Form Definition and 1000 Threads

Let $S_n(k)$ be defined by: $S_n(k) = 1 + 2k+3k^2+...+(n+1)k^n$, where $|k| < 1$ and $n \in \Bbb{N}$. Derive a closed form for $S_n(k)$ and find the limit: $$\lim_{{n}\to{\infty}}S_n(k)$$.

Homework Statement Find the vector form of the equation of the line in ℝ2 that passes through P=(2, -1) and is parallel to the line with general equation 2x-3y=1. Homework Equations None. The Attempt at a Solution vector form: x=p+td general form: ax+by=c --------------------------------...

Homework Statement Find the normal form of the equation of the plane that contains the point P=(0, 1, 0) and has normal vector n=[3, 2, 1]. Homework Equations None. The Attempt at a Solution I know that the general form: ax+by=c since that's written in my textbook. But why is the answer...

Hello! I am reading Geometry Topology and Physics by Nakahara and in Chapter 5.6.4 he defines the canonical (Maurer-Cartan) one form on a Lie group G as: ##\theta : T_gG \to T_eG##. Then he states a theorem in which ##\theta = V_\mu \otimes \theta^\mu##. Both by the tensor product and by the...

Hello all First time on the Math side of this great Forum. I need to do a correlation study. There are 3 methods to measure Form Error of a round metal part. The base line is the traditional measuring method that is laborious but of the highest accuracy. A cnc machine measurement machine...

Hi everyone, I am trying to get familiar with some of the terminology and concepts recurring more or less everywhere in LHC phenomenology. In particular, something that is troubling me a bit is the concept of sudakov form factors, which I don’t find explained anywhere in an understandable way...

We have some type of squash plant growing in a small garden. It remarkable how much it has spread. I would think it advantageous for such a plant to form additional root systems, try as I might I do not see them on this plant. Is that typical of squash and pumpkin type plants, no additional root...

Homework Statement The answer in (algebraic form) is Homework EquationsThe Attempt at a Solution I'm not sure how to simplfy into the form required from here.

I have read differing arguments about the potential utility or negative issues arising due to AI. There are many concerns about the 'control problem' of AI or how to 'sandbox' AI so it can serve our interests and not let a fast takeoff of AI lead to a situation where AI is the sole power around...

Homework Statement The set ##\Bbb{A}## of all the algebraic integers is a subring of ##\Bbb{C}## Homework EquationsThe Attempt at a Solution Here is an excerpt from my book: "Suppose ##\alpha## an ##\beta## are algebraic integers; let ##\alpha## be the root of a monic ##f(x) \in...

I am reading "Introduction to Real Analysis" (Fourth Edition) by Robert G Bartle and Donald R Sherbert ... I am focused on Chapter 7: The Riemann Integral ... I need help in fully understanding an aspect of the proof of Theorem 7.3.5 ...Theorem 7.3.5 and its proof ... ... read as follows: In...

Homework Statement Develop a composite form of Boole's rule for an integral of the form ∫ f(x) dx, where the bounds of integration are from [a,b]. Determine the error bound formula for the composite form of Boole's rule. ∫ g(t) dt = h/45[14g(0)+64g(h)+24g(2h)+64g(3h)+14g(4h)] -...

Homework Statement Express the following in the form x = Re{Aeiαeiωt} (a) x= cos(ωt) + sin(wt) (b) x= sin(ωt +π/4) + cos(ωt) (c) x= 2cos(ωt+π/3) + (√3)sin(ωt)-cos(ωt) Homework Equations cos x = 1/2 e^ix + 1/2 e^-ix sin x = − i/ 2e^ix + i/2 e^−ix The Attempt at a Solution To be honest, I have...

Homework Statement The equation of the line that goes through the points and can be written in general form . What are the values for A, B, and C? Homework Equations Ax+By+C=0 The Attempt at a Solution [/B] I think the answer is x=1. But how do i express that in general form?

Determine the equation of the circle that passes through the point (-4, 1) and its center is the midpoint of the line segment joining the centers of the two circles given below: x^2 + y^2 -6x - 4y + 12 = 0 and x^2 + y^2 - 14x + 47 = 0. Write the equation in standard form. 1. Does the...

I just want to know does any random equation of the form y=f (x-vt) reppresent a wave?? If yes why?? And is this true for all cases or is there any condition?

I have some diamond powder and was wondering how I can turn it back into its hard stone form? What can I do. I'm a hobbyist chemsit/physicist. I just need some help from more knowledge people

$\tiny{s10.03.25}$ $\textsf{Write complex number in rectangular form}$ \begin{align*}\displaystyle z&=4\left[\cos\frac{7\pi}{4} + i\sin \frac{7\pi}{4} \right]\\ \end{align*} $\textit{ok from the unit circle: $\displaystyle\cos{\left(\frac{7\pi}{4}\right)}=\frac{\sqrt{2}}{2}$}\\$ $\textit{and...

$\tiny{hcc8.11}$ $\textsf{Find product $(1+3i)(2-2i)$}\\$ $8 + 4i$ $\textsf{Then change each to complex form and find product. with DeMoine's Theorem}$ $\textit{ok looked at an example but ??}

I am reading Real Mathematical Analysis by Pugh, and he claims that "All mathematical assertions take an implication form a --> b." However, is this really true? For example, the assertion, "There exist infinitely many prime numbers," doesn't seem to take the if-then form.

I´m having a problem with the value of the expression ##F(it)-F(-it)##, found on the Abel-Plana formula, where $$F(z)=\sqrt{z^2 + A^2}$$, with ##A## being a positive real number (F(z) is analytic in the right half-plane). Well, I know the result is ##F(it)-F(-it)=2i\sqrt{t^2 -A^2}##, for...

Homework Statement Using the matricial form of TFD (discret Fourier Transform) calculated : X=Fd(x) for x=(1,0,2j,1-j)^T from K^4 Homework Equations X=wx The Attempt at a Solution I know the "x" matrix and the "w" matrix but i don't know how to find the "ω" variable from the "w" matrix...

In Chapter 70 of Srednicki's QFT he discusses what he calls the index of a representation T(R) defined by Tr(TaR TbR) = T(R)δab I think other places call this a killing form, but I may be mistaken. In any case he discusses reducible representations R = R1⊕R2. He then states (eqn 70.11) that...

Also does bad bacteria(mold) forms on the weight(i.e use to submerged the cabbage) ,on the portion that is above the brine. I use glass paper weight, as I think they only form on food surface. My thought, sauerkraut is safe, as long it is submerged inside the brine and mold formation is only...

i want to know the concave mirros is always real images or any difference then how to calculate thanks and regards Uday

Hi everyone I am reading Sean Carrol's lecture notes on general relativity. link to lecture : https://arxiv.org/abs/gr-qc/9712019 In his lecture he introduced dxμ as the coordinate basis of 1 form and ∂μ as the basis of vectors. I understand why ∂μ could be the basis of the vectors but not for...

Why do the time-evolution operator in quantum mechanics ##\exp{iHt}## and the Gibbs-weight operator in statistical physics ##\exp{-H/T}## have the same functional form? – i.e. both exponentials of the Hamiltonian operator. The Matsubara trick/method just takes this as a fact in thermal QFT; but...

Hi everyone, I am trying to self study some general relativity however I met some problem in the contravarient and covarient basis. In the lecture, or you can also find it on wiki page 'curvilinear coordinates', the lecturer introduced the tangential vector ei =∂r/∂xi and the gradient vector ei...

Hello everyone, I have a complex number problem that i would greatly appreciate some help with. Thanks in advance to anyone offering their time to make a contribution. Q) Write the following in polar form: I have attempted the question (please see my working below) and have been advised that i...

Homework Statement as listed above the question is how many and which three digit NIP can be formed whit the use of prime numbers[/B]Homework Equations nothing currently trying to understand[/B]The Attempt at a Solution well i have found at least 168 primer numbers below 1000 i mean in the...

I've been going through my book learning about differential equations of multiple variables and I have a quick question about differential forms. If you are working a problem and get to the point where you're left with a differential form like ##(y)dx##, does that mean that the change in the...

My book is going through a proof on exact differential forms and the test to see if they're exact, and I'm lost on one part of it. It says: If $$M(x,y)dx + N(x,y)dy = \frac{\partial F}{\partial x}dx + \frac{\partial F}{\partial y}dy$$ then the calculus theorem concerning the equality of...

I didnt know much about black holes and also astrophysics but I want to know some general information about how they form.Well As massive star collapses the matter get so densed, then I guess internal pressure gets so high and as star explodes as supernova (maybe I described it wrong), ın any...

Homework Statement About an endomorphism ##A## over ##\mathbb{C^{11}}## the next things are know. $$dim\, ker\,A^{3}=10,\quad dim\, kerA^{2}=7$$ Find the a) Jordan canonical form of ##A## b) characteristic polynomial c) minimal polynomial d) ##dim\,kerA## When: case 1: we know that ##A## is...

In a barrier potential with sections: I: V(x)=0 (x<-a), II:V(x)=V (-a<x<a) and III:V(x)=0 (a<x) you can write the solution in this form: Ψ(x)=Ae^(ikx)+Be^(-ikx) (x<-a) Ψ(x)=Ce^(ik'x)+De^(-ik'x) (-a<x<a) Ψ(x)=Ee^(ikx) (a<x) and with boundary conditions solve, but why do you can write this...

For example: If I isolated (in a perfect vacuum), one Magnesium atom and one Oxygen atom, and they are close to each other, will they make a bond by themselves, with no outside energy? Does this work with any combinable pair of atoms, and what other pairs of atoms would make a bond by...

If a circle has 6 segments, how many maximum parts which can be formed? I know that 1 segment makes 2 parts, 2 segments make 4 parts, and 3 segments makes 7 parts. Judging by the pattern, is the answer 22? What will the exact picture of the circle be? Thank you very much.

Homework Statement "The complex power spectral density is : S(jw)=(1/(w*(w^2+a^2))*e^j90 Find the form and duration of the signal." Homework Equations Any ideas on how to solve this ?[/B]The Attempt at a Solution What I've managed to calculate so far is: http://tinypic.com/r/zydamr/9

Hi. I was investigating through this week why there are the differential forms, why are they anti-symmetric, why do we have the Jacobian when expressing the volume in a different coordinate system. This was just fantastic! I found all the connections between these topics. And I found that all of...

Specifically how did the stars collapse into themselves. If gravity varies by the distance squared and gas pressure varies by the volume cubed, how could a cloud of hydrogen, or any gas for that matter, collapse into itself? Not to mention the fact that the heat increases which would also...

Homework Statement I guess this is a general question. in the book, it was originally in a normal form, but it was then converted into a double angle form, which I would have to use double angle forumla to compute, so what's the point of simplifying in the first place? Homework EquationsThe...

Homework Statement The question asks me to express the system shown in the image in state-space form, firstly by using the transfer function which was fine but the second part asks to express in state-space form by using the components A,B,C,D as the states x1,x2,x3,x4 Homework Equations U(s)...

Homework Statement I'm fine with the first part. Part b) is causing me trouble http://imgur.com/xA9CG5G Homework EquationsThe Attempt at a Solution I tried subbing in the solution y1 into the given equation, but I'm not sure how to differentiate this, i thought of using integration by parts...

Homework Statement I've used z* to mean z conjugate. Given the equation z + 2iz* = 8 + 7i, express z in the form a + ib. From SQA Advanced Higher Mathematics 2005 Exam Paper Homework Equations n/a The Attempt at a Solution I substituted a+ib and its conjugate in for z and z*, which, after...

Dear all: I am studying the finite difference method for black scholes equation(heat equation form), and my Prof. asked me to use the Crank Nicolson and Compact Scheme to do it. But I cannot reach second order for the space, i.e., x . The error only reach 1st order and sometimes not stable, it...

Hello, so while i was doing my own revision. When putting tan^-1(1/-√3) i receive the answer as -30. however, (im doing demovoier's theorem) in the revision book it says the answer should be 5π/6 . Help to brighten me? thanks.

Homework Statement [/B] "Absorption spectra of gaeous HCl show a line at 3.5 μm which results form molecular vibration (oscillation from interatomic distance)" How do I get the frequency of vibration from this? Homework Equations ? If I knew this, then I could probably solve the problem The...

Homework Statement you are given the standard form z = 3 - 3i Homework EquationsThe Attempt at a Solution so to convert this to polar form, i know that ##r = 3√2## but how do i find theta here? There are so many mixed answers it seems online that I can't tell... i know that ##(3,-3)## is in...

I am working on a lab where we were measuring how magnetic force between two parallel conductors varies with current and the separation between the two conductors. I need to find a formula for the dependence of Force on current to create a fit line on the data in my graph that shows the relation...

I am basically just rewriting a question that was posted on other forums. While watching videos of a MIT lecture on the eigenstates of angular momentum (video: '16. Eigenstates of the Angular Momentum II' by MIT OpenCourseWare) the professor visualized different spherical harmonics for low...