Theorem Definition and 1000 Threads

I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ... I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ... I need some help in fully understanding Theorem 1.4.4 ... Theorem 1.4.4 reads as follows: In the above proof...

In Mathematical Methods for Physicists, Sixth Edition, Page 60, Section 1.11, the Gauss' theorem is written as: In Mathematical Methods for Physicists, Fifth Edition, Page 61, Section 1.11, the Gauss' theorem is written as: Kindly I would like to know please: 1. What is the difference between...

Homework Statement ##(\mathbb{C}^\times,\cdot)/\mu_m\cong (\mathbb{C}^\times,\cdot)## for any integer ##m\geq 1##, where ##\mu_m=\{z\in \mathbb{C} \mid z^m=1\}##. Homework EquationsThe Attempt at a Solution Here is my idea. Consider the map ##f: \mathbb{C}^{\times} \to \mathbb{C}^{\times}##...

Homework Statement Homework Equations Rolle's Theorem: If f(a)=f(b)=0 then there is at least one a<c<b such that f'(c)=0 The Attempt at a Solution $$y=2x^3-3x^2-12x-6~\rightarrow~y'=6x^2-6x-12$$ The function: y': How do i know y' isn't 0 somewhere? if it's continuously descending, so i...

I am reading Micheal Searcoid's book: "Elements of Abstract Analysis" ... ... I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.3 Ordered Sets ... I need some help in fully understanding Theorem 1.3.24 ... Theorem 1.3.24 reads as follows: In the...

Just wanted to know if the work is sound and logical on my paper posted above. I realized I probably should have included notation for the power of the sets. This is my first attempt at theorem proving in Introductory Real Analysis. I realize now that I’m starting into a subject that...

Homework Statement Team 0 and Team 1 have played 1000 games and Team 0 has won 900 of them.[/B] When the two teams play next, knowing only this information, which team is more likely to win? Homework Equations P(X,Y) = P(YlX) x P(X) = P(XIY) x P(Y) (Not Sure) The Attempt at a Solution Hi, I...

So folks, I'm learning complex analysis right now and I've come across one thing that simply fails to enter my mind: the Cauchy Integral Theorem, or the Cauchy-Goursat Theorem. It says that, if a function is analytic in a certain (simply connected) domain, then the contour integral over a simple...

I'm currently carrying out an analysis on waveforms produced by a particular particle detector. The Nyquist-Shannon sampling theorem has been very useful for making an interpolation over the original sample points obtained from the oscilloscope. The theorem (for a finite set of samples) is given...

Homework Statement Why does ##\lim_{n \rightarrow \infty} f(x_n) = f(c)## contradict ##\lim_{n \rightarrow \infty} \vert f(x_n) \vert = +\infty##? edit: where ##c## is in ##[a,b]## Homework Equations Here's the proof I'm reading from Ross page 133. 18.1 Theorem Let ##f## be a continuous real...

Hello! I am a bit confused about the dimensionality of the vectors in Wigner-Eckart theorem. Here it is how it gets presented in my book. Given a vector space V and a symmetry group on it G, with the representation U(G) we have the irreducible tensors $${O_i^\mu,i=1,...,n_\mu}$$ (where ##n_\mu##...

The theorem: Let ##X##, ##Y## be sets. If there exist injections ##X \to Y## and ##Y \to X##, then ##X## and ##Y## are equivalent sets. Proof: Let ##f : X \rightarrow Y## and ##g : Y \rightarrow X## be injections. Each point ##x \in g(Y)⊆X## has a unique preimage ##y\in Y## under g; no ##x \in...

Hello everyone, In Bernoulli's theorem, I understand Potential energy (because of height) and Kinetic energy (because of velocity), but I don't understand pressure [energy]; Is it something like the vibration of molecules and bumping them into each other (in simple words). Any help or simulation...

Cutkosky rule states that: $$2Im \big(A_{ab}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c p^{\mu}_{c}-\sum_a p^{\mu}_{a}\Big)|A_{cb}|^2\hspace{0.5cm} (1)$$ putting ##a=b=p## in Cutkosky rule we deduce the Optical Theorem for ##pp## scattering: $$2Im \big(A_{pp}\big)=(2\pi)^4\sum_c \delta\Big(\sum_c...

I have a digital transmitter from which I collect and save values from. How do I know if I must apply this theorem or not? My values seems fine..

Hi All I normally post on the QM forum but also have done quite a bit of programming and did study computer science at uni. I have been reading a book about Ramanujan and interestingly he was also good friends with Bertrand Russell. You normally associate Russell with philosophy but in fact...

Homework Statement Plate a of a parallel-plate, air filled capacitor is connected to a spring having force constant k, and plate b is fixed. They rest on a table top.If a charge +Q is placed on plate a and a charge −Q is placed on plate b, by how much does the spring expand? Homework...

1. The problem statement, all variables and given/known I don't understand the proof of the following theorem: Theorem 3.1.1 Let ##g_{ab}## be a metric. Then there exists a unique derivative operator ##\nabla_a## satisfying ##\nabla_a\,g_{bc}=0## 2. Homework Equations After some manipulations...

Hello. I am studying Analysis on Manifolds by Munkres. I have a problem with a proof in section 20. It states that: Let A be an n by n matrix. Let h:R^n->R^n be the linear transformation h(x)=A x. Let S be a rectifiable set (the boundary of S BdS has measure 0) in R^n. Then v(h(S))=|detA|v(S)...

Would the shell theorem prevent a binary planet system, with two ideally equal masses, structure etc?

I am reading Micheal Searcoid's book: Elements of Abstract Nalysis ( Springer Undergraduate Mathematics Series) ... I am currently focussed on Searcoid's treatment of ZFC in Chapter 1: Sets ... I am trying to attain a full understanding of Searcoid's proof of the Pairing Principle ... The...

Homework Statement Use Stokes' Theorem to evaluate ∫cF ⋅ dr, where F(x, y, z) = x2zi + xy2j + z2k and C is the curve of the intersection of the plane x + y + z = 1 and the cylinder x2 + y2 = 9 oriented counterclockwise as viewed from above. Homework Equations Stoke's Theorem: ∫cF ⋅ dr = ∫s...

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 7.2 Euclidean, Principal Ideal, Unique Factorization Domains ... ... I need help with the proof of Theorem 7.2.20 ... ... Theorem 7.2.20 and its proof reads as...

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 7.2 Euclidean, Principal Ideal, Unique Factorization Domains ... ... I need help with the proof of Theorem 7.2.14 ... ... Theorem 7.2.14 and its proof reads as follows: In the above proof by Bland we...

Gödel's incompleteness theorem only applies to logical languages with countable alphabets. So it does not rule out the possibility that one might be able to prove 'everything' in a language with an uncountable infinite alphabet. Is that a loophole in Godel's Incompleteness Theorem? Doesn't...

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 3.2 Subrings, Ideals and Factor Rings ... ... I need help with another aspect of the proof of Theorem 3.2.19 ... ... Theorem 3.2.19 and its proof reads as follows...

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 3.2 Subrings, Ideals and Factor Rings ... ... I need help with the proof of Theorem 3.2.19 ... ... Theorem 3.2.19 and its proof reads as follows: In the above proof by Bland we read the following:"... ...

I am reading The Basics of Abstract Algebra by Paul E. Bland ... I am focused on Section 3.2 Subrings, Ideals and Factor Rings ... ... I need help with the proof of Theorem 3.2.16 ... ... Theorem 3.2.16 and its proof reads as follows: In the above proof of $$(3) \Longrightarrow (1)$$ by...

According the the Godel's completeness theorem, a statement in first order logic is true if and only if it can be formally proved from the first order axioms. But what does it mean that a statement is true? Obviously, it cannot be by definition that true means provable in first order logic...

1. Homework Statement . Figure 1 shows a 50 Ω load being fed from two voltage sources via their associated reactances. Determine the current i flowing in the load by: (a) Thevenin's theorem (b) Superposition (c) Transforming the two voltage sources and their associated reactances into current...

We can look at infinitesimal transformations in the fields that leaves the Lagrangian invariant, because that implies that the equations of motions are invariant under this transformations. But what really matters is the those transformations that leaves the action invariant. So we can always...

As a preface to this theorem stated in my text, it states that: "If all the coefficients of a polynomial ##P(x)## are real, then ##P## is a function that transforms real numbers into other real numbers, and consequently, ##P## can be graphed in the Cartesian Coordinate Plane." It then goes on...

Say I have a given problem that states: Does the Intermediate Value Theorem guarantee that the following equation has a real solution between ##(\frac{7}{2})## and ##(\frac{9}{2})##? $$3x^4-27x^3+177x^2+1347x+420=0$$ Now what I want to do is determine the sign of x=##(\frac{7}{2})## and...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some further help to fully understand the proof of part of Proposition 4.2.16 (Jordan-Holder) ... ... Proposition 4.2.16 reads as follows...

I am reading Paul E. Bland's book, "Rings and Their Modules". I am focused on Section 4.2: Noetherian and Artinian Modules and need some help to fully understand the proof of part of Proposition 4.2.16 (Jordan-Holder) ... ... Proposition 4.2.16 reads as follows: Near the middle of the above...

Homework Statement With the stokes' theorem transform the integral ## \iint_\sigma \vec{\nabla}\times\vec{F}\cdot\vec{\mathrm{d}S} ## into a line integral and calculate. ## \vec{F}(x,y,z) = y\hat{i} -x^2\hat{j} +5\hat{k}## ##\sigma(u,v) = (u, v, 1-u^2)## ## v\geq0##, ##u\geq0##...

Here is an article from The Telegraph about triangles in older versions of Stonehenge. (The layout was revised several times). There are several right triangles referred to that are taken as understanding Pythagoras's theorem. The article has drawings. Not sure I buy that they knew A2 + B2 =...

1. The Problem Stament, all variables and given data a 15 kg crate, initially at rest, slides down a ramp 2.0 m long and inclined at an angle of 20 degrees with the horizontal. if there is a constant force of kinetic friction of 25 N between the crate and ramp, what kinetic energy would the...

I bought "Emmy Noether's Wonderful Theorem" by Dwight E. Neuenschwander. After flipping through it, I realized a lot of the math is over my head. For example, multivariate calculus and differential equations. Has anyone else bought this book or really studied how to apply her theorem? I want...

Hi, I have a (probably stupid) question about the Baire Category Theorem. I am looking at the statement that says that in a complete metric space, the intersection of countable many dense open sets is dense in the metric space. Assume that we have the countable collection of dense open sets ##...

In Peskin's textbook section 7.3 The Optical Theorem for Feynman Diagrams(Page233), he said it is easy to check that the corresponding t- and u-channel diagrams have no branch cut singularities for s above threshold. But I can't figure out how to prove it. Can angone help me? Thanks!

Theorem:- For any quadratic function f(x), the mean of the derivative of any two points is equal to the derivative of mean of those two points.Let f(x) be a real valued quadratic function defined as:- f(x)=ax^2 +bx +c Then, f'(x)= 2ax+b Let's consider a interval [i , j] that is defined under...

Where can I find a detailed derivation for GHZ theorem for 3 particles?

Homework Statement Homework Equations Stokes theorem $$\int_C \textbf{F} . \textbf{dr} = \int_S \nabla \times \textbf{F} . \textbf{ds}$$ The Attempt at a Solution I have the answer to the problem but mine is missing a factor of$$\sqrt 2 $$ I can't seem to find my error

I am in trouble with this theorem. I did it from Griffith's electrodynamics but I am not getting the physics of it. So can someone explain it lucidly.

Hello! (Wave) I want to prove that each continuous function $f$ in a closed and bounded interval $[a,b]$ can be approximated uniformly with polynomials, as good as we want, i.e. for a given positive $\epsilon$, there is a polynomial $p$ such that $$\max_{a \leq x \leq b} |f(x)-p(x)|<...

Hey! :o Let $D=\left \{x=(x_1, x_2)\in \mathbb{R}^2: x_2>\frac{1}{x_1}, \ x_1>0\right \}$. We have the function $f: D\rightarrow \left (0,\frac{\pi}{2}\right )$ with $f(x)=\arctan \left (\frac{x_2}{x_1}\right )$. I want to show using the mean value theorem in $\mathbb{R}^2$ that for all...

In Griffith's electrodynamics he writes about poynting's theorem.He considers some charge and current configuration. Then he says that these charges move.Which charges is he talking about and why would they move?

I am reading T. S. Blyth's book: Module Theory: An Approach to Linear Algebra ... I am focused on Chapter 2: Submodules; intersections and sums ... and need help with the proof of Theorem 2.3 ... Theorem 2.3 reads as follows:In the above proof we read the following: " ... ... A linear...

I am reading T. S. Blyth's book: Module Theory: An Approach to Linear Algebra ... I am focused on Chapter 2: Submodules; intersections and sums ... and need help with the proof of Theorem 2.3 ... Theorem 2.3 reads as follows: In the above proof we read the following: " ... ... A linear...