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

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4
votes
1answer
48 views

What is the best time complexity of checking the inequality $a_1x_1 + \cdots + a_mx_m \le K$ to have a non-negative integer solution?

Consider $$a_1x_1 + \cdots + a_mx_m \le K$$ with $a_1, a_2, \ldots , a_m$ and $K$ being integers. I only need to know if the inequality has an integer solution or not. It means that there is ...
1
vote
1answer
41 views

Help with proof that $E(G|a < G < b) \lt E(H|a < H < b)$ for truncated normal distributions

Consider two independent normally distributed random variables with equal standard deviations, $G\sim N (\mu_{G}, \sigma)$ and $H\sim N (\mu_{H}, \sigma)$ that are truncated between points $a$ and $b$....
1
vote
0answers
27 views

Show that the set of algebraic numbers is countable(Proof verification)

I just want to make sure that my proof is good. There is a bijection between $\mathbb{Q}^n$ into $X^n$ given by $\phi : \mathbb{Q}^n \rightarrow X^n = \{ x^n + a_{n - 1}x^{n - 1} + ... + a_0 : a_i \in ...
-1
votes
0answers
14 views

Proof that a binary adder can be used to perform an n-bit unsigned subtraction

A n-bit binary adder can be used to perform an n-bit unsigned subtraction operation $X - Y$, by performing the operation $X + Y + 1$, where $X$ and $Y$ are n-bit unsigned numbers and $Y$ (with a bar) ...
0
votes
1answer
20 views

Proving two punctured domains are conformally equivalent

Prove that $S_1:=\{z:0<\lvert z \rvert<R_1 \}$ and $S_2:=\{z:0<\lvert z \rvert<R_2 \}$ are conformally equivalent. Proof: We need to find an analytic biholomorphic function $f:S_1\to S_2$...
2
votes
0answers
32 views

Show that if $E$ is Jordan measurable then $m(A-B) \leq \epsilon$

I want to show that if $E$ is Jordan measurable then $m(A-B) \leq \epsilon$ where $A \subset E \subset B$. I think I have the right ideas but feel I am missing some details. I'd like some feedback ...
1
vote
0answers
21 views

If $f(x)\geq 0$ for all $x \in [a,b]$ and $\alpha \in BV([a,b])$ is increasing , then $\int_a^bf d\alpha \geq 0.$

This is a proof verification question. Here, $\, f$ is continuous and $\alpha$ is of bounded variation. My only tools are the sums, for a given partition $P = \{a=x_0 < \ldots < x_n = b \}$ of $...
1
vote
1answer
27 views

Is my proof of the additivity property of Riemann integral correct?

Background I am trying to prove the following theorem. Let $f:[a,b]\to\mathbb{R}$ be a bounded function. If $c\in(a,b)$ then show that $f:[a,b]\to\mathbb{R}$ is Riemann Integrable on $[a,b]$ if $...
-4
votes
0answers
74 views

(Collatz Conjecture) Is this a valid proof? [on hold]

This was uploaded to 4chan.org/sci/ by an anonymous user this morning. In the interest of honesty, nobody should claim the prize money if this is a valid proof, as the author appears to not want it. ...
3
votes
1answer
44 views

Find all solutions of $e^{e^z}=1$ in the complex space.

Find all solutions of $e^{e^z}=1$ in the complex space. Attempt: $e^{e^z}=1$. Assuming $e^z$ is a complex number, I will start off solving $e^z=e^{x+yi}=1$: $e^x(\cos y+i\sin y)=1\Rightarrow \sin y=...
0
votes
0answers
21 views

Verification: Closed Set Expands to Fill Space, but Contains No Open Ball $B_\epsilon(0) $?

I have the proof that $C$ closed, convex, symmetric in Banach space $X$ and $\cup_{n \in N \setminus 0} n.C= X $ then $B_\epsilon(0) \subset C$ for some $\epsilon > 0$. I also have the proof for $...
0
votes
0answers
9 views

$ι:U→V$ is an embedding, $Q:=ιι^*$, $L∈𝓛(ℝ^d)$, $Φ∈\text{HS}(U,ℝ^d)$ $⇒$ $\text{tr}LΦ\sqrt Q(Φ\sqrt Q)^*$ doesn't depend on $ι$

Let$^1$ $U$ and $V$ be separable $\mathbb R$-Hilbert spaces $\iota\in\operatorname{HS}(U,V)$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ $u:\mathbb R^d\to\mathbb R$ be twice Fréchet ...
3
votes
1answer
58 views

Continuous functions in the product topology on $\Bbb{R}^{\Bbb{N}}$

I'm trying to prove the following statement: Let $(X, T )$ be a topological space, and let $f : X \rightarrow \Bbb{R^{\Bbb{N}}}$ be a function, where $\Bbb{R^{\Bbb{N}}}$ has the product topology. Let ...
0
votes
1answer
33 views

Increasing sequence and subsequence

I want to prove this statement, If $x_n$ is an increasing sequence and if some subsequence of it converges, then $x_n$ also converges. My proof is that suppose $x_{n(k)}$ is the subsequence that ...
3
votes
1answer
43 views

Proving that $\lim_{x\to 2}\frac{x^2-5x}{x^2+2}=-1$ using the $\epsilon$-$\delta$ definition of a limit

My attempt: $$ \left|\frac{x^2-5x}{x^2+2}+1\right|<\left|\frac{x^2-5x}{x^2+2}\right|< \left|\frac{x^2-5x}{x^2}\right|<\frac{1}{x^2}|x^2-5x|,$$ using the restriction $|x-2|<2$, so $0<x&...
2
votes
2answers
32 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
votes
1answer
53 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 $\...
2
votes
3answers
54 views

$A \Delta C = B \Delta C$, then prove that $A = B$ where $\Delta$ is a symmetric difference operation.

I suppose that if I can prove that every element that belongs to set $A$ also belongs to set $B$ and vice versa and also any element that doesn't belong to set $A$ doesn't belong to set $B$ either and ...
6
votes
2answers
63 views

Proving $\lim_{x\to1}(x^3+5x^2-2)=4$ using the $\epsilon$-$\delta$ definition of a limit

I want to prove that the limit of $f(x)=x^3+5x^2-2$ when $x\to 1$ is $4$. So, I want to show that for any $\epsilon >0$ $\exists \delta_{\epsilon}$ such that for all $x$ that satisfies $|x-1|<\...
0
votes
1answer
28 views

Is it possible for two triangles to be different if the sides of one is equal to another?

I was reading Euclid's Elements E-book I found online and got stuck on this concept. I will just copy what I found to be very absurd. There could still be another different triangle with the same ...
12
votes
0answers
210 views

Derivation of a representation through a vector field

Question: (Exercise 3.4.12 - Sharpe) Let $H$ be a Lie group, $V$ a vector space, and $\rho: H \to Gl(V)$ a representation. Let $U$ be a manifold, $X$ a vector field on $U$, and $h: U \to H$ and $f:...
0
votes
1answer
45 views

Is this proof correct? Lagrange multipliers

Suppose that $f,g : \mathbb{R}^n \to \mathbb{R}$ are $C^1$ functions and $c$ is a regular value of $g$. If $a \in g^{-1}(c)$ is a minimum for $f$ restricted to $g^{-1}(c)$ then there is $\lambda \in ...
0
votes
1answer
19 views

Proof Verification: C closed, convex, symmetric in Banach space X and $\cup_{n \in N \setminus 0} n.C= X$ then $B_\epsilon(0) \in C $.

I have an outline of the proof of this which I've expanded (correctly or otherwise) below, I'd appreciate feedback on it. (I think that C has to be closed in order to assert that $\cup_{n \in N \...
2
votes
2answers
156 views

Prove If a,b, c in N, then lcm(ca, cb) = c lcm(a,b).

Prove: If $a,b,c$ in $\mathbb N$, then $lcm(ca, cb) = c \cdot lcm(a,b)$. Assume $a$,$b$,$c \in \mathbb N$. Let $m = lcm(ca,cb)$ and $n = c\cdot lcm$. Showing $n = m$. Since $lcm(a,b)$ is a multiple ...
0
votes
4answers
36 views

Show that in the factor group $\Bbb Q / \Bbb Z$, there is an element for every $n \in \Bbb N_+$ such that the order of that element is $n$

Task: Show that in the factor group $\Bbb Q / \Bbb Z$, there is an element for every $n \in \Bbb N_+$ such that the order of that element is $n$. Solution: We take a look at the residue ...
-1
votes
3answers
24 views

Does this series converge or diverge? application of root test

Suppose $\sum a_n $ is convergent and $a_n > 0$ for all $n$, does it follow that $\sum \left( \frac{ 1 + \sin (a_n) }{2} \right)^n $ is convergent?? yes Since $\sum a_n$ is convergent, then $\sum ...
0
votes
1answer
24 views

Applying topological definition of continuity to $f(x) = \frac{1}{x}$

I am trying to show that the function $f:\mathbb{R} \rightarrow \mathbb{R}$ defined by $$ f(x) = \left\{ \begin{array}{l} \frac{1}{x}, \, x > 0 \\ 0, \, x \leq 0 \end{array} \right. $$ is not ...
2
votes
1answer
32 views

Doubt regarding a limit which is related to MVT

Let the function $f(x)$ be differentiable and $f'(x)$ be continuous in $\left(-\infty,\infty \right)$ with $f'(2)=14$ then evaluate the limit $$\lim_{x\to 0}\frac{f(2+\sin x)-f(2+x\cos x)}{x-\sin x}$...
3
votes
1answer
74 views

Is there a formal way to show that $X \cap Y \subseteq X$.

The question is in the title. It is trivial that $X \cap Y \subseteq X$. Because $X \cap Y$ only contains elements that are both in $X$ and in $Y$. So every element in $X \cap Y$ is also an element of ...
1
vote
1answer
384 views

A collection of pairwise disjoint open intervals must be countable

Let $U$ be a collection of pairwise disjoint open intervals. That is, members of $U$ are open intervals in $\mathbb{R}$ and any two distinct members of $U$ are disjoint. Show that $U$ is countable. ...
4
votes
1answer
150 views

Confused by a proof about harmonic numbers

I've been puzzled by a step in D'Aurizio's proof concerning the finiteness of a certain subset $J_p$ of $\mathbf{N}$: $$J_p = \{n : p \text{ divides the numerator of } H_n\}.$$ His paper is here: ...
3
votes
1answer
33 views

Proof of XOR properties

I want to prove the following two properties of the Nim-sum/XOR operator $\oplus$ to better understand Nim games. For the position $n = a_1 \oplus a_2 \oplus a_3 \oplus \cdots \oplus a_k = 0$, ...
6
votes
6answers
114 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)(...
3
votes
2answers
36 views

Problems Calculating Fractional Derivative

I have been trying to calculate the fractional derivative of $e^{ax}$ using the Liouville Left-Sided derivative, which states that, for $x>0$ and $0<n<1$, $D^n f(x) = \frac{1}{1-n} \frac{d}{...
1
vote
0answers
21 views

Prove: $\omega(f,P)\leq \omega(f,Q)+2nL*\lambda(P)$

Let $P=\{x_{1},...,x_{k}\}$ and $P\subseteq Q$ a refinement of the partition $P$ which is due to adding one point $Q=P\cup\{y\}$, In this case both partition are the same except of on interval $[x_{i-...
3
votes
2answers
36 views

Minimal perimeter of a triangle

Imagine a triangle with a base $[0, s]$ and a height $h$. ($s, h \gt 0$) For what orthocentre $x$ does the triangle have a minimal perimeter and how long is it? Now, the proof starts with: ...
0
votes
0answers
44 views

Can we prove invariance of dimension directly from the Jordan-Brouwer separation theorem?

Is the following proof correct? Consider spaces $\mathbb{R}^n$ and $\mathbb{R}^m$, where $n<m$, and sphere $S^{n-1}\subset \mathbb{R}^n$. Suppose that we have a homeomorphism $f:\mathbb{R}^m \...
5
votes
0answers
67 views

If $|G|=p^3q^2$ then $\Phi(G)$ is cyclic for primes $p\neq q$.

I have conjectured this result for the Frattini subgroup by doing some calculations in GAP. I think this is even true if $|G|=p_1^{i_1}\cdots p_n^{i_n}$ for $i_j\leq 3$ holds, but I would like to ...
0
votes
1answer
21 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(...
1
vote
3answers
67 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 ...
6
votes
4answers
684 views

I am almost certain the book is wrong on this “proof” of a limit.

Advanced Mathematics by Mingming Chen, Zhengyou Guo Jingxian Yu, Jinqiu Li. Chemical Industry Press pg 28, section 1.4.2 Example 2. Prove $$\lim_{x \to 1} \frac{1}{x-1} = \infty$$ Proof $\;\...
0
votes
0answers
31 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 ...
1
vote
0answers
32 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: ...
0
votes
2answers
562 views

Prove If a set contains more vectors than there are entries in each vector, then the set is linearly dependent

I want to prove this theorem: If a set contains more vectors than there are entries in each vector, then the set is linearly dependent. That is, any set $\{ v_1,v_2,...,v_p \}$ in $\mathbb{R}^n$ is ...
3
votes
3answers
100 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 ...
0
votes
0answers
25 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^{\...
1
vote
0answers
41 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
votes
3answers
37 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
votes
2answers
49 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}...
1
vote
10answers
117 views

A clean proof of $x^2 \geq x$, for any integer $x$

I am trying to prove that $x^2 \geq x$ for any integer $x$. Since we know that for any number $n$, $n^2 \geq 0$ we conclude that if $x \leq 0$ the proposition will hold. Next we must prove that the ...