I’m not asking you to do my homework. This is a legitimate brain burning problem. This problem is a problem that nobody in my campus (not any math teacher) could wrap their heads around. I also asked some random people who like math and nobody could solve it. I’m asking people to prove that my solution is wrong but they can’t. I’m not asking for an alternative solution that is correct, because I acknowledge that there is a correct solution to it and thus I lost 2 points. I know the solution. I’m asking you to prove WHY my particular solution is wrong. Here’s the question: https://ibb.co/pd0bNnx
So I am in the second year of my degree in mathematical physics. Next semester I’ll be taking an introductionary course in quantum mechanics aimed at physics majors. The course won’t be “mathematician”-level rigorous but I want to understand the content on a completely rigorous and proof based level. I have a bit of time during the semester break to get some mathematics under my belt I need for this.
So far I took the following classes relevant to this:
- Two semesters of Linear Algebra and Real Analysis (both “proof based”)
- A semester of Functional Analysis where we coverd Banach and Hilbert spaces, basic operator theory and the spectral theorem for unbounded normal operators on Hilbert spaces
- A semester of Measure and integration theory where we covered basic measure theory, the Lebesgue integral and L^p spaces
- A semester of Abstract Algebra where we covered basic group, ring, and field theory (this won’t be too relevant for quantum mechanics I suppose).
What other topics of mathematics do I need to get a good understanding on introductionary quantum mechanics and it’s mathematical foundations? I guess partial differential equations are important but I don’t know how much I need to know about them. What other “big theorems” like the spectral theorem are used in quantum mechanics I need a thorough understanding of? It’d be awesome if someone could help me!
Sorry if this is more suited to r/learnmath, but I thought there might be a quicker response here. I’m aware that there are great notes which help when reading through baby rudin for example, so I was wondering if there was a collection of these for more textbooks?
Hi everyone! I am considering graduate studies in Statistics and curious about the opportunities to earn statistical research, consulting experience while getting a little case out of it. So far, I can only imagine a grad student works as a TA or work towards the thesis. Please help me learn more information regarding this topic. Thank you so much!
So decision theory is basically the theory of what decisions rational agents make, for different definitions of rational (think game theory, except you usually talk about 1 player or completely cooperative games). One proposal was that rational agents maximize the expected value of a utility function. This means that there was some function from outcomes to real numbers representing how much they “liked” that outcome. When making a decision, the agent figures out the probability that each outcome for each option they could choose, and then they choose the option that maximizes the expected value of the utility of the outcome. This theory was able to explain why people avoided risky financial situations even if on average they would make a net positive of money. Their utility function “cared” more about not losing money than gaining it. This intuition behind why someone would have a utility function like that is the fact that quality of life scales sublinearly with wealth (if you gain $1,000, you’ll probably be rather happy. If you lose $1,000, you might end up homeless).
Anyways John von Neumann and Oskar Morgenstern proved a theorem about this hypothesis called the Von Neumann–Morgenstern utility theorem. They introduced a new concept called VNM-rational. A VNM-rational agent satisfies 4 axioms, stated in the article. The theorem then proved that if an agent is VNM-rational, then there exists some utility function (commonly called the VNM utility function) such that the agents decisions coincide with the decisions that maximize that utility function, even if they are not aware of it. This was surprising, since the axioms for VNM-rationality seemed much weaker than the expected utility hypothesis.
Anyways, enough background. Why I am writing this post? Well, the cool thing is that the VNM theorem is constructive. In fact, for a small number of outcomes, you can efficiently find the VNM utility function of an agent by answering a series of “would you rather” type questions. The activity below tells you how to calculate it.
Note that this activity will only give a completely accurate result if you are VNM rational. (This is not a given, since some people think that being VNM rational is not required to be rational in the regular sense of the word.) Also, this activity is meant for fun; although the results may be interesting, they will not turn you into a hyperrational robot person or anything like that. Feel free to give silly answers to the questions.
List some possible outcomes your life could have. Ideally you’d list all of them, but because that could make the list too long, only list a couple. For example, “creating a robot dinosaur army, taking over southern France, and then dying peaceful” could be an outcome.
Determine which outcome you like the best, and which you like the least. If there is a tie, choose one arbitrarily. However, if you are indifferent between all the outcomes, assign each outcome a utility of 0, and jump to step 4. Otherwise, call the best outcome A, and the worst outcome B. Assign A a utility of 100, and B a utility of 0.
Now, for every other scenario C, determine C’s utility using these questions. (Note that there is a shortcut at the end. It is quicker, but more complicated conceptually.)
(OPTIONAL) Pick any positive real number q and any real number r (it does not matter which ones you chose). For each outcome, replace its utility with qx + r, where x is the old utility.
And you are done! Now, for any decision, figure out what probability you will have for each outcome given each option. The one that maximizes the utility function is the VNM-rational choice, assuming you did the above steps correctly.
Now, some more observations. You probably noticed that during option 4, you could had many different options for q and r. This is because the Von Neumann utility function is not unique. Any q and r will give you a valid one. However, you can get from any Von Neumann utility function to another using step 4. In other words it is unique up to addition of constants and multiplication of positive constants. This seemingly small problem actually has a lot of implications. Utilitarians, for example, propose that we should take actions that maximize the average or total utility across all individuals (or some other aggregate of utilities). The fact that VNM utility is not unique means that you can not simply calculate everyone’s VNM utilities and use that in your calculation. If you want to use VNM utility functions, you have to specify which one to use for each person. (If you are using average utility, you actually only need to be unique up to an additive constant, since the additive constants will not change the ordering of decisions under the average. Multiplicative constants will still change the result though.)
Another thing you probably noticed is that this is not a super practical way of making decisions. First off, your life usually has an extremely large number of outcomes, and second off, calculating how small decisions (like which shoe to tie first) effect their probabilities is basically impossible. Although this activity is mostly for entertainment value, there are some cases ways to make it more practical decision making tool, if you are so inclined. One thing you can do is only use it for one, big, decision at a time, and list the outcomes of the scenario it effects, instead of looking all the way to the end of your life. For example, you could try using these to decide what major to choose based on how it will affect your income (calculating utils of income is much faster than a general set of outcomes, since utility is usually monotonic (but not linear) with respect to wealth). For more complicated decisions, you can use what is known as an Influence diagram, which breaks a situation done into chunks, and then depicts how they are related. These are apparently actually used by large businesses, although I am having trouble finding any specific examples (which would happen either way, since businesses are usually pretty secretive about decision making). They have been widely studied by decision theorists, though.
So, what do you all think? Anyone want to share their results?
I need help identifying a test based on a chart. It’s a 5×5 chart with the same labels on each axis. Where each cell crosses itself, there is a 1. There are no values in the cells past the 1s which fall like a staircase from the top left corner to bottom right. Any idea what type of test I’m looking at
Also this is for a business statistics class at university if that helps
Say I have datapoints scattered across a 2d plane. The data is normally distributed along the x axis, but the mean, standard deviation, and size of the distributions all vary across the y axis. Knowing this, how can I figure out the mean, standard deviation, and size as functions of y? I created a gif to help describe what I’m asking.
By now I imagine most everyone has seen the Numberphile’s video for the Trapped Knight with Neil Sloane of OEIS.org. Link below.
Well I’m curious if folks would like to work together towards implementing a new ruleset. Perhaps the Rook and Queen might be interesting to study also. Here goes the suggested rule change.
So on the infinite spiral, we start at 1 like usual and hop the knight’s pattern to the lowest available tile. But in this variant, we’d combine for the sum of the tile we’re on and the tile we travel to. So the first jump is to 10 from 1, 1+10=11. Here we would cross 11 off wherever it is to be found on the board, and remain to jump from 10. So 1 and 10 can both be returned to still, in fact 1 will always be returnable to because no 2 numbers will sum to it, unless perhaps something at the end game occurs and it travels back to 1 and gets trapped there, then it maybe it might be worth considering that 1 has finished. What does everyone think? Would this be difficult to implement into a program displaying the board with an option to select the size of the board displayed?
EDIT: The only additional rule is no returning to the previous square you just traveled from, obviously.
Also, let’s say to travel to the lowest available you need to still have that the sum available to be crossed off, which comes up once moving from 5, where you’d like to move to 8, except 13 has already been removed, and so therefore you’re stuck hopping to 14, and crossing out 19.
There are actually a few end game variants that are springing into mind, but one step at a time.
Thanks for reading!