darklight03/StackMIAsub huggingface dataset:
In set theory, the notions of set and membership are considered primitive. We only specify some of the properties that we think our primitive notions have using the axioms. Usually, the very first axiom of set theory is the axiom of extensionality which specifies that two sets are equal if and only if they have the same members. My discomfort with specifying this axiom as the first is that we haven't said anything about how the notion of membership relates to the notion of set in any previous axiom. That is we haven't specified that it makes sense to say something is a member of a set. Yet we use this notion in the axiom of extensionality. Why we don't have an axiom that exclusively talks about the membership of things in a set?
I have been searching for an automatic method (a computer program) to evaluate any first-order logic (FOL) formula given some knowledge base. The most common approach to do this is to use PROLOG. The issue with PROLOG is that it employs a subset of FOL which, for example, restricts the use of quantifiers. When searching about this, I have learned that PROLOG uses the resolution algorithm, which requires formulae to be in Conjunctive Normal Form (CNF). I have also learned that there is a process called Skolemnization that can be used to remove the quantifiers from a formula and help convert it into CNF. Therefore, my question is the following: Can the resolution algorithm be applied to any FOL formula? By this, I mean if there exists some algorithm that takes as input any FOL formula and normalizes it (using skolemnization and other methods) into CNF so that resolution can be applied to it?
To "empathize" with someone means to "understand and share the feelings of another" (Oxford dictionary). In science fiction, an "empath" is "a person with the paranormal ability to perceive the mental or emotional state of another individual." (Oxford dictionary) Is there a word to describe the "opposite" (behaviour in nature, not antonym) of an "empath" - someone who can communicate in such a manner that it is very easy to empathize with them because you can easily understand or recognize the feelings behind what they are saying? More details: As pointed out above, an "empath" has the ability to perceive the mental or emotional state of another individual. I am seeking a word (if it exists) that describes the ability to communicate in such a manner as to "create a mental or emotional state" - an emotional connect - to make others more easily relate and empathise with them. (Something that most story tellers do effectively). (Some commenter thinks my question is about Sci Fi. No, I am just looking for a word to describe something in plain English. But it doesn't matter if it is an uncommon and rare word from old English or even a modern slang).
The second law of thermodynamics states that the entropy of the universe increases over time and this has lead to theories like the heat death of the universe and the big rip. What this means in effect is that all matter and energy has an expiration date, beyond which it gets divided accross infinite space and this is irreversable since entropy does not move backwards. The first law of thermodynamics states that energy cannot be created or destroyed. The only explanation this leaves for the universe's existence is that energy has existed since always and never had to be created. And herein lies the contradiction: If energy has been around for an infinite amount of time, why hasn't the heat death or the big rip already taken place? How come we are able to observe the universe in its current fairly organized transitional state unless A. energy was created a finite amount of time ago or B. Energy has existed since always but it can freeze or move backwards entropy-wise allowing us to observe it in its current form. Either way one of the two laws is violated. How do physicists reconcile this paradox?
The other day I wasn't feeling so well so I visited a doctor. He recommended that I take a break and get intravenous therapy so I did. I laid on a bed and the nurse inserted a needle into my left arm which was connected to the pack of liquid containing vitamins. I soon fell asleep. More than half an hour later I woke up. The pack was practically empty. However blood was slowly flowing out of my vein. I frantically called the nurse, who apologized for not catching it earlier. I thought the sole factor that delivers the fluid to my body was gravity, since the pack is hung in the air much higher than my arm. But given that the direction of flow reversed once the pack was emptied, my assumption is probably false. My new theory is that the vitamins are delivered because the pack's pressure is higher than my vein's when there is enough liquid in the pack. However this still doesn't explain why I started losing blood as when the two sides' pressure reaches parity all movement should stop. Since the pack's initial pressure was higher, There should be no change of direction until reaching equilibrium. Why did I lose blood and are there any precautions I can take to prevent this incident in the future?
Many elementary explanations of the Einstein-Lorentz transformation derive it using Einstein's special relativity postulates, combined with some thought experiments considering light sources and observers. However, I have read that these thought experiments often do not hold when subject to rigorous mathematical scrutiny, and that Einstein himself did not use them when making his arguments. Further, it is obvious that Lorentz did not use Einstein's postulates when deriving his transformation because at that time he did not trust in Einstein's postulates. Rather, from what I have read, the actually rigorous way that the transformations were derived is using more complicated math such as hyperbolic geometry. I cannot seem to find a good resource that walks through the derivation in a rigorous way; especially one that does not assume prior knowledge and explains the math being used. Can someone please either provide a rigorous derivation here, or point me to a good resource for me to find it? I am especially interested in learning the approaches of both Einstein and Lorentz.
I have two rectangles of different sizes side by side. I want to scale them both (each maintaining their original aspect ratio) so they each end up with the same height and together equal a specified, fixed width. I would like to find a formula that will work no matter what the sizes of the rectangles are (some may be bigger than the target width, some may be smaller - thus some will have to be scaled up and some down). I found what appears to be a very similar question here, however the only solution provided seemed to imply a universal scaling factor. That doesn't work for my situation because the rectangles need independent scaling. I am also only scaling two rectangles instead of three, and the resulting widths of my rectangles must together add up to a specific width (the other question just didn't want to exceed one). Any help would be incredibly appreciated.
I'm reading Carroll's GR book. I'm able to follow it for the most part, but a couple of paragraphs are a bit hard to decipher: According to the WEP, the gravitational mass of the hydrogen atom is therefore less than the sum of the masses of its constituents; the gravitational field couples to electromagnetism (which holds the atom together) in exactly the right way to make the gravitational mass come out right. What exactly does "couples to" mean? Right now that's just a vague phrase to me that implies gravitational field has something to do with EM - but what's the precise notion behind it? Sometimes a distinction is drawn between "gravitational laws of physics" and "nongravitational laws of physics," and the EEP is defined to apply only to the latter. Then the Strong Equivalence Principle (SEP) is defined to include all of the laws of physics, gravitational and otherwise. A theory that violated the SEP but not the EEP would be one in which the gravitational binding energy did not contribute equally to the inertial and gravitational mass of a body; thus, for example, test particles with appreciable self-gravity (to the extent that such a concept makes sense) could fall along different trajectories than lighter particles. I have no idea what the statement in bold means at all. Could anyone please explain this so that a layman like me could understand?
As we know, the recent Nobel prize was awarded for the creation of attosecond light pulses. I read this excellent answer, describing both how the pulses are created and what applications they have. I understand how the pulses are created by the addition of waves with harmonic frequencies in a classical sense. However, from a quantum mechanical point of the view, light comes as photons with quantised energy, with the energy of each photon related to its frequency. As the attosecond pulses consist of many frequencies, I wonder how they relate to photons. My guess would be that the attosecond wave describes the probability of detecting a photon at a particular location/time, but that the photon detected can have the energy corresponding to any of the constituent frequencies. Is that a correct guess, or is that too simplistic? When we detect individual photons of the attosecond pulse, what frequencies can they have?
First, let me acknowledge there are numerous posts on this question already. The most pertinent to my specific question is probably this one. To restate the problem: "A family has two children. Find the probability that both children are girls, given that at least one of the two is a girl who was born in winter." The solution offered in the text includes this step: "use the fact that {both girls, at least one winter girl} is the same event as {both girls, at least one winter child}". This subtle shift is necessary to reach the correct answer. But it seems to come out of nowhere, right? I wonder if the change was motivated by a desire to be able to invoke independence. Since {both girls} and {at least one is a winter girl} are clearly not independent, while {both girls} and {at least one winter child} are. Hard to get inside the author's head, but that's my best guess. What do you all think?
There are various threads on this site explaining the mathematical details of how, in QFT, position operators are non-relativistic. I can follow some of the math, while some of it goes over my head. But even with the parts of the math I can follow, I have a hard time relating it to anything conceptual -- that is, all the mathematical explanations I've seen have a feel similar to proofs by contradiction in that they show that it's true there are no relativistic position operators, but not why it's true. So, to be clear, what I'm looking for isn't a nonmathematical explanation necessarily, but just an explanation that feels like more than just a bunch of algebraic manipulations -- something that connects all those calculations to the actual objects we're trying to model and their properties. To clarify what specifically I'm confused about, I don't get why bringing in special relativity should mess up the notion of position at all. I understand that position is a relative property, so that if we change reference frames any position operator would be affected. But momentum and energy are also relative, and yet there doesn't seem to be any issue with defining relativistic versions of their operators. What makes position so different that we can't just define it in some given reference frame and apply the Lorentz transform as needed? Does it have something to do with the geometry of spacetime being hyperbolic?
I've read several threads over the past several days talking about how photons don't have wavefunctions in the same way as massive particles do because they don't have non-relativistic limits. If I understood correctly, that's because the usual position operator introduced in introductory QM courses really only applies to non-relativistic theories. The Newton-Wigner operators kept being mentioned as the closest QFT analog to the position operator from non-relativistic QM, so I've been trying to find information on them, but the relevant Wikipedia page is very sparse and vague and everything else I found was very long and technical. All I really want to know is: What's the actual definition of the Newton-Wigner operators? How does it differ from the definition of the position operator from non-relativistic QM? From what little the Wikipedia article did say, I know the Newton-Wigner operators aren't Lorentz covariant, which, if I understand correctly, means they're reference-framed dependent w.r.t. the Lorentz transform. But is that the only difference between them and analogous operator from non-relativistic QM? If so, then why does position in particular often get singled out as being different in relativistic and non-relativistic QM and QFT, when other properties, such as energy and momentum, are also reference frame dependent?
As a caveat, i am not a mathematician but rather a programmer with an amateur interest in patterns, fractals, sequences, data science. That said, i have been following recent developments in aperiodic tiling with interest. I've had an idea for an application for it, but I don't know enough in the field of mathematics to know if: It's obvious, been done, move on It's an interesting idea worth exploring, or It'll never work or be useful, just stop now. The thought is as follows: By definition, aperiodic monotiles don't repeat on the plane, therefore They are effectively a visual representation of an infinite, non repeating sequence. They can be computed relatively easily (e.g. see here), and the next tile can be calculated based on the position of the previous one. Therefore, you effectively have a pseudorandom number sequence, given a seed of the coordinates to start at. Is this worth exploring, or just a rabbit hole leading to a dead end?
Consider a sample of an ideal gas kept in a pouch of some volume. This pouch is then kept in a bigger container of volume V. As soon as we open the pouch then the gas will expand irreversibly in the container. Also consider that this expansion is adiabatic in nature and no energy flows in or out of the container. If we let the system be kept isolated for a sufficient amount of time in which it approximately achieves steady state, then can we say that in this steady state condition every molecule in the gas sample will approximately have the same speed? I thought that the entropy of the system would be maximum in this configuration as the energy is tending to be spread out equally among all molecules. Is this notion of entropy being a measure of the distribution of energy logically correct? Also if it is not correct then is there any mathematical way to find the distribution of the molecular speed of the gas sample?
I would like to know if there are any books on combinatorics and number theory that follow an axiomatic approach akin to that of Sierpinski's General Topology. I have found some books for both subjects that might follow this axiomatic approach but they aren't explicit about it. The books I am referring to are: Aigner's Combinatorial Theory, which one user on Amazon said follows an algebraic approach but I don't know if that is equivalent to an axiomatic approach. Landau's Elementary Number Theory, with the author having written an axiomatic book on Analysis called Foundations of Analysis, which I have a suspicion might be axiomatic. And Sierpinski's Elementary Theory of Numbers by the author I mentioned above with my suspicion being that it also follows an axiomatic approach. I have scoured the internet for such books but to no avail. I haven't found posts on stack exchange that address this question, specifically for number theory and combinatorics, either so I decided to write one myself. I really hope someone here might have found such books. More specifically Brualdi's book on combinatorics states that the multiplication principle is a consequence of the addition principle while Vinogradov's number theory book states that division with remainder is a generalization of the quotient of two integers. I would like the books to follow such an approach, stating what is essential and deriving everything else logically. I don't think I can be any more specific but I hope you get the gist of what I am saying.
This is probably a really common word and I'm having a moment, but it occurred to me the other day that I can't think of the verb that describes the action of sweeping a knife across a pat of butter, which has the effect of scraping butter from the pat onto the knife. It's kind of the opposite of "spreading"; i.e., the action to deposit butter from the knife to the bread/toast/whatever. "Scraping", I suppose, is close...but it doesn't feel right. Intuitively, I feel like scraping involves a fair amount of effort to remove the surface layer, which is usually hard, contrary to the properties of butter at room temperature. "Peeling" or "paring", perhaps, but again I'm not convinced: butter doesn't have a peel and paring is more of a cutting action. The artefact of the process is a "curl of butter", so maybe "curling", but I've never heard that used in this way.
Suppose two particles A and B collide. Consider the position vs time plot of the trajectories of A and B, shown in green and red in the following diagram. The two trajectories "meet" at coordinates (x', t') indicating collision, although according to the Pauli exclusion principle, two particles cannot be at the same position at the same time. Now, it is clear that the trajectories A and B are continuous and differentiable, indicating that both A and B have well-defined velocities throughout their trajectories. But now consider trajectory C formed by the part of the trajectory A before collision, and the part of the trajectory B after collision (at the point of collision, assume trajectory C has the value of trajectory A). Trajectory C is shown in blue: My question is this: is trajectory C continuous and differentiable at the point of collision (x', t')? That is, does it have a well-defined velocity at point (x', t')? My own thoughts on this problem so far is that trajectory C is not continuous at (x', t'), because the left-hand limit of trajectory C is the point on A's green trajectory near (x', t'), while the right-hand limit of trajectory C is the point on B's red trajectory near (x', t'). But I want to understand/formalize/prove this more rigorously.
I do understand that we can't experimentally verify anything we imagine about the interior of a black hole. If we were to apply what we know about the physics of the observable universe and assume that those laws remain valid on the other side of an event horizon, then are the following assumptions at least plausible? Anything that falls into a black hole might experience perpetual free-fall, because everything closer to the singularity will experience a much stronger pull than things farther away. The observer, similarly, will experience a constant acceleration away from the event horizon, so objects that cross it after the observer does will always be moving away from them, from their point of view. Objects within the event horizon of a black hole can't interact with our observable universe, but there doesn't (to me) seem to be any reason why they can't interact with each other. So they could clump together in volumes that are small enough where the tidal forces don't rip them apart. In this way, wouldn't the inside of a black hole look like a universe where everything is constantly accelerating away from each other, but in smaller scales can still bind together through all the normal interactions that we observe on "this side"? I hope this question doesn't break the rules of stackexchange, and I'm totally ready to have all of the above bullet points torn to shreds.
I've encountered a few possible definitions of a "connected ring" and am having some confusion relating them. The first one is defined for any commutative ring: A commutative connected ring has a spectrum which is connected in the Zariski topology. But there is also the concept of a topological ring, where the ring itself (not the spectrum) is endowed with a topology. In this case, it also seems natural to consider a topological ring to be "connected" if the the ring itself is a connected topological space (irrespective of the spectrum). My concrete questions are thus: For a commutative topological ring, is there a relationship between "connectedness" in the spectrum-sense vs in the ring topology-sense? More broadly, is there any relationship between the topologies on the two spaces? They seem unrelated to me. When I read results on ring theory, I'm often confused whether topological statements refer to the ring itself or the spectrum. Take for example the following: Every compact Hausdorff ring is totally disconnected. I'm assuming here that the ring in question is a topological ring, and "compact Hausdorff"-ness is refering to the topological properties of the ring itself. But here, is "total disconnected"-ness referring to the ring topology, or to the spectrum topology that results in the definition of a "connected ring" above? Am I getting tripped up on overloaded terminology here, or is there some deeper connection that I'm missing? Thanks for the help!
I understand that the state space representation is mathematically equivalent to the transfer function representation for linear systems, and that it allows us to solve the corresponding DE by finding the eigenvalues of a matrix. However, for nonlinear systems, the transfer function can only represent a linear approximation, while the state space form can represent the full system. But what's the advantage of using state space form for nonlinear systems, if we can't generally solve them by matrix methods? How does state space representation help us analyze or design nonlinear control systems any better than we could by sticking with the original DE representation? Some background: My impression was that the state space form of linear systems is essentially just syntactic sugar for the final result of transforming a nth order DE into a system of n first order DE's, and writing that system as a single matrix equation. It "hides" the derivatives under the extra parameterization variables. But for nonlinear systems, we can't just get a system of linear equations and write it as a single matrix equation that doesn't explicitly involve derivatives. So I don't see how the state space form simplifies anything for nonlinear systems.
In my school, I learned that when two blocks are placed on the ground with one block above the other, if a force is applied to the lower block, two opposing forces of friction act on it: one from the ground and the other from the upper block's surface. Consequently, according to Newton's third law, the upper block experiences a friction force in the forward direction. However, I have a question regarding this scenario. If the external force applied to the lower block is significantly less than the limiting friction of the ground, the lower block won't be set into motion due to the opposition from the static friction of the ground. In addition, I believe that the static friction of the upper block also plays a role in opposing the motion(as it does when the block do move). Consequently, the upper block should experience an equal and opposite reaction that sets it into motion as well. However, this doesn't seem to happen in reality. What misconception do I have in this situation?