Value Definition and 1000 Threads

On ##x\in{(-1,1)}##, ##x\in{\mathbb{R}}##, ##\forall{n\in{\mathbb{N}}}##, ##x^{|2n|}=O(x^{|2n+1|})## Sugestions? Any answer is wellcome! Greetings, PF

I submitted this solution, and it was marked incorrect. Could I get some feedback on where I went wrong? Let S represent the event that Party A wins the senate and H represent the event that Party A wins the house. There are 4 cases: winning the senate and house (##S \cap H##), winning just...

I have found an answer to all of them (a-e) but I don’t know how to plot the function. Thanks!

Find the problem and solution below; Now the part i do not seem to understand from the given solution is the negative value i.e to be specific ##-0.06## In my understanding we shall have ##5## possibilities with each giving the correct mean value of ##3.48## which implies ##+0.06## from the...

I just read Feynman's The Value of Science and really appreciated his perspective. We generally value art, literature, and poetry but if you want to do science for the sake of science people take offense. It takes years to build a foundation before you can appreciate it, so the general public...

Hello, I noticed while trying to calculate the stardart gravity acceleration of the Earth that I never arrived at the defined value of 9.80665 m/s2 no matter that I calculate it with the equatorial radius, the polar radius, mean radius or the average of the equatorial and polar radius. With what...

Consider the bank identification number ## 237a_{4}18538 ##. Note that ## a_{9}=8 ##. This means \begin{align*} &a_{9}\equiv (2\cdot 7+3\cdot 3+7\cdot 9+a_{4}\cdot 7+1\cdot 3+8\cdot 9+5\cdot 7+3\cdot 3)\pmod {10}\\ &\equiv (205+7a_{4})\pmod {10}\\ &\equiv (5+7a_{4})\pmod {10}.\\ \end{align*}...

$\tiny{s8.5.5.2}$ Find the average value of the function on the given interval. $\sqrt{x}\quad [0,4]$ average value $\boxed{f_{ave}=\dfrac{1}{b-a}\int_a^b f(x) \ dx}$ so with $a=0$ and $b=4$ thus $\dfrac{1}{4-0}\displaystyle\int_0^4 \sqrt{x} \ dx \implies...

Hi A theorem states that if V(x , t) ≥ V0 then <E> is real and <E> ≥V0 for any normalizable state. The proof contains the following line <E> = (ħ2/2m)∫∇ψ*∇ψ d3x + ∫ Vψ*ψ d3x ≥ ∫ V0ψ*ψ Can anybody explain why that inequality is true ? Thanks

This is the code that i wrote Clear["Global`*"] Z = 500; W = 100000; G = 250; H = 100; K = 0.5; T = 30; L = 4000; P = 5; S = 2.5; Y = 1; A = 0.1; V = 2.5; J = 8000; f[x_] := 1/ x {(J*Z*x*(2*Y - x))/( 2*Y) - ((W + T*G) + ((L + T*P)*2*Z*Y*(1 - ((Y - x)/Y)^1.5))/ 3 + (H + T*S +...

Hi, i am really new to scratch programming so I am having difficulties in interpreting how does this code work ? I understand the first block set a = 5. But i don't know what is that " Repeat 10 " block. What does it do? So really stuck at that "repeat 10" so haven't gone further.

So, I have a hamiltonian for screening effect, written like: $$ H=\sum_{k}^{}\epsilon_{k}c_{k}^{\dagger}c_{k}+ \frac{1}{\Omega}\sum_{k,q}^{}V(q,t)c_{k+q}^{\dagger}c_{k} $$ And I have to find an equation for the time evolution of the expected value of the operator ##c_{k-Q}^{\dagger}c_{k}##. I...

I want to know how did author derive the red underlined term in the following Example?

I want to know how did author derive the red underlined term in the below given Example? Would any member of Math help board enlighten me in this regard? Any math help will be accepted.

Hi PF! Take a deck of cards. 26 are red and 26 are black. You draw cards randomly without replacement: if the card is red you gain a dollar, if black you lose a dollar. What's the expected value? My thought process was to look at simple cases and build to a 52 card deck. We know if ##r=0##...

There are 3 cases of getting 2 as smallest values: (1) Taking one card (n = 1) → Probability = 1/4 (2) Taking two cards (n = 2) → Probability = 1/4 x 2/3 x 2! = 1/3 (3) Taking three cards (n = 3) → Probability = 1/4 x 1/3 x 1/2 x 3! = 1/4 Total probability = 1/4 + 1/3 + 1/4 = 5/6 But the...

Find question here, My approach, using cosine sum and product concept, we shall have; ##\cos (A+B)-\cos (A-B)=-2\sin A\sin B## ##⇒\cos D-\cos C=-2\sin\dfrac{C+D}{2} \sin\dfrac {C-D}{-2}## ##⇒-3[\cos(A+B)-\cos(A-B)]=6\sin A sinB## We are given ##A=4θ## and ##B=2θ##, therefore, ##⇒-3[\cos...

My Answer: I am still beginner in this area so it s quite hard for me to understand this one. I am not sure what the output that this question asked me. I thought it might be asked about the value of x1, x2, x3, and x4

To find maximum of ##\frac{dp}{dt} \rightarrow \frac{d^{2} p}{dt^{2}}=0## $$\frac{k}{c}p'(t)(1-p(t)^{c})-\frac{k}{c}p(t)(c.p(t)^{c-1}).p'(t)=0$$ $$p'(t)=0~\text{or}~p(t)=\left(\frac{1}{c+1}\right)^{\frac{1}{c}}$$ For p'(t) = 0 → p(t) = 0 or p(t) = 1 So I have 3 values of p(t), how to know...

Can someone help me understand the answer to this differential? I have the following expression where Now what I can understand the differential of what will be the following?

I have a scatterplot that I'm trying to extract. I found 54 values out of 55. There is one "missing" value, probably because it is overlapped and I can't actually see. I have the MEAN and Stand Dev of the 55 values. Is there a way to reverse find the one that is missing? I mean, is there a...

T=R/C So: 0.003s=R/100μC R=3*10^-7Ω I am really confused with the equations I have to use

Hi, I'm new to the quantum world, and would like to know why mathematically can we say that for mean value of measurment: <T>=<phi|T|phi> ?

I know that friction on a surface has a maximum value. When an object is pushed with a force greater than the maximum friction value, the object will move. But when a free-falling object falls from a height, does air resistance have a maximum value as well?

The following IVP diff(T(x), x) = v/200*(45 - T(x)) + 0.015*(22 - T(x)) where T(0)=39 Describes the tempetatur T in celcius at the time x of a tub filled with water. A tub which is filled with hot water at rate of v l/min. Lets say I am told that a guy takes a 40 min bath, and during those 40...

Hello there, for the above problem the wavefunctions can be shown to be: $$\psi_{n,l}=\left[ \frac {b}{2\pi l_b^2} \frac{n!}{2^l(n+l)!}\right]^{\frac12} \exp{(-il\theta - \frac {r^2\sqrt{b}}{4l_b^2})} \left( \frac {r\sqrt{b}}{l_b}\right)^lL_n^l(\frac {r^2b}{4l_b^2})$$ Here ##b = \sqrt{1 +...

Using the given rule for the ##x_n##, write $$ \sum_l y_l = x_1 + \frac{1}{2} x_2 + \frac{1}{6} x_2 + \frac{1}{3} x_3 + \frac{1}{12} x_2 + \frac{1}{12} x_3 + \frac{1}{20} x_2 + \cdots + \frac{1}{n} x_n $$ $$ = x_1 + \sum_{n=2}^\infty \frac{1}{n(n-1)} x_2 + \sum_{n=3}^\infty \frac{2}{n(n-1)} x_3...

Hi there! I'm doing a project where I will be making 3 ecospheres (ecosphere = enclosed water ecosystem trapped in a jar) using water & substrate taken from 3 different natural sources. I'm hoping that microscopic life will flourish in these jars because my plan next is to study the lifeforms...

By chance, I have read a paper that left me in shambles. I would like you to help me figure out if it makes sense or not. I tried to follow Bridgman's logic, without a complete success. The extraordinary claim is that, if you start with a system consisting of a 1cm^3 copper cube where 2 faces...

Can we have two tangents (two turning points) within the given two end points just asking? I know the theorem holds when there is a tangent to a point ##c## and a secant line joining the two end points. Or Theorem only holds for one tangent point. Cheers

This is a problem created by me. Ok my approach on this, ##2x##=##-\dfrac {7}{2}####y## ##4x=-7y## ##\dfrac {x}{y}=\dfrac {-7}{4}## ##⇒x=-7##, ##y=4## therefore, ##5x+5y=-35+20=-15##. there may be more solutions...your thoughts guys.

How does one manipulate rational absolute inequalities? For example, I want to transform the absolute value inequality ##|x-3|<1## to ##\frac{|x+3|}{5x^2}<A \ ##, for some number ##\text{A}##, to find an upper and lower bound on the latter term using the constraint in the first term, and not...

Thanks

I don't need an answer (although I don't have sadly, it's from a test). I need just a tip on how to start it... i cannot use Taylor in here (##\ln(x)## is not Taylor function), therefore, its only MVT, but I don't know which point I should try... since I must get the annoying ##\ln(x)##...

In non relativistic quantum mechanics, the expectation value of an operator ##\hat{O}## in state ##\psi## is defined as $$<\psi |\hat{O}|\psi>=\int\psi^* \hat{O} \psi dx$$. Since the scalar product in relativistic quantum has been altered into $$|\psi|^2=i\int\left(\psi^*\frac{\partial...

How to prove that \[ \sum_{i=1}^{\infty}\frac{1}{2^{3i}}\left(\csc^{2}\left(\frac{\pi x}{2^{i}}\right)+1\right)\sec^{2}\left(\frac{\pi x}{2^{i}}\right)\sin^{2}\left(\pi x\right)=1 \] for all \( x\in\mathbb{R} \). Using graph, we can see that the value of this series is 1 for all values of x...

Is there any way to answer the question without just calculating it using calculator, maybe manipulating the number using logarithm properties? Thanks

Let the roots of the given quadratic equation be ##x=α## and ##x=β## then our quadratic equation will be of the form; $$x^2-(α+β)x+αβ$$ It follows that ##(α+β)=(r+is)## and ##αβ=4##. We are informed that ##α^2+β^2=6i ## then $$6i=(r+is)(r+is)-8$$ ... $$8+6i=(r^2-s^2)+2rsi$$ solving the...

Find the values of tan-1(1+2i). We can use the fact: tan-1z = (i/2)log((i+z)/(i-z)). Then with substitutions we have (i/2)log((1+3i)/(-i-1)). Then I think the next step would be (i/2)(log(1+3i)-log(-1-i)). Do we then just proceed to solve log(1+3i) and log(-1-i)? I'm just a little confused...

how does the first step use mean value theorem? I don't get it , can anyone explain , thanks.

OK, here once a sketch is done, we have two circles ##c_1## and ##c_2## with centre's ##c_1 (3,2)## and ##c_2 (7,5)## having radius ##2## and ##1## respectively. It follows that the distance between the the two centre's is given by ##L=\sqrt {(7-3)^2+(5-2)^2}##=##5## Now, the least possible...

Using the inequality of arithmetic and geometric means, $$\frac {x+y}{2}≥\sqrt{xy}$$ $$6^2≥xy$$ $$36≥xy$$ I can see the textbook answer is ##36##, my question is can ##x=y?##, like in this case.

My approach is as follows; $$\log_8 N= \frac {1}{2} p$$ $$\log_2 (2N)=q$$ $$→8^{\scriptstyle\frac 1 2} = N$$ $$ 2^q=2N$$ $$2^{\scriptstyle\frac 3 2} =N$$ $$2^q= 2N$$ then from 1 and 2, it follows that, $$2^{q-1.5p} =2,$$ on solving the simultaneous equation; $$q-1.5p=1, q-p=4$$, we get...

I've double-checked my equations and can't find what's wrong. First I calculate the partials: $$f_x = (2x - 2x^3 -4xy^2)e^{-x^2, -y^2}$$ $$f_y = (-2yx^2 + 4y - 4y^3)e^{-x^2, -y^2}$$ By setting f_x = 0 I get: $$x^2 = 1 - 2y^2$$ Then I calculate f_y = 0 $$-2yx^2 + 4y - 4y^3 = 0$$ I plug in the...

Something that has been on my mind a while. If a CCGT generates electrical power at 64% efficiency, and a COP4 heat pump, powered by that generator, generates 4 times the thermal power to electrical power, wouldn't that mean we can get 256% heat out of the fuel's calorific value? Maybe that is...

I tried using the equations above, but I wasn't really able to come up with an intuitive explanation. From my understanding, the electric field vector only varies in the x-y plane while the magnetic field vector only varies in the z-y plane. Also, both vary sinusoidally and both reach extrema...

Hi, PF "It is more difficult to draw the graph of a function whose domain has an endpoint at which the function fails to have an extreme value", states my textbook, "Calculus: A Complete Course" A function with no max or min at an endpoint Let...

summary: you toss six coins. if you have 3 heads, you don't get any money. if you have 4 or more heads, you get the number of heads amount of dollars. if you have 2 or less heads, you toss six coins again and get the number of heads amount of dollars. the number of ways to get 0 head...

\langle \phi_0| \hat{c}_{-k \downarrow} \hat{c}_{k \uparrow}\hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| - \hat{c}_{k \uparrow} \hat{c}_{-k \downarrow} \hat{c}^\dagger_{k \uparrow}\hat{c}_{-k \downarrow}|\phi_0\rangle = \\ \langle \phi_0| \hat{c}_{k...