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

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3answers
47 views

Is my proof True ? ( about Group theory, direct product )

I have a problem. It states that: Let $G$ is a group and $|G|=mn$, $(m,n)=1$. Assume that $G$ has exactly just one subgroup $M$ with order $m$ and one subgroup $N$ with order $n$. Prove: $G$ is ...
1
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0answers
23 views

Alternate Proof of Unique Lifting Property of Covering Spaces

I proved one of Hatcher's propositions on my own and my proof is quite a bit different than his. The Unique Lifting Property says: Given a covering space $p:\tilde{X} \rightarrow X$ and a map $f: ...
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1answer
16 views

Is collection of all functions I-convergent to a point form a ring?

$S$ be a set. $I$ is an ideal of $S.$ $X$ is a topological space. A function $$f: S\rightarrow X$$ is said to be $I$-convergent to a point $x\in X$ if $$f^{-1}(U)=\{ s\in S; f(s)\in U\}\in \mathscr F(...
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0answers
23 views

$\Bbb{R}^{\Bbb{N}}_{\square}$ is not ccc

Consider $\Bbb{Z}^{\Bbb{N}} \subseteq \Bbb{R}^{\Bbb{N}}$. The set $\Bbb{Z}^{\Bbb{N}}$ is uncountably infinite, since $|\Bbb{Z}^{\Bbb{N}}|$ = $|\Bbb{Z}|^{\Bbb{|N|}}$ = $\aleph_0^{\aleph_0}$ > $2^{\...
0
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0answers
29 views

Tietze Extension Theorem - How does the induction work?

I am reading a version of the Tietze Extension Theorem here: https://proofwiki.org/wiki/Tietze_Extension_Theorem There was a Lemma that says: And then it was repeatedly applied: How was the ...
0
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0answers
34 views

Are functors (from small categories) functions?

I am looking for either (1) confirmation that the following is true, (2) the mistake making it false pointed out to me: Let $F:\mathcal{C} \to \mathcal{D}$ be a functor from a small category $\...
1
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0answers
38 views

Is this metric space normable?

Given the function $\rho:\mathbb{R}^n\to\mathbb{R}_+$ defined by $$\rho(x)=\log\left(\frac{\sum\left|x_i\right| e^{\left|x_i\right|}}{W\left(\sum\left|x_i\right| e^{\left|x_i\right|}\right)}\right)$$ ...
2
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3answers
29 views

$(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$

Let $(R,\mathcal m)$ be a Noetherian local ring and let $P$ be a prime ideal of $R$. If $P^2$ is a prime ideal of $R$, then $P=0$. I was thinking to use Nakayama lemma as: $R_P$ is local with $PR_P$ ...
0
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3answers
42 views

Proving if $F^{-1} $ is function $\Rightarrow F^{-1}$ is $1-1$?

Let F be a function from set A to set B. If $F^{-1}$ is a function, then $F^{-1}$ is one to one. Prove: If $F: A \rightarrow B $ and $F^{-1}$ is a function, then F is one-to-one. Proof: ...
2
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1answer
28 views

Integral of Simple Functions converges to Integral of Measurable Function

Let $f$ be a measurable function and $E_{n,m} = \{x : \frac{m}{2^n} \leq f(x) < \frac{m+1}{2^n} \}$. Prove: $$\lim_{n \to \infty} \sum_{m=1}^{\infty} \frac{m}{2^n} \mu(E_{n,m}) \to \int f \, d\mu$...
2
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2answers
27 views

The Greatest Number of Edges on a Bipartite Graph

Let $G$ be a bipartite graph on $p$ vertices. Find a formula in terms of $p$ that determines the greatest number of edges that $G$ could have. Prove that this formula is correct. Let $V$ be the set ...
0
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0answers
12 views

On $\sum_{\substack{\zeta(\frac{1}{2}+i\gamma)=0\\0<\gamma<T}}\prod_{n=1}^\infty \left| 1-\frac{(\gamma\log x)^2}{n^2\pi^2}\right|$ as $O(\log x)$

On assumption that the identity (2) for a representation of $\pi(x)$ holds, see here Two Representations of the Prime Counting Function in this site Mathematics Stack Exchange, and since using the ...
0
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0answers
15 views

Proof regarding little o for vector field

I'm struggling with one part of a proof of a theorem. Let $\gamma(t) : A \subseteq \mathbb{R} \rightarrow \mathbb{R}^m$, with $\gamma \in \mathscr{C}^1(A)$ (hence $\gamma$ differentiable). ...
1
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0answers
20 views

Proof of equivalence between limit of a vector field and limit of a scalar field

I have a doubt with a proof regarding the following implication. Consider $F=(f_1,..,f_m): A \subset \mathbb{R}^n \rightarrow > \mathbb{R}^m$ and $\bar{x}$ a limit point for $A$, then $$\...
2
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1answer
55 views

Check whether my proof is correct or not.

The problem is : If the series $\sum a_n ^2$ and $\sum b_n ^2$ be both convergent, prove that the series $\sum a_n b_n$ is absolutely convergent. Using A.M. > G.M. we have $(a_n ^2 + b_n ^2)/2 \...
1
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2answers
29 views

Alternative proof: show that any metrizable space $X$ is normal - Part 2

This is a follow up to one of my earlier questions I am reading some stuff online and saw a proof as follows Per a comment in Part 1 in linked, We know that $d(C_1,C_2)$ could easily be zero ...
1
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1answer
34 views

Proving that a prime ideal $p \subset R$ yields a prime ideal $p[x] \subset R[x]$

I'm curious as to whether I can have my proof critiqued. Proposition : Let $\mathfrak{p}$ be a prime ideal in a ring $R$. Show that $\mathfrak{p}[x]$ is a prime ideal in $R[x]$. Proof : Suppose $\...
3
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3answers
96 views

“Alternatives” to Natural Transformations

I would like someone to either (1) point out the mistake in what follows or (2) confirm what is said is correct. This would be accomplished by addressing the part in yellow only. The rest of the ...
1
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0answers
22 views

Proof verification: Show that the fixed field is $\mathbb{Q}(\sqrt{3})$

Let $H$ be the subgroup $\{i,\alpha\}$ of $\text{Gal}_{\mathbb{Q}}\mathbb{Q}(\sqrt{3},\sqrt{5}),$ where $i$ is the identity map and $\alpha$ is defined as $\alpha(\sqrt{3})=\sqrt{3}$,$\alpha(\sqrt{5})=...
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6answers
111 views

Proving that ${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $

How can I prove that $${x +y+n- 1 \choose n}= \sum_{k=0}^n{x+n-k-1 \choose n-k}{y+k-1 \choose k} $$ I tried the following: We use the falling factorial power: $$y^{\underline k}=\underbrace{y(y-1)(...
2
votes
5answers
89 views

Monotonicity of the sequence $(a_n)$, where $a_n=\left ( 1+\frac{1}{n} \right )^n$

Define $a_n=\left ( 1+\frac{1}{n} \right )^n$ for $n\geq 1$. I want to show that it is increasing. First, we have $$\frac{a_{n+1}}{a_n}=\left ( \frac{1+\frac{1}{n+1}}{1+\frac{1}{n}} \right )^n\left ( ...
3
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1answer
45 views

$\text{ Proving }\; A \subseteq \Bbb R \text{ A is bounded above} \Rightarrow A^c \text{ is not?} $

Prove: Let $A \subseteq \Bbb R$. Prove that if $A$ is bounded above, then $A^c$, the complement of $A$ is not bounded above. $ A^c = $ those element of the universe that are not in A. $ \Bbb R =$ ...
0
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1answer
56 views

Can I show that $\int_{\gamma(0;r)} \frac{1}{z-a} dz = 0$ when $|a|>r>0$ without using Cauchy Theorem?

I encountered this problem as a previous result of an exercise in a text book way before proving Cauchy Theorem, so I think there must be another way to prove it without it. Show that $\int_{\...
3
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0answers
36 views

Derivative of Exponential map on manifolds

I'm trying to compute the derivative of the map $f:\Sigma\times [0,\delta)\to M$ given by $$f(p,t)=\exp_p tN(p),$$ in $X\in T_p\Sigma$, where $(M^n,g)$ is a Riemannian manifold, $\Sigma\subset M$ a ...
1
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1answer
51 views

Proving that $\sin(1/x)$ is not continuous at 0

Let $$f(x) = \begin{cases} 0 &\text{ if $x=0$}\\ \sin(1/x) &\text{ otherwise} \end{cases} $$ Prove that $f$ is discontinuous at $0$ My proof goes like this: for the function to be continuous ...
4
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0answers
12 views

Proof verification: Cross Ratio

Prove: If $[z_1,z_2,z_3,z_4] \in \mathbb{R} \cup \{\infty \} $, then $z_1,z_2,z_3,z_4$ are either concyclic or collinear. My proof below uses the geometric interpretation of cross ratio. I am not ...
0
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0answers
12 views

proof of preimage of union and intersection of sets

I was learning to proof the following proposition "the inverse image of an intersection or union equals the intersection or union of the inverse image" following these two really good youtube videos: ...
1
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1answer
34 views

Box topology is finer than the uniform topology on $\mathbb{R}^\mathbb{N}$

This time, I wish to show that the box topology is finer than the uniform topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ However, the problem here is that ...
2
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1answer
38 views

Prove that norm $\| y \|=\sqrt{\int_{a}^{b}| y(x)|^2 dx}$ forms a linear space

Prove that norm $$\mid\mid y \mid\mid=\sqrt{\int_{a}^{b}{| y(x)|}^2 dx}$$ forms a linear space. I am troubled at $$\mid\mid y+ \hat{y}\mid\mid\leq\mid\mid y\mid\mid + \mid\mid \hat{y}\mid\mid$$ ...
1
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1answer
27 views

Uniform topology is finer than the product topology on $\mathbb{R}^\mathbb{N}$

I wish to show that the uniform topology is finer than the product topology on countable Cartesian products on $\mathbb{R}$ denoted $\mathbb{R}^\mathbb{N}$ We know both spaces are metrizable: The ...
0
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0answers
40 views

Group $G$ with $ord(G)=319$ is a cyclic group

Let $G$ be a group with $ord(G)=319$. Proove that $G$ is a cyclic group. Answer: $ord(G)=319=11*29=n$, the Euler's totient function gives $\phi(n)=\phi(11*29)=\phi(11)*\phi(29)=10*28$. Since $gcd(...
0
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0answers
33 views

Can do you repeat these calculations combining the explicit formula and Nicolas criterion, on assumption of the Riemann Hypothesis?

I did easy calculations to get for $x=N_k=\prod_{n=1}^k p_k$ the kth primorial, combining the so-called explicit formula$\dagger$ for the second Chebyshev function and Nicolas criterion for the ...
1
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1answer
29 views

How to show continuity of a function with $n-1$ exponentiations?

Say we are given a function $$\Gamma(x)=f_1(x)^{f_2(x)^{\cdot^{\cdot^{f_n(x)}}}}$$ where $f_i,i\in[1;n]$, are continuous functions in their domains. Also assume that the function makes sense, e.g., ...
0
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0answers
32 views

Multiplicative implication of Goldbach's conjecture?

I've recently been thinking what would goldbach's conjecture imply in multiplicative expression: We start by stating Goldbach's conjecture in terms of the powers of a polynomial: $$ f(x)^2 = (\sum_{...
-3
votes
0answers
20 views

Let K be an integer between $800,000$ and $900,000$ so that (Greatest Common Divisor) [closed]

Let K be an integer between $800,000$ and $900,000$ so that,$\gcd(K,271)>\gcd(K,2016)>100$. List all values of K. Need serious help with this!!! Respond asap, please!
3
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1answer
91 views

Is my proof ok? If $\sum u_n$ diverges then $\sum \frac {u_n} {u_1 + u_2 + \dots + u_n}$ also diverges

The question is : If $\sum u_n$ is a divergent series of positive real numbers and $s_n = u_1 + u_2 + \dots + u_n$ , prove that the series $\sum \frac {u_n} {s_n}$ is divergent. I tried my best. ...
2
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1answer
23 views

Describe all smooth surfaces in $\mathbb{R}^3$ with coordinates $(x,y,z)$ such that the pullback of the one-form $\theta:=dy-zdx$ is identically zero.

My question is as the title states: Describe all smooth surfaces in $\mathbb{R}^3$ with coordinates $(x,y,z)$ such that the pullback of the one-form $\theta:=dy-zdx$ is identically zero. Now, ...
0
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2answers
46 views

Check my work: Evaluating $\tan\frac{7\pi}{8}$ using a half-angle formula

I am doing a trig problem involving half-angle identities, and I am not sure if my solution is correct. Can someone please check my work? The question: Find the exact value of $\tan\frac{7\pi}{8}...
0
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0answers
48 views

Calculus, limit at infinity exists, bounded second derivatives

Let $f:[0,\infty) \to \mathbb{R}$ twice differentiable. If $f''$ is bounded and $\lim_{x\to \infty} f(x)$ exists, show that $\lim_{x\to \infty} f'(x) = 0$. Update: So following the link from one of ...
0
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3answers
33 views

How to show that any separable space is CCC

I thought I had the proof of this in my head, but it doesn't sound right on paper. Can someone see if my argument could be improved. Let $(X,\tau)$ be a topological space that is separable, then it ...
0
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0answers
15 views

Linearity of projection of angle

In the book Putnam and Beyond, problem 252 reads as follows: Consider the angle formed by two half-lines in three-dimensional space. Prove that the average of the measure of the projection of the ...
6
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1answer
52 views

Does there exist a multiplicative $f:\mathbb{Q}^+\to\mathbb{Q}^+$ such that $f\neq x\mapsto x^a$ for all $a$?

If we consider the functional equation: $f:\mathbb{Q}^+\to\mathbb{R}$ such that $$ f(xy)=f(x)f(y) $$ for all $x,y\in\mathbb{Q}^+$ I think, I have constructed a solution which is not of the form $x\...
1
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1answer
21 views

In a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact.

Prove that in a metric space $X$, a subset $S \subset X$ is relatively sequentially compact if and only if its closure $\overline{S}$ is sequentially compact. The terms relatively sequentially ...
0
votes
1answer
17 views

Discrete Math Proof: Divisibility equivalence

For all integers $a$, $b$, $d$, if $d$ divides $a$, and $d$ divides $b$, then $d$ divides $(3a+2b)$ and $d$ divides $(2a+b)$. Prove the statement. What Assumptions do I need to make at the beginning ...
1
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1answer
35 views

Discrete Math Proof verification: products of floor

Determine if the following is true or false and provide a proof: $\forall x\in\mathbb{R},\exists y\in\mathbb{R}$ so that $\lfloor xy\rfloor = \lfloor x\rfloor \lfloor y\rfloor$ My attempt: -The ...
1
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0answers
33 views

Finite dimensional separable algebra is étale

Say a $\Bbbk$-algebra is separable if $L\otimes _\Bbbk A$ is reduced for every extension $L/\Bbbk$. Say it's étale if there's an extension $L/\Bbbk$ such that $L\otimes_\Bbbk A\cong \prod_1^nL$. Here'...
2
votes
2answers
85 views

About a proof that $\lfloor x^2\rfloor = \lfloor x\rfloor^2$ for unbounded non integer values of $x$

I am taking a first course in discrete mathematics. The instructor parsed the following question that has the following solution, respectively: Prove the statement: For all positive integers $N$, ...
1
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1answer
44 views

Alternative proof: show that any metrizable space $X$ is normal - Part 1

There is a proof online that shows that all metric spaces are normal. The proof is as follows However, it has the additional baggage of needing to show that $d(x,A)$ is continuous and $U,V$ are ...
1
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1answer
35 views

The set of all real or complex invertible matrices is dense

I'm trying to show that the set of all invertible matrices $\Omega$ is dense over $F=\mathbb R$ or $\mathbb C$. Let $A\in\Omega$ and $C\in M_{n\times n}(F)$. Since $\|A-C\|<\frac{1}{||A^{-1}||}$, ...
1
vote
2answers
19 views

Show that any metrizable space $X$ is regular

This is a quick follow up to another question Show that any metrizable space $X$ is Hausdorff Recall, a topological space $(X,\mathcal{T})$ is regular if we can separate any point $x$ from ...