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timo

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Everything posted by timo

  1. I'll try to express what I understood in mathematical terms. From the title of your thread I was assuming a rotating vector field would be f(x, y, z, t) = (sin(t), cos(t), 0). That is a homogeneous field that rotates around the z-axis. But your first sentence seems to be considering a time-static, rotationally-symmetric field like f(x, y, z) = (y, -x, 0), instead. I do not understand why f(0, 0, 0) had to be (0, 0, 0) in that case (even though it is in my example). Mathematically, the function could have any value. Physically, the magnetic force around a wire would be a counter-example, since it "rotates around the wire" and diverges at the origin (i.e. is undefined). Without looking it up I'd expect that the velocities of circular planetary orbits also diverge at the origin. So I think there are some implicit assumptions you make in your first sentence that you are omitting, e.g. something like "the function is continuous and defined everywhere". Your moving example would probably be something like f(x,y,z,t) = (y-t, -x, 0), I think. So now, time is back in the picture, which is a significant difference. You can quickly solve that for (0, 0, 0) == (y-t, -x, 0) and see that for all (x=0, y=t, z=arbitrary) f(x, y, z, t) = (0, 0, 0). So in that particular example, the answer would be "at each time, there are points at which the function is zero" and "there is no point at which the function is zero at all times". If you constrain the domain of the function, the first statement can become invalid, of course (if the calculated points are outside of the domain). Not sure I really understood what you were asking. But I hope this helps.
  2. Since no one brought up the mainstream answer to that question, yet: Consider the journals that the sources you are citing were published in.
  3. I wonder how different that is from money. Effectively, you seem to get one credit for a task that you can use for someone doing another task for you (providing breakfast). There are a few differences to our current implementation of a monetary system: I assume you image a single world-wide credit currency, you seem to assume there are no fractions of credits, I am not quite sure how tasks/missions are assigned, ... . But most of these differences seem to be disadvantageous compared to our current system.
  4. As Swansont mentioned, Hydrogen from Electrolysis (*) is mostly a storage option for electric energy, either directly or as a precursor step of synthesizing other storage gases/liquids. It has low efficiency but high energy density and low storage costs. Hydrogen is central in future renewable energy systems as the basis of long-term energy storage for the electric system, replacement CO2-neutral fuel for ships and planes, and possibly even for land-based vehicles (batteries and overhead-lines being competing technologies, there). The main drivers that can trigger its breakthrough would be cost reduction or an outright ban of competing technologies (=fossil fuels). Costs for renewable electric power generation (i.e. the "raw material") have come down significantly in the last two decades (Electrolysis from fossil fuels is pointless of course - you could just keep your fuel in the first place). Bottom line: Drivers are the need to replace fossil fuels and the cost reduction in renewable electric power generation. This is nothing new to the scientists or the energy community, and Hydrogen is not the only topic you can get excited about (-> battery technology, low-temperature heating grids, off-shore wind turbines, demand-side management, ...). Hydrogen specifically came into the news recently when the German government released its national Hydrogen strategy (https://www.bmbf.de/files/bmwi_Nationale Wasserstoffstrategie_Eng_s01.pdf). No such strategies exist for any of the other topics I mentioned. Releases like this have a significant impact on the science community and what they talk about, since everyone want to jump on the funding train. Apparently, similar strategy ideas may exist on the EU level (https://www.euractiv.com/section/energy/news/leaked-europes-draft-hydrogen-strategy/). And perhaps that's not even completely coincidental, given that Germany just took over the EU presidency. So to Swansont's question about who is shouting about a Hydrogen economy: The answer could be "the Europeans". (*) Fun fact I learned only recently: Electrolysis is not how Hydrogen is being produced on scale today. Instead, we start with oil, extract the Hydrogen and release the CO2 to the atmosphere . Totally pointless in the context of replacing fossil fuels with CO2-neutral fuels, of course.
  5. A matrix M can be a function that "transforms" points into new ones in the sense of y = M*x, where x is the vector to the original point, y the vector to the point x gets transformed to, * the multiplication of matrix and vector, and M the transformation matrix. Given enough pairs of original points and transformed points, it is possible to reconstruct the transformation matrix. Please note the x and y in this case are not what they were in your post: Here, they are two vectors, while your term "x and y graph" suggests you refer to the x- and y-coordinates. If you have a single point (x,y) or a set of points (x1, y1), (x2, y2), ... then it is not clear what "transforming these points into a matrix" means. In other words: There is no such thing as transforming points into a matrix. Bottom line: It is possible to reconstruct a transformation matrix when enough pairs or original and transformed vectors are given. At minimum, for an N-by-N square matrix at least N pairs of N-dimensional vectors must be given (it does not guarantee success and will fail e.g. if two pairs are identical). The reconstruction is done by solving systems of N equations with N unknowns for N times. [And if I had the time right now I would add the explicit equations for the 2D case - that should be pretty straightforward].
  6. In my experience, the concerns about climate change are the impact on human civilization over the next few hundred years. Not the impact on biological systems over thousands/millions of years. The whole climate change concern issue is a selfish, short-sighted thing of the humies, completely disregarding how great it could be for dinosaurs and turtles in the long run.
  7. Here's some relatively recent ideas about a climate-friendly energy sector: http://energywatchgroup.org/wp-content/uploads/EWG_LUT_100RE_All_Sectors_Global_Report_2019.pdf Spoiler: Just as every group working in the field have said for years (I personally know roughly ten research groups in Germany alone), the authors claim that the major sources of energy should to be wind and solar. The nice feature of this paper is the global scope.
  8. I am not sure I understand the question. I am not even sure that the premise is true (did Alchemy and Chemistry even exist at the same time?). But as a hint to what I guess may be the answer you are looking for: Do you understand why turning lead into gold is not within the scope of Chemistry?
  9. An operator f(...) is linear if f(A+B) = f(A) + f(B) and f(a*A) = a*f(A), with addition and multiplication being the addition of two vectors and their multiplication with a real number, respectively, in your case. Alternate form of the same statements for a matrix M, vectors x, y, and a scalar a: M(x+y) = Mx + My, M(a*x) = a*(Mx). When interpreted as an operator V -> V, matrices are always linear. But it should be straightforward to explicitly show that for your given matrix by starting from one side of the two defining equations and rearranging until you get the other side.
  10. Yes - if the height of people is a normal distribution or at least well-approximated by it (not sure to what extent it is). Very specifically, if you assume a Gaussian with 1.70 m as mean height, and 0.15 as the standard deviation then the chance that a person is 1.90 m or larger (in the region [1.9; infinity)) is roughly 10 %: https://www.wolframalpha.com/input/?i=integrate+1%2F((2+pi)^(1%2F2)+0.15)+e^(-((x+-+1.7)%2F0.15)^2%2F2)+from+1.9+to+infinity (note that I put explicit numbers here - if you replace 1.9 with x you'll see the erf() again in the expression for the solution).
  11. Ctrl-Z just cost me a long post that I put a lot of time in and will absolutely not type in again just to be fooled by my muscle memory. Key points: 1) Consider erf() just as the number-crunching function that computers provide you to calculate integrals over Gaussians. Not much more. The explicit relation between Gaussians and erf() is https://www.wolframalpha.com/input/?i=integrate+1%2F((2+pi)^(1%2F2)+sigma)+e^(-(x%2Fsigma)^2%2F2)+from+A+to+B 2) Here's some Python code to play around if you want. It plots a) A single random variable "number of bugs a programmer fixes each day" b) The resulting "number of bugs fixed per programmer per year", which is a sum of 200 random variables in a) and itself a random variable. Key observation: The distribution looks very different and very Gaussian. c) The probability to "fix at most N bugs per year" which is the same as "fix 0 to N bugs per year" which is the same as "the sum of probabilities to fix 0 ... N bugs per year" which indeed is pretty much the same as the integral over [0; N]. The resulting curve, as a function of N, looks ... *surprise* ... like the error function. import numpy as np import seaborn as sns import matplotlib.pyplot as plt # We visualize the distributions by drawing a large sample of random variables. SAMPLE_SIZE = 10000 def randomFixesPerDay(): # Number of bug fixes per day is a random variable that becomes 0, 1 or 2. return np.random.randint(3) def randomFixesPerYear(): # Number of bug fixes per year is a random variable that is the sum of # 200 (=workddays) random variables (the bugfixes each day) return np.random.randint(3, size=200).sum() # Experimental probability for # bug fixes per day dailyDistribution = [randomFixesPerDay() for i in range(SAMPLE_SIZE)] sns.distplot(dailyDistribution, kde=False, norm_hist=True, bins=[0, 1, 2, 3]) plt.title('Probabiliy of Bug-fixes per Day: 1/3') plt.xlabel('# Bugs') plt.ylabel('Probability') plt.show() # Experimental probability for # bug fixes per year annualDistribution = [randomFixesPerYear() for i in range(SAMPLE_SIZE)] sns.distplot(annualDistribution, kde=False, norm_hist=True, bins=np.arange(150, 250)) plt.title('Probabiliy of Bug-fixes per Year\n(note smaller value on y-axis)') plt.xlabel('# Bugs') plt.ylabel('Probability') plt.show() # Integral [0; x] over annualDistribution looks like error function xValues = np.arange(150, 250) yValues = [len( [value for value in annualDistribution if value <= x])/SAMPLE_SIZE for x in xValues ] plt.plot(xValues, yValues) plt.title('Integral: Probability of fixing [0; N] bugs per year\n(i.e. "not more than N")') plt.xlabel('x') plt.ylabel('Probability') plt.show()
  12. There is only one context in which I ever encountered the error function, and it the same context as here: It is the integral over the (normalized) Gaussian function. The Gaussian, on the other hand, is the most important function in statistics. The reason is the Central Limit Theorem: If you take a variable that is distributed according to some probability distribution, and then take the sum of many of these variables, the probability distribution of the sum becomes increasingly more similar to the Gaussian with an increasing number of numbers added (and the mean value of the sum is the sum of the individual means and the variance of the Gaussian is the sum of the addends' variances). The random walk is a process in which a walker takes a number of independent steps with random length and distance. By the central limit theorem, the resulting total deviation from the original location (the sum of steps) will look like a Gaussian, soon. This Gaussian gives the probability density to find the walker at a certain location. To find the probability to find the walker in a certain region, you sum up the individual probabilities of all locations in this region (i.e. you integrate over this region). When computing this integral, the solution can be expressed in terms of erf(). EDIT: I'm still posting this despite just having received a pop-up saying "someone else posted probably the same thing" 😜
  13. You indeed have an infinite number of points on a finite line segment. The length of a line segment determines if it is finite or not, not the number of points contained (which is always infinite for line segments with non-zero length - and therefore a pretty useless measure).
  14. You correctly state that there may not be a unique answer, since A and B may be multiplied by any non-zero number. That begs the question: What is your point? Contrary to your claim, your thread title does not completely describe the problem. As a rough guess of what you meant: There are numbers that cannot be represented as fractions of natural numbers. The most prominent cases are pi (as in "ratio of circle circumference to radius") and "e" (as in the exponential function).
  15. Random comments on your seemingly random questions: 1) Particle physics is indeed looking at debris to a very large extend. However, people are not looking for new objects in the debris. They look at the content and distribution of the debris and compare it with the predictions of the different mathematical models. 2) The reference to "statements about their encounters" does not refer to particle collisions (caveat: I am interpreting a single sentence out of context here - but modern particle physics did not exist during Einstein's lifetime, anyways). It refers to a key concept in relativity that comparing situations at different locations is tricky. It is not required that the objects in questions are elementary particles that collide. The famous spacefaring twins meeting each other after their space travel (or lack thereof) are would be typical situations that the statement refers to.
  16. I did an interview for a newspaper article a couple of years ago. Me and a PR colleague talked with a journalist for about an hour in the morning. The journalist already had a vague idea about the topic and we essentially had a chat about it. The journalist then sent me the draft article in the evening. I sent him correction proposals that were all accepted. Nothing spectacular on this level. Still, there were a few takeaways from this experience: 1) It is the journalist's story. When you get the interview request you feel very important and in the center of things. But in reality you are just helping the journalist to write the article. This also why I said that I sent "correction proposals". The journalist is not required to send you the story beforehand or get your permission for publication. 2) I originally sent elaborated explanations to my corrections, explaining in which context the statement would be correct and when it would me misleading. I ran my corrections through our head of PR. He said a memorable sentence in the sense of: "that guy is a poor devil, a freelancer being paid per article written. Just send him corrections that he can accept or decline and don't cause him extra work". Point is: As a scientist you may be excited about the topic, and of course you expect the journalist to be exited, too. In reality, to the journalist your topic could as well be an orphan kitten that has been adopted by a dog: It is a story that gets the next article. 3) We had prepared lots of great diagrams but the journalist insisted on a photo of me, instead. My face is completely irrelevant for the science and even inappropriate for the fact that our results were achieved by a team. It is a manifestation of the journalist rule "no news without a face". Since I experienced this from the producing side, I often find myself re-discovering this: When a minister proposes something (that his employees worked out), when the director of a research institute is asked an expert opinion about a topic (which he bases on the work of the people actually doing the science - his employees) or -a current example of discussion in my family an hour ago- when the main discussion of the climate conference is how Greta Thunberg looked at Donald Trump.
  17. There are two approaches here, the formal one and the brain-compatible one. 1) Formally: Realize that there are hidden coordinate dependencies. You are probably looking to construct a function p(y). Since y is the coordinate at the lower side, you have p(y) at the lower edge and p(y+dy) at the upper. This is (possibly) slightly different from p(y) (because of the displacement dy). If you call the difference dp, then p(y+dy) = p(y) + dp(y) = p + dp (note that dp can and will be negative). 2) Brain compatible: Put the equation first and then define the variables: The forces on top and bottom should cancel out. There force up is the pressure force p*A from below. The force from up is the pressure force p2*A from above the small fluid element plus the weight dw of the small fluid element. Since we are talking about infinitesimal coordinates, and since p(y) should be a function, it makes sense to say that p2 = p+dp, which you can then integrate over.
  18. In my opinion, the content you listed is below the minimum required for AI (not really sure what "Data Science" is, except for a popular buzzword that sounds like Google or Facebook). More precisely: Apply these topics to multi-dimensional functions and you should have the basis of what is needed for understanding learning rules in AI. However: All of the content you listed is the minimum to finish school in Germany (higher-level school that allows applying to a university, that is), even if you are planning to become an art teacher. And Germany is not exactly well known for its students' great math skills. The course looks like a university level repetition of topics you should already know how to use, i.e. a formally correct way of things that were taught hands-on before. I do not think a more rigorous repetition of topics will help you much, since you are more likely to work on the applied trial&error side. Bottom line: If you are already familiar with all the topics listed, I think you can skip the course. If not, your education system may be too unfamiliar to me to give you any sensible advise. Btw.: University programs tend to be designed by professionals. So if a course it not listed as a mandatory, it is probably no mandatory.
  19. By this standard, physics is not very strange most of the time.
  20. The equation is not particular to Compton scattering. It is the relation between momentum and energy for any free particle (including, in this case, electrons). I am not sure what you consider a derivation or what you skill level is. But maybe this Wikipedia article, or at least the article name, is a good starting point for you: https://en.wikipedia.org/wiki/Energy–momentum_relation
  21. There is no law of conservation of mass. Quite the contrary: The discovery that mass can be converted to energy, and that very little mass produces a lot of energy, has been a remarkable finding of physics in the early 20th century. The most well-known use is nuclear power plants, where part of the mass of decaying Uranium is converted to heat (and then to electricity). The more modern, but from your perspective even more alien view is that mass literally is a form of energy (I tend to think of it as "frozen energy"). In that view, you can take the famous E=mc^2 literally. There is a law of conservation of energy, but energy can be converted between different forms. In your example, it is converted from kinetic energy of the photons to mass-energy of the electron and the positron (and a bit of kinetic energy for both of them). Note that the more general form of E=mc^2 is E^2 = (mc^2)^2 + (pc)^2 with p the momentum of the object - it simplifies to the more famous expression for zero momentum. I say this to make the connection to your other question where you asked about this equation.
  22. This post is a bit beyond the original question, which has been answered - as Psi being a common Greek letter to label a wave function and wave functions being used to describe (all) quantum mechanical states. I do, however, have the feeling that I do not agree with some of what your replies seem to implicate about superposition, namely that it is a special property of a state. So I felt the urge to add my view on superposition. Fundamentally, superposition is not a property of a quantum mechanical state. It is a property of how we look at the state - at best. Consider a system in which the space S of possible states is spanned by the basis vectors |1> and |2>. We tend to say that [math] | \psi _1 > = (|1> + |2>)/ \sqrt{2} [/math] is in a superposition state and [math] | \psi _2 > = |1>[/math] is not. However, [math] |A> = (|1> + |2>) / \sqrt{2} [/math] and [math] |B> = (|1> - |2>) / \sqrt{2} [/math] is just as valid as a basis vectors for S as |1> and |2> are. In this base, [math] | \psi _2 > = (|A> + |B>)/ \sqrt{2} [/math] is the superposition state and [math] | \psi _1 > = |A>[/math] is not. There may be good reasons to prefer one base over the other, depending on the situation. But even in these cases I do not think that superposition should be looked at a property of the state, but at best as stemming from the way I have chosen to look at the state. Personally, I think I would not even use the term superposition in the context of particular states (a although a search on my older posts may prove that wrong :P). I tend to think of it more as the superposition principle, i.e. the concept that linear combinations of solutions to differential equations are also solutions. This is kind of trivial, and well known from e.g. the electric field. The weird parts in quantum mechanics are 1) the need for the linear combination to be normalized (at least I never could make sense of this) and 2) that states that seem to be co-linear by intuition are perpendicular in QM. For example, a state with a momentum of 2 Ns is not two times the state of 1 Ns but an entirely different basis vector. Superposition in this understanding almost loses any particularity to QM. Edit: Wrote 'mixed' instead of 'superposition' twice, which is an entirely different concept. Hope I got rid of the typos now.
  23. To turn rotational energy into electric energy you can indeed use dynamos, as you already assumed. Or better: Dynamo-like devices. The general term seems to be https://en.wikipedia.org/wiki/Electric_generator. Essentially, i.e. from a physics perspective, electric generators move around magnets in the vicinity of looped electric wires ("electro-magnets"). This induces a current in the wires. Doing this moving around in a controlled way generates a controlled current. Sidenote: I thought the general concept of a dynamo is a "turbine". But according to Wikipedia that refers to the moving part, only. Still, turbines are so closely related to electric power generation that looking up that concept may be relevant, too.
  24. Thermodynamics, in its general meaning, is always equilibrium Thermodynamics. Calculations of processes assume that the systems go through a series of equilibrium states during the process, which is called a quasi-static process. Reversibility is not required for studiot's entropy change equation in the first post. The equation is fully applicable to bringing two otherwise isolated systems with different temperatures into thermal contact, which I will use as an example: In the theory of thermodynamic processes, both systems' states change to the final state through a series of individual equilibrium states (*). Because of their different temperatures and conservation of energy (and because/if the higher-temperature system is the one losing heat to the colder) the sum of entropies increases. In the final state, both systems can be considered as two sub-volumes of a common system that is in thermal equilibrium. Since the common system is in equilibrium (and isolated), it has a defined entropy. Since entropy is extensive, it can be calculated as the sum of the two entropies of the original systems' end states. As far as I understand it, removing barriers between two parts of a container is essentially the same as bringing two systems in thermal contact. Except that the two systems can exchange particles instead of heat. The two systems that are brought into contact are not isolated - they are brought into contact. If one insists on calling the two systems a single, unified system right after contact, then this unified system is not in an equilibrium state (**). And I believe this is exactly where your disagreement lies: Does this unified, non-equilibrated state have an entropy? I do not know. I am tempted to go with studiot and say that entropy in the strict sense is a state variable of thermal equilibrium states - just from a gut feeling. On the other hand, in these "bring two sub-volumes together"-examples the sum of the two original entropies under a thermodynamic process seem like a good generalization of the state variable and converges to the correct state value at the end of the process. (*): I really want to point out that this is merely a process in the theory framework of equilibrium Thermodynamics. It is most certainly not what happens in reality, where a temperature gradient along the contact zone is expected. (**): In the absence of a theory for non-equilibrium states this kind of means that it is not a defined thermodynamic state at all. But since there obviously is a physical state, I will ignore this for this post.
  25. I assume you refer to my magnet example: The force between two magnets depends not only on their location, but also on their orientation. Take two rod magnets NS in one dimension which are one space (here I literally mean the space character) apart. In the case NS NS their attract. If one is oriented the other way round, e.g. NS SN, they repel each other. On other words: Their force does not only depend on their location, but also on their orientation (the equations in 3D are readily found via Google, but the common choice of coordinates may not obviously relate to what you describe). So for calculating forces or energies of magnets, you need their orientation as an additional parameter. This orientation can be expressed as a unit vector (and to relate to my first post: since this is a geometric and not an integration topic, unit vectors are better suited than angles).
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