darklight03/StackMIAsub huggingface dataset:
I have a doubt in this question, Diagram given below In the Question it is asked that "calculate the final velocity of the block in the figure" and in the solutions it is given that the work done by normal force exerted by the surface is not considered when applying the work-energy theorem but according to the proof of the work-energy theorem, the "'net' external force X displacement upto which the force was applied is equal to the change in kinetic energy of the object." (could use integration by taking the variable force to be constant for infinitely small time intervals) but by this definition normal force should also be added to calculate the net external force on the object as it also always acts in opposite directions to gravity (with different magnitudes obviously) So why is the normal force not considered here? Is the question/solution wrong? If it is correct, why? And what is the mathematical proof of not considering the normal force when applying the work-energy theorem?
Most introductory logic textbooks that I have skimmed through in a while, keep the terms 'sentence' and 'expression' undefined. I would intuitively see Earth is round. Why didn't Harry come to the party? Come here, Harry! x lives in Norway. all as sentences. However, 'x lives in Norway' is usually not taken to be a sentence in logic because it contains a variable. It rather gives sentences after variables have been replaced by constants. It seems to me that the term 'expression' has a broader meaning and sentences are special cases of expressions. But still what exactly is an expression? Would arh ahfb hghd udh be an expression? Would 'arh', 'udh', etc. be expressions? If 'arh ahfb hghd udh' is an expression , would it be a sentence? Is it necessary for an expression to have a meaning? All these troubles imo are stemming from a lack of definition of terms 'sentence' and 'expression'. How exactly can we define these terms?
So, as I have read and even been taught by my teachers, sign convention in trigonometric functions is based on the location of the respective x and y points denoting the coordinates of a particle going around a circle. Although I am kind of sure that I am right but I still want to confirm this one thing: When determining the slope of a graph, we encounter both obtuse and acute angles (the angle made by the tangent of the graph at a particular point on it with the x axis). So, is it just a coincidence that tan(a)(the angle made by the tangent) is positive in case of acute angles and negative in case of obtuse angles in both this graphical sense as well as tan(a) for a point on a circle? Because if a is obtuse(graph), that would mean that the quantity in the y axis is decreasing and if it is increasing, the angle is positive. So, I just want to ask that here tan(a) just happens to be equal to the slope of the graph right? I checked for other trigonometric functions and well sin(a) seems to be negative if the angle is obtuse in the graphical sense but positive in the circular sense. Edit: I have added a picture for clarity about my definition of graphical and circular sense.
I want to learn calculus by myself, I searched a lot on internet as well as math stack exchange for suggestions for best calculus books, I know a lot of famous books like Stewart Thomas but I do not like the way it is presented. I want to learn rigorous mathematics from books like Apostol Spivak Courhant Serge-Lang, but I am not sure if I can take them, the level of proofs and logic they need, I am afraid if I buy them and not able to solve because high advanced level they are, some motivations is needed and also guide me if I need some more books as prerequisite to understand Apostol Spivak Courhant I am reading books like how to solve it by George Polya, if you can suggest some books which refine logic and thinking please suggest me. I am also someone who enjoys mathematics but not want to learn just basics. its more like dream if I one day able to read and understand books like real analysis by Walter Rudin THANK YOU
I am trying to write down a complete/detailed definition for the parity symmetry. Symmetry as a concept is different in mathematics and in physics. There are also many other concepts which differ in their use in physics and mathematics i.e : symmetry group, discrete/continuous group, continuous and discrete symmetry etc. I am trying to consider the following concepts and phrases for the definition: Parity symmetry. Parity transformation. Invariant property (invariance and symmetry while similar they also differ in use) parity symmetry group. discrete symmetry group. Then: In physics: "Parity symmetry it describes the invariance of a system, it's properties, under the parirty transformation, a spatial transformation, represented via the symmetry group of parity, a discrete symmetry group." Is my definition, while including the above listed phrases accurate? Also, can someone show me the fact that there is a group for parity? I.e elements, neutral element, inverse etc? In other words, which are the elements and the operation in the symmetry group of parity?
As we know, two bodies undergo radiative heat exchange due to each emitting a spectrum of light according to its temperature (blackbody radiation). When one body is hotter than the other, it emits a higher magnitude of light at each frequency. Thus the heat flow is from the hotter body to the colder body, even though light is exchanged in both directions. My question: as the photoelectric effect showed even a large magnitude of light of the 'wrong' frequency can't eject an electron, has the experiment been done where a large magnitude (high amount of W, i.e. not according to blackbody radiation) of low-frequency light was directed to an object to observe if the large amount of low-frequency light heats the object? Or is it not possible for other reasons?
Suppose you trying to sell the idea of the Yoneda embedding to perhaps a rather mixed bunch of students (so you can't presuppose too much mathematical background). Still you can say Think of a group-as-a-category (one object, all the arrows isomorphisms). Then applying the Yoneda embedding theorem we get ... [a bit of chat] ... hey, Cayley's Theorem. Think of a poset-as-a-category. Then applying the Yoneda embedding we get ... [a bit more chat] ... hey, the familiar result that a poset is isomorphic to a certain bunch of subsets of its objects (upper sets) ordered by inclusion. Those are in fact the usual textbook offerings. But what third or fourth examples of such embeddings (not requiring the fully caffeinated Lemma) might work as equally accessible? Or maybe not quite as accessible but more interesting??
(Look at the picture) Let's assume there is an horizontal plane impacted by diagonal airflow with components from coming downwards and ahead. If we say that the airflow is fully deviated by the horizontal window (which isn't allowed to move through space), the outcoming airflow has only an horizontal component greater than the one it had before. So, if the wing generated a force in order to change airflow direction, the airflow generated an opposite force on the wing. According to this theory as long as the horizontal component of the airflow isn't reduced, the wing is pushed both upwards and forewards, even though the airflow comes from ahead (which is counterintuitive). Is that correct or am I missing something, since I would normally think the wing must be pushed backwards?
I'm finally closing some gaps in sound waves, so forgive me for lots of questions. In metal, it's said sound travels fastest. The reason is molecules are tightly packed(more dense) in metals than in air lets say, and collisions with each other will be fast as each molecule doesn't have to travel long before it collides with another one, hence it travels faster. Though, I wonder even though it will be faster, we should have disadvantage here which is the signal created by source won't be much weak than it would be in air. My logical thinking is since molecules are tightly packed, it will be harder to transfer energy from particle to next particle(more tightly packed, harder to vibrate the same frequency as created by source). So even though sound travels fast in metal, it should give us weaker signal in the end than in air. Would this logical thinking be correct somehow?
I'd like to know how the parallel transportation behaves in non-Levi-Civita connections and how does one realize it formally. I know that parallel transportation along some piece-smooth curve is defined through moving by geodesics: say, in case of conformal connection, one transports a vector along a short geodesic close to a piece of the given curve preserving the angle at each moment of moving, so, for example moving by a horizontal line in the flat realization of the Lobachevsky plane (where geodesics are semicircles and vertical lines) one gets that a vector rotates uniformly. But how does one understand the parallel transport in general (of a general connection given on a principal bundle)? So, are there any axioms of the parallel transport except being an isomorphism between model spaces in close points on a smooth manifold and "additivity: moving along one piece of a curve and then the second" and "inverse: going in both sides" axioms?
I heard that if a dish of mercury is heated by a moving flame placed under it, the mercury will spin around - Sanderson [Ivan T. Sanderson, biologist and paranormal researcher] then goes on to make the basic observation that a circular dish of mercury revolves in a contrary manner to a naked flame circulated below it, and that it gathers speed until it exceeds the speed of revolution of said flame. Now, as in this question Why is mercury magnetic? , if an electric current is passed through the mercury in the presence of a magnetic field, the mercury will spin around, but I can't find any reference to a version with a flame. Is this true/would this theoretically be true? If so, why? Or was the person who said this just misremembering the electric current version?
While calculating the electric potential at a point near charged bodies such as a uniform ring, hollow shell and solid sphere, iv'e seen that the potential at a point is equal to: V = KQ/D(avg) Here D average denotes the average distance between point of potential measurement and the elements of the object. It is somewhat like the distance between the "Centre of charge of the body" and the point of measurement. This seems to work out for my few cases which ive tried. It also sometimes works for electric field at a point by replacing D with D squared, but fails in some cases such as with a uniform ring. My doubt: Is this approach of calculating "Centre of charge" always applicable for potential at a point? Why so? And if yes, why does it fail for electric field strength at a point?
I read several times about global warming leading to more exteme weather events, i.e. flooding, droughts and even winter storms occuring at higher rates and with more intensity. So, higher temperature supposedly leads to an increase in the variance of the probability distribution of the weather. This is not obvious at all. Naively, one might think that climate change would just shift the temperatures by several degrees. So, is there any physical mechanism, explaining the increase in the variance of the weather? I thought about the equilibrium vapor pressure of water at higher temperatures but I did not come useful conclusions. Btw, a standard physical model is the damped harmonic oscillator. Is there any similar model to demonstrate the effects of globabl warming or meteorolgy and climate change? I'm thinking of something like a closed box with stone islands and water under some radiation.
Suppose a player wishes to move on a square lattice graph without diagonals. Vertices on this graph have a chance (fixed or time-dependent, e.g. cumulative density of X~Po(t) with range n to infinity) to become a "point of interest". Upon visiting such vertex, it will revert to a "normal" vertex, which can become a "point of interest" again. The player wishes to visit all "points of interest", but travelling through the least number of edges. The Elevator Algorithm seems to solve a simplified version of this problem, where the "player" can only travel in two directions. My guess is that, if the player starts from the center of the lattice graph, the optimal choice would be either try to walk over all the vertices exactly once for multiple rounds (Hamiltonian path maybe?) or to stand still. However, this guess only works for the fixed-chance scenario. Thus, I wish to know how an optimal strategy would be derived, given a certain number of lattices are already "points of interests", especially when vertices convert to "points of interests" with a time-dependent chance.
I was recently reading about the Hafele-Keating experiment and asking, how does time in the plane which has flow westwards could have passed faster than on the surface of the earth if the frame of reference was here. It was then I realised that the frame of reference in the experiment was chosen to be the center of the earth. But now I'm asking what would have happened, if one really chose the frame of reference to be on the surface of earth. Then, the plane should have had a velocity greater than that of the inertial observer on the surface of the earth in both direction (in one even greater due to earth's rotation) and time should have passed in both situations slower with reference to the observer on the surface of the earth. I'm not sure if these assumptions are correct, as I've just started to learn SRT on my own.
Isobars are atoms (nuclides) of different chemical elements that have the same number of nucleons. According to the https://en.wikipedia.org/wiki/Mattauch_isobar_rule if you have two adjacent elements on the periodic table have isotopes of the same mass number, one of these isotopes must be radioactive. The decay can happen by positron emission, Electron capture of Beta decay. When electron capture occurs, there will be a hole in the first electron shell that will quickly be filled by an electron from a higher shell, giving of what's called a https://en.wikipedia.org/wiki/Characteristic_X-ray Something I've realised though is that if the Isobar with more protons is the heavier Nuclide, but the mass difference is less then the Characteristic X-ray of the lighter Nuclide, then it would be impossible to decat via capture of an electron from the innermost shell. Are there any Nuclides where this is the case?
This may be a very basic question, please excuse my lack of knowledge but I don't seem to understand the concept of anti-matter gravity. Upon research, many sources align with the conclusion that anti-matter reacts to gravity similarly to matter. ie. that it space-time warps around its mass. If we consider a particle/anti-particle pair we would expect and conclude that there is no gravity (warp in space-time) by this. ie. Just the vaccum of space. However, if we consider them as separate particles, then their corresponding gravities -pulling spacetime in the same direction, would add together (since gravity is not reversed for anti-matter). We would expect to increase the total gravity as they approach each other (greater density of mass in space-time). However, as we know, a pair would have no gravitational effects. It's hypothetical but to clarify; consider a plane vacuum with one particle and one antiparticle beginning to collide. Disregarding their electrical attraction, they each warp space-time in the same direction due to their masses. However, after the collision (as a pair) there is no gravity. So, how would their gravitational potentials dissipate and react as they approach each other and collide.
I'm trying to create a cheap concave lens effect for a class I'm doing. It seems like a convex lens starts to create a similar effect anywhere passed twice it's focal length. It also makes everything upside down, but that's okay. Are the effects of a convex lens a demonstrational equivalent to a concave lens once you pass double the focal length? Ignoring the upside down aspect. This is for a hand made kaleidoscopes library program to give you some idea of the stakes. EDIT: I set up a magnifying glass and walked some distance away from it. What I see through it is: upside down, in focus, and inclusive of a wider view than what it blocks. I can see an entire shed within the lens, while it does not block me from seeing the shed around it.
Applied Force is our label for a contact force that a person exerts. When an applied force acts at an angle, it is actually a combination of two forces: normal and friction. The component of the applied force that is perpendicular to the surface is a normal force, and the component parallel to the surface is a friction force. I have two questions; When the applied force is at an angle then forces of friction and normal are between the surface of the object and the hand, not between the object and the ground so can we say that the parallel component of applied force is responsible for the acceleration of the object? (Is it okay to say that it is the friction force between our hand and the object that accelerates the object)? What will happen if an applied force acts on a thread also can we say that the applied force is the combination of normal and friction force between my hands and thread if so then tension force will be also called a combination of normal force and friction force or if I am wrong then please correct me.
I believe this question would have been asked before, but not like this. The popular answer to this question is that the slide-release action of a bow sets up vibrations in the strings, of which ultimately only the resonant frequencies would survive. Plucking a guitar string sets up a transverse oscillation on the strings at its resonant frequencies. But when you play a violin, you slide a bow over the strings. We know that the strings first attach to the bow, then release once the static friction is overcome, and this process repeats again and again. But how does this process ensure periodicity? We don't hear a discontinuous noise from a violin; rather we hear smooth continuous notes. How do the strings know when to catch the bow, and when to release them ?
Bell's therorem seems to disprove localism because measuring, let's say spin of an entangled electron, seem to communicate the measurement to it's another pair instantaneously. But isn't another thing possible? Maybe the electrons are not communicating anything and instead the two instruments which are measuring the electrons "know" at what angle the other instrument is measuring the other electron and that is a part of the measuring process of instrument and thus the Bell's equality is violated but local realism is still valid. Here the two instruments and the people who are performing the experiments are in some kind of weird sync where they cannot measure arbitrarily in any direction but instead the measurement angles are predetermined before hand and the instruments already know the angles with of each other which are used to measure, hence the spin directions can be correlated. This would mean that Universe is deterministic but also locally real. Is this at least theoretically possible? If it is possible wouldn't it be more saner theory to adopt rather than throwing local realism away? Why? Why not?
In "Gravitation" by Misner, Thorne and Wheeler the authors pose the following puzzle: The metric perturbation of the wave changes the scale of the distances slightly, but also correspondingly changes the scale of time. Therefore does not any possibility of any really meaningful and measurable effect cancel out? And they also give an answer: The widened separation between the geodesics is not a local effect but a cumulative one. [...] When one investigates the separation of geodesics [...] over a large number of periods he finds a cumulative, systematic, net slow bending of the rapidly wiggling geodesics toward each other. This small, attractive acceleration is evidence in gravitation physics [...]. Is LIGO measuring only a net acceleration (or maybe a net displacement as a cause of the net acceleration) of the mirrors instead of a wiggling? The LIGO collaboration states the opposite. On the FAQ page (https://www.ligo.caltech.edu/page/faq) you can find Back and forth the waves pass through (interfere with) each other as the arms themselves change length, causing light interference that ranges fully between totally destructive to totally constructive. In other words, instead of nothing coming out of the interferometer, a flicker of light appears. And how could we obtain the frequency of the gravitational wave when averaging over a large number of periods? But if I misunderstood the statements in the book "Gravitation", what is the correct solution of the puzzle stated in the beginning?
Consider a simple gas (or fluid) within a box at thermal equilibrium. I manage to give a kick to one particle within the gas, such that it acquires some momentum. After some time, it should be expected that this localized energy (or momentum) input would be just distributed such that Boltzmann distribution is recovered. But how does this recovery take place ? The simplest mode of transport of this momentum disturbance could be for example a sound mode, that is, a sequence of kicks in the same direction of the initial kick is generated starting from the kicked particle. But I can also imagine that after the first kick, momentum is transferred in random walk slow manner. Here the momentum disturbance would be tracing a some kind of random trajectory instead of a linear trajectory as it would be for a sound mode. How to know which kinds of modes are taking place (roughly depending on kick strength, density, temperature and so on..) ?
In this post Reconstructing isomorphisms via the bijection between the corresponding posets of subobjects I asked for the possibility of constructing an isomorphism via the order-preserving bijection between the corresponding posets of subobjects. In the case of the category of sets just work with the singletons to define the bijection - I'm working with classical NGB set theory - Now, an interesting counterexample has been exhibited in the aforementioned post. However, I've another question. Well, which are the categories sharing the same property as set (i.e. to give rise to an isomorphism between the corresponding posets of subobjects). I was thinking to toposes. Might it be a good way or are there also counterexamples coming from toposes? Actualy, there exists a paper by Barr and Diaconescu entitled "Atomic Toposes" (https://www.math.mcgill.ca/barr/papers/atom.top.pdf). Well, I think that what I'm searching for is related to the fact that the subobject lattice is atomistic, i.e. every element may be expressed as the join of its atoms. It is property related to my problem?
If something is said and people understand it, sounds like a word to me. If people didn't understand it, I would think they would rather say, "what do you mean?", than making it a word after a definition is explained. Or are there exceptions I'm not thinking of? Dictionary or not, if people understand something, even if just locally, it's a word. Of course, there is some irony in people seemingly claiming to have a mastery of language who do not know what the word 'word' means. One example may be this David Cross bit I've heard, where he said 'irregardless' isn't a word, even though it's been used for a long time and is understood. However, it is considered nonstandard in at least some dictionaries. There may be cases where something incorrectly thinks that a valid word isn't part of standard English even though it actually is. I would still include this in the scope of the question, because they they think it's wrong, whether they are right or not. If someone is making a mistake, if it's still understood, it'd still be a word, for that moment, at least. A more common example may be the use of "ain't". (Kids exclaiming 'that's not a word' out of disbelief would be an exception, but I think it's a rather different usage.)