Form Definition and 1000 Threads

This is a bit confusing...conflicting report from attached wolfram and symbolab. Which approach is correct?

For this, Does someone please know how setting ##P_1## and ##P_2## true makes the CNF true? If I see ##P_2## true, then it ##(true + false)## since it is negated. Therefore, should they be setting ##P_1## true and ##P_2## false? Many thanks!

Now i just need some clarification; we know that quadratic equations are equations of the form ##y=ax^2+bx+c## with ##a,b## and ##c## being constants and ##x## and ##y## variables. Now my question is... can we also view/look at ##x=y^2+2y+1## as quadratic equations having switched the...

For this, Does someone please know how do we derive equation 9.9 from 9.8? Do we take the limits as t approach's zero for both sides? Why not take limit as momentum goes to zero? Many thanks!

Zero does not have an inverse. And y=0 does not have an inverse. Does the wave form y=0 for all x have an inverse?

This is the question as it appears on the pdf. copy; ##z=2\left[\cos \dfrac{3π}{4} + i \sin \dfrac{3π}{4}\right]## My approach; ##\dfrac{3π}{4}=135^0## ##\tan 135^0=-\tan 45^0=\dfrac{-\sqrt{2}}{\sqrt{2}}## therefore, ##z=-\sqrt{2}+\sqrt{2}i## There may be a better approach.

This is basically a physics problem but I will try my best to highlight the mathematics behind it. Suppose I have two functions: $$T(z,B)=\frac{\text{z}^3 e^{-3 A(\text{z})-B^2 \text{z}^2}}{4 \pi \int_0^{\text{z}} \xi ^3 e^{-3 A(\xi )-B^2 \xi ^2} \, d\xi },$$ $$\phi(z,B)=\int_0^z...

Looking at the proof of the Schwarz inequality in Margenau and Murphy, you will see what I attached. Gamma is asserted to be positive (OK). Given that the usual "quadratic form" solution would read "-(B+B*) .....". The sign does not seem correct to me as shown. In a fact B+B* = 2Re(B) and...

Besides the energetic preference (lower threshold, and more phase space above)? Antineutrino capture is a weak process, so it can and does change quark flavour. p+ν=n+e+ is actually uud+ν=udd+e+ that is u+ν=d+e+ But given enough energy (like cosmic ray neutrinos), do antineutrinos also get...

Proof: Suppose that the only prime numbers of the form ## 6k+1 ## are ## p_{1}, p_{2}, ..., p_{n} ##, and let ## N=4p_{1}^{2}p_{2}^{2}\dotsb p_{n}^{2}+3 ##. Since ## N ## is odd, ## N ## is divisible by some prime ## p ##, so ## 4p_{1}^{2}\dotsb p_{n}^{2}\equiv -3\pmod {p} ##. That is, ##...

In the following%3A%20https://pubs.rsc.org/en/content/articlehtml/2013/sm/c3sm00140g?casa_token=3O_jwMdswQQAAAAA%3AaSRtvg3XUHSnUwFKEDo01etmudxmMm8lcU4dIUSkJ52Hzitv2c_RSQJYsoHE1Bm2ubZ3sdt6mq5S-w'] paper, the surface velocity for a moving, spherical particle is given as (eq 1)...

I am looking for experimental nucleon form factors from electron-nucleus scattering. Is there any compliations or tables? In 'The proton charge radius ', H. Gao and M. Vanderhaeghen, Rev. Mod. Phys. 94, 015002 (2022), p. 24, there is 'world data on the proton and also the neutron'. In 'Form...

Please help me in understanding the history of physics regarding the atomic or non-atomic, say continous, structure of physics. In my years at school I grew up with physics of ultimate simplifications. Everything was a point "particle", like the moon revolving around the earth. Like the apple...

Can four (or more) electrons form a completely antisymmetric joint spin wave function？

logyx + logxy = 3/2 Solution $$\begin{align*}\log_{ y }{ x } + \log_{ x }{ y } &= \frac{ 3 }{ 2 } \\ \log_{ x }{ y } &= \frac{ \log_{ y }{ y } }{ \log_{ y }{ x } } \\ \log_{ y }{ x } + \frac{ 1 }{ \log_{ y }{ x } } &= \frac{ 3 }{ 2 } \\ \left(\log_{ y }{ x } \right)^ { 2 } + 1 &=...

My goal is to numerically solve (finite elements using FEniCSx software) Ohm's law ##\vec J = \sigma \vec E##, where ##\vec E = -\nabla V##, ##\vec J## is given (the current density is given on some boundaries), and ##\sigma## is algo given (the electrical conductivity). The problem is solved...

Hi I was just wondering about the suvat formulae and a question popped into my head, which I'd like someone to try and explain the reason as to why please. So I know that when we have a formula such as F=ma or v = u + at, you can evaluate the magnitude of both sides and arrive at a scalar...

Everywhere on the net, there is no image that the expansion has a departing point, through an expansion in spherical form... Why is that?

I learned that for a bilinear form/square form the following theorem holds: matrices ## A , B ## are congruent if and only if ## A,B ## represent the same bilinear/quadratic form. Now, suppose I have the following quadratic form ## q(x,y) = x^2 + 3xy + y^2 ##. Then, the matrix representing...

$$\begin{bmatrix} 4& 3 \\ 6 & -2 & \\ \end{bmatrix}= \begin{bmatrix} 1& 0 \\ a& 1& \\ \end{bmatrix}⋅ \begin{bmatrix} b& c \\ 0& d& \\ \end{bmatrix}$$ ##b=4, ab=6,⇒b=1.5, d=-6.5, c=3## $$\begin{bmatrix} 4& 3 \\ 6 & -2 & \\ \end{bmatrix} = \begin{bmatrix} 1& 0\\ \dfrac{3}{2} & 1 & \\...

Hello everyone, I was looking at the light matter interaction Hamiltonian and I worked out a simple calculation where I was surprised to see that I had to introduce an explicitly non-local vector potential if I want to go further: $$\langle\psi|...

Proof: Suppose for the sake of contradiction that the only primes of the form ## 8k-1 ## are ## p_{1}, p_{2}, ..., p_{n} ## where ## N=16p_{1}^2p_{2}^2\dotsb p_{n}^2-2 ##. Then ## N=(4p_{1}p_{2}\dotsb p_{n})^2-2 ##. Note that there exists at least one odd prime divisor ## p ## of ## N ## such...

Square matrices are closed under addition and their own form of multiplication, but in general do not commute. What algebraic structure then describes this, along with polynomials of matrices and allows us to amend with other operations, such as differentiation or integration defined on these...

Hey all, I was wondering if there was an equivalent closed form expression for ##\Gamma(\frac{1}{2}+ib)## where ##b## is a real number. I came across the following answer...

When I observe a qubit's state, decoherence happens such that I find the qubit in a particular state. After I cease observing a qubit's state, what physical process causes a fresh superposition of states to develop? Is zero-point energy at least a contributor?

Can there be NaClO sold in solid form? If yes how common is it and its comparison in practical use to the most common liquid form?

When I solve the equation sometimes the answer in the answer keys is different but the same. Why do they do that? For example: After solving the equation I got 1/√2 which is the same as √2/2 because we multiplied it by √2/√2. Is there any good explanation why the book writer mathematicians like...

The attached picture below is the note I solved halfway through. Please tell me the entire process of getting to the correct answer.

Hello! I have some electrons produced from a 3D gaussian source isotropically inside a uniform electric field. The electric field guides them towards a position sensitive detector and I end up with an image like the one below (with more electrons on the edge and fewer as you move towards the...

Hi, In my calculus book,I found this vector form of line equation in space (bold means vector): Given point (x1,x2,x3) lies on line L & v=<a,b,c>, then equation of line is : r = <x1,x2,x3> + t <a,b,c> with t any number. Now, my question if I plug any number for t, then result will be vector...

I was reading another article when this headline from June 22 caught my attention. Collisions hint that four neutrons form a transient isolated entity https://www.nature.com/articles/d41586-022-01634-x An experiment firing helium-8 nuclei at a proton target has generated evidence that four...

Hello, I would like to know, if there's a closed form solution to the following problem: Given a sum of say, 3 sines, with the form y = sin(a.2.PI.t) + sin(b.2.PI.t) + sin(c.2.PI.t) where a,b,c are constants and PI = 3.141592654 and the periods in the expression are multiplication signs, what...

Actual statement: Proof (of Mr. Tom Apostol): We will do the proof by induction on ##n##. Base Case: n=1. When ##n=1##, the matrix of T will be have just one value and therefore, the characteristic polynomial ##det(\lambda I -A)=0## will have only one solution. So, the Eigenvector...

I have a equation with a double sum. However, one of the variables in one of the sums comes from a stochastic distribution (Gaussian to be specific). How can I get a closed form equivalent of this expression? The U and Tare constants in the equation. $$ \sum_{k = 0}^{N_k-1} \bigg [ \big[...

clear; lambda = 3e-2; x = 4 * pi/lambda * linspace(eps, 15, 100000); T = 5e-3; t = [0:0.001e-3:T] ; % 0.1:1e-3:0.1+T]; u = 3; a = 4*pi/lambda * u; for i = 1:length(x) Z(i) = sum(-((cos(a.*t) - cos(x(i).*t)).^2 + (sin(a.*t) - sin(x(i).*t)).^2)); end % Z1 = csc((a+x)/2) .*...

For part (a); $$\int e^{3y} \,dy=\int 3x^2\ln x \,dx$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}+k$$$$\frac{e^{3}}{3}=e^3-\frac{e^3}{3}+k$$$$\frac{e^{3y}}{3}=x^3\ln x-\frac{x^3}{3}-\frac{e^3}{3}$$$$e^{3y}=3x^3 \ln x-x^3-e^3$$ You may check my working...i do not have the solution.

Antarctica's only active volcano shows how CO2 allows volcanoes to form persistent lava lakes at the surface https://phys.org/news/2022-05-antarctica-volcano-co2-volcanoes-persistent.html

Proof: Suppose ## n>1 ## is an integer not of the form ## 6k+3 ##. Then we have ## n=6k ## for some ## k\in\mathbb{Z} ##. Thus ## n^{2}+2^{n}=(6k)^{2}+2^{6k} ## ## =36k^{2}+2^{6k} ## ## =2(18k^{2}+2^{6k-1}) ##...

Considering two interacting particles in 3d, the corresponding Hilbert space ##H## is the tensor product of the two individual Hilbert spaces of the two particles. If the particle interaction is given by a potential ##V(\mathbf r_1 -\mathbf r_2)## ,what is the corresponding potential operator...

Proof: ## 17=2^4+1 ## Now I'm stuck. How should I find the correct answers to this problem?

Hi everyone! Hope your week is going well. I'm an ex-physics and math student, now getting my PhD in mathematical biology, and I've recently come back to the subjects because I miss them and feel like it'd be fun to get proficient in some of this again. I've been mostly working on building my...

Proof: Suppose ##a=8^n+1 ## for some ##a \in\mathbb{Z}## such that n##\geq##1. Then we have ##a=8^n+1 ## =## (2^3)^n+1 ## =## (2^n+1)(2^{2n} -2^n+1) ##. This means ## 2^n+1\mid 2^{3n} +1 ##. Since ##2^n+1>1## and ##2^{2n} -2^n+1>1## for all...

Proof: Suppose a=n^4+4 for some a##\in\mathbb{Z}## such that n>1. Then we have a=n^4+4=(n^2-2n+2)(n^2+2n+2). Note that n^2-2n+2>1 and n^2+2n+2>1 for n>1. Therefore, every integer of the form n^4+4, with n>1, is composite.

For part (a), ##z##=##\dfrac {3+i}{3-i}## ⋅##\dfrac {3+i}{3+i}## ##z##=##\dfrac {4}{5}##+##\dfrac {3}{5}i## part (b) no problem as long as one understands the argand plane... For part (c) Modulus of ##z=1## and Modulus of ##z-z^*##=##\frac{6}{5}i##

Proof: Suppose p is a prime such that p=n^2-4. Then we have p=n^2-4=(n+2)(n-2). Note that prime number is a number that has only two factors, 1 and the number itself. Since n+2>1 for ##\forall n \in...

Proof: Suppose p is a prime such that p=n^3-1. Then we have p=n^3-1=(n-1)(n^2+n+1). Note that prime number is a number that has only two factors, 1 and the number itself. Since n^2+n+1>1 for ##\forall...

There is a problem from a Russian textbook in classical mechanics. Consider a scalar equation $$\ddot x=F(t,x,\dot x),\quad x\in\mathbb{R}.$$ Show that this equation can be multiplied by a function ##\mu(t,x,\dot x)\ne 0## such that the resulting equation $$\mu\ddot x=\mu F(t,x,\dot x)$$ has...

Dear Everybody, I am confused by ##1/n C##, where C is a cantor set in base 3 and ##n\geq2##. I can understand the construction of the normal Cantor set. How do I comprehend this set with this extra condition. Do I multiply the set with ##1/n## or not? Thanks, Cbarker1 mentor note...

I'm having trouble trying to calculate how the answer below was achieved from an example i have seen, see below: 208L0 - 2.5L90 x 27.42L36.9 which is then calculated to 255.12L-12.4. I have tried converting everything to rectangular form, subtract where required and the convert back to polar...

I grok, am NOT asking about, the answers below. Rather, how can I calculate the final answer DIRECTLY, without division? I don't know why my Latex isn't rendering here? Please see https://math.codidact.com/posts/285679. Orange underline 1. Unquestionably, $\color{#FFA500}{4 \times 3/2} = 3!$...