All Questions
Tagged with proof-verification or solution-verification
45,395 questions
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How should I show that $x_{0}(\tau)$ and $x_{1}(\tau)$ satisfy the given equations?
Homework problem: Consider the differential equation (which is a modified van der Pol equation) $\ddot{x}+\epsilon(x^4-1)\dot{x}+x+\epsilon\alpha^2 x^3=0, x(0)=a>0, \dot{x}(0)=0, 0\leq\epsilon<&...
- 167
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0
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23
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Need an elegant solution for a geometric probability problem
Consider a square $S \subseteq \mathbb{R^2}$ in the first quadrant with unit side length such that one of its vertices coincides with the origin and the sides of $S$ are parallel to the coordinate ...
- 385
1
vote
1
answer
39
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Proof by contradiction validity. [duplicate]
Is the following proof by contradiction valid?
Theorem: Let $a$ and $b$ be real numbers. If $ab=0,$ then $a=0$ or $b=0$ (or both).
Proof: We will prove this by contradiction. Assume that $ab=0$ and $a\...
1
vote
1
answer
37
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Is this simple proof of Euler's Formula correct, given my definitions of e^z, sin(z), and cos(z)?
I was wondering if the provided proof of Euler's Formula is valid, or if it is a circular argument.
First, my definition of the exponential function is as follows:
$$e^z = \lim_{n\to\infty} \left(1+\...
- 41
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0
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42
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If $a,b\in\mathbb{Z}^+$, then $a^3\mid b^2$ implies $a\mid b$ [duplicate]
After writing this solution, I wanted to see if I could locate a post which might corroborate my attempt. The closest (in approach) I could find was Kell Berliner's solution (2nd soln from the top ...
- 319
1
vote
1
answer
22
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Abstract algebra question primary module
Suppose that $M$ is an $R$-module.
I will from now on simply use $M$.
Suppose that $N$ is an $R$-submodule such that $N \subseteq M$.
Define (N:M) as the set $\{r \in R | r \cdot M \subseteq N \}$.
...
- 163
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48
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Where is the mistake in this application of Ramanujan’s Master Theorem?
Given $a<0$, $x>0$, we have:
$$e^{ae^x}=\sum_{n=0}^\infty\frac{ a^ne^{nx}}{n!}$$
$$=\sum_{n=0}^\infty\frac{ a^n}{n!}\sum_{m=0}^\infty \frac{ n^mx^m}{m!}$$
$$=e^a\sum_{m=0}^\infty e^{-a}\sum_{n=0}...
- 390
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0
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32
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Independence of $p$-adic valuation and absolute value $v_\infty$
There is a theorem in a book I am reading:
(Indepedence of Valuations) Let $p_1,\cdots,p_n$ be prime numbers and $a,a_1,\cdots,a_n\in\mathbb Q$. For any $\varepsilon>0$ and positive integers $l_1,\...
- 185
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56
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Alternative proof for absolute convergence implying convergence
After some time away from maths, I'm revisiting real analysis. At the current moment, I'm going through sequences and series, and reach the well known result about absolute convergence of a series ...
- 2,320
6
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1
answer
145
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Can this locus be an ellipse?
Let $A=(-1,0)$ and $B=(2,0)$ be two points on the $x-$ axis. A point $M$ is moving in the xy-plane in such a way that $\angle MBA=2\angle MAB.$ Find the locus of $M$.
My Attempt:
$\tan2\theta=\frac{2\...
- 8,783
-3
votes
1
answer
55
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Need a detailed proof of odd and even functions [closed]
is f(-x)=-f(x) the same as -f(-x)=f(x)
Just curious if it is or isn't, thanks! I want to know the proof for it if possible, basically, if it is why, and then if it isn't why not?
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1
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30
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Prove that the homomorphism induced by a retraction is an epimorphism
If $r:X\rightarrow A$ is a retraction and $a_0\in A$, then $r_*:\Pi_1(X,a_0)\rightarrow\Pi_1(A,a_0)$ is an epimorphism.
Proof: By definition, $r_*$ is a homomorphism. We now show that $r_*$ is ...
-3
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2
answers
61
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Solution of infinitely nested radical [closed]
I was experimenting $√√√√...(∞$ times$)$ =?
It turns out as $√x=x⇒x²-x=0⇒x(x-1)=0=>x=0,1$ where, x=0 is rejected because the expression given is > 0 (Is this incorrect reaßon? If not, is there a ...
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Work verification: absolute extrema of the function $z = z(x, y)$ implicitly defined by the equation $z^3 + z(y+1) - x = 0$ [closed]
I'd like to have my work on the following problem verified:
Prove that the equation $z^3 + z(y+1) - x = 0$ uniquely defines an implicit function $z = f(x, y)$ for $y > -1.$ Find the absolute ...
- 363
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1
answer
32
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Lebesgue integrability wrt to a measure space and its completion
Let $(X, \Sigma, \mu)$ be a measure space and $(X, \overline{\Sigma}, \overline{\mu})$ be the corresponding complete measure space. Prove that $f \in L^1(X, \overline{\mu})$ if and only if there ...
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45
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Prove that the integer sequence $a_1,a_2,...,a_p$ is an AP iff there exists a partion of the set of natural numbers into $p$ disjoint sets
'Let $p$ be an odd prime number and consider the following sequence
of integers: $a_1, a_2,\cdots, a_{p−1}, a_p$. Prove that this sequence is an arithmetic progression
if and only if there exists a ...
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0
answers
36
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General Relativity, Wald, exercise 4b chapter 2
Suppose we have n vector fields $ Y_{\left(1\right)},\ldots,Y_{\left(n\right)} $ such that at every point of the manifold they form a basis for the tangent space at that point .
I have to prove that:
$...
- 79
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1
answer
38
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Unsure about soundness of proof about transitivity.
sorry in advance for any imprecision, missing tags etc but this is my first question.
I'm reading Daniel J. Velleman 'How To Prove It' and i'm having a hard time understanding how to write a sound ...
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0
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31
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Checking the rigor of a trigonometry proving question [closed]
Attached are the questions and the suggested solution for a trigonometry question. Upon considering the solution to part (iii), I find this use of the given identity intuitive and admit to having ...
1
vote
0
answers
41
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Finding $x$ at intersection of $a \equiv bx\mod M$ and $c \equiv bx\mod N$ [duplicate]
I have two equations $a \equiv bx\mod M$ and $c \equiv bx\mod N$ in which I know $a, b, c, M$, and $N$. I am trying to find $MN > x \geq 0$ ($x$ is non-negative). Further, $M$ and $N$ are odd and ...
- 685
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1
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88
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+50
Continuity of an Integral map with Lebesgue measure, maximum and minimum
Can anyone check what I've done in solving this problem and help me with the remaining tasks? Thank you
Given $f:\mathbb{R}^n\to (0,\infty)$ that $f\in L^2(\mathbb{R}^n,\mathcal{L}^n)$ show the map ...
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0
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51
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Verification of proof: Probability of winning equals probability of losing in a biased coin game
Problem:
We're covering probability in lecture one day, and you volunteer for a demonstration. The professor shows you a coin and says he'll bet you one dollar that the coin will come up heads. ...
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1
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1
answer
178
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What are some proofs of the Pythagorean theorem that use imaginary numbers?
What are some proofs of the Pythagorean theorem that use imaginary numbers?
There seems to be a lack of such proofs found on the internet, hence my question.
I will post my attempt, as an answer ...
- 30k
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0
answers
35
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Algebraically determining the domain of $\frac{x-a}{x+a}\geq0$.
I am studying Introduction to Calculus and Analysis I by Richard Courant, SECTION 1.1e, page 12, Question 2. (b) (iv).
The question is as follows: Determine the interval in which $\frac{x-a}{x+a}\geq0$...
- 167
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0
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51
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Why does implicit differentiation fail when using it to solve an equation?
Suppose that I have two real-valued functions
$f(x):=\cos(\frac{\pi}{2}x)$ and $g(x):=x^2 - 1$
and I want to find their intersection points when both are graphed on the Cartesian plane.
For each ...
- 83
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0
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18
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Proof that all possible connected subsets of the number axis are exhausted by these nine intervals.
I am studying Introduction to Calculus and Analysis 1 by Richard Courant, SECTION 1.1e, page 12, Question 2 *(a).
The Question is as follows:
An interval may be defined as any connected part of the ...
- 167
3
votes
1
answer
66
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How should I show that $A(\eta)$ satisfies the equation $2\frac{dA}{d\eta}+2A^3\bar{A}^2-A=0$?
Homework problem: Consider the modified van der Pol equation $\ddot{x}+\epsilon(x^4-1)\dot{x}+x=0, 0<\epsilon<<1$, with initial conditions $x(0)=a$ and $\dot{x}(0)=0$, where $a$ is a constant....
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88
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A closed set with empty interior
How do I prove that the set
$L_n = \{f \in C([0, 1]), \exists \, x_0 \text{ such that } \forall \, x \in [0, 1], |f(x) - f(x_0)| \leq n|x - x_0|\}$
is closed with empty interior? Furthermore, how do ...
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39
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Jordan forms of this matrix
Suppose $n \geq 3$. Let $M \in \mathbb{C}^{n \times n}$ be such that $M^3 = I_n$. What are the possible Jordan canonical forms of $M$?
This is my solution. From $M^3 = I_n$ I can deduce the minimal ...
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0
answers
73
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Showing that $\mathbb{Q} [x,y] / (xy)$ is not a free $\mathbb{Q} [x,y]$-module [duplicate]
I'm trying to show that $\mathbb{Q} [x,y] / (xy)$ is not a free $\mathbb{Q} [x,y]$-module. Does the following seem like a complete solution? Just checking that my understanding of things is correct.
...
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0
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25
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If $u, v \in \text{End}(V)$ are unitary, then $|\det(u + v)| \leq 2^n$
Let $V$ be a finite dimensional vector space over the complex numbers. If $u, v \in \text{End}(V)$ are unitary, then $|\det(u + v)| \leq 2^n$.
I'm not sure if my solution to this problem is correct. ...
- 345
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0
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48
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Concerning union and intersection of some subsets of $\mathbb{R}$
Question: Is the following claim true or false?
$$\bigcup_{n=1}^\infty\left(1+\frac1n,2-\frac1n\right)\subseteq\bigcap_{n=1}^\infty\left(1-\frac1n,2+\frac1n\right)$$
Is the union on the left equal to ...
- 6,937
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45
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Proving Extreme points in a convex set are not convex combinations.
If S is convex set. Prove that a point $u$ in S is an extreme point of S if and only if $u$ is not a convex combination of other points of S.
I need to prove that a point $u$ is an extreme point iff ...
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112
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If $1_V + v$, $1_V + v^2$ and $1_V + v^3$ are unitary, then $v=0$.
Let $V$ be a finite dimensional vector space over the complex numbers
and $v \in \text{End}(V)$. If $1_V + v$, $1_V + v^2$ and $1_V + v^3$ are unitary, then $v=0$.
This is my solution, but I'm not ...
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Terence Tao 3.4.1 proof checking [closed]
{Exercise 3.4.1}
Let $f : X \to Y$ be any function, and let $V$ be any subset of $Y$. Prove that the forward image of $V$ under $f$, thus the fact that both sets are denoted by $f^{-1}(V)$, will not ...
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2
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61
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My solution for $\lvert x-a\rvert<x-b$ disagrees with graphical evidence.
I am studying Introduction to Calculus and Analysis by Richard Courant, Section 1.1e, page 12, Question 1. (b).
The question: "Using signs of inequality alone (not using signs of absolute value) ...
- 167
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0
answers
29
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Limit of Normalized Integral Involving $x f(x)$ for $f \in L^1(0, \infty)$
Let $f \in L^1(0, \infty)$, meaning $f$ is an integrable function on $(0, \infty)$. I am trying to prove that:
$$
\lim_{n \to \infty} \frac{1}{n} \int_0^n x f(x) \, dx = 0.
$$
What I used:
For any $\...
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Equality relates to directional derivative in Differential geometry
I'm currently at 3rd chapter of Lee's Intro. to Smooth Manifold and I spent quite a few hours on verifying the following equality on page 52
If $v_a\in \mathbb{R}^n_a$ for some $a\in \mathbb{R}^n$ ...
- 1,117
2
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2
answers
103
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Find all integral domains $R$ such that $x^{4} = x$ for all $x \in R$.
At the moment, I am not even studying ring theory but I was thinking of the following question.
Question: Find all integral domains $R$ such that $x^{4} = x$ for all $x \in R$.
My initial thought was ...
- 512
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1
answer
56
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+50
Concerned about the proof of the Model Existence Lemma not using the $(\exists \text{I})$ rule anywhere
TLDR: where in the below proof of the Model Existence Lemma do I need to use the $\exists$ introduction $(\exists \text{I})$ rule?
Trying to fully write the proof of the Model Existence Lemma using ...
- 5,339
3
votes
1
answer
81
views
Operators on a Hilbert space with equal spectrum
Suppose $T_0$ and $T_1$ are two bounded operators on a Hilbert space $H$ $(T_0,T_1\in B(H))$ with $\sigma(T_0) =\sigma(T_1)$.
I want to show that there exists a bijective $*$-homomorphism between $C^*(...
user1446697
2
votes
0
answers
62
views
equivalent definition of $\liminf$
I have to prove the equivalence between these $2$ definitions:
$\liminf_{x\rightarrow x_0}f(x)=\liminf_{n\rightarrow \infty} f(x_n) \forall x_n\rightarrow x_0 $
( where $\liminf_{x\rightarrow x_0}f(x)...
- 21
1
vote
1
answer
85
views
using dominated convergence theorem for this series
$$
\mbox{I have to find}\quad
\lim_{m\ \to\ \infty}\sum_{n\ \neq\ m}\frac{m^{2}}{n^{2}\left(m^{2} - n^{2}\right)}
$$
I thought I could use dominated convergence theorem using that
$$
\frac{m^{2}}{n^{2}...
- 541
2
votes
0
answers
15
views
Defining function for domain with smooth boundary
The following is a part of exercise 3 from Chapter 1 of Steven G. Krantz' book "Geometric Analysis of the Bergmann Kernel":
Let $\Omega\subset \mathbb{R}^N$ be a domain. Suppose that $\...
- 523
1
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0
answers
51
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Proving/disproving $ \int |f|^q \, d\mu= \sum_n a_n^q \mu(E_n). $
Let $(X, \Sigma, \lambda)$ be a measure space. We consider the measurable function $f = \sum_{n=1}^\infty a_n \mathbf{1}_{E_n}$, where $a_n \geq 0$ and $\{E_n\}_{n=1}^{\infty}$ is pairwise disjoint ...
user1490909
0
votes
1
answer
56
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need to prove that $DSPACE(O(2^n)) \neq EXP$
this question is from my computational complexity HW.
I'm not sure if my solution is correct:
If $DSPACE(O(2^n)) = EXP$, than we can take language $ L \in DTIME(2^{2^n})$ which not in $EXP$ (from the ...
2
votes
2
answers
109
views
Prove that tensor product is associative
I wished to prove that tensor product is associative in the sense that
Given three finite-dimensional real vector spaces $X,Y,Z$, there exists a unique linear isomorphism $F:X\otimes Y\otimes Z\...
- 511
2
votes
4
answers
113
views
Find the eigenvalues of linearly dependent $A$ without using characteristic polynomial. And compute $A^{2024}$
$$
A = \begin{bmatrix}25 & 11 & -16 \\ 50 & 22 & -32 \\ 75 & 33 & -48 \end{bmatrix}
$$
a) Find the eigenvalues of $A$ without calculating $\det(A - \lambda I) = 0$
b) Find the $...
- 33
1
vote
0
answers
45
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After proving that there exists at least one irrational number between two rational numbers, show that there are infinitely many.
Is my solution sufficient to solve the problem?
I am attempting to provide a solution for SECTION 1.1a, page 2 Q1.(b) of Introduction to Calculus and Analysis I by Richard Courant. The question: &...
- 167
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votes
1
answer
36
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Proof verification for translation is continuous in $L^1$
We want to show that the translation with respect to $t$ is continuous for $L^1$ functions. Specifically, it suffices to show that $$ \lim_{t \to 0} \int_{\mathbb{R}} |f(x + t) - f(x)| \, \mathrm d m =...
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