So we know a function f is differentiable if there is best linear approximation for it. That is we have $u(x_0 + h) - u(x_0) - ah = h\phi(h)$ where $\phi(h)$ goes to zero as h goes to zero.
here $a$ is ofcourse the derivative at $x_0$
So the above is exactly what this is saying right ?
And yeah I mean, I guess I'm happy with how things are panning out, though there should at least be more classes like his, where it's a fully rigorous multi class
@EricStucky I read A&M and solved first 4 chapters I put my solutions in pdf, but it is good. I have both A&M and Eisenbudd as hard copies I am planning to finish both this year. But first I am gonna do A&M then Eisenbudd.
I'm trying to get my academics in order, damin, so I've been rather short on time for the last month. In particular, I'm asking this because I'm flailing around really hard to convince myself to study algebra (and Eisenbud is not helping :/)
This looks like a question that could be posted on the main site. I really don't know a method of solving that, that isn't trial-and-error or similar or involves using a computer, at least for now. (I really don't know that much at all)
Like I said before, I don't know that much, and that includes complex numbers in exponents, I am aware of e^i*pi = -1, but that is the limit of what I know
I need help in understanding Tension. Research effort: Read the topic through three books (Mechanic Part 1, Principles of Physics, Laws of Motion) but still have some doubts in my mind.
@Abcd The two T's are perpendicular because they arise from two different stresses, one from the block and one from the hand. They happen to be equal in ($magnitude$ not $direction$) due to the fact that as the string is treated as massless, the net force on any differential element must be zero from newton's first law.
@Abcd I don't understand what the second question means.
@Abcd The forces shown there are the forces exerted on the pulley, from the physical situation, there must intuitively be a force on the pulley. We can be assured that the force is T in magnitude as in a massless string the only force it exerts by virtue of contact is tension. We know its directions as tension is always directed along the string.
@Abcd Consider a small part of the string, forget it being massless for now. If it moved with an acceleration 'a' , it would then satisfy the equation $f_{net}=ma$ (Newton's second law, not the first, I keep getting the order mixed up). Allowing the mass 'm' to reach zero (negligible) requires that $f_{net}=0$
I think you would understand if you looked at which objects exert the forces on what.
I fail to understand why there's tension("pulling force") on the pulley also (and that too equal in magnitude to the "other tension" due to the block), when it's massless. @TimTheEnchanter
@Abcd Firstly, if the pulley is frictionless and the rope just slides on it, it doesn't have to be massless. Secondly the tension exists because the rope is wrapped about the pulley, but it is balanced by forces acting on the pulley exerted by whatever it is the system is attached to (table, floor , etc).
> The suspension of PBMC cells was collected from the culture flasks into a sterile tube and centrifuged for 5 minutes at 200 g. (I wonder if one must use a space to separate the number from the g letter)
Nevermind, solved, I was trying to find a knot theory pdf that was linked here a while ago
user84215
9:38 AM
Please come into Practicing MathJax and share your knowledge and skill by posting worth noting points and sophisticated examples to improve this room and also to survive this room from being deleted or frozen.
@Abcd "In physics, tension, as a transmitted force, as an action-reaction pair of forces, or as a restoring force, is a force and has the units of force measured in newtons (or sometimes pounds-force). The ends of a string or other object transmitting tension will exert forces on the objects to which the string or rod is connected, in the direction of the string at the point of attachment."
@Abcd The second $T$ in the picture is actually the tension along the string which acts opposite to the force the pulley applies on the string, consider the mass $m$ to be the point of attachment of the string, I think.
I think the picture is a little poorly drawn. The two $T$'s should be forces in two different reference frames (reference frame of the mass, reference frame of the pulley).
I think you're taking the terms massless and frictionless too literally, its neither (truly) massless nor frictionless, but the mass and friction coefficient are both low enough to be neglected.
@TimTheEnchanter yeah, actually, this is a much better explanation.
@Abcd Tim has the right explanation. The net force on the pulley has to be zero (it's massless), but you can still have various forces applying to it, as long as their algebraic/vector sum is 0.
@Abcd Yep, pretty much every way we represent things is for our convenience.
In fact strictly speaking the force arrows (like semi said) should originate from the block and the pulley as those are the points of application, but for the purpose of clarity they seem to have been shifted.
Is thinking about whether to post an MSE that is based on the investigation yesterday thus asking about what is essentially a $\aleph_{a},a > 1,GCH=TRUE$ based number system...
Well, recall that yesterday I am investigating a continuum decimal base number system
$$s=\int_0^{\infty} \frac{a(x)}{10^x}dx$$
and at the end of the analysis, found that the set full of these elements are isomorphic to the Laplace transform of any function evaluated at 1
$$F(1)=\int_0^{\infty}f(x)e^{-x}dx$$
Another thing observed is how for a countably infinite place decimal number system restricted to places less than 1, the image of such map is the interval $[0,1]$, but for the continuum case in base $b$ it is $[0,\frac{b-1}{\ln b}]$. Thus logically the next step is to ask:
For a base $b$ number system of length $|S|=\aleph_{a},a > 1$ what will $u$ be in the image $[0,u]$ of the map
$$\sum_{x\in S}f(x)b^{-x}$$
and whether there exists a least $a$ such that $u$ is unbounded
The issue is that right now, the question is too preliminary and sketchy to be made a nice question on MSE thus I need to think
well, the issue is that, equal by the value they evaluated to, or equal by the structure (e.g. all inverse powers of n) ? There seemed to be so many possibilities
because there are many (uncountably many?) series that can be evaluate to the same value
and there are at least countably many series that has similar structures (e.g. the collection of series with summands of the form $\frac{1}{n^p}$ for each prime $p$)
For more concrete example, the collection of all alternating series that sum to 0 forms a class based on their sum or their structure
What can be said about an infinite series $\sum^{\infty}_{n=1} a_n$ with $a_n \neq 0$ for all $n$, whose sum is zero ? Does such a series exist ? If yes, can you give an example ?
Can't think of any good way to integrate it except to expand the two exponentials using taylor series
The issue is that with operators as part of the integrands, it heavily reduces the possible ways to simplify it unless there is some symmetry in A,B,C such as commutativity can be exploited
Let $A$ be an $n \times n$ matrix. Then the solution of the initial value problem
\begin{align*}
\dot{x}(t) = A x(t), \quad x(0) = x_0
\end{align*}
is given by $x(t) = \mathrm{e}^{At} x_0$.
I am interested in the following matrix
\begin{align*}
\int_{0}^T \mathrm{e}^{At}\, dt
\end{align*}
f...
@Secret Is asking conceptual questions/ the interpretation of a particular question allowed? I mean am I allowed to ask "why something is so" in a problem, without wanting the solution from anyone?
If A,B,C are independent of x, then perhaps one can integrate by parts to split this up into two operator exponential integrals (which based on my basically nonexistent knowledge on linear operators in functional space, might mean convergence need to be handled carefully though if the operators are bound, it will be easier and more similar to the finite case)
If A,B,C are also functions of x, then I think I am out of luck, other than picking a basis and integrating each entry term by term
If A,B,C are both functions of x, is unbounded and has no nice structure, then I think at my current …
Another thing about the integrand is that $e^{xA}Be^{(1-x)C}$ is really tempting to somehow turn it into the conjugate $e^{xA}Be^{-xA}$ (or better $e^{xA}Be^{(1-x)A}$ so that Duhamel's formula can be exploited somehow...) by multiplying a suitable factor of $e^K$
I now think the question is not to be solved with formal manipulations of the integral and use of the formula he provides, but rather you have to do actual operator theory in order to show the formula he wants to use
and here it is necessary to know what kind of operators $H$ and $V$ are
usually $H$ should be self-adjoint with spectrum bounded from below, if $V$ is continuous and self-adjoint then the statement is true I think
(true in the sense that one has pointwise convergence of the integral to the funciton on the left)
From the context of his question, I think H is the Hamiltonian and V is some perturbation potential. The Hamiltonian is usually self adjoint but I am not very sure about the pertubation potential V cause it can be anything smooth
(that said, I don't remember whether one allows very discontinuous V. That term only needs to be small)
Indeed, these are my thoughts also, in physics Hamiltonians are bounded from below and self-adjoint, but I think pertubation potentials need not be bounded operators usually
you can do it for the case $H, V$ bounded from below I think
I wrote an answer but I dont know enough about dissipative crap to know whether or not $H+V$ has spectrum bounded from below if $H$ and $V$ have it, so I just treated the case $V$ being a bounded operator^^
hello, I'm studying the expected value in probability, there is a formula $ E[X] = X \times P_{X} (x) $ and the formula (for $ a $ as a constant number) E[a] = a;
so $X\times P_X(x)$ is just another notation for the expectation value of $X$ or does it define something that is conceptually different but is equal to it?
are being multiplied, like as you have 5 outcomes, 1,3,6,6,7 and you wanna count the number of them, you say : 1(1) and 1(3) and 2(6) and 1(7) and the expected value or mean will be : (1+3+12+7) / 5
for probabilities, we don't know what out come is gonna happen and how many times each is gonna happen so instead of the number of outcomes we multiply the probability of each : x . p(x)
-------------------------------------------------------- there is a formula $ E[X] = X \times P_{X} (x) $ and the formula (for $ a $ as a constant number) E[a] = a;
so we can say that E[X=a] = X.P(x) X =a, P(x)=P(a) so E[a] = a.p(a) but we had E[a] =a, what happens to p(a)?
the expectaion $E$ has as argument always a function that takes the $X$s and returns numbers. For example $E[X^2]=\sum_i x_i^2 P(x_i)$, $E[a]=\sum_i a p(x_i) = a\sum_i p(x_i)=a$ if $a$ is a constant
when you want to calculate $E[X=a]$ you need to ask yourself: what is the function $X=a$?
this function is $1$ when $X=a$ and it is zero else
so the formula $E[f]=\sum f(x_i) p(x_i)$ will tell you that you get $\sum_i (x_i = a) p(x_i) = p(a)$
why the function "$X=a$" is $1$ when $X$ is $a$ and $0$ else?
the notation $X=a$ for a function is not standard, you could also take to to be $a$ when $X$ is $a$ and $0$ else, but choosing the above convention seems the most natural to me, since the expression is asking "is $X=a$?", to which the answer is either yes (1) or no (0)
also here https://ch.mathworks.com/matlabcentral/answers/68621-is-there-a-place-that-has-matlab-projects-that-increase-in-difficulty-so-that-i-can-practice-my-skil @SimplyBeautifulArt
What can be said about an infinite series $\sum^{\infty}_{n=1} a_n$ with $a_n \neq 0$ for all $n$, whose sum is zero ? Does such a series exist ? If yes, can you give an example ?
Let $A$ be an $n \times n$ matrix. Then the solution of the initial value problem
\begin{align*}
\dot{x}(t) = A x(t), \quad x(0) = x_0
\end{align*}
is given by $x(t) = \mathrm{e}^{At} x_0$.
I am interested in the following matrix
\begin{align*}
\int_{0}^T \mathrm{e}^{At}\, dt
\end{align*}
f...
