Unit Definition and 1000 Threads

The problem is: 1. Assume ONE BOX W*W*H1 filled with a dielectric material with dielectric constant eprs. The vertical walls are assigned periodic boundary conditions and the other two faces are Floquet Ports (One TE mode and one TM mode). 2. Assume a transmission line that is L1 long with a...

I was thinking about how CCD's and CMOS's almost exclusively come in multiples. To be clear, what I am referring too is how each chip is fabricated to feature many pixels. Does the fundamental unit of a CCD/CMOS have a name "commercially"? Something like a "single pixel camera"? Looking at...

Got an email today proposing naming the unit of momentum for Emmy Noether. It would be either the noether or the emmynoether: some people feel it is important to emphasize her gender. My understanding is it would not be SI, but US-only. Rereading the enail, it might not even be MKSA; I suppose...

We know that if cartesian coordinates ##(x,y)## (see figure alongside) are rotated to ##(x',y')## about the origin by an angle ##\theta## counter-clockwise as shown, the rotated coordinates are given by $$\begin{align*}x'&=\cos\theta \;x+\sin\theta \;y\\ y'&=-\sin\theta \; x+\cos\theta \; y...

Define a u-coboundary operator u-d on a cubic p-cochain f as cup product multiplication U from the left with a unit 1-cochain: (u-d)f = 1Uf. Because 1U1=0 and associativity of U, we have (u-d)^2=0. What is the relation of u-d and the standard coboundary operator d? Are they the same?

i tried integrating the stuff but it didn't work what to do

Hi PF Forum, How are you guys. I hope you guys are doing fine. I often come a statement like this. This earthquake is as strong as 1 million atomic bombs. This volcano is as strong as some million atomic bombs. And now in this video. Chicxulub impact "... this impact carried more than 10...

Hi, I struggle to calculate the unit of the rate constant that is [m^3/sec] rate constant of excitation by electron impact is as follows: where, e=charge of electron [Coulomb], me=mass of electron [kg], ε=energy [eV], σe(ε)=cross section for electron impact [m^2], f(ε)=electron energy...

One normally sees that the definition for conventional current as defined as the amount of positive charge that passes a point over unit time. However, why could we not define conventional current as the amount of positive unit charge that passes a point over unit time. I added in unit there...

Questions regarding Primitive Unit Cell (and what I think the answer are, correct me if I am wrong) 1. Can there be more than one Primitive Unit cells for the same crystal? yes, Wigner Seitz cell always will exist. There can be other primitive Unit cells along with Wigner Seitz too. But...

My AC has begun to emit a rather foul smell, which has led me to geek out on the inner workings of its indoor section. For example, how does the condensed moisture make its way to the drain pan? I am assuming that the condensation happens on the fins of the evaporator. But in that case, what...

**Problem:** Find parametric equations for a simple closed curve of length 4π on the unit sphere which minimizes the mean spherical distance from the curve to the sphere; the solution must include proof of minimization. Can you solve this problem with arbitrary L > 2π instead of 4π? There...

A poster on another forum has fitted a large number of 230V 50Hz 24W nominal LED tubes, apparently designed to be plugged straight into the old fluorescent fittings. He has found, by clamp meter measurement, that the new LED lights appear to be consuming 80W, not the claimed 24W. A number of...

I'm seeking some reassurance I've understood the question since there are no answers given in the book and I'm getting a bit tied up in knots. What I am struggling with most is there are three per unit values for impedance already noted on the diagram without qualifying what ##Z{base}## is or...

Evaluate the Fourier series $$\frac{1}{\pi^2}\sum_{k = 1}^\infty \frac{\cos 2\pi kx}{k^2}$$ for ##0 \le x \le 1##.

HI, I am studying linear algebra, and I just can't understand why "Unit vectors u and U at angle θ have u multiplied by U=cosθ Why is it like that? Thanks

Hi, I've just calculated the orbital precession for the earth using the sigma formula of general relativity. $$ \sigma=\frac{24 \pi^{3} R^{2}}{T^{2} c^{2}\left(1-e^{2}\right)}=\frac{24\pi^3×1.5×10^{11}}{3×10^7×3×10^8(1-0.0034^2)}=0.012 $$ What is the unit of the result? Degrees per century or...

Hey guys, I got an XRD data for my sample and want to determine its density. This requires finding lattice parameters. However, I'm not certain about the crystallographic system of my sample. How can I determine the type of the unit cell my sample has from XRD data (I can use GSAS II if...

How do you convert ##1.66 \times 10^{-7} s ## to ##ns##? I know the answer is ##166 ns = 160 \times 10^{-9} s## however what is the thought process? Many thanks!

How do I calculate the unit normal vector for any metric tensor?

There are units of measurement smaller than our eyes can see, but that we use in science. But, what is the smallest one that we can see and work with regularly? I mean...on a traditional ruler, it'd be a millimeter, right? But, we can obviously go smaller - even if a ruler doesn't mark it...

Hi all I have acoustic noise level value in dB vs 1 dyne/sq cm e.g. -10 db vs 1 dyne / sq cm. I need to convert it into dB ref 1upa unit. How i will convert it? Thanks & Regards

A hollow rectangular tube with cross-sectional area of 1 barn and a length of 1 megaparsec will hold a quantity of material of approximately 2/3 of a teaspoon.

suppose you write, clockwise, n numbers (or "units", doesn't matter) in a circle. you then color, clockwise, each k-th number. you do this until you've colored all n numbers, or until you've reached an already colored number. let x be the number of colored numbers. i've figured that if...

How to transform density unit in natural units $MeV^4$ to SI units $kg/m^3$, Here's my trial: ##MeV^4 = (10^6)^4 ~ eV^4 = 10^{24} ~ eV^4 ##, ## eV = 1.6 * 10^{-19}~ kg~ m^2 / sec^2, ## ##MeV^4 = 10^{24} ~ 1.6^4 * 10^{-40} ~ kg^4 m^8 / sec^8 ## This is not simply ##kg/m^3##! Any help how to...

Hopefully the symbols I am using are standard. I will define them upon request. I have a theorem that says, given a difference equation \left ( \sum_{j = 0}^m a_j E^j \right ) y_n = \alpha ^n F(n), we can define a polynomial function \phi (E) = \sum_{j = 0}^m a_j E^j such that \phi (E) y_n =...

Homework Statement:: The SI definition of unit of time says the following. "The second is the duration of 9192631770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom." Relevant Equations:: None I know an...

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Not sure how to show that because ##\vec{v} = |v|\hat{v} = 3|e|\hat{e}##, but since ##\vec{e}## is a unit vector we know ##|e| = 1## so our equation now becomes ##\hat{v} = \frac{3\hat{e}}{|v|}##. So, we're left to the task of showing that ##|v| = 3## in order to conclude that ##\hat{v} =...

I need a justification that ##|\dfrac {dr}{dt}|##=##\dfrac {ds}{dt}## cheers guys... all the other steps are easy and clear to me...

I looked at this question and i wanted to ask if we could also use; ##C## =## c_2 ##(##-\dfrac {3}{2}i## +## j - 3k)## ... cheers This problem can also be solved by using the approach of cross product ##A×B##...

I tried to looking at lower-dimensional cases: For ##n=2## we have $$(x(t),y(t))=(cos(t),sin(t))$$ For ##n=3## we define two orthogonal unit vectors ##\vec{a}## and ##\vec{b}## that are orthogonal to ##\vec{u}##, leading to $$(x(t),y(t),z(t))=(cos(t)(a_1,a_2,a_3)+sin(t)(b_1,b_2,b_3))$$ It seems...

Hydrogen production rate is typically measured in current density, but what is the area in which the current is divided by to find such a value?

Divergence formula $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}} \frac{\partial}{\partial q^{j}} (A^{j} \sqrt{G})$$ If we express it in terms of the components of ##\vec{A}## in unit basis using $$A^{*j} = \sqrt{g^{jj}} A^{j}$$ , we get $$\vec{\nabla} \cdot \vec{A}= \frac{1}{\sqrt{G}}...

So when evaluation the cross product of the velocity of the charge and the unit vectors associated with the point I am getting v x r = j x [ i + j]. Well j x j is 0. j x i = -k, but yet the answer is positive. Why is this?

Good morning to all of you. For my diploma thesis I have to compare the combined production of electricity and heat from a biomass gasification unit and from a biomass combustion unit. So I would like to ask you if you know data or refer me to papers so that I can compare these 2 technologies...

I need an equation to graph a sine wave that act like a unit circle but only positive numbers. so I need it to be 0 at 0, A at 90 , 0 at 180, A at 270, 0 at 360, and A at 450 and so on and so on... Now I know sin(0) is 0 in degrees and sin(90) 1 and I know if you Square a number is...

I am doing a learning project by writing a simulation that includes capacitance and current flow amongst capacitors that may potentially be in parallel. I don't care about certain details yet - dissipation factor, frequency dependent effects, temperature. Tiny capacitences within diode junctions...

(a) I did (7.07*4.1)-(-7.03*3.94)=56.7 with this method I got this answer correct in my first attempt. (b) This where I seem to have gone wrong. I used a · b = (axbx +ayby) then I used a = sqrt(ax2+ay2) to get a single number for the answer. Filling in the numbers 7.07*-7.03 + 4.94*4.1 =...

For example is this correct : 19icap.4(-i cap) = 76(i.-i)= 76 Or is it , take - out. Then -76(icap.icap)= -76 Is it -76 or 76 ?

The unit vector r roof points in the direction of increasing r with phi fixed; phi roof points in the direction of increasing phi with r fixed. Unlike x roof, the vectors r roof and phi roof change as the position vector r moves. What I was thinking of the image is Although, I was thinking why...

Now, i am extremelly confused about all this thing. More preciselly, i can't understand how 1.29 was obtained. It was used the A representation? How do he uses it? There is something to do with the canonical basis?

Ok, so basically my task is to calculate the apparent diffusion D based on experimental data, which is kind of easy, BUT the problem is with unit of b... Gamma is gyromagnetic ratio and can be expressed as: (s*T)-1 or MHz/T --> but I am not so sure, teacher did not specify Delta is the pulse...

Hi, what is a unit vector? I mean, it is ##\hat{A}=\vec A/|A|##. A dimensionless vector with modulus (absolute value) one, I've read somewhere. So, dimensionless with modulus. Isn't that a contradiction? I mean, absolute value regardless dimension? Am I out of context?. ##\Bbb R^3## is a...

I am trying to check if an expression is dimensionless. If it is, then I have done things correctly. However, I am stuck on how to deal with a (Debye^2) term. How can I break it down to find out if it cancels out with the other units I have left? I know this is probably a trivial question...

Hi, I was working through a filter design problem and got stuck on a concept. Scenario: Let us say we have the following pulse transfer function and the sampling frequency is ## f_s = 50 \text{Hz} ##. G(z) = \frac{1}{3} \left( 1 + z^{-1} + z^{-2} \right) The zeros of the transfer function...

Is there a function that takes positive values only in the unit ball not including the boundary points defined by the set ##\{x^2+y^2+z^2<1\}##, and ##0## everywhere else?

Hey! :giggle: Make a drawing for each of the values of the angle below indicating the angle at the unit circle (in other words: $\text{exp} (i \phi )$) and its sine, show cosine, tangent and cotangent. Give these four values explicitly in every case (you are allowed to use elementary...

Hey! :giggle: Let $\lambda\in \mathbb{R}$ and \begin{equation*}a=\begin{pmatrix}1 & 2 &-1& \lambda & -\lambda \\ 0 & 1 & -1& \lambda & 2\\ 2 & 2 & 1 & 1 & 3\lambda-1 \\ 1 & 1 & 1 & \lambda & 5\end{pmatrix}\in \mathbb{R}^{4\times 5}\end{equation*} (a) Let $\lambda=1$. Determine a Basis...