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

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1answer
41 views

Full binary tree proof validity: Number of leaves $L$ and number of nodes $N$

I'm working through the full binary tree proofs for a blog post I'm writing and I want to make sure I'm not missing anything. This particular proof focuses on relating the number of total nodes $N$ to ...
1
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0answers
36 views

Divergence Theorem Problem

Let $f,g:R^3\setminus\{0\}\to\mathbb{R}$ be $C^1$, let $\Omega=\{(x,y,z):x^2+y^2+z^2=1, z\geq 0\}$, $\Lambda=\{(x,y,z):0<x^2+y^2+z^2\leq 1, z\geq 0\}$ and $T=\{(x,y,z): x^2+y^2=1, z=0\}$. Show that ...
5
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2answers
168 views

Can there be only one extension to the factorial?

Usually, when someone says something like $\left(\frac12\right)!$, they are probably referring to the Gamma function, which extends the factorial to any value of $x$. The usual definition of the ...
3
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1answer
85 views

Is there a shorter way to prove that $\int_{0}^{1}{\sum_{j=0}^{2n-1}x^{2j}\over (1+x^{2n})(-\ln{x})^{1\over s}}dx…?$

Is there a shorter way to show that $$\int_{0}^{1}{\sum_{j=0}^{2n-1}x^{2j}\over (1+x^{2n})(-\ln{x})^{1\over s}}dx=\Gamma\left(s-1\over s\right)\sum_{i=1}^{n}{\sqrt[s]{2i-1}\over 2i-1}\tag1$$ ...
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0answers
43 views

Proof review - All natural numbers can be written as unique product of primes.

P(n): All natural numbers greater than 1 can be expressed as a unique product of primes where order doesn't matter. By strong induction: Base Case n=2: 2=2*1 so base case holds as this is a unique ...
2
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3answers
164 views

On the proof that $\sum\limits_{k=0}^{n-1}\frac {a^k}{(1+a^k x) (1+ a^{k+1}x)}=\frac 1 {1-a} \left( \frac 1 {1+x} -\frac {a^n}{ 1+a^n x }\right)$

Question:- Find the sum to $n$ terms of the following series $$\frac{1}{(1+x) (1+ax)} + \frac{a}{(1+ax) (1+a^2 x)} + \frac{a^2}{(1+a^2 x) (1+a^3 x)} + \cdots$$ My solution:- First of all I found ...
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1answer
17 views

Sandwiched sequence converge to same limit in $\omega_1$

I am stuck on a question that might need a trick to crack, any help is appreciated Problem statement Let $(a_n), (b_n)$ be sequences on $\omega_1$ as a topological space, such that $a_n \leq ...
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2answers
35 views

Prove that inverse of matrix A is symmetric

Suppose $A^T = A$ is a real, $n$ by $n$ matrix. We want to show that $A^{-1} = (A^{-1})^T$, that is, the inverse is symmetric. $A^T = A$ $ (A^T)A^{-1} = A A^{-1} = I $. Thus $A^{-1}$ is the right ...
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3answers
46 views

How to show that the set of all $(x,y)$ such that $3x^2 + 2y^2<6$ is an open set.

How can it be proved that the set of all $(x,y)$ such that $3x^2 + 2y^2<6$ is an open set? I tried to prove directly the aforementioned statement. Without success I tried to prove that the image ...
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0answers
28 views

A “sloppy” proof of Neyman's factorization theorem

Could you please explain why the attached proof is called "sloppy"? What is wrong with it? Thank you!
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0answers
20 views

After the midnight when all trhee clock-hands are in same direction (superposed) again?

in order to solve this question: "After the midnight when all trhee clock-hands are in same direction (superposed) again?" that appears to be simple, probably is, but i could not give a answer ...
2
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2answers
90 views

If $A \subseteq B$, does $\mbox{dist}(x,\partial A) \le \mbox{dist}(x, \partial B)$ hold for all $x \in A$

Let $A,B \subseteq \mathbb{R}^n$ with euclidean metric. Furthermore let $$ \mbox{dist}(X, Y) := \inf\{|x - y| : x \in X, y \in Y\}. $$ Does the following implication hold? $A \subseteq B \implies \...
3
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0answers
37 views

Find all possible Jordan Canonical forms and the eigenspaces dimensions of a nilpotent matrix

$A$ is a $10X10$ matrix. $$\operatorname{rank}\left(A^2\right)=2$$ $$\operatorname{rank}\left(A^3\right)=0$$ So from this I know that: All the eigenvalues are $0$ (so i'll just talk about the ...
1
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1answer
23 views

Verify my proof: For an invertible $A$ , $V\in F^n$ is an invariant subspace $V$ of $A$ $\iff$ $V$ is an invariant subspace $V$ of $Adj(A)$

$$ v_0,v_1\in V $$ $$ V \text{ is an }A\text{-invariant subspace} $$ $$ Av_0=v_1 $$ $$\iff$$ $$ A^{-1}Av_0=v_0=A^{-1}v_1 $$ $\text{Using }A^{-1}=\frac{1}{\operatorname {det}(A)}\operatorname{Adj}(...
3
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0answers
41 views

Different result on inequality to prove Riemann-Lebesgue lemma

Riemann-Lebesgue lemma. Assume $h(x)$ is continuous on $(-\pi,\pi]$. Then $$\int_{-\pi}^{\pi}h(x)\sin(nx)\mathrm dx\to 0\text{ and }\int_{-\pi}^{\pi}h(x)\cos(nx)\mathrm dx\to 0$$ as $n\to\...
1
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1answer
54 views

Real Analysis, Folland Problem 2.4.42, counting measure with convergence in measure

Problem 2.4.42 - Let $\mu$ be counting measure on $\mathbb{N}$. Then $f_n\rightarrow f$ in measure if and only if $f_n\rightarrow f$ uniformly. Attempted proof - Suppose that $\mu$ is a counting ...
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1answer
26 views

Is the unit square with dictionary ordering second countable?

I'm conflicted: If we consider the set $\{x\} \times (0,1)$, for $ x \in [0,1]$, these are open in the unit square, uncountable and disjoint, but what about open intervals of the form ((a,b), (c,d)) ...
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3answers
75 views

Full binary tree theorem proof validity?

I'm reviewing some of the theorems that make up the Full binary tree theorem and want to make sure my proof for how the number of internal nodes $I$ is related to the number of total nodes $N$ is ...
5
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3answers
161 views

How can prove that $\int_{0}^{\infty}\sum_{i=0}^{j}(-1)^{i +j}{j\choose i}{e^{-x^{i +a}}\over x}dx={j!\gamma\over a(a+1)(a+2)\cdots(a+j)}?$

Where $\gamma=0.5772156...$ is the Euler's constant and $j$ is an integer, $j\ge 1.$ $a\in \Re$ How can I show that $$\int_{0}^{\infty}\sum_{i=0}^{j}(-1)^{i +j}{j\choose i}{e^{-x^{i +a}}\over x}...
1
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2answers
105 views

Proof verification affine curve not isomorphic to plane curve

I'm trying to prove that the affine curve $X\subset\mathbb{A}^3$ given by $\alpha:\mathbb{A}^1\to\mathbb{A}^3$, $t\mapsto(t^3,t^4,t^5)$, is not isomorphic to a plane curve. Here is what I've done: it ...
1
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3answers
41 views

Prove that there exists n consecutive composite numbers

I'm asked to prove that there exists n consecutive composite numbers. This is what I've come up with. $$n! + 1 = (1 \cdot 2 \cdot 3 \cdot 4 \cdot 5 \cdot \dotsc \cdot n) + 1 $$ is a prime number ...
2
votes
2answers
34 views

Induction Sum of all odd Numbers

Show that $\sum_{k=1}^{n}(2k-1)=n^2$ Beginning: n=1 $\sum_{k=1}^{1}(2k-1)=(2*1-1)=1=1^2$ Let $\sum_{k=1}^{n}(2k-1)=n^2$ be true, then for n=p+1 $\sum_{k=1}^{p+1}(2k-1)=(p+1)^2$ has to be true too....
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2answers
25 views

For each ideal $I$ of $B$, there seems to be a corresponding morphism $f/I :A/f^{-1}I \rightarrow B/I$. Is this right?

(All my rings are commutative with $1$.) Suppose $f : A \rightarrow B$ is a morphism of rings. Then for each ideal $I$ of $B$, there seems to be a corresponding morphism $$f/I :A/f^{-1}I \rightarrow ...
0
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1answer
22 views

How to prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference?

how can i prove that a diference between the same component of two vectors is less than or equal to the norm of the vector diference? i mean supose that $A=(a_1,...,a_m)$ and $B=(b_1,...,b_m)$ both ...
1
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1answer
24 views

Show that $|\lambda_i(A)|<1$ iff $|\lambda_i(\beta A)|<1$ $\forall \beta: |\beta|\leq 1$

Here $\lambda_i(A)$ is the $i$-th eigenvalue of the square matrix $A$. I would like to know if these two inequalities are equivalent. I assumed they are (please correct me if I am wrong). So I tried ...
4
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0answers
22 views

Show that if $f(x,t)$ and $f_x(x,t)$ are continuos on $D$ then $F'(x)=\int_c^d f_x(x,t)\mathrm dt$

I want to check a proof. This is about Riemann integration. Show that if $f(x,t)$ and $f_x(x,t)$ are continuos on $D$ then $F'(x)=\int_c^d f_x(x,t)\mathrm dt$ Definitions: $D=\{(x,t):x\in[a,b]...
1
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1answer
69 views

If $|\sum_{k=1}^{n}\frac{\sin(k\theta)}{k}|<M$, is it true that $|\sum_{k=1}^{n}\frac{\sin(k\theta)}{k^{3}}|<M$?

It's proven that $\sum_{k=1}^{n}\sin(k\theta)/k$ is uniformly bounded for all $\theta\in\mathbb{R}$ and all $n\geq 1$. So there exists a $M>0$ such that $$\left|\sum_{k=1}^{n}\frac{\sin(k\theta)}{k}...
1
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1answer
22 views

verify the meaning of the closure of a set in a finite/non-metric space.

My motivation behind this question is to better under the fundamental concepts of topology in their general sense. I understand what a set closure when dealing with real numbers and a euclidean ...
2
votes
1answer
37 views

Is this Adjoint Operator Self Adjoint?

I'm helping some students study for their qualifying exam, and I wanted to double check my interpretation of a question. Suppose we define the operator $L$ on $H=L^2\left([0,\infty)\right)$ so that ...
3
votes
1answer
50 views

If a theory $A$ can prove $B$ consistent relative to $A$, and $A$ is consistent, does $B$ have to be consistent?

Let's say we have two sets of axioms $A$ and $B$ such that $\mathsf{ZF} \subseteq A \subseteq B$, and from $A$ we can prove that if $A$ is consistent, then $B$ is consistent as well (that is, $A \...
2
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1answer
40 views

Proof verification about inverses of linear mappings.

In order to prove the following statement: "Let $F: U\to V$ be a linear map, and assume that this map has an inverse mapping $G\colon V \to U$. Then $G$ is a linear map." In Serge Lang book he ...
1
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1answer
20 views

For every $v\in V$, determine if it belongs to some negative cycle in $G$

Let $G=(V,E)$ a directed graph with a weight function $w:E\to\mathbb{R}$. For every $v\in V$, determine if $v$ belongs to some negative cycle. Obviously we need to utilize Bellman-Ford algorithm for ...
0
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1answer
39 views

Alternate solution for Halmos exercise of elementary algebra.

I solved the exercise 2 of this pdf of Halmos: http://math.slu.edu/~srivastava/Halmos.pdf Show that $a^{2} + b^{2} = 10a+b$ has no solutions if $a$ and $b$ are digits and $a \neq 0$. Suppose that ...
0
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0answers
34 views

Let m = $min(\omega_1)$. Show that {m} is clopen in $\omega_1$

Let $(W, \leq)$ be an uncountable well order, then $\omega_1 = \{\alpha \in W: pred(\alpha)$ is countable } where $pred(\alpha) = \{y \in W: y < \alpha \}$ I have a couple of thoughts and I was ...
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2answers
1k views

Solving an equation, need help with explaining why one solution cant be true

I have solved the equation $$\sqrt{\frac{2}{x}}-\sqrt{\frac{x}{2}}=\frac{1}{\sqrt{2}}$$ where $x$ is $x=1$ and $x=4$. My question is why can't $x=4$? I understand that the equation does not hold if I ...
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2answers
22 views

Eliminate the parameter given $x = \tan^{2}\theta$ and $y=\sec\theta$

$x = \tan^{2} (\theta)$ and $y = \sec (\theta)$ knowing that $\tan^{2} (\theta) = (\tan (\theta))^2 = \dfrac{\sin^{2}\theta}{\cos^{2}\theta}$ and that $\sec(\theta) = \dfrac{1}{\cos(\theta)}$ $\to$ ...
4
votes
5answers
96 views

Let $f(x) = 5x+9$. Show that $\lim \limits_{x \to -3}f(x)=-6$

Let $f(x)=5x+9$. Show that $\lim \limits_{x \to -3}f(x)=-6$ A couple of questions about showing this and proving this. As I'm working through the problem I don't understand how I proved or showed ...
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0answers
47 views

If $\,\Delta_{\scriptsize\mathbb{R}^n}u = \lambda u\,$ and $\,u(x)=u(x+y)$, then $\,u_{y}(x):=e^{2 \pi i \left\langle x,y\right\rangle}$

Related to Analysis on Manifolds via the Laplacian page $52$, I would like someone explain to me why if we have a function $u$ such that $\,\Delta_{\mathbb{R}^{n\,}}u = \lambda u\,$ and $u(x)=u(x+y)$ ...
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4answers
73 views

Proper presentation for proof of $y = x^3 - 3x + 1$ having only irrational roots.

Considering the assertion: The polynomial $x^3 - 3x + 1$ has no rational roots, the following is a proof by contradiction: Let a root of $x^3 - 3x + 1$ be written in the form of $\frac{p}{q}$ where ...
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0answers
23 views

Is my proof of the compound angle trigonometric identities correct?

Please consider this diagram below: Consider the circle to be the unit circle(ignore the numbers on the axes): $BE=\sin\alpha$ , $AE = \cos\alpha$ $CF=\sin\beta$ , $AF=\cos\beta$ Since $\angle CGF=...
2
votes
1answer
20 views

Degree of a field extension for three fields

Given field extension $L/K$ and fields $E_1,E_2$ with $$(1)\ K\subset E_1\subset L,\ [E_1:K]=n_1$$ $$(2)\ K\subset E_2\subset L,\ [E_2:K]=n_2.$$ If $\gcd(n_1,n_2)=1$ then $K=E_1\cap E_2$. Proof: ...
0
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1answer
22 views

Prove that the arithmetic-geometric mean inequality holds for any list of numbers whose length is a power of 2

I am self-studying and currently reading How to Prove it by Velleman. I tried to prove the above by induction (I proved that this holds true for $n=2$), but I think my proof is wrong. I only started ...
0
votes
1answer
47 views

Is $X^p - t\in \mathbb{F}_p (t) [X]$ separable over $\mathbb{F}_p (t)$?

Is $X^p - t\in \mathbb{F}_p (t) [X]$ separable over $\mathbb{F}_p (t)$? I am trying to understand what are in these two structures. My thought is that, if we look at the derivative of $X^p - t$, we ...
1
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0answers
51 views

Check whether my solution holds good or not.

A sequence $\{x_n\}$ be such that every subsequence of it has a further subsequence converging to $1$ then show that the main sequence will converge to $1$. My work : Let us try to prove it by the ...
9
votes
1answer
110 views

Any hints on how to prove that $\ln{1\over 2\sin\left({90\over \pi}\right)}=\sum_{n=1}^{\infty}{(-1)^{n-1}B_{2n}\over 2n(2n)!}$?

How do you prove that $$ \ln\left(1 \over 2\sin\left(1/2\right)\right) = \sum_{n = 1}^{\infty}{\left(-1\right)^{n - 1}\,B_{2n} \over 2n\left(2n\right)!}\ ?\tag1 $$ where $B_{2n}$ is a Bernoulli ...
2
votes
0answers
26 views

Proof verification: Show that the Frobenius map is surjective.

I would like to prove the following but I would like someone to check my proof. For an algebraically closed field $K$ with characteristic $p$, the Frobenius map $F(x) = x^p$ is surjective What I ...
1
vote
2answers
32 views

eliminate the parameters

Given: $x = \frac{1}{2} \cos(\theta)$ and $y = 2\sin(\theta)$ Part a) solving the first one for theta: 1) multiply both sides by $2$: $$2x = \cos(\theta)$$ 2) divide both sides by $\cos (\...
2
votes
0answers
50 views

Factoring $x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$ over $\mathbb{Q}$

For a quntic polynomial to be reducible to the following form over $\mathbb{Q}$: $$x^5+B x^4+C x^3+D x^2+E x+F=(x^2+a x+b)(x^3+p x+q)$$ We need to match the coefficients ($a=B$ obviously, so we ...
0
votes
0answers
13 views

Show that $\| *\|_p$ represents a norm on $C([0, 1]).$

A few weeks ago, I had to work on the following excercise: Assume, $$\|f\|_p := (\int_{0}^1 |f(x)|^p)^{1\over p}$$ with $f \in C([0,1])$ and $p \in [1, \infty)$. Show that $\| *\|_p$...
2
votes
0answers
19 views

Exploring congruences and identities involving Mersenne primes and the terms of Lucas-Lehmer test

When I was exploring congruences$\dagger$ involving the terms $S_k$ defined in Lucas-Lehmer test (this reference is the Wikipedia, but the main reference is Crandall and Pomerance, Prime Numbers: A ...