timo

The professor is preparing you for the next step where he/she says "and now assume 2 and 5 were any arbitrary value: all steps I did still hold true, so...": "When Tv1 = av1 and Tv2 = bv1: If a!=b then v1 and v2 are not colinear". Either that or for some other reason or for no reason whatsoever.

I share MigL impression that this forum is dominated by non-science topics. And not in a good sense, where people talk about other hobbies or exchange baking recipes or funny pictures of their cats. But in the sense that most activity goes into preaching/trolling (religion or culture war stuff) and people taking the bait. The problem is not only the OPs but also the people reacting to nonsense posts. Phi speaks of discussions that "appear to be a good time". And I can understand that from a moderator's perspective. But I find it a bit embarrassing. I feel this trend and the inverse relation between quality of a thread and the number of reactions it gets has been there for a long time, like 10+ years. And it is one of the reasons why my activity here has been declining to the point that ... well, my last post was in 2021. In fact, not completely coincidentally one of my last posts here contains a rant about that situation I an not sure if the "good old times" really actually existed at all. If not, then it could just be that your interests change, an everyone else feels everything is fine. It would be interesting if we could get some objective measure of this. But let's assume the trend exists: I do not think that "our interests have changed" is a likely reason - at least not on the forum level. Granted, people's interests and life situations do change. One of my drops in activity was when I started my PhD and the topics I was interested in became too narrow for a discussion forum (and also not Sci-Fi enough - not sure I saw a thread about Statistical Physics on this forum ever). Plus, much better opportunities to discuss these interests opened up. But you would expect that people dropping out are replaced by new people. I think the trend may be a symptom of something else: Forums are dead and people have moved on to other forms of communications. That's just a thought of course. I don't have data to verify it.

Joigus already mentioned is implicitly, but I think it's worth pointing it out explicitly: When it comes to the exact integral, the different methods with equal-width rectangles that approach dx->0 do approach the same limit (with some exceptions that are not relevant here). And this common limit is called "the integral". For many important functions, e.g. polynomials, we know how to compute the limit exactly. And we don't even care about the rectangle construction in these cases, and just jump to the known solution - which does not depend on the exact rectangle-method that has been used. Now: When it comes to functions for which we do not have a known solution, we often have to fall back to what is called "numerical integration". In this case, we use a single, small dx, but we do not take the limit dx->0. Then, we brute-force the approximation by summing up all the individual rectangle results (computers are very good at doing stupid, repetitive tasks very quickly). In this case, the method you propose (f(x0)/2 + f(x1)/2) is indeed considered superior over the simplest approximation (f(x0)). The calculation I showed in my previous post still holds, but N now is a fixed number that does not become arbitrarily large. In practice, the method you proposed is usually the simplest choice for numerical integration that someone with a bit of knowledge about numerical integration will use. Numerical integration routines integrated in programming languages or software libraries will often use even more complicated rules to calculate each rectangle (arguably not even a rectangle, but still dx-sized segments and a representative mean function value for each segment). Bottom line: Don't worry if you don't understand everything in this post. My point is: Your idea about improving the rule to calculate the integral is actually very good. It does not matter much for the definition of the integral (well .. it does in the sense that the definition of the integral would be broken if it gave a different result). But for numerical integration on a computer, your idea is actually very relevant.

Try to formulate your collection of mathematical symbols in English and then try to understand what it says. Then, adapt it according to the other situation. Translate back into mathematical symbols after that.

Yes, both methods approach the same limit. In this case, you can explicitly write that down: Assume you integrate from 0 to 1, and you split the range into N intervals of equal length. In the first case, your integral approximates as [math]I_{1, N} = \frac 1N \sum_{i=0}^{N-1} f(i/N) = \frac 1N \left( f(0) + f(1/N) + f(2/N) + \dots + f((N-1)/N) \right)[/math]. In the second case, your integral approximates as [math]I_{2, N} = \frac 1N \sum_{i=0}^{N-1} \frac 12 \left( f(i/N) + f((i+1)/N) \right) = \frac 1N \left( \frac 12 f(0) + f(1/N) + f(2/N) + \dots + f((N-1)/N) + \frac 12 f(1) \right) [/math]. If you compare the terms, you notice that [math]I_{1, N} - I_{2, N} = \frac 1N \frac 12 (f(0) - f(1) ) = \frac{f(0) - f(1)}{2N}.[/math] So whatever finite numbers f(0) and f(1) are, the difference between the two ways to approximate the integral becomes tiny when N becomes large enough. Btw: This editor is horrible: Preview should preview the rendered tex, not show me the raw tex I typed for different screen sizes. I want my editor from ten years ago back.

I was already chuckling when you posted this, and the thread seems to prove this part of your prediction wrong. In my experience, it is the non-scientific content that get the most attention on sfn. Probably because it is easier to respond to. I certainly put less effort into this post than into science-related posts. Possibly even less than into my one-liner that is the first reply in this thread when I thought this was a genuine question.

I helped my colleagues to move a Volkswagen E-Up for ~50 km between two cities about eight years ago - highway in one direction, smaller roads in the other. The car was used in a field test in the city I lived in, so it made sense for me to just go from/to work by car instead of by train. I drove on a cold but typical German winter day. Turning on the heating approximately halved the remaining range, and I ended up turning heating on and off periodically during the trip. I found that experience quite impressive back then, because I was not aware of such basic issues as heating before. I imagine that there is room for improvement when it comes to thermal insulation, and that newer cars perform better (the E-Up is an electric variant of a combustion engine design, so heating may not have been a big issue in the design). But that's speculation.

I agree that a lot of information is readily available. In fact, I propose that young, critical and open-minded scientists should use these information to answer the questions they might have.

Perhaps reading up on Climate Change would answer some of these questions to the young, critical and open-minded scientist.

In most cases, two days to reply to an email is not considered much (and neither would a week).

I felt the urge to quote this. Fun fact: A Greek influencer named Archimedes became famous for his obsession for lifting heavy objects with little force by using a lever.

I also think the bandwidth to transfer the satellite images that show the whole earth being flooded was very limited back then 📧. On topic, in case it wasn't clear by now: Since all the water for flooding already is on earth, and already pushes as weight on the ground (including the ice), you should expect no significant effect on the stability of the ground when it rains. Also, as studiot said, if all ice melted, the water would not cover all of the land. Here's the first Google hit I found regarding this: https://www.nationalgeographic.com/magazine/article/rising-seas-ice-melt-new-shoreline-maps (seeing the maps it is kind of funny that Australia is one of the few coal power fans in the world).

Where would the water for the rain come from?

What you talk about seems more like a method for calculation to me than an actual measurement. You can indeed calculate the volume of a body by taking a surrounding volume and then subtracting the parts of the surrounding volume that does not belong to the body (in your language: the volume filled with gap values). This may in some cases be an efficient method, e.g. for a block of stone with a cylindrical hole. In the general case, if you have a generic way to calculate the amount of gaps then you could probably use that method to calculate the volume of the body in the first place. This, as you already mentioned, is the tricky part. The most generic way to approach this is dividing a volume in tiny blocks of simply-calculated sub-volumes and use these blocks to approximate the total volume from the sum of these blocks. For example, for your stone you could use small cubes and use laser scans to determine if a cube contains rock or not. If you make these blocks finer and add a few Greek letters, then you get what in Mathematics is called "integration", which is the theoretical basis for such measurements. In many semi-geometric structures you can get good results by gluing together and substracting known geometric structures (as in the case with the block with a cylindrical hole).

I never heard that mentioned as an issue for actual projects. But yes, maybe. How, specifically, do you think that lack of water is a problem for solar power projects?

Or you take the engineering approach and just read-off provided numbers: https://globalsolaratlas.info/map. If I remember correctly, the tool even has a "mark an area and integrate" functionality. I'd like to say something constructive here. But I find it hard to make out what this thread is about, or to add anything meaningful on this very vague level. I mean: Yes, solar panels generate electricity. Yes, you can put them on rooftops. And yes, there is lots of sun in the equatorial regions. And to Swansont's post: Yes, there are problems in detail. Transport and storage are somewhat generic problems, and they are at least easy to handle - any scenario calculation in the planning phase will implicitly include them. Rather specific problems to solar power in the Saharan regions seem to be sand, corruption and a perception of modern-day colonialism when rich white guys try to tell Africans what they should be doing. The idea of exploiting the solar power opportunities in the Sahara region is obviously not new. My personal favorite idea in "think big" is a world-grid with a solar power belt around the whole equator, btw. In Germany, the Desertec initiative was very well known. They planned to generate electric power in Africa and export it to Europe (sounding like modern-day colonialism: check). To my knowledge, the project died in 2014 when most major industry partners quit. I don't know why it failed, but the common rumors are about drop in renewable energy generation costs within Europe and worries about generating your power in regions that are considered politically unstable (-> Arab spring and the civil wars that followed and are still ongoing).

When it comes to experiments on black holes, it gets a bit tricky. You cannot produce them in the lab, and none of these things exists naturally on earth. In fact, we had put lots of efforts into even detecting anything in the universe that we concluded/agreed must be a black hole. While double-checking if even that has happened, yet, I ran into an article claiming that 2019 was the first time you could get an image of a black hole (https://www.sciencemag.org/news/2019/04/black-hole). Don't pin me on the accuracy of that statement, but my point is: It is already pretty challenging to detect black holes in the first place, so you cannot expect these objects to be as well researched experimentally as, e.g., a laser diode. I don't think there is any accepted idea how something could evade the event horizon of a black hole. I am not sure if even tunneling would work, but that is certainly an interesting mathematical question. Today, I think the idea of Hawking radiation is a widely accepted mechanism of how black holes emit energy (in addition to the normal jet emissions). I've never bothered to read up on in, but popular scientific depictions of it describe it as an effect happening outside of the event horizon. As for white holes: Formally, they appear if you allow the distance-to-center coordinate to become negative, which creates a second volume of spacetime with similar properties as the original one, except that nothing can enter the event horizon (whereas before nothing could exit from there). You can fantasize about this being a white hole, the exit of a black hole. And you can fantasize about making this passable, which is then called a worm hole. Usually, you'd have exactly one white hole exit for a black hole. But well, maybe something interesting happens if you make the radius coordinate a complex value. I don't quite see how that would relate to the CMB. Also, keep in mind that the CMB is pretty well understood and, in contrast to black holes, experimentally measured with an extreme precision (https://en.wikipedia.org/wiki/Cosmic_microwave_background#/media/File:Cmbr.svg) - with the caveat that it is only measured within a single solar system.

There are lots of answers to that question. For some, the answer is that there is a pedophile ring drinking children blood in the backroom of a pizzeria (*). If you haven't found your answer in this thread you should invest the time to be a bit more specific about the question. (*) Rumor has it that this actually is some people's answer about the question what climate science is about, too.

I understand that you are saying that light travels faster than sound or nerve signals in the body. From that you jump to "I think that the point when the action is created is ‘ground zero’ and might be faster than light as it takes up no time. It’s hard to explain but I think you get the idea". Well, ... I do not get the idea. I even had to look up the term "ground zero". But it did not help to know that it means a point on the surface that a bomb explodes over. I do not understand what you are trying to say, so it is hard to give constructive advise here. I think it could make sense to think if "the point where the action is created" really has a speed, and what that would be. If it has no speed, then you cannot compare it to the speed of light. Note that "speed" in this context means distance traveled per unit of time, not "time it takes for something to happen".

The quoted part of the text you provided seems correct to me. In particle physics, there is a concept of a parton. A parton is the thing that does the core interaction when a proton is shot at something else in a particle collider ("core interaction" is the part of the process with the highest energy, the one that you draw Feynman diagrams for to describe it). Experimental physicists have a very pragmatic approach to these partons: They define a probability to get a certain parton (a quark or a gluon) with a given momentum from the proton. These probabilities can be taken into account when simulating/calculating collider events. The probability function is called the parton distribution function (PDF). These PDF can be measured in experiments, and they also contain heavier quarks. Just Google for them yourself, the first hit I found (no guarantee for quality) is figure 1 of http://www.scholarpedia.org/article/Introduction_to_Parton_Distribution_Functions. So from the perspective of someone doing particle physics experiments, it is probably correct to say that a proton contains all kinds of stuff. There are a lot of reasons I can think of why that could be missing the big picture (is this an effect of perturbation theory? How is the probability to heavier quarks related to the CKM matrix? How is the remnant of the interaction, the Underlying Event, handled? ...) but I lack both time and skill to write about this. Bottom line: Saying that a proton is more complicated than three quarks held together by a gluon pit is correct. There is at least one mathematical model that describes it as a magical box containing random objects which may well be the most-used mathematical proton model in the world. I agree with the author that specifically to understand LHC physics, "it is three quarks" is not enough. I don't think that "plus zillions of gluons and quark, anti-quark pairs" is the key to enlightenment, either.

I find it really hard to get a coherent picture of the opening post. I see three different aspects that trigger different tones of response, some of which are pretty redundant with the replies given already. So maybe I'll just briefly touch all three aspects to show why at least I have problems with getting a clear picture of this thread. First, there is the relatively long explanation about sums or averages not giving the full information about the individual components that contribute to them. That is correct, mathematically trivial and well known to everyone working in any field of complex systems. It is also pretty banal, and applies to pretty much every science or society related number you ever hear in the TV news: the gross-domestic product, the number of Covid-19 infections, salaries in the IT sector, the time that kids spend on social media, ... . Now, admittedly, there are a lot of people who, for different reasons, appear to limit the discussion of a topic essentially to these numbers. So for this aspect of the opening post I am torn between a sarcastic "great work, Sherlock" and an honest "it is great that you are aware that this one number is not the full picture". I think the relative volume of the sub-optimal example pushed a few people towards the former reaction. Second, there is the aspect of the specific role of an average temperature in climate science, or more specifically its role in the climate change debate. For me, this would be a great topic of debate and learning. I worked as a scientist in a somewhat related field for several years, and still my understanding of it is very basic and with a lot of "that's how I imagine it is". I'll not formulate a coherent story for this post, but just throw in a few imho relevant pieces: In the context of the greenhouse gas effect the average temperature is a very sensible, experimentally-measurable observable with some weaknesses (energy stored in the oceans). Climate scientists don't model average temperatures but create sets of future scenarios for the evolution of complex systems. The evaluation of these scenarios cannot be reduced to a single number that tells you how good or bad the scenario is. What you can do is group your scenarios according to some meaningful parameter, see what typical scenario effects are for that parameter, and then have some delegates barter about how bad you want it. Remember: The problem with climate change is not the increase in the mean temperature, but increase in extreme weather conditions, change in habitability on the planet, the self-enforcing mechanism (loss of reflective ice, melting of permafrost, methane emissions from the oceans), and possibly a bit of land loss from rising sea waters. And finally, there is the third aspect of the opening post which really turns me off: The first half of the first sentence and the last sentence. Thanks to them, the post with potential for an interesting discussion comes in a wrapping that says "troll, ignorant or political agenda inside" to anyone with a bit of experience in social media. So despite giving the OP a huge benefit of doubt with the time I put into this post I don't want to leave them without comment: 1) "Climate scientists are concerned with deviations in the average global temperature": No, they are mostly not. Type "climate science" into Google and check out what they do. 2) "Has there been any research in this area [of what is really going on]?": Yes. There is a complete scientific discipline called Climate Science that is concerned with these questions.

C++ supports state-of-the-art random number generation. So it would be easiest to use a c++ compiler for the code (I expect that it should compile the C parts just fine) and pick an rng that is provided by the language standard library. So: Pick up to one: http://www.cplusplus.com/reference/random/ . I do not expect that the choice of the rng matters for this case. But it is a good habit to never run a Monte-Carlo simulation without a good rng, and including one is really easy in most programming languages.

a) The variable is a map (in Python language: a dictionary or dict). Maps are pairs of unique keys and possibly non-unique values. Typical cases are lists of usernames (as keys) and the users' passwords (as values for the keys). In your case it is a map of string keys that each have an integer value. b) The curly brackets are a way to define dicts in Python. You should be able to literally type in a dict like "{'a':0, 'b':1, 'c':2}". Here, the case is just a bit more complicated because the term is { <elements are generated by a for-loop> }. c) The colon separates key and value in a dict, as in my example from case b). The loop generates key:value pairs a:i for all i in range. d) The statements are the a:i. The loop runs from 0 to len(a)-1. In this case from 0 to 5. e) As mentioned before, keys in a map are unique. So there can only be at most one value for the key 'a'. The usual behavior for maps and dicts is that setting d[key] = value will either create a key with the given value in d. Or, if the key already exists, it will keep the key and set the value to the new one given. In that context, it is at least the behavior I would expect from the loop: At i=0, it sets ind['a']=0. And then at i=4 it overrides the existing value with ind['a'] = 4.

Or you call your counter variable "numberOfLinesProcessed" or "linesProcessed" and reconsider if you really need to comment the increment. In my experience, certain circles tend to dislike in-code commenting except for cases where something unexpected happens or a non-obvious choice has to be explained (and both case classes should be avoided). And in recent years I have come to adapt that way of programming. That said, "comment a lot, even in-code" is probably still a good advice for beginning programmers, who I assume are the target audience for this thread. As for a tip I'd have: If in doubt, try it out. Simple examples: What values does a loop over "range(10)" cover? Is "x[1]" the first or the second element in an array x? What is the result of y = A*x, where A is a 2D-array/matrix and x is a 1D-array/vector? You can read the manual. But it takes time and you need to be certain that you understood it correctly. And once you leave language features and go to library-features you also have to implicitly assume that the library behaves according to its documentation (i.e. that it does not have a bug). Or you can just assume something and hope everything goes well. Or you can just try it out and discover the answer for yourself, which is often faster than reading manuals and produces less errors than just assuming some behaviour. And most importantly: It's the most fun variant.

Kind of my favourite joke at the moment, mostly because my work reminds me of it so often. Hope I did not actually have it from this thread: A mathematician, a physicist and an engineer are given a small red rubber ball and the task to determine its volume. - The mathematician measures the circumference and then calculates radius and volume using the relations of a sphere. - The physicists dumps the ball into a measuring container with water and measures the amount of water that is displaced. - The engineer goes looking for the industry norm for "small red rubber balls".