darklight03/StackMIAsub huggingface dataset:
About a year ago, I came across a really cool property of the envelope curve of a parabola that I couldn't prove. I'm posting it now for help: If we have a straight line and a circle that belong to one plane, then the enveloping curve of the parabola whose focus is a moving point on the circumference of the circle and whose guide is that straight line forms two parabolas that can be drawn with the four information: the perpendicular from the center of the circle on the line is the axis of symmetry of the two parabolas, the distance between the guides of the two parabolas is equal to the length of The diameter of the circle, the center of the circle is a common focus of the two parabolas, the straight line is midway between the two guides. If you start from a circle that does not share the line at any point, the segment will leave traces of two parabolas inside each other, as shown in the picture But if you start with a circle cutting the straight line, leave the traces of two intersecting parabolas at two points on this straight line. But if a circle starts touching the straight line, then the traces of one parabola touching the straight line will be left I was hoping to prove it myself, but unfortunately my level of proof does not allow me. Please do a complete proof that discusses the three cases, please, and thank you. Also, is this feature previously discovered or is it new, please attach a reference if it was previously discovered
My question probably sits more on the applied mathematics side. There is a spring mass system, for the case where the mass moves into the spring (vertically downwards) the spring experiences compression and the second order equation for a spring mass system will provide a solution. However, when the mass moves vertically upwards the lets say that the spring cannot experience tension and thus the mass moves freely without the effect of the spring. It is pretty easy to develop the differential equations for each of the two scenarios individually. However would the following procedure presented a decent manner of modelling the system as a whole: Lets say we apply a displacement as an initial condition which causes compression in the spring. Thus initially the differential equation of the spring governs the system. At each iteration the results are reviewed to check if the spring experiences tension. if the spring has tension then the initial conditions for the last iteration of the spring mass system is used as initial conditions for the differential equation which presents free motion. The above mentioned points will work in both directions, thus if the free motion differential equation is in use and the system begins to move vertically downwards the initial conditions from the last iteration will be used as initial conditions for the spring mass system. I am not looking for a coded solution for this since it can be done easily on Python and Matlab using the ode solvers. Would the presented method provide relatively accurate results for the system?
As a rule of thumb, vapor condensation usually happens at the interface between the system and the heat reservoir. Now, according to my analysis below, it is the only way for vapor to condense, which implies the near impossibility of condensation in the bulk of vapor. : As the temperature drops, within the vapor system, whose thermodynamic behavior can be described by a canonical ensemble, energy is favored over entropy. That is, the energy should escape from the system to the heat reservoir to create a large entropy overall. This means that the system has the tendency to evolve into the low energy configuration when the temperature drops. Thus, for such a phase transition to take place, the passage of energy from the system to the reservoir is crucial. The case for vapor is unique in that its only access to the heat reservoir is on the surface, while for other systems, such as the Ising model, it has access to heat reservoir (the phonons, for instance) within the bulk so that we can see "bubbles" of broken symmetry, domain walls, forming inside the bulk. At the interface between vapor system and the heat reservoir, the mechanism for such a transfer of energy is the conversion of kinetic energy of vapor into the phonons of the reservoir. In the bulk of vapor, however, there is no direct way for the energy to be transferred. Classically, the energy released by forming a droplet can only be carried to reservoir by vapor. This process is extremely inefficient, and its explicit mechanism is obscure. Quantum mechanically, it should be possible that the release of energy is in the form of radiation, where the "radiation background" plays the role of a heat reservoir. Clearly, these two passages of transferring energy to the reservoir should be negligible compared to the mechanism taking place at the interface. So, in any realistic cases, condensation of droplets never happens in the bulk without impurity. It remains to be checked whether such an analysis is physically well-founded and consistent with the experiment.
I am using ZFC as a tool to demonstrate my problematic logic. In zfc we construct a proof system for zfc in zfc (a simulation of a proof is what I mean); we will call it inner proof system. We establish that if there is a proof in this inner proof system that concludes A then A holds. Turing machines can be formalized in zfc with the notions of halting and not halting. The whole question relies upon the argument that the zfc can prove that for every halting turing machine on a certain input there exists a proof of this in the inner system (which I am not entirely sure it is provable in zfc without further assumptions). A turing machine that takes an input (and treats it like an incoding of a turing machine and also as an input) and goes through all possible proofs (in the inner proof system) that conclude that the input halts on itself or does not halt. If such a valid proof was found our machine does the opposite of the proof's conclusion (meaning that if the proof demonstrated halting the machine would enter an infinite loop and if the proof demonstrated not haulting the machine would halt). Under the assumption that zfc is consistent it holds that this turing machine does not halt on itself. This is a construction used to demonstrate the first and second incompleteness theorems in zfc for computer scientists. Assuming the consistency of zfc, the statement that the aforementioned turing machine does not halt on itself cannot be proven by zfc and thus cannot be proven by the inner proof system. Assuming the consistency of zfc, according to the completeness theorem there exists a model of zfc where the statement: "the aforementioned turing machine does not halt on itself" is true and another with this statement false. My problem is with the model in which it holds that this statement is false. This means that the relevant turing machine halts. It follows that there is a proof of that in the inner proof system. It follows that the inner proof system is inconsistent. I know there is something wrong with this logic but I cannot pinpoint it (obiously because I did not formalize this argument sufficiently). Where in this sketch of a proof does the argument fail (for example because a statement is not directly provable from zfc without certain assumptions)? Note: my backgroung is in computer science
"For classical (non-quantum) systems, the action is an extremum that can never be a maximum; that leaves us with a minimum or a saddle point, and both are possible." The above statement is an excerpt from the "Introduction" (preface) of the book "THE PRINCIPLE OF LEAST ACTION - History and Physics" by ALBERTO ROJO & ANTHONY BLOCH. I want to know whether the "For classical (non-quantum) systems, the action is an extremum that can never be a maximum" aspect of that statement is true because it looks pretty definitive. (definitive in the sense certain or assertive) Note: Now I know there are a lot of related questions that look like this, but not any of them looks for a direct and definitive answer for this direct and definitive question, most are descriptive questions for descriptive scenarios and most answer's given are describing particular scenarios, incomplete ones or ones that asserting irrelevance of such question's for actual path determination as we only seek stationary action not whether that is minimum, maximum or inflexion point. (This is intended as to why this should not be labelled as a duplicate, not as a judgement on other questions or their answers as they serve their intended purpose. It is however important to differentiate between the scope of this question and other similar questions. I hope the Phys.SE community will respect the original poster's judgement on the relevance and uniqueness of their own questions unless there is overwhelming evidence to say otherwise.). I have already browsed similar questions as indicated by the system and have not found any definitive question or definitive answer. This definitive question clearly expects a definitive answer, so I hope it will remain a question, not a duplicate.
I am an A-level physics student, and I've been taught that temperature is the average kinetic energy of a particle. So when gas particles are heated, they move faster. This makes sense as an airplane traveling faster does make the nearby air warmer when measured from the plane. Say I release a box of room temperature atmospheric pressured gas in the vacuum of space. Assuming there is no gravity, all the gas particles will be traveling in a straight line as they won't be bumping into other particles. And since space is only a few degrees above absolute zero, the gas particles will cool down after they transferred a lot of their "thermal" energy via radiation, and thus should slow down (lower temperature = lower speed). Looking at a single gas particle this breaks the conservation of momentum, as it is quite literally slowing down to nothing. So what's going on. Now imagine there is indeed gravity in space, the gas particles will eventually start traveling and be accelerated toward a source of gravity. So its kinetic energy increases and so does its temperature (higher speed = higher temperature). First of all it is heated up by nothing without any sort of heat transfer taking place, and would there be any distinction between temperature and speed? Why does this only seem to apply to gas and not solids - a fast moving car wouldn't look hotter would it (ignore friction with air). So can anyone point out where the chain of logic breaks down because it is not making any sense.
Many electromagnetic interactions are modeled as exchanges of a real photons: e.g. an excited electron can relax and emit a photon. Somewhere else, a photon and an electron can interact, "consuming" the photon and leaving the electron in a more excited state. Electromagnetic radiation is modeled as a flow of real photons. The flux of energy radiated through a surface over a time is the sum of all photons' energy that pass through that surface. Other electromagnetic interactions are modeled as exchanges of virtual photons: e.g. the electrostatic force between two electrons is mediated by virtual photons exchanged between them. I understand that these virtual particles are not "real" in the sense that they can't be measured directly, and are just a representation of whatever actual physics our models approximate. They "exist" transiently, in infinitude. In a vacuum field, virtual photons pop in an out of existence, with their total energy and momentum summing to zero. Can electro/magnetostatic fields be modeled as flows of virtual photons to/from on their sources? In the absence of a field, the "positive" and "negative" virtual photons cancel out in energy and momentum. In the presence of a charge, are positive virtual photons flowing one way, and negative virtual photons in the other, relative to the charge? What would the net energy and momentum flux through some surface around e.g. an electron be? Part of why I ask: radiation pressure is quantized in the sense that each individual photon imparts momentum on an object, instantaneously, yet it's difficult for me to imagine the situation for static charges where they accelerate continuously.
When calculating area of a hole in an irregular surface for water flow calculations, what defines that area? I need to calculate the amount of water passing through holes in surfaces in a fixed time, knowing the pressure on either side. This is mostly a straight-forward process (discounting the discharge coefficient, but that is a different matter), but the formulas depend on the area of the hole. What I know about the hole is its boundary. If the surface is planar, this is obvious enough and can be calculated via Green's theorem. But when the hole is in a non-planar surface, it becomes much more problematic. What does the area of the hole even mean? The area of the surface? The hole is exactly where the surface does not exist. Outside the hole there is some surface, which I could presumably find information about (though at a serious increase in complexity), but where the hole is, there is nothing. I cannot even be sure how the surface where the hole was cut was shaped. That information was lost with the hole. "Yeah, there was a kilometer long capped tube there! Good thing that was where the cut-out needed to be." Besides, the water doesn't know how this mythical surface was shaped either, so its behavior will not be influenced by that. Whatever definition of area is appropriate here, it cannot be dependent on the exact shape of some surface. The areas of all possible surfaces filling the hole is clearly bounded below, so there is some minimal area. Presumably this would be the best choice. That is a mathematical rather than physical question, but I want to be sure I am properly handling the physics instead just jumping on an idea.
What is electric current? An electric current is a flow of charged particles, such as electrons or ions, moving through an electrical conductor or space. It is defined as the net rate of flow of electric charge through a surface. How do we generate electric currents? We know charge is the property of a particle due to which it can interact with electric fields and experience electric forces. We use this property of charge to have electric currents; we connect a conductor with a potential difference (battery) so there an electric field sets up in the conductor causing the free-charged particles (electrons) to drift opposite to the direction of the electric field. Hence we get an electric current. But electrons have another property too i.e mass, which is the property of a particle by which it can experience Gravitational forces. I need help in designing a setup to produce electric current using the property of gravitational mass of electrons. So if we have a long conductor wire and a Gravitational field is switched on such that it's direction is along the length of the wire then electrons (which are free) will experience a Gravitational force and we should have a flow of charges, the electric current. But there can be some problems in this: The gravitational forces are way more weaker than the electric forces. As free electrons move away from one end of the wire, they leave this end positive, so a mean electric attractive force acts on the free electrons tending to prevent their further movement. Possible solutions and crux Electrons in a conductor are loosely bound to the nucleus and at any temperature they will have enough thermal energy so that they can be free and do continuous random motion like gas molecules in a container which is also called brownian motion. And our sir has told us that brownian motion ceases at absolute zero temperature. If we connect the wire from top to the earth then the charge vacancy can be fulfilled by charge flow from the earth. So above thought experiment suggests generating current using gravity instead of electric potential difference. Some study materials I don't thought that I will need to do it but now I felt the need to do this after an answer asking for "how one would switch on a gravitational field" and there can be more like that comments in future maybe. So here is a Wikipedia page about thought experiment
The way I work on homework questions, especially for analysis and topology, might be a little different (or maybe not). I would remember the questions and think of them when I run, take a shower, and during other periods when I don't need to use my brain; later I will write out the solution without much thinking. This typically works well even on harder questions that my professor expects more than two hours to solve. However, sometimes I forgot about what I thought before, and my notes for insights, which are random diagrams or phrases on a scratch paper hidden in a stack of papers, are normally either not found or incomprehensible, so I have to rethink the questions. When I actually write out the proof, it is very readable, but as an undergraduate, I have many other time commitments, so I don't have enough time to completely elaborate or type out what I'm thinking. In that case, how can I record my thoughts quickly so I will be able to reproduce them? Also, sometimes I would try different approaches, and typically most of them do not work, so how can I record these attempts and not go into loops? What is frustrating is sometimes I go back again and again on a "branch" that is seemingly close to the answer but is very unrelated and other times I spent a too short amount of time on something that is close to the solution. Is there a way to prevent this? For mathematical research, I would imagine the complexity dramatically increases. For my research project (as an undergraduate working in mathematical optimization), my professor typically introduces to me some very nice lemmas, which help me to prove the optimal goal. However, if I'm working on complicated questions by myself, how can I keep track of all the methods I tried and not worry about forgetting about them the next day? Of course, typing everything out is a solution, but how exactly can we type something that is just vague intuition in our brain instead of proofs for lemmas and theorems?
In machine learning sometimes we build models using hundreds of variables/features that we don't know (at least at first) if they might have a relation with the target. Usually we find that some of them do and others don't. Some of them even have a true relation that we couldn't think of at the begining. Once we build a first model, sometimes we have an idea to include a new variable that we know it has a natural relation with the target and that we didn't think of at first. Sometimes this new variable is sparse, though, what means that it's constant or null for the major part of the data. The problem then is that to use the information that the new variable has, we need to find in some node a cut point of that variable that reduces the loss_function more than all the other cut points of the other variables. However, a sparse variable usually doesn't reduce the loss_function a lot because the major part of the data ends on the same side and only a very few part of the data goes to the other side. Also, when we have that amount of variables, statistically we find cut points in other variables which are not related to the target that reduce more the loss_function for those data points in that node. Not because there is a true relation, but just because of statistics. This ends in overfit and also not being able to use the prediction capacity of our new variable. In this circumstances, what can we do to extract the value of our new variable?
In Einstein's original thought experiment involving "a (very long) train running along a [straight] railway embankment", of essential importance appears the prescription that "[E]very event which takes place along the [railway track] line also takes place at a particular point of the train." The constituents of the train are surely distinct from the constituents of the railway embankment and track (especially since they are supposed to have moved wrt. each other; each constituent of the train and each constituent of the track segment under consideration have separately met each other in passing). Is it consequently correct to say that each separate event involving train and embankment/track has one particular constituent of the train and one particular constituent of the track as participants in this event, such that this particular pair uniquely identifies the event ?, and that each such event has two distinct parts, namely one distinctive part attributable to the participating train constituent (which is characterized by the train constituent indicating being met, in passing, by the track constituent; perhaps with additional characterizations), and another distinctive part attributable to the participating embankment/track constituent (being vice versa foremost characterized by the track constituent indicating being met, in passing, by the train cconstituent) ? Note that the event-parts in question are not presumed to be separate from each other, or resolvable, in a geometric (spatial) sense. My question is not whether and how certain sets of (finely-resolved) distinct events may be considered and addressed as one (coarsely-resolved) event; nor whether any one actual particle (or even several) may be considered fully contained in any spacetime region of finite spatial extent. My question is rather conceptually: Whether (at all) and (if so) How to reconcile speaking of an event as "having distinguishable parts", as described above, while also speaking of an event as "a point in spacetime", or "a point of spacetime" with the understanding that: "A point has no part(s)." ?
I'm currently taking this college calculus course, and this exercise has stumped me. It is in German, but hopefully what it's asking is fairly clear. To summarize, the problem first asks me to prove that the preimage of an intersection of a family of sets is equal to the intersection of a preimage of a family of sets, where f: A -> B and Ui is a family of subsets of B. To do this, I just gave the general example of x being an arbitrary element of A, which means that x is an element of the preimage of the intersection, which means for all values of i f(x) is an element of Ui, which means x is an element of the intersection of the preimage of Ui. Image of my answer cause I can't figure out MathJax: Then the problem asks to show that for a family Vi, a subset of A, an image of the intersection of a family of sets is not in general equal to the intersection of an image of a family of sets. Yet, for every example formula and sets I can conjure up, I always seem to prove that the two sides are in fact equal. No amount of coaxing and complaining on Bing or ChatGPT causes them to bring an example up either - they just keep giving examples where the two sides are equal. So, is this a typo from my professor and this expression actually is generally correct, or am I missing something? What family of sets and functions would make these not equal? Thank you!