Tagged Questions

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

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2
votes
2answers
18 views

Did I prove this correctly?

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
votes
1answer
11 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
17 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
votes
0answers
11 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
21 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
votes
1answer
15 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
vote
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
1answer
22 views

Verification of Proof that 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 primes. When G has a trivial center it means subgroup Z(G)={e}. If a group is of order pq then the ...
1
vote
1answer
24 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. ...
-1
votes
0answers
25 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
18 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
19 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
votes
0answers
22 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
vote
1answer
37 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
26 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
11 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
16 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 ...
0
votes
1answer
27 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
votes
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
54 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
votes
0answers
22 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
21 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
24 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
29 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
14 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
27 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
25 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
39 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
42 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
50 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 ...
3
votes
1answer
28 views

Eulerian circuit with no isolated vertex is connected

This is my first question (ever), and I am pretty new to math. So I ask for patience and understanding in advance. So this is the proof I came up with: Consider $G = (V,E). $ By definition of ...
1
vote
2answers
27 views

Well-defined functions on equivalence classes

Could someone please explain how to develop a proof and the reasoning behind the following: My professor said that to determine if a function is well-defined, we check to see if equivalent elements ...
3
votes
1answer
36 views

Verification of Proof that if f(x) is continuous and periodic then it is uniformly continuous on the reals.

Suppose f is defined on all reals. Then there is a positive p s.t. f(x+p)=f(x) for all x. This is my proof: Assume f is continuous on [0,p] then it is uniformly continuous on [-p,p]. Then for x,y ...
0
votes
1answer
26 views

Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff $(a_n)_{n=m'}^{\infty}$ does.

Is my proof correct? Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers. Let $c$ be real number. and let $m' \geq m$ be an integer. Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
1
vote
2answers
35 views

Prove (by induction?): If $A \subset \mathbb{N}$, $4 \in A$ and $n+1 \in A$ whenever $n \in A$, then $\left\{n \mid n \geq 4 \right\} \subset A$.

Prove: If $A \subset \mathbb{N}$, $4 \in A$ and $n+1 \in A$ whenever $n \in A$, then $\left\{n \mid n \geq 4 \right\} \subset A$. So for the base case, I did $n = 4$, so we have $4 \in A$ by ...
2
votes
2answers
45 views

Prove that, if $\{u,v,w\}$ is a basis for a vector space $V$, then so is $\{u+v, v+w, u+v+w\}$.

I'm trying to prove the following statement: In a vector space $V$ over a field $\mathbb{F}$, if $\{u,v,w\}$ is a basis for $V$, then $\{u+v, u+v+w, v+w\}$ is also a basis. $$\underline{\text{My ...
0
votes
0answers
27 views

Show that both $*$ and $.$ operation are same.

Let $G$ be a topological group with identity $x_0$. Let $\pi_1(G,x_0)$ is a fundamental group with the usual $*$ operation. If we define $(f.g)(s)=f(s)g(s)$ $\forall s\in [0,1]$ $\forall f,g\in ...
1
vote
0answers
31 views

If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \to \sqrt{x}$

If $(x_{n}) \rightarrow x$, show that $\sqrt{x_{n}} \rightarrow \sqrt{x}$ for $x > 0$. Let $\epsilon > 0$ be arbitrary, want to find $N \in \mathbb{N}$ such that $n \geq N \Rightarrow ...
-8
votes
0answers
60 views

Cracking the RSA by The Riemann Zeta Function [on hold]

How to generalize the following Riemann Zeta Function? Here we have the visualization of the Riemann Zeta Function 3D Plot . We can observe clearly that all zeros are on the critical line. Although, ...
0
votes
0answers
8 views

Primality Criterion for Specific Class of Proth Numbers

Is this proof acceptable ? Theorem : Let $N = k\cdot 2^n+1$ with $n>1$ , $k<2^n$ , $3 \mid k $ , and $\begin{cases} k \equiv 3 \pmod {30} , & \text{with }n \equiv 1,2 \pmod 4 \\ k ...
2
votes
0answers
42 views

Proof of the existence of a partition of unity

I am trying to understand the proof that partitions of unity on a smooth manifold exist. To this end, I would like to post the proof in my own words here and kindly request that someone read it and ...
0
votes
1answer
29 views

Use induction and Pascal’s identity to prove that if $n > 1$, then $C(n, 1) = C(n, n-1) = n$

So, I did the base case and I get: BASE CASE $C(2, 1) = C(2,1) = 2$. $2 = 2 = 2$. Base case holds true. Inductive Step This is where I'm not exactly sure what to do, using Pascal's rule. I have ...
1
vote
3answers
26 views

Prove that a sequence has a limit

Sequence $a_{n}$ satisfies $|a_{n}| \leq n$ for all $n \in \mathbb{N}$. Let sequence $b_{n} = \frac{a_{n} + 5}{n^{2} + a_{n}}$, prove that $b_{n}$ has a limit, and find it. I know that $b_{n}$ has a ...