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

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2
votes
2answers
65 views

Is it necessary that $A$ has $m-1$ eigenvectors corresponding to $\lambda$, which are orthogonal to vector of all ones

If $A$ is real symmetric matrix and has an eigenvalue $\lambda$ with multiplicity $m$, Is it necessary that $A$ has $m-1$ eigenvectors corresponding to $\lambda$, which are orthogonal to vector of all ...
1
vote
1answer
63 views

Theorem 4.6 from baby Rudin

Theorem 4.6. In the situation given in Definition 4.5, assume also that $p$ is a limit point of $E$. Then $f$ continuous at $p$ if and only if $\lim \limits_{x\to p}f(x)=f(p)$. Proof: Let $f$ ...
2
votes
1answer
115 views

Integrability of Thomae's Function on $[0,1]$.

Consider the function $f: [0,1] \to \mathbb{R}$ where f(x)= \begin{cases} \frac 1q & \text{if } x\in \mathbb{Q} \text{ and } x=\frac pq \text{ in lowest terms}\\ 0 & \text{otherwise} ...
0
votes
0answers
27 views

Boundary Value Problem Verification

I have this boundary value problem and I was wondering If I was solving it correctly (showing all necessary working) or could you show me a better way of showing my working. $$ X''(x)=0, ...
3
votes
0answers
38 views

Prove that $2\mid x$ and $5\mid x$ if and only if $10\mid x$

I have to do it without using Fundamental Theorem of Arithmetic. Can someone check my work? Prove if $2\mid x$ and $5\mid x$, then $10\mid x$. Let $x \in \mathbb{Z}$. Suppose $2\mid x$ and $5\mid ...
2
votes
0answers
64 views

Showing a norm inequality

Suppose $x \in \mathbb{R}^n$, show $||x||_2 \leq \sqrt{n} ||x||_{\infty}$ attempt: We have $||x||^2_2 = |x_1|^2 + ... + |x_n|^2 $. We know there is some $k$ such that $|x_k| = \max_{1 \leq j \leq n ...
0
votes
0answers
65 views

Prove that probability distribution function is continuous at a point

Let $X$ be a random variable on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Show that $F_X(x)$ is continuous at $x = x_0$ if and only if $\mathbb{P}(X = x_0)= 0.$ ...
6
votes
2answers
234 views

Number of Solutions in this Number Puzzle

Today in math class, a number puzzle arose where you would have to find a 10 digit number, where each digit describes the number of other digits in the number, for example: $$\text{0 1 2 3 4 5 6 7 8 ...
2
votes
3answers
60 views

Prove $\arctan(x+y)<y+\arctan(x),x\in \mathbb{R}, y>0$

Prove $\arctan(x+y)<y+\arctan(x),x\in \mathbb{R}, y>0$ Using Lagrange's mean value theorem, $$f(u)=\arctan(u)$$ In the first case, $x>0$ $$x=a,x+y=b$$ $f(u)$ is continuous and ...
1
vote
3answers
62 views

Show that $\langle 5, x^2+x +1 \rangle$ is maximal ideal in $\mathbb{Z}[x]$.

Here is my try, of which I'm rather skeptical. Let $I$ an ideal such that $\langle 5, x^2+x +1 \rangle \subset I \subset \mathbb{Z}[x]$. Because of the containment, there must be some $\alpha \in I$ ...
2
votes
2answers
67 views

Is my induction proof of $2^{n} > 2n+1$ correct?

Hello I am wondering if anyone can conform that the method I use in the following proof is valid. If not please inform me/ point me in the right direction. It is a very basic question, i.e. to prove ...
0
votes
2answers
69 views

A question about conic section (ellipse).

I am asked to solve the problem what is the center of the ellipse with vertex $V_1=(1,3)$ and focus $F_1=(1,0)$ and eccentricity $e=1/2$. My answer is due to the following analysis and computation: ...
10
votes
4answers
340 views

Countable choice and term extraction

The constructive Axiom of Countable Choice (ACC) is widely accepted due to its computational content. It states that: $$ \forall n\in \mathbb{N} . \exists x \in X . \varphi [n, x] \implies \exists f: ...
0
votes
0answers
30 views

Open covers and $(n,\varepsilon)$-separating/ spanning sets: proving three inequalities

In Peter Walters' book An Introduction to Ergodic Theory, one can find the following corollary (p. 174 in my edition). At the end of this thread, I tried to prove it. It would be great if you could ...
0
votes
2answers
66 views

Using linear algebra to find constants in the equation of a circle which passes through given points.

Find constants $a ,\ b ,\ c \ $ such that the equation of the circle, $x^2+y^2+ax+by=c$, contains the points $(6,8)$, $(8,4)$, and $(3,9)$. Use the points to create a system: ...
1
vote
0answers
27 views

Over what subset of $\mathbb{N}$ is the deficiency $D(x) = 2x - \sigma(x)$ a weakly multiplicative function?

This is an offshoot of this MSE question which was posted earlier today. Let $\mathbb{N}$ be the set of natural numbers (i.e., positive integers). We call $\sigma(x)$ the sum of the divisors of $x$. ...
1
vote
1answer
41 views

If $g(x) = x^2 + 3x + 1$ and $f \circ g = g\circ f$, then $f$ and $g$ intersect on the line $y = x$.

Let $f:\mathbb R\to \mathbb R$, $g(x) = x^2 + 3x + 1$ and $f∘g=g∘f$. Prove that $Cf, Cg$ and the line $y=x$ have at least one common point. My solution: $fog=gof$ then $f=g$, $Df = Dg = ...
1
vote
1answer
63 views

Binary Operation with Cayley Table

I am asked to write out a Cayley table of a binary operation $\ast$ on the set $$ S = \{1, 2, 3\}$$ for which there is no solution for $\ast$ in $S$ to the equation $1 \ast x = 2$. Here is my ...
3
votes
0answers
51 views

Exercise on limits: is this proof correct?

Prove that $$\lim_{x\rightarrow0}x^\alpha \ln{x}=0$$ for every $\alpha>0$ This is my attempt: Let $g(x)=x^x$ $f(x)=x^\alpha$ $h(x)=g\circ f(x)=(x^\alpha)^{(x^\alpha)}$ $k(x)=x^{(x^\alpha)}$ ...
1
vote
1answer
87 views

Galois group of $X^3-10$ over $\mathbb{Q} (i\sqrt{3})$

This is Exercise 1(d), page 320 in Lang's Algebra. I think that the Galois group is in this case simply the trivial group $\{Id\}$. Why? Let: $$f(X)=X^3-10.$$ Then the splitting field of $f$ is ...
3
votes
1answer
145 views

Which of the following identities are true? Justify your answer - n! = O(4^n)..

Which of the following identities are true? Justify your answer a)$n! = O(4^n)$ b)$4^n = O(n!)$ If I let $n = 0$ then $(0)! = O(4^0) \implies 1 = O(1)$ so this is true I think? $4^n = O(n!)$ In ...
1
vote
1answer
45 views

Is a proper morphism between quasiprojective varieties projective?

I'm currently reading Shafarevich's Basic Algebraic Geometry 1. In it (page 59), he makes the following definition: ...a wider class of maps $f: X \rightarrow Y$ between quasiprojective varieties, ...
0
votes
2answers
67 views

Is my understanding of this proof correct?

Given the definition: A real number $x$ is said to be positive if $0\lt x$ and negative if $x\lt 0$, and the Axioms of order (Trichotomy, Transitivity, Monotonicity of addition and multiplication), ...
1
vote
0answers
48 views

Application of Radon-Nikodym

Let $\nu$ and $\mu$ be two $\sigma$-finite measures on $(X,\mathcal{M})$ and assume that RN holds, with $w = \frac{d\nu}{d\mu}$. Show that, for each non negative measurable function $g$ it holds $$ ...
3
votes
1answer
112 views

Triangle Inequality Equality Conditions

I am looking for the conditions on two complex numbers $z_1$ and $z_2$ such that $$|z_1+z_2|=|z_1|+|z_2|$$ Letting $z_n=a_n+ib_n$ and using $|z_n|^2=a_n^2+b_n^2$ yields ...
3
votes
6answers
350 views

Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative?

Let $(a_n)$ be a convergent sequence of positive real numbers. Why is the limit nonnegative? My try: For all $\epsilon >0$ there is a $N\in \mathbb{N}$ such that $|a_n-L|<\epsilon$ for all ...
2
votes
2answers
171 views

Examples of monotone functions where “number” of points of discontinuity is infinite

We know that if $f:D(\subseteq \mathbb{R})\to\mathbb{R}$ be a monotone function and if $A$ be the set of points of discontinuity of $F$ then $\left\lvert A \right\rvert$ is countable. Where ...
2
votes
4answers
60 views

Trying to prove $\sum_{i=1}^{N} i^3 = (\sum_{i=1}^{N} i)^2$

I'm trying to prove $\sum_{i=1}^{N} i^3 = (\sum_{i=1}^{N} i)^2$ but I got stuck along the way. This is what I have so far: The base case is true when $N =1$. Then for the inductive step I did: ...
0
votes
0answers
13 views

PRNG Improvements

Purpose This is the (somewhat) mathematical representation of an algorithm for a pseudo random number generator. It uses mostly linear math and generally is not very complex, but then again - I'm not ...
0
votes
2answers
63 views

Identify the error - Discrete math

I'm having problems trying to identify the error in this proof in the question below: Let $u$, $m$, $n$ be three integers. If $u\mid mn$ and $\gcd(u,m) = 1$, then $m = \pm1$. If $\gcd(u,m) = 1$, ...
0
votes
2answers
52 views

Question about proof for why every partial order on a nonempty finite set has a minimal element

The proof goes as follows: Proof. Let $R$ be a partial order on a set, $A$. For any element, $a ∈ A$, let $g(a)$ be the set of elements “less than or equal to $a$”, that is, ...
7
votes
2answers
129 views

Is this a sound demonstration of Euler's identity?

Richard Feynman referred to Euler's Identity, $e^{i\pi} + 1 = 0$ as a "jewel." I'm trying to demonstrate this jewel without recourse to a Taylor series. Given $z = cos\theta + i sin\theta\; |\;|z| = ...
2
votes
0answers
41 views

Is the topological entropy of a continuous map $T\colon X\to X$ zero if $X$ is a finite topological space?

Let $X$ be a finite topological space and $T\colon X\to X$ continuous. As the title already suggests, I am wondering if the topological entropy of $T$, denoted by $h(X,T)$, then is $0$. As far as I ...
3
votes
1answer
88 views

Finding the Number of Subfields of the Splitting Field of $x^{35}-1$ over $\mathbb{F}_8$

Let $E$ be the splitting field of $x^{35}-1$ over the field $\mathbb{F}_8$. Determine $|E|$ and the number of subfields of $E$. Attempt: I am confident that I computed $|E|$ correctly, but I am ...
1
vote
1answer
38 views

Question about a proof of FTA in A classical Introduction to modern number theory

I just started to work on A classical introduction to modern number theory by K. Ireland and M. Rosen and I have a question. In the first chapter, they prove the following lemma: Every nonzero ...
0
votes
2answers
60 views

Proof verification - $\sqrt[3]{2}$ is irrational

This might be a duplicate, but I have not found one on this site. This is my proof and I was wondering if this proof depends on it's conclusion. Here it is: Assume $\sqrt[3]{2}$ is rational. Then ...
3
votes
0answers
52 views

Can every mathematical proof be seen as the verification of some algorithm's action?

Put another way: Can every mathematical proof be reformulated to be about some class of Turing Machines? Example Any proof of the existence of infinite prime numbers is equivalent to the statement: ...
1
vote
4answers
67 views

Not understanding the beginning of the proof for $\log x < x$

I was reading some of the proofs for $\log x < x$ and I noticed a few of them have the proof start off with: "Let $g(x) = x - \log x$". Then they find its derivative to show if the function is ...
2
votes
2answers
67 views

Odd perfect numbers must have an odd number of proper divisors

Theorem: If an odd perfect number exists, then it has an odd number of odd proper divisors. $(1)$ Assume that an odd perfect number exists. Call it $n$. $(2)$ $2$ is not a divisor because $n$ is ...
6
votes
2answers
91 views

$a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$.

I was curious if there are quadratic equations where $a,b,c\in \Bbb Z$ and $a\cdot b\cdot c$ is a root of $ax^2+bx+c$. So trivially if $c=0$, $a$ and $b$ can be arbitrary, and if either $a$ or $b$ is ...
1
vote
0answers
40 views

Proving the continuity of the function $f(x)=x^\alpha$ through power series

The textbook I'm studying on presents the following two theorems: $a^x=\sum_{n=0}^{\infty}\frac{(\ln{a})^nx^n}{n!}$ The function $f(z)=\sum_{n=0}^{\infty}a_nz^n$ converges for every $z\in\mathbb{C}$ ...
6
votes
1answer
35 views

Verification for a block-determinant evaluation, and some further thoughts

First, I want some verification for the validity of my approach for this det evaluation question: If $A,B\in M_n(K)$, $K$ is a number field (in the sense that $\Bbb Q$ is the smallest possible ...
2
votes
1answer
61 views

Problem 20 chapter 3 from baby Rudin

Suppose $\{p_n\}$ is a Cauchy sequence in a metric space $X$, and some subseqeunce $\{p_{n_i}\}$ converges to a point $p\in X$. Prove that the full sequence $\{p_n\}$ converges to $p$. Proof: ...
3
votes
1answer
55 views

Prove $x^n < n^n 2^x$

Given that $$x < 2^x$$ is always true, use it to prove that $$x^n < n^n2^x$$ Here are the steps that I've taken so far: Reduce $$x < 2^x$$ to $$\log(x) < x$$ Then $$x^n < n^n2^x$$ ...
1
vote
1answer
27 views

Show that this piece-wise function defines a differentiable solution

Show that $y(x) = \begin{cases}-x^4 & x < 0, \\ x^4 & x \geqq 0 \end{cases}$ defines a differentiable solution of $xy'=4y$ for all $x$, but is not of the form $y(x)=Cx^4$.
2
votes
2answers
53 views

$\{a_n\} \to a$ iff $\limsup_{n \to \infty} \{a_n\} = \liminf_{n \to \infty} \{a_n\}$

It is clear that if $$\limsup_{n \to \infty} \{a_n\} = \liminf_{\to \infty} \{a_n\},$$ then $\{a_n\} \to a$, since we can just squeeze the terms in the middle. I understand that to prove the ...
1
vote
1answer
40 views

The martingale $M_t,\mathcal{F}_t$ is a martingale with respect to the filtration $\mathcal{F}_{t +}$

Let $M_t$ be a right continuous martingale with respect to the filtration $\mathcal{F}_t$. Can we say that $M_t$ is a martingale with respect to the filtration $\mathcal{F}_{t+}$? Attempt: We know ...
1
vote
1answer
42 views

Algorithm and top-points.

Problem: For an array $A[1],\dots,A[n]$, with $n\geq 3$, it holds that $$A[i+1]>\frac{A[i]+A[i+2]}{2},\qquad i\in \{1,2,\dots, n-2\}$$ That is, it holds that $$A[2]>\frac{A[1]+A[3]}{2},\dots, ...
1
vote
1answer
22 views

Proof Verification - $\exists a \in S (a\ge S_a)$

I wanted to prove that $\exists a \in S (a\ge S_a)$ where $S$ is an finite set of real numbers with order $n$ and $S_a$ is the average of the set. This is my proof so far: Assume $a_i = a_k, i,k ...
1
vote
0answers
10 views

Is the function $Z(t,q)$ progressively measurable?

The excercise is taken from Stroock and Varadhan Multidimensional diffusion processes chap 1 page 44 is the following To the first part, I reasoned as follows: Consider the function $F(q_1,q_3) = ...