Proof Definition and 999 Threads

I am reading J. J. Duistermaat and J. A. C. Kolk: Multidimensional Analysis Vol.II Chapter 6: Integration ... I need help with the proof of Theorem 6.2.8 Part (iii) ...The Definition of Riemann integrable functions with compact support and Theorem 6.2.8 and a brief indication of its proof...

I have went about this problem many different ways but cannot seem to come up with the answer. I am essentially trying to prove the formula provided in the ss of the problem.Could someone help me and tell me if I am approaching this wrong?

I recently trying to learn General Relativity by first scraping the surface on ScienceClic's general relativity playlist, and then I stumbled upon a video where it said that we actually move through spacetime on a constant speed of c, and then I remember about time dilation because how speed on...

I am reading Multidimensional Real Analysis II (Integration) by J.J. Duistermaat and J.A.C. Kolk ... and am focused on Chapter 6: Integration ...I need some help with the proof of Proposition 6.1.2 ... Proposition 6.1.2 reads as follows: Definitions and text preliminary to the Proposition...

Poking around on the internet has not helped me. Penrose references Hawking and his 1996 book and I have ordered that, but I suspect my progress through that book will be slow. I have read that the assumptions include an energy condition, which I assume is expressed as a restriction on the...

Proof: Suppose gcd(a, b)=d. Then we have d##\mid##a and d##\mid##b for some a, b##\in## ##\mathbb{Z}##. This means a=md and b=nd for some m, n##\in## ##\mathbb{Z}##. Now we have lcm(a, b)=##\frac{ab}{gcd(a, b)}##...

Proof: First, we will show that gcd(a, 0)=abs(a). Suppose a is a nonzero integer such that a##\neq##0. Note that gcd(a, 0)##\le##abs(a) by definition of the greatest common divisor. Since abs(a) divides both a and 0, we have that...

Proof: Suppose for the sake of contradiction that gcd(a, b) \neq 1. Then there exists a prime number k that divides both a+b and ab. Note that k divides either a or b. Since k divides a+b, it follows that k divides b. Thus, this is a...

My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof? Beneath is my proof graded 2/5.

My tests are submitted and marked anonymously. I got a 2/5 on the following, but the grader wrote no feedback besides that more detail was required. What details could I have added? How could I perfect my proof? Beneath is my proof graded 2/5.

How can we be sure that a system on the scale of atoms can be described by a single scalar field or the wave function ##\psi##. I don't just want to do shut up and calculate, maybe using a wave function and then putting it through the time evolution of the Schrödinger equation works, but why...

Attempt of a solution. By the Rank–nullity theorem, $$ \dim V=\dim Im_{F}+\dim\ker\left(F\right) \Rightarrow n=1+\dim\ker\left(F\right) \Rightarrow \dim\ker\left(F\right)=n-1. $$ Similarly, it follows that $$\dim\ker\left(G\right)=n-1.$$ This first part, for obvious reasons, is very clear. The...

In my approach, i made use of change of base; i.e $$x-y=\frac {log_b n}{log_b a} -\frac {log_b n}{log_b c}$$ $$x-y=\frac {log_b c ⋅log_b n - log_b n ⋅logba}{log_b a ⋅log_bc}$$ and $$x+y=\frac {log_b n}{log_b a} +\frac {log_b n}{log_b c}$$ $$x-y=\frac {log_b c ⋅log_b n + log_b n ⋅logba}{log_b a...

Hi, PF In a Spanish math forum I got this proof of a right hand limit: "For a generic ##\epsilon>0##, in case the inequality is met, we have the following: ##|x^{2/3}|<\epsilon\Rightarrow{|x|^{2/3}}\Rightarrow{|x|<\epsilon^{3/2}}##. Therein lies the condition. If ##x>0##, then ##|x|=x##...

So, this problem I sort of get conceptually but I don't know how I can possibly rewrite (idX)∗ : π1(X) → π1(X). Does this involve group theory? It's supposed to be simple but I honestly I don't see how. Again, any help is greatly appreciated. Thanks.

Hello! Lately, I've been struggling with this assignment. (angle brackets represent closed interval) I figured out that: a) union = R intersection = {0} b) union = (0, 2) intersection = {1} I asked my prof about this and she explained to me that it should be shown that if a set is an...

Let ##X## be a bounded subset of ##\mathbb{R}## with infinite cardinality. We consider a countably-infinite subset of ##X##. We write this set as a sequence to be denoted ##\{a_n\}_{n\in\mathbb{N}}##. Now define ##A## to be the set of points in the sequence with the property that for each...

I need proof how find average height of half circle? Lets say pressure distribution is half circle with Pmax = radius,I must find average/resultant pressure..

Hello, PF This is Theorem 8 of Chapter 4 of the ninth edition of Calculus, by Robert A. Adams: "Existence of extreme values on open intervals". I have an alternative and easier proof, based on epsilon-delta arguments, but it's not mine, and I don't understand it completely. The fact is that...

If a linear space ##V## is finite dimensional then ##S##, a subspace of ##V##, is also finite-dimensional and ##dim ~S \leq dim~V##. Proof: Let's assume that ##A = \{u_1, u_2, \cdots u_n\}## be a basis for ##V##. Well, then any element ##x## of ##V## can be represented as $$ x =...

This is my work.

Hi All, is there a proof to Arquimedes Principle and the expression for the buoyancy force? In case there is a proof, may we refer to this as Arquimedes theorem? (Buoyancy force = density of the fluid x acceleration of gravity x submerged volume) Best Regards, DaTario

Please view the picture of my work which I've uploaded.

From the paper https://people.csail.mit.edu/rivest/Rsapaper.pdf Can someone explain the green highlight to me please? Sorry that I can't type much because this is the final week. Thanks.

Given ## a,b,c,d,e,f \in \mathbb {R}, ad - bc \neq 0 ##, if ##(x_1,y_1)## and ##(x_2,y_2)## are pairs of real numbers satisfying: ## ax_1 + by_1 = e, cx_1 + dy_1 =f ## ## ax_2 + by_2 = e, cx_2 + dy_2 = f ## then ## (x_1,y_1) = (x_2,y_2). ## Here is my attempt at a proof, I have gotten stuck...

Part 1 ##\left\| \vec{y} \right\|^2 \leq \left\| \vec{y} \right\|^2## and since ##\lambda \in \left[ 0,1 \right] \Rightarrow \lambda^2 \leq \lambda## So ##\lambda^2 \left\| \vec{y} \right\|^2 \leq \lambda \left\| \vec{y} \right\|^2 ## Part 2 ##\left\| \vec{x} \right\|^2 \leq \left\| \vec{x}...

The above article gives lots of evidence to support the claim we are living in a simulation. I know this is usually considered hypothetical, but in the article they give physical explanations that fit topics discussed in this forum. Please read and give your opinion

Dear readers, Maybe someone can enlighten me on the understanding of the proof given by the Michelson–Morley experiment on the special relativity. Just as introduction to detail the setting: There are 2 coordinate systems A and B. A stands still and B moves with the velocity v along one of...

Could someone explain why ##[a][x_0]=[c]\iff ax_0\equiv c\, (mod\, m)##? My instructor said it came from the definition of congruence class. But I am not convinced.

Although a function cannot have extreme values anywhere other than at endpoints, critical points, and singular points, it need not have extreme values at such points. It is more difficult to draw the graph of a function whose domain has an endpoint at which the function fails to have an extreme...

https://slideplayer.com/slide/10708471/ This is the context I am talking about. What contradiction occur here? We begin by telling that there is a Turing machine H that solves the halting problem. So how does this contradicts? Can you tell me about that? What contradiction occur here? We...

https://slideplayer.com/slide/10708471/ This is the context I am talking about. What contradiction occur here? We begin by telling that there is a Turing machine H that solves the halting problem. So how does this contradicts? Can you tell me about that?

Please review my proof. Is this correct or not?

Hi, PF "It is more difficult to draw the graph of a function whose domain has an endpoint at which the function fails to have an extreme value", states my textbook, "Calculus: A Complete Course" A function with no max or min at an endpoint Let...

The question arises the way Goldstein proves Euler theorem (3rd Ed pg 150-156 ) which says: " In three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point"...

Hello everyone, I am preparing to write an exam project in college about special relativity, however i am missing the critical experiment to prove that it is true. I thought about using the life time of muons, but i don't have a scintillator to detect them, so unless a standard geiger counter...

Hi, I have this problem and its solution but i know what right size is, but i don't understand what left size (delta*w(z, z)) is equal to

Show sqrt(3), sqrt(5), sqrt(7), sqrt(24), and sqrt(31) are not rational numbers.

Please check/confirm my work of this proof and tell me if it's correct or not. Thank you.

The discriminant for a monic polynomial having only the free term { a0 } (and the monic degree term of course) is: Δn:0 = σ0 nn a0( n - 1 ) where n is the degree of the polynomial, and σk is a sign (i.e., +1 or -1) while the discriminant for a monic polynomial having only the x1 term (and...

Prove that if ##X## is a topological space, and ##S_i \subset X## is a finite collection of compact subspaces, then their union ##S_1 \cup \cdots \cup S_n## is also compact. ##S_i \subset X## is compact ##\therefore \forall S_i, \exists## a finite open cover ##\mathcal J_i=\{U_j\}_{j\in...

[This is a reference request.] I'm dissatisfied with the "proofs" I've found so far. E.g., in Kayser's review from 2008, in the paragraph following his eq(1.4), he assumes a propagation amplitude Prop##(\nu_i)## of ##\exp(-im_i \tau_i)##, where "##m_i## is the mass of the ##\nu_i## and...

Gluons are supposed to have precisely 0 rest mass. However, gluons are always colour confined into hadrons with binding energies of hundreds of MeV. How is gluons´ lack of rest mass proven? Presumably through some symmetries, or lack of some processes. Which kinds of asymmetries and processes...

We have Rayleigh's dissipation function, defined as ## \mathcal{F}=\frac{1}{2} \sum_{i}\left(k_{x} v_{i x}^{2}+k_{y} v_{i j}^{2}+k_{z} v_{i z}^{2}\right) ## Also we have transformation equations to generalized coordinates as ##\begin{aligned} \mathbf{r}_{1} &=\mathbf{r}_{1}\left(q_{1}, q_{2}...

Theorem: Let ## f(x), g(x) \in \mathbb{F}[ x] ## by polynomials, s.t. the degree of ## g(x) ## is at least ## 1 ##. Then: there are polynomials ## q(x), r(x) \in \mathbb{F}[ x] ## s.t. 1. ## f(x)=q(x) \cdot g(x)+r(x) ## or 2. the degree of ## r(x) ## is less than the degree of ## g(x) ## Proof...

I need help actually creating the proof. I've done the scratch needed for the problem, it's just forming the proof that I need help in. Bar(a+bi/c+di)= (a-bi) / (c-di) Bar ((a+bi/c+di)*(c-di/c-di)) = ((a-bi/c-di)*(c+di/c+di)) Bar((ac+bd/c^2 +d^2)+(i(bc-ad)/c^2+d^2)) =...

Let ##S_n## denote ##\{1,\ldots,n\}##, where ##n\in\mathbb{N}##. Recall that the ##\textrm{sgn}## function maps a permutation of ##S_n## to an element in ##\{1,0,-1\}##. We want to rework the definition of ##\textrm{sgn}## because it is not sufficient for some proofs about determinants. For...

My trouble is being to show A must be of rank 2. Any ideas?

I'll try to phrase this as clearly as possible but my use of terminology might need to be refined. That may be what ultimately comes of this thread, but hopefully the question as I phrase it will make enough sense. I'm not necessarily asking that a proof be provided, rather, I am interested to...