$U,V$ are finite dimensional vector spaces. $W$ is a given vector space. Let $T\in L(U,V), S\in L(V,W)$ be linear transformations then it is to be shown that $\rm dimrange \,ST\le\min (dimrange\, T, dimrange \,S).$
Proof: For any $\rm x\in range \,ST, \exists y\in U$ such that $\rm STy=x$; and h...
How is prime notation treated for multivariable functions? So if I have x = f(a, b) and a = z(c, d) and c = t ^ 2 and d = 3t when I do x prime what is it derived with respect to? the variable t?
Given $a_n>0$.
If Series $\sum \sqrt\frac{a_n}{n}$ converges then $\sum a_n$ converges?
My approach:
Ihad used atmost every method like 1)AM and GM inequality
2) cauchy Swartz inequality
3) by definition of convergent series and so on..
I am not able to prove this statement.
I am thinking now thi...
Yeah I'm pretty new to this, probably don't use the conventional semantics. One more question? If I have functions w, x, y, z, u, v and t (my actual problem rn the teacher gave) and if w is related to x, y and z and if the three functions x, y and z are related to u and v and if the two function u and v are related to t, when w prime is written, does that mean dw/dt?
And also, would that be equal to (I will express partial derivative as pd) pdw/pdx * pdx/pdu * du/dt + pdw/pdx * pdx/pdv * dv/dt + pdw/pdy * pdy/pdu * du/dt + pdw/pdy * pdy/pdv * dv/dt + pdw/pdz * pdz/pdu * du/dt + pdw/pdz * pdz/pdv * dv/dt
So $\dfrac{dx}{dt} = \dfrac{\partial x}{\partial u}\dfrac{du}{dt} + \dfrac{\partial x}{\partial v}\dfrac{dv}{dt}$. So you should have six terms when you put everything together. It looks like you do.
@Silidrone That's what I want you to understand. Take $$\frac{dw}{dt} = \frac{\partial w}{\partial x}\frac{dx}{dt} + \frac{\partial w}{\partial y}\frac{dy}{dt} + \frac{\partial w}{\partial z}\frac{dz}{dt}.$$
What I have, is that A and B are events, and the subscripts indicate the order they occur, but the lecture notes given by my teacher simply proceed to apply this formula repeatedly without further definitions
I shot that formula here in the faint hope that maybe someone would recognize something similar somewhere
i know it's absurd to ask this with this little information
@TedShifrin Okay now I understand it with a help of a visualizer. But basically that's three terms and with chain rule, each will yield two terms, right? So in total I'd have 6 terms, correct?
Yes, @Silidrone, which is probably what you wrote down, but I seriously can't begin to read it. But I'm suggesting how you can think through it systematically.
@shin So the point is that the events cannot be simultaneous? ... I guess your teacher is trying to set this up to apply Bayes Formula, which is the standard approach.
But while using chain rule for each one of them (x, y and z) I will be partially deriving with respect to u and v which will also yield a derivation with respect to t, right?
Yes. It's like layers of an onion. As I said, think through how you did the derivative of $\sin^2(\sqrt x)$ in beginning calculus. First you take $\sqrt$, then you take $\sin$, then you square. So the chain rule peels those layers of the onion.
Yes, @Silidrone. It's correct. I never taught my students to do those tree diagrams. But it's telling you to do what I told you to do if you look at it right.
I just chopped it off below the $w$ to do the beginning.
I am utterly unsure, I'm guessing not given the fact we have the indication that these are a succession. At this point it's divination (and I will be graded on my divination skills, yay!) Instead of an explanation, we immediately get an example of an application
Suppose $P(A) = 6/15$ and $P(B) = 7/15$, and suppose $A, B$ are independent. Then, $P(A\cap B) = \frac{6}{15}\frac{7}{15} + \frac{6}{15}\frac{7}{15} = \frac{28}{75}$
and....... it is garbage! the full expression is $P(A \cap B) = P(A_1 \cap B_2) + P(A_2 \cap B_1) = \frac{6}{15}\frac{7}{15} + \frac{7}{15}\frac{6}{15}$. don't worry I recognize this is meaningless without further explanation (which i do not have, ha!)
yes there's a text, but the class notes aren't sourced and don't reference the text
@shintuku Mathematical STatistics and Data Analysis by Rice is one of the stalwarts when it comes to non-measure theoretic probability theory. And based on what you wrote above would be at the level you need for your course.
IDK what that nonsense is of the full expression you were given as an example, but as Ted said...beyond garbage
When I taught probability, I unearthed a free text or two that seemed reasonable. One is Bertsekas & Tsitsiklis (from engineering at MIT). Another is Grinstead & Snell.
One of my favorite elementary applications of the Bayes formula in probability is to understanding false negatives/positives in drug/disease testing. People have no understanding of or intuition for this.
I'm of the belief that the reason these folks act the way they do to an extent is because of their lack of understanding of things. So because these ideas have a higher level of complexity they tend to try and "simplify" them to concepts they understand which leads down a deleterious road.........I just wanted to try and shoehorn in deleterious into my convo.
I have a qustion about making the idea of a question I'm doing in you P-set ted, "formalish". It is in regards to the question of maximizing the product of one with the square of the next and the cube of the last with the constraint that the three numbers sum to 12.
So informally I have that there is a parabola opening up downwards. In the question prior I showed that if I have a closed unbounded set, $f$ being continuous, and $f(x) \to \infty$ while $\|x\| \to \infty$ with $x \in A$ that there exists a minimum. I don't see why I couldn't apply that here with the right manipulations: so $f(x) \to -\infty$.
Evening (adjust for you time zone preference). I've always found https://math.stackexchange.com/a/4245115/12879 to be a tickling answer to a cute problem. I wrote a straightforward computer implementation of it. I can't make heads or tails of the results.
@AkivaWeinberger I'd go with marinate. It always has a preferable outcome to me.
I do the regular standard deviation of my samples. The deviation is always too low? What kind of distribution of the samples would give a too low standard deviation. In conclusion how would I go about make a decent, math.stack question about my concerns, confusions and ailments?
My last result was e = 2.79231 +-0.0123527 Also, another strange thing is the running time is desperately hard to analyze. I recently started to introduce parameter time_of_day.
Cool. Thanks for rubberducking me. The method proposed by Jean-Claude seems to consistently overestimate e.
@geocalc33 Well |log2| is just a count of the number of binary digits. |Log10| is just a measure of how long it takes to write a number with a pencil. The natural just scares me, so it probably involves infinity and very large exponents.
