Expression Definition and 999 Threads

If a, b, and c are positive real numbers and a² + b² + c² = 3, what is the minimum value of the expression [1/(1+ab)] + [1/(1+bc)] + [1+(1+ac )]? Usage: A² + B² + C² ≥ AB+AC+BC Answer: 3/2

Howdy all, Let's say we have, in general an expression: $$ \int f(x) g(x) dx $$ But in through some machinations, we have, for parameter ##a##, $$ \int f(x) g(x) dx = \int f(x) g(a) dx $$ ...can we conclude that ## g(x) = g(a) ## ???? Thanks

My working follows here, Let ##(x-2)[Ax(x+3)+Bx] + (x+3)[Cx +(x-2)]=0## ##(x-2)(x+3) \left[Ax+\dfrac{Bx}{(x+3)}\right]+(x+3)(x-2)\left[\dfrac{Cx}{(x-2)} +1\right]##=0 ##⇒Ax+\dfrac{Bx}{(x+3)} = -\left[\dfrac{Cx}{(x-2)} +1\right]## ##Ax(x+3)(x-2)+Bx(x-2) = -[Cx(x+3)+(x-2)(x+3)]## On comparing...

The first part of this question asks for the maximum amplitude produced, which I found to be 12.86 Pa. I was able to set up the expression for the combined wave equations. However, I am struggling to understand how to set up and solve an expression to find the distance I could move one of the...

What is the meaning of the expression procedural grounds in the text: Butler attended Boise State University for a year.[9] In 1991, Butler was convicted of assault during his freshman year of college.[5] His appeal was unsuccessful on procedural grounds, as a judge ruled that Butler's defense...

TL;DR Summary: ##(1+ \frac1x)^2 - (1-\frac1x)^2## ##(z+2)^2 -5(z+2)## Upon simplifying the first I get ##\frac4x##. So isn’t the first expression fractional? Upon simplifying the second I get a Quadratic expression.

I simply just wrote down the definition of ##\Gamma'^a_{bc}##, and inserted the transformations of ##g'^{ad}##, ##g'_{dc,b}##, and the like terms. After some rearranging and cancelling out, $$\Gamma'^a_{bc}=\frac{\partial x'^a}{\partial x^e}\frac{\partial x^f}{\partial x'^b}\frac{\partial...

If the distance between the centres of two molecules is σ, then imagining a a cylinder with radius σ the number of molecules can be given by πσ²cn where c = average velocity. So mean free path can be given by λ = c/πσ²cn = 1/nπσ². But do I derive it from exp(-x/λ)?

In the d(i) part, we are asked to write an equilibrium constant. It was mentioned in the chapter that solids and liquids will not be added in the equation. So, I think that there is not going to be any Expression for this reaction as all the reactants and products are liquids. Is my answer right?

I'm reading Semi-Riemannian Geometry by Stephen Newman and came across this theorem: For context, ##\mathcal{R}_s:Mult(V^s,V)\to\mathcal{T}^1_s## is the representation map, which acts like this: $$\mathcal{R}_s(\Psi)(\eta,v_1,\ldots,v_s)=\eta(\Psi(v_1,\ldots,v_s))$$ I don't understand the...

Cohen Tannoudji pp 215 pp 225 pp 223 From above we can say that there exists a velocity operator ##\mathbf v=\frac{\mathbf p}{m}## ,whose eigenvalues are the observed values of velocity. 1. I've seen multiple times that we can't define velocity in quantum mechanics, but here I find that...

I know we're supposed to attempt a solution but I'm honestly super confused here. I think the second an third terms of the del equation can be cancelled out because there is only an E field in the r hat direction, so no e field in the theta and phi directions. That leaves us with ##\nabla \cdot...

I would like to arrive at the following expression for the quantity ##o_{\ell}## ( with "DM" for Dark Matter ): ##o_{\ell}=b_{s p}^2 C_{\ell}^{D M}+B_{s p}## with Poisson noise ##B_{s p}=\frac{1}{\bar{n}}(\bar{n}## being the average number of galaxies observed). the index "sp" is for spectro...

In first part,since every block of 4 consecutive symbol contain at least 2 a's The answer in notes is given (aa(a+b)(a+b)+a(a+b)a(a+b)+a(a+b)(a+b)a+(a+b)aa(a+b)+(a+b)a(a+b)a+(a+b)(a+b)aa)+ But this wont be true since if we choose aabbbbaa which is possible according to the above regular...

As the slide says, it's not 50-99 characters so what it is? Source: https://www3.cs.stonybrook.edu/~pfodor/courses/CSE307/L02_Programming_RE.pdf Also, please tell me how to learn regular expressions? Recommend some university/any courses to take for learning regular expression for backend...

Is there a way to obtain equation 9.42 (I is current, j is current density, and sigma is conductivity) in the following image (from Modern Electrodynamics by Andrew Zangwill, the part on electromotive force) besides using V=IR and substituting the line integral of j/conductivity for V? The...

I try involving differentiating ^2 but I get an expression of different proportionality.

So, I've recently played around a little with the Gamma Function and eventually managed to find an expression for the Beta Function I have not yet seen. So I'm asking you guys, if you've ever seen this expression somewhere or if this is a new thing. Would be cool if it was, so here's the...

hi ! I'm having a lot of trouble simplifying my expression for one of my homework questions. I know someone asked about this homework problem already, but the answers didn't really help me figure out how to simplify it.. I really have no idea what steps to take, and I've even consulted all my...

14) here i tried 3x^3 + 3/8x^3 to to get 27/8x^3 but the answer is 3/2 x^3, why? 15) here i did 9x^3/2 - 1/x^3/2 to get 9x^9/4 - 1/x^3/2 but that's not in the right form, how do i do it correctly?

The first law of Thermodynamics states that the change in Internal energy is equal to the sum of Heat gained or lost by the system and work done by the system or on the system . $dE=Q-W$...(1). In an Adiabatic process ,Q=0 . Therefore $dE=-W$ . Now...

Hello everyone, In trying to better understand how sailboats work, how they can sail upwind (not directly), how the go faster than the wind speed, I have been thinking about the magnitude of the wind force and its equation: $$F_{wind}= \frac {1}{2} \rho_{air} A v_{wind}^2$$ Instead of ##...

Problem statement : Let me copy and paste the problem as it appears in the text : Attempt 1 (from text) : The book and me independently could solve this problem. I copy and paste the solution from the book below. Attempt 2 (my own) : The problem should afford a solution using the second idea...

Hi, I wonder if I am on the right way """""""""""""""""""""""""""______"""______ F = X000 . ( M1 + ( M2 . X002 ) . X003

From the wikipedia page for Fibonacci numbers, I got that the matrix representation for closed-form expression for Fibonacci numbers is: \begin{pmatrix} 1 & 1 \\ 1 & 0\\ \end{pmatrix} ^ n = \begin{pmatrix} F_{n+1} & F_n \\ F_n & F_{n-1}\\ \end{pmatrix} That only works...

R=(01+010)* For it I made the below nfa which i believe seems correct. Plus the tutorials that I am following also make sure it’s correct. Q0 is initial state(forgot to mention in figure). R=(01)*+(010)* But idk how to convert this to NFA What will be languages accepted by this NFA? Won’t...

R=(01+010)* For it I made the below nfa which i believe seems correct. Plus the tutorials that I am following also make sure it’s correct. Q0 is initial state(forgot to mention in figure)...

but the solution site says otherwise-: https://cyberzhg.github.io/toolbox/nfa2dfa I feel correct mine also, but not sure. Check it please.

Question: Galileo released a metal ball from rest so that it could roll down a smooth inclined plane. The time t taken to roll a distance s was measured. He repeated the experiment, each time recording the time taken to travel a different fraction of the distance s. Write an expression for the...

My book states the answer to this problem is . I have gotten very close to the answer. My problem is my Tc/th and tc/tm are flipped compared to the solution. I feel like I am missing something in my algebra but can't see where I am going wrong. Could I get some help identifying where my...

I am trying to obtain this graph from this expression but somehow I cannot obtain it. In the same article it is given that $$r_s = r_v/c$$, $$r = R_c\theta / cos(\beta)$$ and $$C = (\log(1+c)-\frac{c}{1+c})^{-1}$$ So for Virgo cluster my values are, ##R_C = 15 Mpc##, ##r_v=2.2 Mpc##, ##c=4##...

$$n_\vec{k} = \omega a^2(\vec{k})\tag{1}$$ One way is to write the inverse Fourier transforms of the terms above. So, eqn (1) becomes $$\int\mathrm{d}^3x\ n(\vec{x})e^{-i\vec{k}\cdot\vec{x}} = \omega \int\mathrm{d}^3x^\prime\ a(\vec{x^\prime})e^{-i\vec{k}\cdot\vec{x^\prime}}...

Here's what I've done: Mentor note: replaced icode tags with code=python and \code pair. def valid_braces_placement(s, L): if len(L)==0: return False string = '' for element in L: string = string + str(element) D = ['+','-','*'] return...

A rocket of initial mass m0 is launched vertically upwards from the rest. The rocket burns fuel at the constant rate m', in such a way, that, after t seconds, the mass of the rocket is m0-m't. With a constant buoyancy T, the acceleration becomes equal to a=T/(m0-m't) -g. The atmospheric...

I've been struggling, no matter what I try to do I can't come up with a solution. The answer is in situ, but in my mind it would work perfectly fine. The only thing we would do is just use an RNA probe complementary to the gene we are trying to detect and it should light up the moment the gene...

Suppose a program computes or reads-in a symbolic mathematical expression like ##2x^2 - xy + y##. What's an effective way to cause the program to implement the expression as a function (e.g. implement ## f(x,y) = 2x^2 - xy + y##) when the programmer doesn't know in advance what the expression...

Given a singular matrix ##A##, let ##B = A - tI## for small positive ##t## such that ##B## is non-singular. Prove that: $$ \lim_{t\to 0} (\chi_A(B) + \det(B)I)B^{-1} = 0 $$ where ##\chi_A## is the characteristic polynomial of ##A##. Note that ##\lim_{t\to 0} \chi_A(B) = \chi_A(A) = 0## by...

simplify $\left(\dfrac{a^2b^3-2a^{-3}b^3}{2a}\right)^2=$ OK this could get confusing quickly but I don't think we want to square it first

Hi PF In my textbook, the Spanish 6th edition of "Calculus", by Robert A. Adams, at Chapter 3.2, Theorem 1 states: If ##x>0##. $$\dfrac{d}{dx}\ln{x}=\dfrac{1}{x}$$ PROOF For ##x>0## and ##h>0##, ##\ln{(x+h)}-\ln{(x)}## is the area of the shaded part (...) Regard the shaded area at Figure...

Hello! I found this formula in several places for the total angular momentum of a particle with intrinsic spin 1/2 and angular momentum l=1 in the non-relativistic limit: $$\frac{1}{\sqrt{4 \pi}}(-\sigma r /r )\chi$$ where ##\sigma## are the Pauli matrices and ##\chi## is the spinor. Can someone...

k(q_1-q_2)/(R^2+x^2)^3/2

$\tiny{8.aux.27}$ Simplify the expression $\dfrac{{\cos 2x\ }}{{\cos x-{\sin x\ }\ }} =\dfrac{{{\cos}^2 x-{{\sin}^2 x\ }\ }}{{\cos x\ }-{\sin x\ }} =\dfrac{({\cos x}-{\sin x})({\cos x}+{\sin x\ })}{{\cos x}-{\sin x}} =\cos x +\sin x$ ok spent an hour just to get this and still not sure suggestions?

eg ##\cos (\sin x)## Asking this question out of curiosity.

See attachment. I don't understand the solution given by David Cohen. 1. Note: x^2 is nonnegative for any real number x. This is because any value for x when squared is positive. Yes? 2. If x is greater than or equal to 0, then I can say that -2 - x^2 is negative in value, right? 3. What...

See attachment. I don't understand the solution given by David Cohen. I am sure this is a shortcut explanation. I don't like shortcut explanations. 1. What in the problem indicates that x > 1? 2. What in the problem indicates that x < 2?

I have an expression of Matter Angular power spectrum which can be computed numerically by a simple rectangular integration method (see below). I make appear in this expression the spectroscopic bias ##b_{s p}^{2}## and the Cosmic variance ##N^{C}##. ## \begin{aligned}...

Hello, I would like to know the right expression for the expression of variance of Shot noise in spectroscopic probe. Sometimes, I saw ##\sigma_{SN,sp}^{2} = 1/n_{sp}## with ##n_{sp}## the average density of galaxies, whereas my tutor tells me that ##\sigma_{SN,sp}^{2} = 1/n_{sp}^{2}## , so I...

I'm not quite sure where to start. If someone could help me I would very much appreciate it.

Derive an explicit expression for all ##f\in\langle q\rangle\subseteq \mathbb{Q}[x]##. I think it's doable and was wondering if there is a published formula?

I took a stab at the question, but I don't think I did it right. I know that Work = Change in Energy thus, Work = final energy - initial energy Because there is no energy at the final position, then final energy = 0 (I'm not sure if this is correct). So I got the equation W = 0 - (1/2kA^2 +...