Proof Definition and 999 Threads

Curious about proving that ##\sum_{m=2}^\infty \sum_{n=2}^\infty 1/n^m ## = 1 ran this in Matlab and n,m to 2:1000 =0.9990, and n,m 2:10000 =0.9999, so it does appear to converge to 1

Hello everybody! I have a question regarding the first step of the quantistic proof of the Goldstone's theorem. Defining $$a(t) = \lim_{V \rightarrow +\infty} {\langle \Omega|[Q_v(\vec{x},t),A(\vec{y})]| \Omega \rangle}$$ where ##|\Omega\rangle## is the vacuum state of the Fock space, ##Q_v##...

Hello. In chapter 3 (Quantum Black Holes) of this book... https://www.amazon.com/dp/069116844X/?tag=pfamazon01-20 ...Stephen Hawking writes... "The no-hair theorem, proved by the combined work of Israel, Carter, Robinson and myself, shows that the only stationary black holes in the absence of...

Can anyone help me with the Proof of Parseval Identity for Fourier Sine/Cosine transform : 2/π [integration 0 to ∞] Fs(s)•Gs(s) ds = [integration 0 to ∞] f(x)•g(x) dx I've successfully proved the Parseval Identity for Complex Fourier Transform, but I'm unable to figure out from where does the...

I am reading the Tipler and Mosca textbook and am on the part about massless strings. I understand that in real life a string has more tension at the top than the bottom because the top part has to support a greater mass of rope. However, in other examples such as pulling a sled with a rope I...

Can someone please direct me to ,or show, a proof that a Consistent and Sufficiently Strong AFT is not decidable. It presumably involves the Diagonal Argument, but I can't figure out how to apply it. Many thanks.

Simple linear regression statistics: If I have a linear relation (or wish to prove such a relation): y = k x where k = constant. I have a set of n experimental data points ...(y0, x0), (y1, x1)... measured with some error estimates. Is there some way to present how well the n data points shows...

Reading this piece with a number of proofs of the divergence of the harmonic series http://scipp.ucsc.edu/~haber/archives/physics116A10/harmapa.pdf and this example states: 'While not completely rigorous, this proof is thought-provoking nonetheless. It may provide a good exercise for students...

What I basically want to ask here is, about the process of forming mathematical truth/theorem. This seems like a bit broad question, but I have this specific query. We all know that Euclid started with his basic postulates or what we may call axioms, and common notions. Now did he form those...

Homework Statement We've been given a set of hints to solve the problem below and I'm stuck on one of them Let f:[a,b]->R , prove, using the hints below, that if f is continuous and if f(a) < 0 < f(b), then there exists a c ∈ (a,b) such that f(c) = 0 Hint let set S = {x∈[a,b]:f(x)≤0} let c =...

Problem: Let $\phi(x), x \in \Bbb{R}$ be a bounded measurable function such that $\phi(x) = 0$ for $|x| \geq 1$ and $\int \phi = 1$. For $\epsilon > 0$, let $\phi_{\epsilon}(x) = \frac{1}{\epsilon}\phi \frac{x}{\epsilon}$. ($\phi_{\epsilon}$ is called an approximation to the identity.) If $f \in...

So I've taken this Linear Algebra class as an elective. So there's stuff that is so obvious and logically/analytically easy to prove but I honestly don't understand how to prove them using the standard way. So what should I do about this ? And I really like linear algebra so I don't want to mess...

Homework Statement Alright guys this took me an hour because I am really, really... i don't want to say it. I'm about to pop a gasket if my argument is not logical. I couldn't find any info on the latex primer on floor and ceil symbols so i apologize in advance show that if n and k are...

Dear Everyone, $\newcommand{\Z}{\mathbb{Z}}$Suppose the set is defined as: $\begin{equation*} {(\Z/n\Z)}^{\times}=\left\{\bar{a}\in \Z/n\Z|\ \text{there exists a}\ \bar{c}\in \Z/n\Z\ \text{with}\ \bar{a}\cdot\bar{c}=1\right\} \end{equation*}$ for $n>1$ I am having some trouble Proving that...

Homework Statement Let a and b be integers, and let m be a positive integer. Then a ≡ b (mod m) if and only if a mod m = b mod m. Homework EquationsThe Attempt at a Solution By definition a ≡ b (mod m) => m| (a-b) mx = a -b => mx + b = a => b = a mod m b = a - mx => b = m(-x) + a => a = b...

Hi, I'm currently working on showing the relation of quantum fidelity: The quantum “fidelity” between two pure states ρ1 = |ψ1⟩⟨ψ1| and ρ2 = |ψ2⟩⟨ψ2| is given by |⟨ψ1|ψ2⟩|^2. Show that this quantity may be written as Tr(ρ1ρ2). I've been following the wikipedia page on fidelity but can't...

I'm reading about the LU decomposition on this page and cannot understand one of the final steps in the proof to the following: ---------------- Let ##A## be a ##K\times K## matrix. Then, there exists a permutation matrix ##P## such that ##PA## has an LU decomposition: $$PA=LU$$ where ##L## is a...

Homework Statement Let P(n) be the statement "In every set of n horses, all of the horses in the set have the same colour." Base Case: We must prove that P(1) is true. If our set only contains one horse, then all horses in the set have the same colour. Inductive Step: Let m ≥ 1 and assume P(m)...

Show that(A-B)-C=A-(BUC)

Homework Statement Attached are notes from class. Can someone please explain what happens to (-x(n+1)) in step 3 to step 4. Not sure why it goes away. Thanks! Homework EquationsThe Attempt at a Solution

Homework Statement Show that the only subspaces of ##V = R^2## are the zero subspace, ##R^2## itself, and the lines through the origin. (Hint: Show that if W is a subspace of ##R^2## that contains two nonzero vectors lying along different lines through the origin, then W must be all of...

Dear Everyone, Here is the question: "Prove that if $k$ divides the integers $a$ and $b$, then $k$ divides $as+bt$ for every pair of integers $s$ and $t$ for every pair of integers." The attempted work: Suppose $k$ divides $a$ and $k$ divides $b$, where $a,b\in\mathbb{Z}$. Then, $a=kt$ and...

i want to know about stirling approximation. why $$lnx! = xlnx - x$$

Homework Statement Use proof by contradiction to show there is no rational number r for which r^3+r+1 = 0 Homework EquationsThe Attempt at a Solution Assume there is a rational number r for which r^3+r+1=0. Then r = (a/b) with a,b ∈ℤ and b ≠ 0, and a/b is in lowest terms Then a/b is a root...

Homework Statement Let a, b ∈ ℝ. Prove that: If |a| = |b|, then a = b or a = -b. Homework EquationsThe Attempt at a Solution [/B] I am having difficulties with beginning this proof. Would it make sense to have: Case 1: b≥0 and Case 2: b≤0?

Homework Statement Let W be a subspace of a vector space V, let y be in V and define the set y + W = \{x \in V | x = y +w, \text{for some } w \in W\} Show that y + W is a subspace of V iff y \in W. Homework Equations The Attempt at a Solution Let W be a subspace of a vector space V, let y...

Homework Statement If 3n+2 is odd, then n is odd Homework EquationsThe Attempt at a Solution assume 3n+2 is even, show n is odd 3n+2 = 2f => 3n+3 = 2f + 1 => 3(n+1) = 2f+1 so 3*(n+1) is odd form. Only odd*odd gives you another odd, so n+1 is odd. That means n is even. We have a...

According to you this theorem is correct? Exercise 1.2 * Proof that ##\sqrt{x}## isn't a rational number if ##x## isn't a perfect square (i.e. if ##x=n^2## for some ##n∈ℕ##). In effect, if ##x=\frac{25}{9}##, so ##x## isn't a perfect square, then ##\sqrt{x}=\sqrt{\frac{25}{9}}=\frac{5}{3}##...

Hi, in this link https://math.stackexchange.com/questions/2605752/proof-log-convex-implies-convexity , there is a theorem that labelled with yellow. I do not know the name of theorem and would like to know it's derivation. Could you please help me?

Which axioms (at minimum) would have to be invoked so the following expression holds: (x = y) ----> [(y=x) <---> (y=y)] ? All help appreciated, am

Recently I started looking back at some basic mathematical principles, and I started thinking about the theorem of corresponding angles. It's such a basic idea that it seems obvious on an intuitive level, but despite that (or possibly because of that) I can't think of a good way to formally...

Homework Statement The chapter of the book this exercise is found in is titled Proofs Involving Negations and Conditionals. The problem is as follows: Suppose that a and b are nonzero real numbers. Prove that if ## a < \frac{1}{a} < b < \frac{1}{b} ## then ## a < -1##. Homework Equations...

This is not homework. I am reading a book: "The art of infinite: The Pleasure of Mathematics" and pages 119-120 give a proof of the Euler Line theoram: the circumcenter, centroild and orthocenter of a triangle are always colinear (see the attached files). 1. Homework Statement Page 119 shows...

Homework Statement Hi I am looking at this derivation of differential equation satisfied by ##\phi(z)##. To start with, I know that such a disc ##D## described in the derivation can always be found because earlier in the lecture notes we proved that their exists an ##inf=min \omega ## for...

Hello all! I've got this problem I'm trying to do, but I'm not sure what the best way to approach it is. It's obvious that there can only be 2 dimensions, because there's only two linearly independent vectors in the span. However, what would be a good way of using the inequalities to prove...

Homework Statement Hi I am looking at this proof that , if on an open connected set, U,there exists a convergent sequence of on this open set, and f(z_n) is zero for any such n, for a holomorphic function, then f(z) is identically zero everywhere. ##f: u \to C##Please see attachment...

Homework Statement Prove: if 0<a<b then a^(1/n) < b^(1/n) Homework EquationsThe Attempt at a Solution I've already proved a^n < b^n in another problem. So I have Assume a^n < b^n => \sqrt[n^2] a^n < \sqrt[n^2] b^n => a^(1/n) < b^(1/n)

The theorem is as follows: All finite dimensional vector spaces of the same dimension are isomorphic Attempt: If T is a linear map defined as : T : V →W : dim(V) = dim(W) = x < ∞ & V,W are vector spaces It would be sufficient to prove T is a bijective linear map: let W := {wi}ni like wise let...

Homework Statement I'm currently working through Spivak independently and have reached the problems at the end of ch. 1. The problem is: Prove that if 0 < a < b , then a < \sqrt{ab} < \frac{a+b}{2} < b Homework Equations Spivak's properties P1 - P12 The Attempt at a Solution I was...

Hi, I have to show that if ##f \in L^1(ℝ^n)## then: $$ ||\hat f||_{C^0(ℝ^n)} \le ||f||_{L^1(ℝ^n)}$$ Since ##|f(y)e^{-2 \pi i ξ ⋅y}| \le |f(y)|##, using the dominated convergence theorem, it is possible to show that ##\hat f \in C^0(ℝ^n)## but now I don't know how to go on. Thanks is advance.

https://imgur.com/a/oSioYel I am trying to understand this proof, but am tripped up on the part that says "Consider the action of ##G## on ##\operatorname{Syl}_2(G)## by conjugation." My question is, how is this a well-defined action if ##\operatorname{Syl}_2(G)## is not normal? Isn't this...

https://imgur.com/a/jThCPLA I'm trying to understand the proof here, and there is just one point that I get tripped up on. In the last paragraph, I'm not seeing exactly why ##K\cap H < H## based upon our choice of ##y##. Could someone explain?

Recently I came up with a proof of “ for a nth degree polynomial, there will be n roots” Since the derivative of a point will only be 0 on the vertex of that function,and a nth degree function, suppose ##f(x)##has n-1 vertexes, ##f’(x)## must have n-1 roots. Is the proof valid?

Homework Statement Show that if ##0 \leq a < b##, then $$\frac {b^{n + 1} - a^{n + 1}} {b - a} < (n + 1)b^n$$ Homework Equations None that I'm aware of. The Attempt at a Solution Proof (Induction) 1. Basis Case: Suppose ##n = 1##. It follows that: $$\frac {b^{1 + 1} - a^{1 + 1}} {b - a} < (1 +...

Homework Statement One way to establish which transitions are forbidden is to compute the expectation value of the electron’s position vector r using wave functions for both the initial and final states in the transition. That is, compute ∫ΨfrΨidτ where τ represents an integral over all space...

Homework Statement 2. Relevant equation Below is the definition of the limit superior The Attempt at a Solution I tried to start by considering two cases, case 1 in which the sequence does not converge and case 2 in which the sequence converges and got stuck with the second case. I know...

I've been working on this one for a little bit, and I know I really just need to use the chain rule to solve it, but I can't seem to figure out how to set it up properly. Probably a dumb question, but I could really use some help on this!

<Moderator's note: Moved from a technical forum and thus no template.> for every natural n there exists natural k. and numbers={a0,a1,a2,...ak}∈{0,1}. so that n=i=0n∑ ai2i I will assume n=k, i know that if n is even then a0 =0. so if i assume it is true for n that is Even: n+1=i=0n+1∑ ai2i...

I am working on a proof problem on ordered field from a textbook, which lists additive and multiplicative properties similar to the ones here: The followings are what I was able to come out -- I just wanted to make sure that they are acceptable: (a) By the multiplicative inverse property...

Homework Statement Prove Rijkl= k/R2 * (gik gjl-gil gjk) where gik is the 3 metric for FRW universe and K =0,+1,-1, and i,j=1,2,3, that is, spatial coordinates. . Homework Equations The Christoffel symbol definition: Γμνρ = ½gμσ(∂ρgνσ+∂νgρσ-∂σgνρ) and the Riemann tensor definition: Rμνσρ =...