x(0)=x_0$ Show that the period ...
0
votes
2answers
26 views

Prove that $\text{rank}(A) = \text{rank}(A^T)$ using SVD

The title pretty much says it all, I need to prove that $\text{rank}(A) = \text{rank}(A^T)$ using SVD. It seems quite trivial, but I'd like to hear a second opinion. My thoughts are exposed below. ...
0
votes
1answer
41 views

Limit proof for rational function $\frac{1}{x}$

A while ago I posted another one like this with a incorrect approach, please see this one! Is this an accurate proof for limits for the function $\frac{1}{x}$ $\displaystyle \lim_{x\to1} \frac{1}{x} ...
4
votes
1answer
43 views

Adjacency matrix and connectivity proof

Let $G$ be graph on $n$ vertices, $A$ its adjacency matrix, and $I_{n}$ the $n\times n$ identity matrix. Prove that $G$ is connected iff the matrix $(I_{n} + A)^{n-1}$ has no 0s. My proof: If the ...
2
votes
0answers
43 views

Whats wrong in this proof of $10$ is a solitary number?

Friendly numbers are two or more natural numbers with a common abundancy, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same abundancy form a friendly ...
1
vote
0answers
58 views

Prove that module has finitely many elements

Let $p$ be a prime number. Consider the subring $U:= \mathbb{Z}[1/p]$ of $\mathbb{Q}$ and define the $\mathbb{Z}$-module $M:=U/ \mathbb{Z}$ (1): Show that any $\mathbb{Z}$-submodule of $M$ that is ...