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Questions tagged [solution-verification]

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Prove/show that $\epsilon_{ijk}(\hat a_j \hat b_k-\hat b_k \hat a_j)=2\epsilon_{ijk}(\hat a_j\hat b_k)$

I am simply trying to understand where the factor of $2$ comes from in the RHS of $$\epsilon_{ijk} \left[\hat a_j, \hat b_k \right]=\epsilon_{ijk}\left(\hat a_j \hat b_k-\hat b_k \hat a_j\right)=\...
FutureCop's user avatar
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Brezis' exercise 6.2.1: If $E$ is reflexive, then $T(B_E)$ is closed in norm topology

I'm trying to solve an exercise in Brezis' Functional Analysis Let $E$ and $F$ be Banach spaces and $T:E \to F$ a bounded linear operator. Let $B_E$ be the closed unit ball of $E$. Then $T(B_E)$ is ...
Akira's user avatar
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1 vote
0 answers
26 views

Brezis' exercise 6.1: a compact operator in $\ell^p$

I'm trying to solve an exercise in Brezis' Functional Analysis Let $E:= \ell^p$ with $p \in [1, \infty]$. Let $(\lambda_n)$ be a bounded sequence in $\mathbb R$. We consider a bounded linear operator ...
Akira's user avatar
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1 answer
45 views

Let $\alpha \in \mathbb{C}$ and $\alpha\neq 0$, then there exists two unique complex number whose square is $\alpha$.

Let $\alpha \in \mathbb{C}$ and $\alpha\neq 0$, then there exists two unique complex number whose square is $\alpha$. Let $\alpha=re^{i\theta}$. Observe that $e^{i\theta}=\cos (\theta)+i\sin(\theta)=\...
Remu's user avatar
  • 745
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0 answers
25 views

Rudin's PMA, Chapter 1 Appendix Proof of Theorem 1.19 Step 6 - Proof of (M4)

Rudin skipped the proof of Step 6. I want to know if my proof of (M4) is correct. Thanks in advance! The (M4) in the axioms for multiplication states that A field $F$ contains an element $1 \neq 0$ ...
κεωινυανγ's user avatar
1 vote
2 answers
62 views

But can't the $\pi$th root of $1$ also be $\frac1{e^i}$?

So I was bored, and decided to do some math for fun. I came up with $$\text{Find all solutions to }\sqrt[\pi]{-1}\text{ other than just }e^i$$which I thought that I might be able to do. Here is my ...
CrSb0001's user avatar
  • 1,307
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0 answers
10 views

Proving the Equivalence of Function Continuity and Sequential Compactness of Graph in Metric Spaces

Let $(X, d_X)$, $(Y, d_Y)$ be metric spaces and $f: X \to Y$ be a function. The metric $d_{X \times Y}((x_1, y_1), (x_2, y_2)) = d_X(x_1, x_2) + d_Y(y_1, y_2)$ defines the Cartesian product $X \times ...
Herrpeter's user avatar
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0 answers
22 views

Showing that $\varinjlim (L_i \otimes M_i) \cong \varinjlim L_i \otimes \varinjlim M_i$

I am currently self-studying category theory and I was trying to solve the following problem Given a filtered set $I$ and a ring $A$, if $(L_i, f_{ji})$ and $(M_i, g_{ij})$ are inductive systems of ...
Julia's user avatar
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1 vote
0 answers
27 views

Proof: Let $x$, $y \in Q$, $y > 0$, $x > 1$. Then there is an integer $n$ such that $x^n < y ≤ x^{n+1}$.

I want to know if my proof of the following result is correct: Let $x$, $y \in Q$, $y > 0$, $x > 1$. Then there is an integer $n$ such that $x^n < y ≤ x^{n+1}$. Proof: Suppose I have proved ...
κεωινυανγ's user avatar
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Need help with using fitch system for this proof: Given ¬q, (¬p⇒(¬q⇒¬r)), (s∨r), (s⇒t), and (p⇒t), prove t. [duplicate]

I am having difficulty using the Fitch proof system for the following proof: Given ¬q, (¬p⇒(¬q⇒¬r)), (s∨r), (s⇒t), and (p⇒t), prove t. I'm able to use the following rules: Reiteration, Negation ...
JCKing87's user avatar
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0 answers
29 views

Showing $I_1$ is a metric, checking one property

I want to verify if my proof regarding the $d(x,y) = 0 \iff x=y$ property of a metric space is correct. I have seen some other approaches to this here, but I want to check if my use of the ...
Richard K Yu's user avatar
-2 votes
0 answers
27 views

How do we prove that D = [0, 1] × [0, 1] is closed [closed]

This is my homework problem and I am struggling hard for it. Please help me!
Phan Minh Nghĩa's user avatar
-2 votes
0 answers
9 views

If my domain is convex polygon and what values can be for parameter a and b for my characteristic element size h in Hilbert space H^1

If I am Assuming Ω is a convex polygonal domain and u ∈ H^m(Ω) be the solution to (∇u, ∇φ) = (f, φ) on V = H^1(Ω) enter image description here I want to approximate this using parametric H^1-...
Buxior Engineer's user avatar
1 vote
1 answer
29 views

If $f:X→X$ is a bijection then is $g(x):=\begin{cases}f(x),\,\text{if }x∈Y\\x,\,\text{otherwise}\end{cases}$ with $Y∈\mathcal P(X)$ a bijection too?

Let be $f$ a bijection from a set $X$ to $X$ so that for any $Y$ in $\mathcal P(X)$ let's we put $$ g(x):=\begin{cases} f(x),&\text{if }x\in Y\\ x,&\text{otherwise} ...
Antonio Maria Di Mauro's user avatar
0 votes
2 answers
40 views

limit of $\int_{[0,n]\times \mathbb[0,n]}e^{-(x^3+y^2)}dxdy$, as $n\to\infty$

I have to compute the limit as $n\to\infty$ of the following: $$\int_{[0,n]\times \mathbb[0,n]}e^{-(x^3+y^2)}dxdy$$ To study the integrability I have though to say: $$\int_{[0,n]\times \mathbb[0,n]}e^{...
jin's user avatar
  • 169
1 vote
0 answers
63 views

Exact value of $\sin(10°)$ : is my method a good one?

So some time ago, I asked this question on finding the exact value of the sine of ten degrees. While I did get the wrong answer, I was wondering if I would be able to get the correct value of the sine ...
CrSb0001's user avatar
  • 1,307
0 votes
2 answers
101 views

Prove/disprove: Let $G$ a simple infinite group $\implies$ $G$ doesn't have a proper subgroup of finite index.

I'm writing this question for proof verification and also since I didn't find the already existing answers very clear: If $G$ is an infinite simple group then any proper subgroup of $G$ has infinite ...
AsiMathStudent's user avatar
0 votes
1 answer
57 views

Maximize product of two sines given sum of angles

Say we have an angle $c$. Prove that for two angles $a$ and $b$ such that $a + b = c$,$$\sin{a}\sin{b}$$ is maximized when $$a = b = \frac{c}{2}$$
codexistent's user avatar
0 votes
0 answers
35 views

Every vector space $X\ne 0$ has a Hamel basis. Look for upper bound for the chain.

Every vector space $X\ne 0$ has a Hamel basis. Proof. Let $M$ be the set of all linearly independent subsets of $X$. Since $X\ne 0$, it has an element $x\ne 0$ and $\{x\}\in M$, so that $M\ne \...
NatMath's user avatar
  • 572
0 votes
1 answer
46 views

Math for fun: Determining the angle between matrices $A$ and $B$ with three columns and rows

So I was bored, as there were no math problems on Youtube that I thought that I might be able to solve, let alone any math videos, so I decided to start to do some determining the angle between two ...
CrSb0001's user avatar
  • 1,307
3 votes
1 answer
65 views

Help me verify whether my solution of this geometry problem is correct

Consider the following problem. The Problem Point $B$ lies on the segment $AC$. A tangent line is constructed from point $A$ to the circle of diameter $BC$, intersecting it at point $M$. $K$ is the ...
Rusurano's user avatar
  • 578
-1 votes
1 answer
69 views

Prove $ST = 0 \implies ST^* = 0$.

EDIT - checking if it is better now So I have some proof and I'm not sure on one of the transitions. In this same assignment we had to prove the following: Let V a Hermitian vector space and $T:V \to ...
AsiMathStudent's user avatar
1 vote
0 answers
40 views

Theorem 6 Corollary 2, Section 5 of Hungerford’s Abstract Algebra

If $f:G\to H$ is a homomorphism of groups, $N\lhd G$, $M\lhd H$, and $f(N)\lt M$, then $f$ induces a homomorphism $\bar{f}: G/N\to H/M$, given by $aN\mapsto f(a)M$. $\bar{f}$ is an isomorphism if and ...
user264745's user avatar
  • 3,414
0 votes
0 answers
40 views

Theorem on convergence of series

Theorem. If $\sum_{n=1}^{\infty}a_n$ converges absolutely and $(b_n)$ is a bounded sequence, then $\sum_{n=1}^{\infty}a_nb_n$ converges absolutely. Proof. Suppose that $\sum_{n=1}^{\infty}a_n$ ...
RataMágica's user avatar
-1 votes
0 answers
24 views

Prove that the introduction rule of implication can be derived from the rules of conjunction and negation.

The introduction rule of implication states that if, by assuming $A$, we can derive $B$, then we can derive $A\rightarrow B$. $Proof$. Suppose $A$ proves $B$. Now suppose $A\land\neg B$. By ...
lightyourassonfire's user avatar
1 vote
0 answers
20 views

Existence of a stochastically dominating random variable

Is anything wrong in the following reasoning? Suppose that $X_1$ is a real-valued random variable such that $E[|X_1|^t] \leq \gamma$ for some $\gamma \in (0, \infty)$ and some $t > 0$. From Markov'...
Aurelien's user avatar
2 votes
0 answers
55 views

Real analysis problem (solved one direction, struggling with the other one).

I'm trying to solve the following problem: Given a non-constant sequence $a_n$ such that for all $n\in \mathbb{N}$, $\left|a_{n+1}-a_n\right|<q^n$ ($q>0$), prove that $a_n$ necessarily ...
Blabla's user avatar
  • 169
0 votes
1 answer
35 views

Proving two formulations of the Axiom of Choice are equivalent (existence of choice function vs selecting from pairwise disjoint sets)

I've been trying to prove the following two formulations of the Axiom of Choice are equivalent: Formulation 1: Given a non-empty set $A$ of non-empty sets, there is a function $f$ that maps each $x \...
EyeballWitch's user avatar
1 vote
0 answers
24 views

Determine the almost sure limit of the sequence $(m_{n})_{n \in \mathbb{N}}$ for each $p \in \mathbb{N}$

Consider $X_{n} \sim \operatorname{Exp}(\lambda_{n})$ for each $n \in \mathbb{N}$, where $(\lambda_{n})_{n \in \mathbb{N}} \subset \mathbb{R}$ with $\lambda_{n} > \varepsilon$ for all $n \in \...
clementine1001's user avatar
4 votes
1 answer
107 views

Probability of seeing HTTH before THTH in coin flips

I'm doing a past paper question and trying to use Doob's Optional Stopping to find the probability that for independent identical $(X_n)$ uniform on $\{0,1\}$ we see the pattern $a = (1,0,0,1)$ before ...
George's user avatar
  • 399
1 vote
0 answers
74 views

Proving $\prod_{k = 2}^{\infty} \frac{k^3 - 1}{k^3 + 1}=\frac{2}{3}$ more rigorously

Problem: What is $\prod_{k = 2}^{\infty} \frac{k^3 - 1}{k^3 + 1}$? My Approach: We can rewrite as $$\prod_{k = 2}^{\infty} \frac{k^3 - 1}{k^3 + 1} = \left(\prod_{k = 2}^{\infty} \frac{k - 1}{k + 1}\...
subarashi's user avatar
2 votes
1 answer
79 views

Law of Large Numbers for a Brownian Motion

I am self-learning introductory stochastic calculus from A first course in Stochastic Calculus by L.P.Arguin. The part(c) of the below exercise problem on the time-inversion property of Brownian ...
Quasar's user avatar
  • 5,184
0 votes
0 answers
67 views

Let $A$ be some interval and $f,g: A \to \mathbb R$. Assume $f,g$ are uniformly continuous and bounded. Prove/disprove $fg$ is uniformly continuous.

So I have sketched some proof and would love some feedback: We'll divide the proof into two parts: First assume $A = (a,b)$ for some $a,b \in \mathbf R$. $f,g$ are uniformly continuous in $A$ $\...
AsiMathStudent's user avatar
1 vote
0 answers
16 views

endomorphism and square root self-adjoint operator

Let $F=\mathbb{R}$ or $F=\mathbb{C}$, let $(V,\langle.,.\rangle)$ be a finitely generated $K$-vector space with scalar product, let $f \in \operatorname{End}_{F}(V)$. Show that: If $f$ is self-adjoint ...
Marius Lutter's user avatar
0 votes
1 answer
34 views

Time and Work Question - Replacement of people over the course of work

It takes 10 men and 20 women to complete a project in 48 days. After the first day, a woman was replaced by a man and it took them 45 more days to complete the work. If 10 men and 20 women had worked ...
Fin27's user avatar
  • 925
1 vote
1 answer
58 views

Need to find a value of $\alpha$ for which some integral is negative

To motivate the problem: I am learning QM and want to prove that every attractive potential admits a bound energy eigenstate. Using $\psi_{\alpha} (x) = \left( \dfrac{\alpha}{\pi}\right)^{1/4} e^{-\...
weirdmath's user avatar
0 votes
0 answers
33 views

If $\Delta f(x) = f(x+1)-f(x)$ and $\Delta_n f(x) = \Delta \left( \Delta_{n-1} f(x) \right)$. Prove $\Delta_{n+1} P(x) = 0 \quad \forall x \in R$

I am asked the following question: Question: Given that $\Delta f(x) = f(x+1)-f(x)$ and $\Delta_n f(x) = \Delta \left( \Delta_{n-1} f(x) \right)$, prove that for any nth degree polynomial $P(x)$ $$\...
bru1987's user avatar
  • 2,405
2 votes
1 answer
88 views

Find the sum value: $\sum^{50}_{k=1} \frac{k}{1+k^2+k^4}$

Find the sum value: $$\sum^{50}_{k=1} \frac{k}{1+k^2+k^4}$$ Attempt: = $$\sum^{50}_{k=1}\frac{k}{1+k^2+k^4+k^2-k^2}= \sum^{50}_{k=1} \frac{k}{(k^2+k+1)(k^2-k+1)} $$ Expanding into partial fractions, ...
Assandra Lakal's user avatar
0 votes
0 answers
31 views

Showing that exterior measure and exterior measure defined using rectangles are equivalent

I was working through an exercise in Stein's 3rd Princeton Lectures in Analysis book and was looking for some clarification regarding my approach and its validity. The problem is precisely showing ...
kodiak's user avatar
  • 201
1 vote
1 answer
31 views

How to compute the derivative of the maximum of multiple linear functions?

Define $$ s(x) = \max(xa + b, \max(xa + c,d) + e, f) $$ where $a,b,c,d,e,f > 0$ real numbers and $x > 0$. I am trying to differentiate $s$ w.r.t. $x$. Is the following formula true? $$ s'(x) = a\...
Vicky's user avatar
  • 857
1 vote
1 answer
77 views

Proving $\mathbb{R}$ is a subfield of $\mathbb{C}$: Is preservation of operations necessary?

I'm reading Rudin's book where he defines the complex field, initially as ordered pairs $(a,b)$. I'm trying to prove the assertion that $\mathbb{R}$ is a subfield of $\mathbb{C}$. I unfortunately ...
Mathematical Endeavors's user avatar
2 votes
1 answer
250 views

Proof attempt for a weaker form of the Collatz Conjecture

I am kind of new to this problem and I tried solving it with open mind. Please don't be judgmental, this is what I got. Let us assume, for the sake of contradiction, that the Collatz conjecture is ...
Yoav Alhindi's user avatar
0 votes
1 answer
40 views

Find the determinants of A by row reduction to echelon form.

When I try to solve this question, it keeps ending with reduced echelon form, not to echelon form, where all diagonals are zero. And, I get detA = 10. Is this correct? Thank you.
Alex's user avatar
  • 19
2 votes
3 answers
101 views

If $\mathbb{V}$ is a vector space, is it always true that $\mathbb{V} = \text{span}(\mathbb{V})$?

This question have just came to my mind after starting to study some linear algebra. I tried to write the following proof: Let $\mathbb{V}$ be a vector space. ($\subseteq$) Let $v \in \mathbb{V}$. We ...
Sir Newbie's user avatar
2 votes
0 answers
66 views

Incorrect proof: $\frac32-\frac{\pi^2}6=\sum_{k=1}^\infty\frac{(k+1)!B_{2k}}{(2k)!}$

I incorrectly proved that $$\frac32-\sum_{k=1}^\infty\frac{(k+1)!B_{2k}}{(2k)!}=\frac{\pi^2}6$$According to Wolfram Alpha, the LHS is approximately $1.34096$, while $\frac{\pi^2}6\approx1.64493$. Here ...
Kamal Saleh's user avatar
  • 3,852
5 votes
0 answers
47 views

Do unit quaternions admit a continuous square root function?

Let $G$ be a multiplicative group. A function $\text{sqrt}: G \to G$ is a square root function iff, for every $x \in G$, $(\text{sqrt}(x))^2 = x$. The unit real numbers, namely $\pm 1$, doesn't even ...
Dannyu NDos's user avatar
  • 1,619
0 votes
1 answer
24 views

$N$ coins placed randomly on vertices of $M$-sided regular polygon. Find expected value and var. of the number of sides with coins on both vertices.

There are $N$ coins, $N \geq 1$. We place them randomly on vertices of $M$-sided regular polygon, $M \geq N$, in every vertice there is at most $1$ coin. We call X the number of sides with coins on ...
mathman12's user avatar
  • 575
1 vote
2 answers
65 views

If $\lvert a-b \rvert <\epsilon$, with $a$ and $b$ constants and $\epsilon >0$, then prove that $a=b$

I already know the part when if $a\neq b$, then we can say $\epsilon=a-b$ which leads to contradiction. So I decided to try it considering the cases: If $\lvert a-b \rvert <\epsilon$, with $a$ and $...
Roma_Rayado's user avatar
7 votes
1 answer
172 views

Math for fun: Finding the exact value of $\sin(10\unicode{xB0})$

$\def\a#1{#1\unicode{xB0}}$ So I was looking through the homepage of Youtube to see if there were any math problems that I thought that I might be able to solve when I came across this video by ...
CrSb0001's user avatar
  • 1,307
0 votes
0 answers
34 views

Prove: Every affine transformation is two stretches and an isometry

Prove, using synthetic geometry, that every affine transformation of the plane is an isometry and two stretches. Note: This is a direct consequence of the polar decomposition from linear algebra. ...
SRobertJames's user avatar
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