Theorem Definition and 1000 Threads

We talked about Fibonacci numbers, and I wondered: Can any natural number be construed by a sum of unique Fibonacci numbers? My guess was yes, and a C program I wrote confirms that to be up to about 2,000, but that's of course is no proof. The best semi-proof I could come up with is that the...

Homework Statement If $$C = 1+cos\theta+...+cos(n-1)\theta,$$ $$S = sin\theta+...+sin(n-1)\theta,$$prove that $$C=\frac{sin\frac{n\theta}{2}}{sin\frac{\theta}{2}} cos\frac{(n-1)\theta}{2} \enspace and \enspace S = \frac{sin\frac{n\theta}{2}}{sin\frac{\theta}{2}}sin\frac{(n-1)\theta}{2}$$...

Hi Guys, at the moment I got a bit confused about the notation in some QM textbooks. Some say the operators should be symmetric, some say they should be self-adjoint (or in many cases hermitian what maybe means symmetric or maybe self-adjoint). Which condition do we need for our observables...

Homework Statement (y4 - 5y2 + 2y - 15) / (3y - √(2)) The answer says (2√(2)/3)-(1301/81)...

Surface S and 3D space E both satisfy divergence theorem conditions. Function f is scalar with continuous partials. I must prove Double integral of f DS in normal direction = triple integral gradient f times dV Surface S is not defined by a picture nor with an equation. Help me. I don't...

Homework Statement No actual problem, thinking about the telescoping series theorem and Grandi's series For reference Grandi's series S = 1 - 1 + 1 - 1... Homework Equations [/B] The telescoping series theorem in my book states that a telescoping series of the form (b1 - b2) + ... + (bn -...

Hello! (Wave) We say that the space $\Omega$ satisfies the exterior sphere condition at the point $x_0 \in \partial{\Omega}$ if there is a $y \notin \overline{\Omega}$ and a number $R>0$ such that $\overline{\Omega} \cap \overline{B_y(R)}=\{ x_0 \}$. Let the function $\phi \in...

Homework Statement Show that Dx∫f(u)du = f(x) Where the integral is evaluated from a to x. (Hint: Do Taylor expansion of f(u) around x). Homework Equations None The Attempt at a Solution I have ... = Dx(F(u)+C) = Dx(F(x-a)+C) = dxF(x) - dxF(a) = f(x)-f(a). My problem is that it should be...

Homework Statement Prove that $$ \lim_{x\to 0} \sqrt{x^3+x^2}\; \sin\left(\frac{\pi}{x}\right) = 0 $$ using Sandwich theorem Homework Equations Sandwich Theorem The Attempt at a Solution Now we know that sine function takes values between -1 and 1. ## -1 \leqslant...

Hi, this is a newbee question. Does the Fundamental Theorem of Calculus supply a visual (graphical) way of linking a function (F(x)) with its derivative (f(x))? That is, the two-dimensional area under a curve in [a,b] for f(x) is always equals to the one-dimensional distance F(b)-F(a)? If...

Hola, I tried to give a proof of this theorem and then check it against the one given by my book(Fasano, Marmi - Analytical Mechanics); I feel like mine seems reasonable and pretty intuitive, but the one on the book is a bit different and I don't really understand it completely, so I'd like to...

Homework Statement In the first and second photo , it's stated earlier that the C is the boundary of surface on xy plane , but in the question in the 3rd picture , it's not stated that the C is on which surface , so , how to do this question ? For ∫F.dr , i am not sure how to get r , coz i am...

Hey guys, I am supposed to solve for voltage and current on R4 using Thevenin's theorem here. Values are following: U = 100V R1 = 310 R2 = 610 R3 = 220 R4 = 570 R5 = 200 Now, I know I need to solve for Rth and Vth, so I put the circuit into simulation and figured out voltage on R3 is equal to...

Hello everyone! I am reviewing the derivation of the Virial Theorem from an introductory Astrophysics book (Carroll and Ostlie's) and found a step I couldn't follow. I've attached a photo of the step. Can anyone explain how Newton's Third Law brings about eqn 2.41? I don't see how that first...

Homework Statement I understand the proof of the implicit function theorem up to the point in which I have included a photo. This portion serves to prove the familiar equation for the implicit solution f(x,y) of F(x,y,z)=c. My confusion arises between equations 8.1-4 and 8.1-5 when it is stated...

Homework Statement i can't find the normal vector here . In my book , outwards vector is . (Refer to photo 1 ) The question is in photo 2 , i am aksed to use stoke's theorem to evalutae line integral of vector filed But , now the problem is i can't express z in terms of y and x . Can anyone...

I learned in Analytical Mechanics: "Emmy Noether's theorem shows that every conserved quantity is due to a symmetry". The examples I learned where conservation of energy as symmetry in time and conservation of momentum as symmetries in space. Now I wonder, do universal constants are also due to...

Homework Statement [/B] ##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane. What is $$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$Homework Equations If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...

Consider the lagrangian of the real scalar field given by $$\mathcal L = \frac{1}{2} (\partial \phi)^2 - \frac{1}{2} m^2 \phi^2 - \frac{\lambda}{4!} \phi^4$$ Disregarding snail contributions, the only diagram contributing to ## \langle p_4 p_3 | T (\phi(y)^4 \phi(x)^4) | p_1 p_2 \rangle## at...

Homework Statement Consider the beam shown in (Figure 1) . Suppose that a = 15 in. , b = 8 in. , c = 1 in., and d = 4 in. Determine the moment of inertia for the beam's cross-sectional area about the x axis...

Homework Statement Find the solution of the following integral Homework Equations The Attempt at a Solution I applied the above relations getting that Then I was able to factor the function inside the integral getting that From here I should be able to get a solution by simply finding the...

1. Homework Statement i attached the problem statement as an image file Homework Equations p(x) = (x-c)q(x) + r The Attempt at a Solution i've simplified it down to ((x-1)^114) / (2^114)(x+1). is there a practical way to approach this besides long division? wolfram alpha gave an extremely...

I just suppose the Bell's Ansatz for the result of measurement to be $$A (\theta,\lambda) $$ Now the parameter lambda could be anything : -a physical quantity like the polarization angle of the incoming photon -the coordinate of a 'world' - the whole wavefunction. ... In the case of the...

I'm having a little trouble starting on this problem. Can someone help? I was trying to solve it out, but I just ended up telling how to draw the center. Attached is the problem:

Hello again, everyone. Have a multivariate calculus question this time around. If anyone can point me in the right direction and help me see where WebAssign finds me wrong, it would be greatly appreciated. 1. Homework Statement Homework Equations ∫∫ScurlF ⋅ dS = ∫CF ⋅ dr The Attempt at a...

Homework Statement I am trying to express ##T(\phi(x1)\Phi(x2)\phi(x3)\Phi(x4)\Phi(x5)\Phi(x6))## in terms of the Feynman propagators ##G_F^{\phi}(x-y)## and ##G_F^{\Phi}(x-y)## where ##G_F^{\phi}(x-y) =\int \frac{d^{4}k}{(2\pi)^{4}}e^{ik(x-y)} \frac{ih}{-k.k - m^2 -i\epsilon} ## and...

<Moderator's note: Moved from a technical forum, so homework template missing> Sorry i have one question to ask how to check the v.dl part in this problem i cannot do this problem as it is too hard to integrate the equation How did griffith get this long-horrible equation(see the orange...

I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of the proof of Theorem 1.4 ... ... Theorem 1.4 reads as follows: Questions 1(a) and 1(b) In...

I am reading Matej Bresar's book, "Introduction to Noncommutative Algebra" and am currently focussed on Chapter 1: Finite Dimensional Division Algebras ... ... I need help with some aspects of the proof of Theorem 1.4 ... ... Theorem 1.4 reads as follows: Questions 1(a) and 1(b) In the above...

The thread I wanted to post my question on got closed. Recapitulating: The best (simplest) account I have found to date for the Bell inequality (SPOT stands for Single Photon Orientation Tester): Imagine that each random sequence that comes out of the SPOT detectors is a coded message. When...

1. The problem statement, all variables and given/ You must push a box up an incline plane (the angle being constant : a), to a person waiting to receive it, who is a distance of h(constant) vertically above you. Though the slope is slippery, there is a small amount of friction with kinetic...

I don't know how to start proving this theorem, so can someone please help? I need to prove that the circumcircles all intersect at a point M. Thank you! Miquel's Theorem: If triangleABC is any triangle, and points D, E, F are chosen in the interiors of the sides BC, AC, and AB, respectively...

Homework Statement The moment of inertia for a perpendicular axis through the center of a uniform, thin, rectangular metal sheet with sides a and b is (1/12)M(a2 + b2). What is the moment of inertia if the axis is through a corner? The answer is given as this was a powerpoint lecture and it...

Homework Statement Find the remainder of ##4^{87}## in the division by ##17##. Homework Equations Fermat's Little Theorem: If ##p## is prime and ##a## is an integer not divisible by ##p##, then ##a^{p-1} \equiv 1 (\mod \space p)## or equivalently, ##a^p \equiv a (\mod \space p)## The...

Hello! I am a bit confused about the first Sylow theorem. So it says that if you have a group of order ##p^mn##, with gcd(n,p)=1, you must have a subgroup H of G of order ##p^m##. So, if I have a group G of order ##p^k##, there is only one subgroup of G of order ##p^k## which is G itself. Does...

Hello! (Wave) We consider the following problem. $$Lu=f(x) \text{ in } \Omega \\ u|_{\partial{\Omega}}=0$$ I want to show that if $c(x) \leq -c_0<0$ in $\overline{\Omega}$, then it holds that $\min\{ 0, \frac{\min_{\Omega}f(x)}{-c_0}\}\leq u(x) \leq \max_{\Omega} \{ 0...

Liouville's theorem states that the total time-derivative of the distribution function is zero along a system trajectory in phase-space. Where the system follows a trajectory that satisfies the Hamilton's equations of motion. I have a hard time getting an inuitive understanding of this...

Homework Statement Homework Equations Green's theorem The Attempt at a Solution DO I first parametrize? For 1st part, I have 3 parametrizations, which I can then find the normal vector, and use in the integrals?

Hawking area theorem says that area of black hole generally never decrease. Penrose process says that energy can be extracted from black hole. Energy extraction will decrease mass? if yes then if mass is decreased then will area also decrease? I am confusing things here :(

According to DeMorgan’s theorem (break the bar and change the sign), the complement of ܽa⋅b+c⋅d is a'+b'⋅c'+d' Yet both functions are 1 for ܾܽܿ abcd 1110. How can both a function and its complement be 1 for the same input combination? What’s wrong here?I honestly have no idea. I mean, shouldn't...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ... I am currently focused on Garling's Section 1.7 The Foundation Axiom and the Axiom of Infinity ... ... I need some help with Theorem 1.7.4 ... and in particular with...

Homework Statement (See attachment: if it doesn't work, see below for poorer formatting)[/B] Theorem 6.1 (Centered Formula of Order O(h2)). Assume that f ∈ C^3[a, b] and that x − h, x, x + h ∈ [a, b]. Then (3) f (x) ≈ f (x + h) − f (x − h) 2h . Furthermore, there exists a number c = c(x) ∈ [a...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis ... ...At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...

I am reading D. J. H. Garling: "A Course in Mathematical Analysis: Volume I Foundations and Elementary Real Analysis" ... ... At present I am focused on Chapter 1: The Axioms of Set Theory and need some help with Theorem 1.2.2 and its relationship to the Separation Axiom ... ... The...

Why is it possible that ∫∫∫ V f(r) dV ≠ 0 even if f(r) =0

Obviously ##\mathbb{R^2}## is convex, that is, any points ##a,b\in\mathbb{R^2}## can be connected with a line segment. In addition, ##f## is differentiable as a composition of two differentiable functions. Thus, the conditions of mean value theorem for vector functions are satisfied. By applying...

The population mean and standard deviation are given below. Find the required probability and determine whether the given sample mean would be considered unusual. μx̄ = μ = 12,749 σ = 1.2 n = 35 For the given sample n = 35, the probability of a sample mean being less than 12,749 or greater...

I am wondering is someone could comment on a question I have recently answered. I have attached the question and my answer. Apologies for not following the standard procedure of Latex but there are drawings associated with this question. I answered section A and my results are written on the...

Homework Statement Using paper, pencil and the Virial theorem, calculate the position uncertainty (an estimate of the vibration amplitude) of the H atom in its ground state C-H stretching mode. In more precise language, calculate the bond length uncertainty in a C-H bond due to the C-H...

Homework Statement Using the maximum power transfer theorem, find the value of R which will result in the maximum power being delivered to R. Homework Equations P_max= (V_oc)^2/4(R_th)) R_L=R_th P_RL=(V_th/(R_th+R_L))^2*R_L The Attempt at a Solution I have no clue where to even begin with...