I would typically use the term "electric field" like so: "[...] therefore, it has a high electric field". If I wanted to hyphenate this term with a compound word, how would I do so? Someone mentioned that "Electric-field-induced voltage" is appropriate, but I believe "Electric field-induced voltage" is (no hyphen in between electric and field added). Which is it?


Every explanation and example uses a t-bar. But what if the "bar" was a propeller? I can understand there being no difference if it's a vacuum, but I would think the aerodynamics would change in a gaseous environment.


When we equate (with opposite sign) coulombic force to the centrifugal force for a electron revolving in the orbit. Basically we are equating a conservative force with the non-conservative force. Can we equate the two different categories of force. What is the reason for this?


I have rotation curve data (radius vs rotational velocity) for some specific galaxies. How can I estimate the dark matter density for those specific galaxies by using the rotation curves? Or is there such a method to do it?


If a semiconductor laser is operated in a DC condition, the injected current is higher than the threshold, that is to say, the gain will be larger than the overall cavity loss. So, the stimulated emission will be built up to dominate the photon density of cavity. Is it correct that photon will be generated unlimitedly to get high output power?


I am trying to model a pv system and I want to simulate different angles, and to calculate values of different angles. I am asking if there is any software that allows me to design the figure below and calculate angles.


The difference between "I can fix the computer" and "I could fix the computer" (present tense) is that the latter expresses more reluctance or doubt. What is this type of sentence called? I can relate it to conditional mood, with an implied condition: "I could fix the computer [when I feel like it / if I know how]"


I found in John Bell's book "A Primer of Infinitesimal Analysis" such interpretation of time representation: But I can't fully understand this interpretation. Can someone explain easily what John Bell means here? Thanks.


If a water molecule is made up of at least one tritium atom, when the tritium atom(s) decay radioactively, what happens to that water molecule? Does it bond to one or two Helium atoms? Is there any chemical or nuclear equation that defines this process?


I was assigned a presentation on Hypercorrection yet I have absolutely no idea what it is (the definitions I found were really vague). Please help me grasp the basis of it and kindly provide some examples/cases. Thank you so so much!


I am dealing with matrix lie groups, Like GL(n) or subgroups of GL(n). Does geodesics of these matrix lie groups with left invariant metrics are related to one parameter subgroups? Can someone offer some references or examples for me for further study?


According to The book Statistical mechanics by Pathria, The Bragg-William approximation states that "the energy of an individual atom in the given system is determined by the (average) degree of order prevailing in the entire system rather than by the ( fluctuating) configurations of the neighboring atoms" What is the reasoning behind the above statement?


I want to create my own Journal template, and I would like to use other people's templates in https://www.overleaf.com/latex/templates/tagged/academic-journal Can I import a class and modify it to my liking? Is this legally ok?


Let's say someone says: Take me to the store right this second. I want to quote the person directly but incorporate it into my sentence my changing "me" to "him" as follows: He said to "take [him] to the store right this second". Is this the correct use of the brackets to change the quote?


Imagine a vertical line representing the lamppost, a point moving away from the base of this line representing the skateboarder, and a line from the top of the lamppost to the skateboarder representing the light ray. The shadow of the skateboarder is the line from the skateboarder perpendicular to the lamppost, intersecting the light ray.


Can we apply the same formula of the "vector triple product" and "vector scalar product" for complex vectors (that is, vectors with complex entries)? Clearly, I assume we calculate the cross product with the complex vector formula, and that we also calculate the inner product with the complex vector formula (either antilinear to first argument, or antilinear to second argument).


Can we safely say that: Even without assuming Einstein's postulates, experiments have shown that measuring the average speed of light in the vacuum on a closed path always produces the same result: the constant c? Do we need to specify that the measure must be performed in an inertial frame or possibly something else, or can we state this principle in general?


I'm Looking for a book of rings of formal power series, I would like to study the construction of polynomial rings but i did not find a good book yet. Many books give the wrong definition and despite I didn't learn yet ,I know that is wrong.


below screenshot of the same image showing (left) the image as desired in a generic image viewer and (right) after inclusion with XeLaTex and includegraphics. i remember having to deal with something similar years ago but googling did not bring anything relevant to light. anybody know how to get XeLaTex/includegraphics to not change the image background?


I'm trying to do a diagram of forces. I tilted my rectangle with the right angle as well as the vector pointing upwards. How do I make my Weight vector? I want it to point down and to start at the center point of the tilted rectangle, but i'm not getting how. Here's the code.


Could someone help please! I have to find in analytical way (self-calculate) the eigenvalues and eigenvectors of this system. I also have to write the code in matlab by using tfestimate function. Then compare the results.


Wearing swimsuit should be mandatory. Can this be used as an instruction? I suggest Wearing swimsuit is mandatory. But I want to prove why "should be" is not correct in this sentence. Can someone help??


What is the equation-of-state for the photon gas ? Remarks: Here Photon gas = Black body radiation. For more background see How does radiation become black-body radiation?, Does thermal energy include the energy of thermal radiation as part of its definition?.


Do low energy cosmological/relic neutrino-antineutrino pairs annihilate to produce photons at all? Their energy is presently too low to produce electron-positron pairs but there should be an indirect, suppressed path to produce photons. Is there an estimate of the rate or time to annihilate?


Since the problem of quartisection of a disc into four parts of the same area by chords coming from the same point can be solved via Dottie number as shown on this picture: I wonder, whether the problems of trisection and quintisection of disc also can be solved using Dottie number.


Do you know how can one extend this trick to find partial fractions coefficients when the roots of the denominator are repeated? From now, I'm just interested in the cases when the roots are algebraic.


I was reading through Josef Albers' interaction of color (a classical art color theory book), where he makes this point: If I see a substance, or pigment of a particular color, doesn't that mean I'm seeing reflections of primarily that wavelength that corresponds to that color? I don't get this point that it consists of all other colors like white.


I know that a space with an inner product are subset of normed spaces, which are subset of metric spaces, which are subset of topological spaces. My question is: What is the relationship of a probability space with all these other spaces?


We generally know that plasma turbulence creates magnetic field (here, I have assumed that it happens through the dynamo effect). it is also affected by magnetic fields. Is it possible that plasma turbulence could be fluctuated by the earth's magnetic field, if so, does it have any applications (like help generate more plasma turbulence through the magnetic field).


I understand that Toponium is only theoretical because the T quark decays too quickly. My question is this, would a right handed T quark live long enough to form a meson? If not, how long would it take to change from a R to L handed quark and then decay versus form a "stable" meson?


Hi to everyone. I don't understand why the diagram in which final antimuon and muon are exchanged is not possible (so having in the upper part positron and muon, downwards electron and antimuon). It seems in all diagrams I seen that in the upper part I must have only antiparticles, and in the lower part only particles (or vice-versa). Is there a specific rule?


Many times in question papers we can see that there's a question on "how death is romanticized in xxxx poem/ story?" My question is how do we determine that the writer has romanticized death or suicide, what are the indications?


If I have an unbiased ellipsoid with a known covariance, is there a way I can use that covariance matrix to transform all known points on the given ellipsoid to trace a sphere instead?


I use the integration of zotero and overleaf to store the references I use. However, now that the bibliography file is very big, it takes a long time to add every single reference to the .bib file. Is there a way to speed this up? Perhaps a way in which only some files are uploaded?


I am trying to draw a diagram below using Tikz or mathcha. As for Tikz I do not know how to position a node relative to other two or more ndoes. As for mathcha, I do not know how to connect a line with a text or math box.


"Modeless" is widely used which is not in the dictionary such as "Modal and Modeless Dialog Boxes". Why they do not use "modaless"? If "Modal" is the adjective of "mode", so "Modeless" is the opposite adjective of it. What is the difference between "model" and "mode"


A passing ship creates a large wave. As it moves shoreward, the wave encounters an anchored yacht which rises up. If the yacht is prevented from falling back to its pre-wave floating position, has the wave transferred an amount of energy (equal to mgh) when it reaches shore?


Does air in the atmosphere suffer friction in some way due solely to the planet's rotation? I mean, if you took a rotating planet with an atmosphere (not being influenced or heated by its star, its nucleus or any other element), would the air get some friction only due to the planet's rotation? Does this slow down the planet's rotation?


I use IguanaTex to insert formulae in PowerPoint. However, the generated formulae look like this . That is, the box is very long. I do not know why and how to fix this. I attach my code here .


Is this a cleft sentence or a "preparatory it"? It was my tablet which I didn't realize had fallen off the bed and on the floor. Is it correct? I think it should leave out the structure "I didn't realize."


Does glass undergo internal damage in a similar way to wood when struck hard, even if it doesn't visibly break, and is this damage accumulated gradually over time, or does it occur as a binary response, unlike wood.


This is some sort of attractor I have obtained in the digital oscilloscope. The oscilloscope is connected to a chua circuit of my own design. My real question is whether that the phase plot I have obtained on the oscilloscope is a strange attractor or not? Or is it some other attractor?


I noticed in the first exercise itself that one requires knowledge of the Fundamental Theorem of Arithmetic. What introductory books gives the proof of the Fundamental Theorem of Arithmetic? It'd be nice of the book also explained how to extract roots and find the highest common factors of big numbers alongside proofs of the basic geometric formulas-Pythagoras, surface area of a sphere, etc.


Are there any ways to prove if irrational numbers are really irrational and the digits of decimal does not repeat? Or if maybe irrational numbers are actually rational which the repetition of digits are so far apart we see them as 'irrational'.


In the standard model of particle physics, there are four fundamental forces/interactions, each governed or conveyed by its respective fundamental particle: Strong force: quark/gluons Weak force: neutron Electromagnetism: Photon Is gravitation hence subject to the existence of an analogous fundamental particle, the hypothetical graviton? If so, what, if any, empirical evidence suggests it exists?


The sentence was this: In his process of dying; the children, if they are still not independent of his guidance, shall perish. The expression 'are still not' does not seem grammatically correct. What could be an alternative way of framing this sentence using 'still'?


There are some candidates for tetraquarks, pentaquarks or even now sexaquarks/hexaquarks. But, I wonder, what are the highest molecular quark states taking into account QCD? Are there heptaquarks, octaquarks, nonaquarks, decaquarks...stable enough to be found in colliders or Nature?


Suppose U is a subspace of V such that V/U is finite dimensional. V/U is the quotient sapce, namely the set of all affine subsets of V parallel to U. I think we cannot show that V is finite dimensional, but I am confused without any idea.


I want to visualise how various orbitals are arranged in a multi electron atom. And main confusion Im having is how large or small are certain Orbitals. So is the picture attached giving a good estimate of relative sizes? (I got this form Wikipedia.)


I have a problem similar to the one described in this question. However, additionally to having the size given in the file, I would want to give it specifically in axis units (assuming equal axis). How could I achieve that?


If the strong interaction force between the quarks that make up protons & neutrons get stronger with distance, and if the gluons enabling that force split up if enough energy is applied to them, giving rise to new quark-antiquark pairs, how can atomic fission work? How can atomic nuclei be split up if new quark pairs are created when gluons snap?


Not sure if this counts as a ``high quality post" but I'd appreciate any advice on the LaTeX template/ package source that I can create a callout with: Here's an image of the desired output:


Pasted below is a screenshot from this super interesting website: https://ciechanow.ski/bicycle/ Are the blue and green forces in the diagram below an action/reaction pair? If so, how does the process of equilibration described in the text make sense? If not, how would the drawing be changed to show the relevant reaction forces?


I wish to determine Plucker coordinates of Infinite dimension Sato Grassmannian. I've been reading the original paper by Sato and other related works, but couldn't understand few things Do Plucker coordinates of infinite dim Sato Grassmanian also fulfill Plucker relations ? If yes, how do we go about determining those Plucker relations ?


So any gravitational field will have a gradient, no? But an accelerating object does not experience any gradient of force. So you should be able to tell if you are in gravity or accelerating by measurement. Right? Is it even possible to experience constant gravity?


moment is equal to force x distance but why is is it so. why is it so that greater the distance less the effort needed? i tried understanding it mathematically but it just went over my head. if any body can explain it in any other way ,i would appreciate it.


I am looking for a good book or review on the treatment of a charged particle interaction (an electron typically) with magnetic, electric and electromagnetic fields. Ideally with both classical and fully quantum treatments.


I am studying Automata theory and in its introduction , there is a statement about conditionals, if A then B is logically equivalent to A only if B. But if B is true and A is false , then if A then B holds true. Then, what about A only if B ? Will it also be true?


I know that when a material is magnetized, the electrons of the atoms get aligned in the same way, they point to the same direction. But then progressively they demagnetize. So, how can permanent magnets mantain their magnetism? What property do they have to do this?


Is there a word for the other party when you do not meet your agreements? I would say something like "duped", but I don't want to give the impression it was unfair, let alone planned. I suppose something along the lines of "let down", but snappier and less sympathetic. I would say something like 'jilt', but it's not marriage specific.


Consider a one dimensional chain of N classical point masses interacting with neighbor harmonic forces. Is it possible to find initial conditions (positions and velocities) such that non-periodic (irregular) oscillations can be observed ? If yes, what is the relative size phase space that leads to this ? i.e how likely is to observe this ?


How can I stop vim-latex from opening my preamble files open loading the main-file.tex? (It opens all preamble files that the mainfile calls at the beginning in a separate buffer which for me is unnecessary).


The negative mass Schwarzschild metric has no event horizon. Why isnt there a particular radius in which spactime flows outwards at the speed of light? This would imply a region of the solution for which exterior particles could not cross.


When beginning a sentence with an apocopated word, does one capitalise the first letter printed--ie, the second or later letter of the word? Or does one just let the apostrophe stand for the absent capitalised letter? Thank you.


I am looking for a reference (preferably a textbook so that additional preparation material is handy) that calculates the exit time of a Wiener process from a region bounded by sloped lines. Thank you, Gary


At low photon energies, the probability of the photoelectric effect to occur increases, but the probability of the photoelectric effect to happen also increases when going towards most inner shells like K shell but inner shells require much more photon energies to be broken, so isn't there a contrast here? How is this EVEN possible?


We know that the vertical anemometer, just like the cup/vane anemometer, works by measuring how fast the cup/fan rotates because of the wind. But how could we possibly know that the fan rotation is due to vertical winds, and not due to the horizontal winds? Wouldn't the horizontal winds affect the readings of this anemometer? Why, or why not?. image credits: link


In this sentence: Some of the sensitive topics that have dominated xxx talks in the past now seem further down the agenda. What is the meaning of "down the agenda" in this sentence?, What is the usage of "down"? Is it used as a verb?


Looking at the Wikipedia entry on the topic: Superior Highly Composite Numbers It is true of all numbers listed there. Does this commonality continue forever? Put another way: Are all SHCNs factors of all larger SHCNs?


I have to present in class "The classical version of the Frobenius theorem". Here is the classic version of the book "Foundations of Differentiable Manifolds and Lie Groups" (Frank W. Warner). I want more references on this theorem with example and applications of the PDE. Could you suggest me some, please?


I am self-studying the book Lebesgue Integration on Euclidean Space by Frank Jones. There is no doubt that this is a great textbook on real variable functions, but the large number of unanswered examples interspersed in the main text makes me look very struggling. So where should I find the errata or exercise answers about this book?


My understanding is that the dimensions of objects change relative to an observer so that a moving cube would no longer be cube shaped and its volume might differ. However, what if that object was absorbing radiation based on its surface area: would a stationary observer see a different amount of heating than a person within the cube? And would this be a paradox?


People have extracted so much underground water that it affects the Earth's balance. Most relative clauses start with a relative pronoun, such as "that." What is the subject in the relative clause above? Is it "that" or "it?"


When I'm reading about color spaces on wikipedia, I noticed that it only concerns about the power spectral only distribution of a illumination without caring about the phases. I wonder if some frequency components of the light would cancel out, why only using power distribution is enough? If two illuminations are not coherent, then how exactly would their power distributions merge?


Is there any literature related to Slip boundary conditions of friction type for magnetohydrodynamic equations? As there is some results for Navier-Stokes equations with slip boundary conditions of friction type but no result I can find for MHD. Can we use the slip boundary conditions of friction type for MHD specially for its weak solutions. Especially, for weak solutions of MHD equations.


In a chemical reaction, I have studied that a reverse and forward reaction occurs every time and neither reactant nor product can be zero. My question is: if the forward reaction and the backward reaction take place at a particular temperature then how can both the backward and forward reaction take place every time simultaneously?


We know that when an electron jumps from shell to shell it produces light waves which produce the sensation of vision to our eyes. But can anything be said about the colour of an atom in particular. Can anything also be concluded about the colour of any subatomic particle.


Title says it all, but some more examples could be: like vs love pleasure vs euphoria carelessness vs apathy etc. ...essentially words that carry more weight than their counterpart despite having similar definitions. I hesitate to call them synonyms- though that may not be incorrect to say, it isn't the term I'm looking for here.


On my way of satudying Heisenberg matrix mechanics, I get blocked by formulas engaging derivations with respect to a matrix arguments. My question is the following : Is there any mathematically rigorous definition of deriving a matrix-valued function with respect to one of its matrix argument ? thanks in advance.


Pretty straightforward question. Does there exist a theoretical upper bound for the acceleration of a point mass? If not, what is the greatest acceleration of a point mass ever recorded, or that will be reached by new technology in the next few years?


From linguistics, I saw typology. By dictionary, typology is the study of or analysis or classification based on types or categories. So I was wondering if there are some relationships and differences between typology and type and category theories? Is typology the application of type and category theories in specific areas?


I would like to have a line with both equations and text, but the math environment takes over. What can I do so that I can include text in the equation display ?


We have a metal sphere completely immersed in a liquid of density A by means of a string. Since the sphere is exerted by the buoyant force due to the liquid the sphere must also exert an equal and opposite force on the liquid. So wouldn't that contribute to the pressure at the bottom ? (Assuming no atmospheric pressure)


Are photons created in Spontaneous Parametric Down-Conversion (SPDC) intrinsically entangled? If, yes, why to construct all complicated interferometers (Mach-Zender, Sagnac) around it? Is it critical that the photons not be entangled in any degrees of freedom other than the one encoding the quantum information? thank you Vlad


Professor Brian Greene covers a unique angle into the twin paradox by having the stationary observer and the rocket release light beams towards each other in fixed intervals according to their proper time. Here is a screen grab from a demonstration from his course at World Science U: Have any experiments been carried out to replicate what has been shown here?


Where can I read about Valleytronics? Are there any good books or review papers on this topic? And which are the most influential papers? The question was also posted at the Matter Modeling site.


I know that the Ihara zeta function being the same for two graphs does not necessitate an isomorphism between the graphs, but say instead the zeta functions were the same and the zeta functions of their Galois covers were the same. What could we say then? What if the same follows for the intermediate covers?


While I was reading dot and cross products I stumbled upon the well known fact that the dot product yields a scalar and the cross product yields a vector. All I want to know is why. Why does the dot and cross products always yields Scalars and vectors respectively. And also, what led us to define the products the way they are defined?


What does the existence of CTC imply for the Causal Structure of the spacetime? Can a strongly causal spacetime have any CTC[without fluctuating the metric]? Is there any such example? Can someone share please? I'd really appreciate if so.


Good morning everyone! I'm pretty new to reledmac package and, although i read the manual several times i still can't figure out how to put the /pstart number on the outer margin of the page (like i can set line number in the inner margin). Here's an example of what i mean:


I just learned the definition of point-free topologies (also known as pointless topologies). Is every point-free topology expressible as a pointwise topology (so the notion of point-free topologies is nothing but an alternative way to look at existing pointwise topologies), or are there point-free topologies that are not pointwise topologies (so that point-free topologies is indeed a "proper" generalization)?


enter image description here hi everyone I have this data of wind direction in radians from -pi to pi (picture is added). any idea which stochastic differential equation (SDE) can represent it? i need an SDE that will produce approximatly this behavior.


Texlive/pdflatex used to produce documents I really liked the look of, which was like this: I have since reverted to the defaults: Is there any way to look in the .out/.log/etc files that produced the first pdf to try to reproduce that look?


Is there some classic example of an universally injective ring morphism which is not a faithfully flat morphism? I was not able to find it in any commutative algebra book and neither around here.


How will the absorption spectrum of a doped semiconductor look if we consider both the dopant as well as the exciton energy levels? And qualitatively, what is the difference between the formation of an e-h pair corresponding to an exciton energy state, and similarly, another e-h pair corresponding to any of the dopant energy states?


The white x-ray spectra (bremsstrahlung radiation) when plotted for different incident electron energies in the same graph, looks very similar to the Black-body radiation curve except for starting from a minimum wavelength. Firstly, what is the governing equation for x-ray spectra which explains such curve, and secondly how is it related to the BB curve.


The question is in the title. I've done some reading and seen that assuming AC, existence of such a function is guaranteed since the cardinals are well-ordered. Can we explicitly construct such a function?


Form what I understand a necessary condition for Riemann integrable function is it must be bounded, and a sufficient condition can be that the function is if a bounded function has finitely many discontinuities. Can someone provide an example where it showed that Riemann integrable function MUST be bounded?


To get some insight on the zero locus of the cubic curve, I've tried a couple of online visualisers on Google and mostly failed to generate the plot (run time errors...) except Wolfram Alpha which gave me the following; Is this visualisation correct??


What is the precise definition of a dust solution in general relativity? If the Einstein tensor of a metric has only the first diagonal term non-zero, it that sufficient for that solution to be called a dust solution?


In a state of quantum coherence, if only a part of a macroscopic object interacts with the environment, does the non-interacting part also have an effect? Does the part that interacts with the environment affect the rest of the part that does not interact with it, so that behavior becomes classically unconditional?


I would like to study string-theory on my own and would like to know the knowledge pre-requisite on mathematics and physics I should master before embarking on my journey to study string theory. Thanks


I have a system of nonlinear equations and what define if it is concave or convex. probably it is solved by using Hessian matrix, but I don't know how to do this. Someone can help me?


Consider this locus Now a property of this is that the angle between the tangent at any point B and the line joining origin to B is constant . My question is why does this happening? I was able to prove this but don't understand why this happens.


What is the mechanism and the reasons of this heat release? I know that the molecules of oxygen are solvated and the hydrogen bonds are rearranged. But for what reasons are heat released and what does that concretely mean? Does it mean that the mean kinetic energy of the molecules is a bit increased?


This is the sentence: The auto-negotiating device will link at the speed of the non-negotiating device, and set its port to half-duplex mode. Can someone explain on what is being referred here when "its" is mentioned? Its = Auto-negotiating device? Its = Non-negotiating device? Which does it indicate and why?


A perfectly inelastic central collision of two equal relativistic particles whose kinetic energies are equal to their resting energies results a single relativistic particle (and nothing but it). The mass of the resulting particle is greater than the sum of the masses of the particles that collided, but why and how?


We were asked to determine how many square roots can you construct with the root spiral before overlapping. I attached an image of the root spiral in question and an other image wich contains my work on the problem. Any help is greatly appreciated!


I mean we know proton and electron posses charged that are opposite to each other and they emit electromagnetic radiation perpendicular to each other. My question here is where does this charge comes from at the first place . What process is there that provide charge to these subatomic particles ?


I want to add specific pages from a PDF as backgrounds to different pages in my LaTeX document, creating a new command for each desired image. How can I do this? I'm somewhat new and can't find how to do it.


I have read that despite strong connections between prime factorization and DLP an algorithm for the former does not imply the latter directly. But I was reading about the number field sieve and it seemed like the bottleneck was identifying smooth norms. Wouldn't an ultrafast prime factorization algorithm achieve that?


i know that they arn't in the normal sence connected but how is it that when one particle is observed spinning up the other one "starts" spinning down. It can't have always been spinning down.


Toby calls Jetta ugly and gets embarrassed when Sarah calls him out and tells him that he should be nice to her because Jetta's done nothing to him. Days later, Toby calls Sabah a bully for responding to the hate she received from another classmate.


I am a scientist not specialized in plasma physics who has little knowledge in transport phenomena. Please, what are the mechanisms responsible for dissipation in collisionless and collisional fully ionised plasmas? Many thanks.


I read in a physics book that Without clocks of extreme accuracy, the Global Positioning System (GPS) that is now vital to worldwide navigation would be useless. What would be effects on navigation of less accurate clocks in the context of GPS?


There are a few "question words" that mirror their answer words- When/Then Where/There What/That Who/Thou (might be stretching it here..) Do these words have origins where this makes sense, or is this a coincidence?


How can make style of theorem like this?.Honestly, I took this picture from the internet because I couldn't do it in latex, and I modified it to show what I want to do in the environment of the theorem.


What we learn about physics can usually be divided into two parts: study of Particle or Wave. We later learn wave-particle duality, etc. Particle is somewhat intuitive, but what's so special about a wave that makes it fundamental? Why not other things?


I am using the textbook 'Introduction to Solid State Physics' to learn about 'PARAMAGNETIC SUSCEPTIBILITY OF CONDUCTION ELECTRON' (Book by Charles Kittel) Consider the diagram when a field is applied I dont understand how we can have total energy which is negative. In the case of the 'parallel to field spin', there is section that shows negative total energy


I know that we can have complex inner products defined such as Conjugate linear to first argument: SUM(conj(lhsElement), rhsElement) Conjugate linear to second argument: SUM(lhsElement, conj(rhsElement)) Can we have a complex inner product defined to be conjugate linear to both arguments, such as SUM(conj(lhsElement), conj(rhsElement))?


I am writing a manuscript. There is a sentence my advisor and I keep editing. The sentence: We did not detect significant effects of lineage, initial length, nor status on individual growth rate. Would it be more appropriate to write the above sentence with "or" instead of "nor"? If so, why? And why do I have to use "or?


Suppose I have hexagons that like the ones below but I know the area and the points of each hexagon that represent a cuboid of dimensions g,h,d. How can I find the values for g, h, and d? Any pointers on finding a solution?


In the electromagnetic field, there are NxN points on the incident plane. By changing the phase of the electric field at these points, the desired electric field distribution is generated in the target field, that is, the amplitude and phase of the electric field are simultaneously satisfied. If we can use the LM algorithm, how do we represent the Jacobian matrix?


I am currently very interested in texts, books, and papers that combine topology and logic. I recall reading many years ago a journal article that talked about both topology and the axiom of choice. I would be very glad to see some references.


I'm looking for references which provide good surveys and histories of results related to Gateaux and Frechet derivatives and their relationship to Asplundness and RNP. Specifically, any papers or books which explain the underlying ideas and geometry, rather than simply proving results or leaning heavily on other unexplained existence results as a black box.


Although I say "Listen to reason," if I insert "voice of," do I use "Listen to the voice of reason" or "Hear the voice of reason," or can I use either? The topic came up in a class I was giving distinguishing between TO HEAR as a sense verb (not deliberate, involuntary) and TO LISTEN as a deliberate, voluntary verb.


Can we say that an object at rest on the surface of the Earth accelerates in the direction of the ct-basevector, as viewed by an observer standing on Earth? Obviously, there is no acceleration in the space direction.


Below is the simplex tableau in optimal format. I supposed to have identity matrix, but instead I got negative values. This means my solutions are negative. How to proceed further in order to find optimal solution to a problem? Simplex Tableau in optimal format


Is "stump" for 'leg' a good choice when the goal is to pick a slang word that is commonly understood by Brits? Or, would, for instance, "pin", or any other synonymous slang word, be a better choice for that reason?


I tried reading the English translation of Bourbaki's theory of sets. and I am confused, there is little that makes sense to me. I have experience with Naive set theory, so I was looking forward to learning ZFC and surrounding schemes with some rigour but this is too complex and inconsistent.I am looking forward to some dialogue where I get to know something.


Symploce is two or more sentences / phrases sharing the same beginning and end, but a different middle, while I'm looking for two or more sentences / phrases sharing a middle but different beginnings and ends. Such as in the question: A is to B, as C is to what? And Roses are red, violets are blue


I know that the probability of the electron tunnelling out of the orbital is non-zero, but my question is strictly wrt the energy-time uncertainty at zero Kelvin. Is that probability non-zero as well?


There are certain functions that can be non-differntiable but are still solution to a "differential equations". This is fine, as a non-differentiable function can still have an integral. The main question is the following: What does it mean by taking the Laplace transform of Non-differntiable functions.? Isnt it a violation of some rules here and there?


I had my purse stolen during my stay in London. In this sentence, "stolen" is used insted of "steal". In my childhood, I had a fire break out. In this sentence, "break out" is used instead of "broken out". Why??


Ou high-school professor told us that a time varying magnetic field in a cylindrical region produces a sort of circular electric field which is Non conservative in nature, because Electric field varies radially, however, I fail to understand why that is the case.


Knowledge hoarder and information hoarder refers to a person who keeps knowledge/information only for themselves,I am looking for a term that can be used to a person who gathers knowledge/information,like a common hoarder but they hoard knowledge or information


Why do we get a null point in potentiometer or meter bridge. I mean the battery is connected so current should keep flowing through the circuit. So why do we get a point where no current is flowing as the battery is connected


Is usage of "thereof" in a title: Confirming feasibility of attack and investigating improvements to range and reliability thereof correct? My intuition says it is not, but it would be a pity if this is the case as I don't want to make the title overly verbose.


Can I have another phrase for the sentence/phrase; He is a very easy person to scare. Please I need this because I am writing a story which I'm going to produce when I'm finished...


Is objectual a word? I could not find it in Merriam Webster. I am trying to use it in a sentence like this: A phrase signifies the objectual nature of thing in question. Would I be stretching the grammaticallity of the word object?


What is the word for doing something where said thing is a protest against what you are doing? Example, posting a facebook status that pokes fun at people copying and pasting statuses and sharing them however by posting this status you're doing the very thing you're speaking against.


I have seen a video on YouTube on channel Vsauce about the flat earth belief. There he claims that walking to the edges of this flat earth would be difficult. Something like going up an acclivity. Is this claim true and if yes why? Any answer containing maths would be useful.


I was reading mobility in current electricity and there it was written that the direction of conventional current due to positive and negative current carriers is in the same direction but I cannot understand how and why?


I am currently starting to work on LED but I am confused how can I calculate the external quantum efficiency for my LED? like I know the basic idea, I want to know what experimental setup should I used for my LED and how? In my Lab we have Raman spectrometer with CCD detector.


I understand that matrix is a linear transformation of a vector space and matrix multiplication is applying one transformation over another. I couldn't get a geometric intuition as to why multiplying matrix by its transpose give covariance matrix.


Is there a way to run the --shell-escape command for the LuaLaTex engine in Texshop? I know there are editable TEX and Latex commands in the Texshop settings under Engine->pdfTeX settings, but I do not see anything for the LuaLaTeX engine.


In the Born-Oppenheimer approximation, the effective potential energy, is the potential energy that an electron gains when considering all the inter-particle interactions in a molecule? Said in another way, which particles do experience this effective potential? Is it correct to say that the effective potential is the intrinsic molecular potential energy?


Has the double slit experiment been executed with a proton beam? Electrons and photons seem to get all the attention. What about a particle with positive charge, and one said to be composed of multiple charged entities?


Can someone please explain how the boundary points on a circle are also extreme points? If i take a point inside the circle and one point on the boundary, I can take a convex combination of those two to get a third point on the boundary?


I did part a and b but for part c does the fact that when those three vectors are added gives the zero vector does it mean that they are all perpendicular to eachother? So therefore they would all have the same cross product?


If we have a sealed piston-cylinder with gas inside and we heat it from the outside, the temperature changes. At the same time, if we look at it as a control mass system,the volume increases as the gas expands. Mass remains same. Hence specific volume changes too. So how can temperature and specific volume always be independent?


Can someone give me a strict defenition of a constant sheaf. And I need a nice reference for supermanifolds. Second, concerning the commutation of the pullbacks with the restrictions, how cna I check that in the special case where each of the sheaves is a subsheaf of the sheaf of continuous functions, then the commutation is indeed satisfied.


As generally known, FWD cars are more stable than RWD, but I don't know why and I couldn't find the reason. Can anyone explain me with physics concept? Additionally, I also want to know why driving experience of RWD is generally much better(comforter) than those of FWD.


The compression of a gas produces an increase in temperature and therefore greater kinetic energy of the particles due to the exchange of momentum between the piston wall. Is this transferred energy also capable of producing a sound wave?


Where does the interior Schwarzschild metric come from? How is it derived and why does it have NOT a singularity? Would it mean that the singularity is only apparent and for those out of the black hole (who are ruled by the usual, exterior metric)?


What is the word that means the area of work where someone has experience and knowledge? It should be similar to major, but major doesn't include the work/training expertise that you may have acquired beyond college.


I am planning on doing some personal studies and some poster research for some future conferences in Homological Algebra, does there exist a current list of outstanding problems in the field of Homological Algebra?


If the eigenvalues of a matrix all have positive real part, must the matrices diagonal entries all be positive? This is true of symmetric matrices, however I'd like to know if this result holds more generally.


I have asked to find the condition such that the lines joining focii of an ellipse don't subtend right angle at any point on ellipse.. pllz tell me the condition which i have to apply btw the two lines.. whether in slope format or another.. I am a high school student and suffering problem in this question plzz help..


Consider this sentence. X studied fault structure using method A. Y studied fault structure using method B. However, the geometrical complexity of a fault remained poorly understood. Do I have to use 'remains poorly understood' or 'remained poorly understood'?


When calculating the presence of soliton or anti-soliton in the extreme dimerization polyacetylene SSH model, we say that in the case of open-boundary condition and odd number of atoms, we must have an edge soliton. But why is that the case? I understand that we must have a soliton, but why is the edge mode preferred over the bulk case?


How do I merge cells irregularly in tables? I need to do this stuff for homework and don't feel like printing it out, writing it by hand, and scanning it back in to upload. I could just hide the borders on the cells above the Answer column but then text wouldn't flow into that area. I want a real non-rectangular cell.


What is the origin of stretch as it is used in the following sentence? We should eat before the final stretch. In this context, final stretch is used to mean 'last segment', or 'the effort needed before the work will be done'. Is this use present in dictionaries?


What's the difference between the term "parameter" and the term "argument" in TeX jargon? Are outermost enclosing pairs of matching curly braces of undelimited arguments to be considered components of these arguments? Are delimiters of delimited arguments to be considered components of these arguments?


A colleague has written: If you are encountering any more problems please don't hesitate to reach out. Reading this, I believe it should phrased: "If you encounter any more problems..." but I cannot describe why - the language tools I have tested say the original is "clunky" or have no complaints. A) Is the original technically incorrect? B) If so, why?


I know that free ultrafilters are defined in contrast to principal/fixed ultrafilters. Nonetheless, is there some categorical way to view the use of the word "free" here (e.g. some pair of adjoint functors), or some universal property that free ultrafilters satisfy?


If object is at rest relative to the Earth's surface on a frictionless surface. What is the effect of Earth rotation and orbiting on an object? Does object rotate and orbit with Earth and stays at same position due to gravitational force. Or object stays same location and earth rotate underneath it?


Shwezigon Pagoda's appearance always makes me feel powerful. And it also made me consider how mighty Bagan and our nations were back in those days. Or should I be using "and also it" here? Does it make a difference?


I read the octet and decuplet baryons symmetric and anti-symmetric wavefunction from 'Riazzudin and Fiazzudin's book. But I am little confused about the wavefunction of neutral sigma and Lambda baryon. What is the exact form of the wavefunction of baryons?


There exist sites such as https://shouldiblamecaching.com/, https://isitdns.com/, and https://isitchristmas.com/ that use the domain name to ask a question and show a simple yes or no answer. Is there a term that can be used to refer to this category of websites?


Is there a term for a "union" or "meeting" between Kings? An example would be: four different kings go to a meeting in one's realm to discuss politics. (Though I really don't know if it was a common practice or has a name.)


My friend said that it is 'boae constrictor' since 'boa' is in Latin. But I told him that first of all, it would be more correct as 'boae constrictor(s)', but the actual correct form is 'boa constrictors'. Which one is it?


In a question on a different SE site the title is as follows Writing the introduction section of an academic journal while the question is about writing the introduction section of a paper that will be published in an academic journal. What is this rhetorical device (i.e. saying "the journal" while instead referring to an article inside the journal) called in English?


In general relativity, there is a definition of the surface gravity that is associated with the Killing vectors, is there a similar definition for the interior of stars/compact objects? I would expect that if I measure the surface gravity inside and near the surface of the star and outside it should be the same.


Want to know if this sounds OK. Here I am asking a friend if it's OK for me to come over to her place to return her book. Hi, Was thinking to come return your book today, would you be home? Thanks


If I have a spool of thread with a force pulling to the right on a rolling cylinder, how can I mathematically determine the direction of the friction? . .-----------> F . . . . . . . . . . . . . . . . ^^^^^^^^^^^^^^^^^^^^^^^^


What is so different between the molecules that have already evaporated but below the critical temperature than those that exist after reaching the critical point? If there are gases above critical temperatures that cannot be converted back to liquid, are there any critical liquids in the sense that they cannot be frozen back to solids?


could someone recommend some references where the problem of constructing an accurate map of the Earth is tackled from the point of view of general maps between manifolds (so something like this and not like this) ? Thanks!


Considering the hyper-fine-structure configuration of the H-atom, why is its stable, low energy state (or the "flipped-back" state) the state when the spins of the electron and the proton point in the same direction, ie are parallel (and not antiparallel, as is sometimes stated in the literature)?


For Vacuum fluctuations based QRNG source, how will the linewidth of the laser source affects the shot noise. If we decrease the linewidth of the laser source, will we get a better output.


I am doing a quantum mechanics question involving the positivity of the norm. So I'm using the fact that the norm will be greater than zero but i want to apply an operator onto the ket on one side of the equation, is it a legal operation to apply the operator to the zero on the other side and say its still zero?


I'm working in ZFC and wondering which axioms can be considered constructive. I find Axioms of Pair, Union, Empty Set, Infinite, Power Set and Replacement to be clearly constuctive. Choice and Regularity/Foundation are non-constructive. I've read here that constructive Zermelo-Fraenkel considers Extensionality. How is Extensionality constructive?


For example, someone spoils a show or says something is poorly written. Originally you may not have thought or noticed that but suddenly your view starts changing to become similar or the same as their view.


It's said that Neutirnos can only make up a tiny fraciton of dark matter. So why can't Dark matter be mostly made up of Neutrinos? Why can't there just be a huge number of them? I suspect myself that neutrinos are so light that they end up being "too fast". But I'm not sure myself.


Three forces P, Q and R act along the sides BC, CA, and AB respectively of a triangle ABC in order to keep the system in equilibrium. When will the resultant force touches the inscribed circle.


I was looking at my neice's second grade note book in which was the question " Where did the four cows live? " but her mam corrected it to "Where did the four cows lived?" I think it should be live as we use first form of verb with did. Am i wrong?


Are there any colours that our human eye cannot comprehend but other animals can see? The ability to see colours is the property of our eyes. For example an average dog would see less colours than us. So does this imply that there could be colours which the human mind cannot comprehend while some animals can see them?


If we put some blocks on a piston and the piston moves downwards giving pressure to the gas, will the pressure by external mass(blocks) on piston and pressure by piston on gas be the same?


Spanish speakers use 'basket,' for basketball, 'smoking' for black tie and 'freaki' for geek. They also use 'camping' for camp site and 'parking' for car park, but the participles retain the same meaning. The latter are loan words, but I wonder whether it's accurate to use the term 'loan word' if that word undergoes a significant shift in meaning.


I have been reading about Hackenbush recently and have learned that the surreal numbers can be represented using RGB Hackenbush. I am having a hard time understanding On, Off, and Oof. What Hackenbush games represent these numbers?


Fascinated by the font of this journal, and found out it's ITC New Baskerville Semi Bold. How to add this font to the overleaf tex, the closest package I could find is baskervillef and librebaskerville.


There should be opposing viewpoints so that discussion can take multiple turns AND people can finally reach a solution that neutralizes both viewpoints. Should there be a comma before "and" in this sentence because both are independent clauses?


My question is not why this is different than in the near field where it was measured by Hertz. I am asking for a source with a description of who measured it in the far field and how.


I just recalled the previous winter that I had stood in front of our heater to get warm. I then observed that the heat that was going up had a shadow on the wall behind the heater. I suspect those were the shadow of the air molecules. But I can't prove.


Fermions with no electric charge may carry magnetic moments e.g., the neutron. Since particles with magnetic magnetic moments interact, they're expected to scatter off each other electromagnetically. How does QED describe such a scattering?


As capacitance is the ability to store charge therefore we say conductors can show self-capacitance but why can't insulators? we can store charge on insulators as well so why can't they show capacitance?


I am teaching an introduction to pure maths module. Are there any really good resources on relations? I want something that will have an impact and motivate the learner. I am not really looking for artificial engineering examples but some bona fide real applications.


The energy is not Lorentz invariant quantity since energy depends on the state of motion. Then how come the physics is the same in every inertial reference frame when every inertial frame measures different energy?


I'm new to capillary forces and I found this statement: I found Jurin's law or the Young-Laplace equation, but it's always about capillary tube and I would know how to find this expression. Regards


I was curious as to why the "Law of Reflection" is only a law and not a principle. Are there any specific conditions or circumstances where it is not followed by chance? If so, how so? Note: I am not discussing about or referring to anything in special theory of relativity, however the reader may offer information on it as well.


I am trying to input the non numbered section bibliography in my summary in a document using lyx, but it has been appearing only if I number the section. Is there an alternative ?


Why is it that the water which constantly passes through the core of a nuclear reactor doesn't become radioactive? Despite passing so closely to the active nuclear fission reaction, as gamma particles are capable enough to penetrate thick surfaces. It ends up being converted to steam. What actually is going on in it?


From our point of view in our Galaxy, his center, Sagittarius A, is a super massive black hole. At event horizon of this singularity, gravity stop time. Is there any reason why time would not run in reverse, on the other side?


What is the maximum number of operations that can be defined on Cbits (like mechanical switches) and Qbits (like quantum states)? If they are different how will the ratio change if we limit ourselves to Reversible circuits? Edit: By operation I mean that a n-bit state remains an n-bit state


If objects gain mass as they approach the speed of light, is it possible that as a photon "attempts" to exceed the speed of light it gains enough mass to interact with the higgs field immediately slowing it back down until it no longer has enough mass to cause an interaction?


Propositional logic and first-order logic are theories in ZFC. The principle of structural induction is a theorem of ZFC. Is that what justifies the usage of structural induction in propositional logic and first-order logic? By justifying I really mean proving, that is, tracing back to axioms. Meaning of structural induction: Sturctural induction.


When Austin Powers says this phrase? Is it considered to be somewhat of an exception to normal use, like with, Myself is often used where I or me might be expected, Or was it more abnormal than that, and was just a joke? I'm not sure of other instances where it's used commonly in a similar manner.


According to Special theory of relativity, magnetic field is equal to electric field if we see it in a frame of electron (in the frame of electron) but why we have to see it in accelerating frame? Why can't we just see in stationary frame?


A lot of sources stop expanding the taylor approximation at the second order Hessian matrix for a multivariable function. I'd like to include the third order taylor expansion. Does anyone have the matrix equation for this and have any sources talking about it? Is there a name for the third order derivative matrix?


From what I understand, the voltage and current waves propagate down the input transmission line, hit the nonlinear oscillator, and then bounce back toward the generator. How are gates such as the X or the Hadamard encoded into these waves?


HOM is a two-photon interference effect where temporally overlapped identical photons coming perpendicular to a beam splitter must leave it in the same direction. How is momentum conserved in this process? Initially, the total momentum of the photons is only on the x-axis, but after the beam splitter, it has a y-axis component. Does some momentum pass to the beam splitter itself?


Electric charges are of two types - positive and negative charges.electrons are negatively charged and protons are positively charged. If the term electric means electrons and electric charge means negative charge why do we include protons? How can a charge be positive? Please correct me if I'm wrong . Thank you


Which are common protocols to measure the light transmittance of glass (example: a window)? I have searched for references and officials standards without success. Which tools can be used to achieve this measurement ?


Which sentence is more grammatically correct? Welcome to a new way to travel Welcome to a new way of travel (We are promoting our airport service which allows one to avoid stepping foot inside the airport to get to their commercial flight. )


So I want to write my notes where there are lists connected with vertical lines. I want the exact format displayed as in the below photo where in the nested lists the "main" line is stretched. Also I want this design to be added in the preamble. Thanks!


I have long been curious about a particular English (in parts of GB) phrasing habit. For example: Oh she's lovely, she is. That's a nice one, that is. You should keep doing that, you should. I am curious what that repeat is called, how it came about, and anything else I can find out.


I recently discovered that if is considered a preposition in contemporary grammar. Is "when" considered a preposition as well? And in the following sentence, is the chunk introduced by when considered an adjunct, or as integral in that it provides the referent for "it"? I like it when you say "indubitably".


I am interested in learning about differential graph theory or differential operators on graphs, something related to what E. Bautista introduced in his answer here. Can one suggest a textbook in which such topics are discussed? (The quoted answer refers to some papers, but I prefer a book covering a comprehensive treatment of the topic.)


I am working on a paper on how pivot rule selection influences the time needed for LPSolve to solve a problem. I have decided to write some teory about different algorithms, but i can't find anything about the "First index" algorithm. Anybody know what it is, maybe know anything I could use as literature about it?


In this image, power is used to apply voltage to the charged grids. But if no voltage is applied, would ions still be accelerated by Coulomb forces to create thrust? If not, why?


I found these two common English words (snarky and smarmy) that seem like forms of a literary device. However, unlike onamatopoeia, the comparison is not made with sound, but rather it is more from the "visual" images/memory that they evoke. What is this called in terms of "literary technique" (if not onamatopoeia)? "Synesthesia" perhaps?


I am finishing a book with a bunch of complex TikZ embedded pictures, some with complex path decorations and so. I wonder what is better, to include the whole list of usetikzlibrary{} at the top of the LaTeX doc or specify the needed ones inside each tikzpicture, yet repeating


This is more a vocab question than anything else but what "type" of thing are injective, surjective, bijective. I fully understand what these words mean but I'm looking for a word to describe the type of properties these might be. Basically "odd and even" is to "parity" as "injective, surjective, bijective" is to what? Thank you all!


Anyone have anything? I'm writing a "proof" for something and it came up. I'm stupid (as username suggests) and it would take way too long for me to solve it, and I wouldn't want to delete it since I want to just write QED in place of the proof for the solution


I'm looking for a good noun to describe a character who can use Comakinesis, which is hair manipulation, but "Comakinesis-user" isn't good enough. Also, I can't find a word to describe hair manipulation in a magic user word ending in 'mancer' so I'm wondering if kinetic ability names have any kind of word form like that?


Past Perfect can't be used on its own, can it? I mean, if there is a sentence without context, just a sentence on its own, and there is no clause with predicate in Past Simple in it. Is it grammatically correct to use it in American English? Like in this example : John had already given his present.


I've seen loads of cars that skid during a curve due to taking it with a high velocity. I can guess that it has to do with the grid with the road and maybe something to do with centifugal forces?


I have heard a couple of people use intriguement casually but there is nothing online except for a listing in the Urban Dictionary: intriguement the feeling of being intrigued He was nearly as old as her father, much to Martha's intriguement. So, is it an actual word?


When using the proof environment provided by the amsthm package, you get an empty square box at the end of your proof as the default QED symbol. Question: Is there a way to make it automatically put "QED" for some proofs but not others? Essentially, can one create a second proof environment which puts a different symbol at the end?


I try to write this equation on overleaf and numbering in second line (the same as in the picture) but I couldn't. Another problem is when I use align or equation, the prime in max' function goes to the center but not the right of max. Help me!!


I am looking for a "thick" book on complex analysis, which also has historical details such as the motivation behind it. Which problem lead to the creation/discovery of a certain topic. E.g. what is the reason behind fundamental groups how is it in connection with topology. Thanks in advance.


When calculating the displacement of a rolling body do we just calculate the displacement due to Vcom in a particular time t or additionally need to consider also the displacement that may be produced because of the rolling motion.


I saw the excellent answer here: Did the Big Bang happen at a point? but I have a hard time imagining the initial state. If the distances between all points in the universe were zero at the Big Bang, how is it not a single point in space?


Relation between elements of essential spectrum and the eigenvalue of a self adjoint operator. In particular, is there any way we can say that the element of essential spectrum is an eigenvalue of infinite multiplicities of a self adjoint operator. Any idea or if there any article related to this concepts or any book will be helpful.


Is there any way to reasonably restrict control points of cubic bezier curve so it's oscilating circle will never have radius smaller than r? Bezier curve with it's oscilating circle I need to find the conditions that control points have to meet, so curvature of the bezier is never too large


I used a reference manager for my bibliography and added personal notes to all my refernces to organise them. Now using bibtex (bibliography style unsrt) these notes appear in my bibliography. How can I exclude the notes from my bibliography without having to alter every entry in my reference library?


Is there some interpretation for the eigenvectors of the Laplacian of a (directed) weighted graph? Normally, the spectrum of a Laplacian informs us about connectives, is this also the case for a weighted graph?


I have found lots of numbers that can be done two ways but can't seem to find any that are done three or four ways. I believe there would be a way to use complex numbers to help with this.


from wikipedia (if you say it's false I believe you): A series of independent states (for example, a series of coin flips) satisfies the formal definition of a Markov chain. However, the theory is usually applied only when the probability distribution of the next state depends on the current one by definition, the next coin flip should depend on the current one, no?


I don't feel so confident in the sign convention on this example, wouldn't a clockwise angular velocity mean negative sign not positive? Why is my answer and the books answer positive, and it seems like it should be negative to insinuate clockwise direction?


Just like the title, I really can not understand how brownian-motion can be nonmonotonous. I thinck the probability of a brownian-motion moving in same direction in a really short interval can be a positive number.


I've got this message from a quite respectful software library: A value is trying to be set on a copy of a slice from a DataFrame. Shouldn't it be something in passive voice, like "The value is being tried to be set.."?


It is well known that for every monad T, we can consider the Eilenberg-Moore-Category and get an adjunction which induces T, similar for the Kleisli-Category. But since every adjunction induces a Comonad as well, my question is what are the Comonads in the Eilenberg-Moore-Category resp. the Kleisli-Category that are induced by T in this way?


I can't, for the life of me, understand chain rule on conditional distributions of random variables. I can't see the pattern. Like, how do I find out what things like these are: P(A,B|C) P(X,Y,Z|W) P(U|V,W) P(M|N,O,P) P(H,I|J,K)


Recently I have been studying conic -sections in my school . The problems given in the course are good but I am looking for any resources where I can find new theorems and strategies to solve the problems of comics. I want a resource which can provide Olympiad-like geometric problems on conic sections. So please can anyone suggest any good resource for such problems?


My doubt lies in the fact that gravitational waves are produced even before the merger so I kindly ask an expert to make me a clear picture of the source of energy of grav. waves before the merger and the energy after the merger that we are told is gained by the vanished masses of the merging objects.


In other words, would this construction be proper? And if so, is there a name for it? The court then turned to negligence, which issue it easily resolved. (As opposed to ". . ., which it easily resolved.") Aha. I missed this when I first looked at MW. I think it's the third definition under the first section: https://www.merriam-webster.com/dictionary/which


if we covered all the deserts with solar plates , would that be the solution to the energy crisis ? i mean put a solar plates in ALL the deserts of the earth to harvest all solar power


Much like McGyver meaning "resourceful," Einstein meaning "intelligent," Savant meaning "gifted..." I remember hearing and using a name to refer to someone as a charismatic go-getter. Another word that comes to mind is "maverick."


As long as the time dilation is about what the observers think or see, there would be no great difficulty. But the assertions that the clocks run faster or slower cannot be justified.


Power at the receiving antenna is given by the Friis transmission formula, which says that the received power is proportional to the wavelength squared. But energy carried by an electromagnetic wave E=hc/lambda which is inversely proportional to the wavelength. Shouldn't they both have the same relationship with wavelength?


What are the long metal things in stores like Walgreens that hold things like gummy bears or nuts or hair clips? The metal things are straight and then at the end they curve up, and they are connected to the wall to carry things without using shelves.


Why when a person is standing still on a floor we say that the force that this person acts on the floor equal to it's weight? , Isn't the weight the force that the earth pulls the person and acts only on the person, how then this force goes to the ground ?


My teacher told me that in anode ray experiment the electron were emitted from the gas present inside the vacuum tube and the ionized electron form cathode ray and ionized gas form anode ray. So I wanna know thus atoms present inside the vacuum tube were fully ionized and only their nucleus remain? So for my bad English.


From a point just north of the equator, A straight line to the Magnetic North would be through the earth. If a compass was turned on it's side, would the north pointing arrow point toward the ground along that straight line?


I know that currently dark matter and dark energy are separate things, not related and one not deriving from the other. But if both are included in a generalized gravitation theory, the picture can change. Then there may be a correlation between the dark matter and dark energy, and they may then have a common origin.


thank you all. I can translate from language to another manually. Are there any way to translate a lyx document, for example, from English to Arabic in the same way as in a Word document such that the mathematical expressions remain as they were? I want do this automatically by a lyx function if possible.


I have a hexagon with interior with the opposite sides being identified. In order find the identification space I did the following cutting and pasting. My question is : Is it correct way to do the identification? I have heard cutting is not a continuous operation. That's why I am confused. Could anyone give me some suggestions in this regard? Thanks for your time.


Modulational instabilities occur in waves which occur in nonlinear system. A linear stability analysis is done to examine the dynamics in the short-time regime to observe the structures which spontaneously form. What methods are there for examining the structures in the nonlinear regime?


The focus on the exterior of a building alone will lead to the increase in construction cost, making such an endeavour unjustified. I want to know about the grammatical structure of this part: making such an endeavour unjustified.


All chapters begin a new page by default. I want to remove this break between two short chapters. By the nopagebreak doesn't work. What can be the cause? Is there another way to do that? I don't give all my code there, but I can write some details if needed.


I already posed this question Negative energy solutions in Klein-Gordon and Dirac equations but I am not satisfied with the answers. Trying to be very sharp: does Klein-Gordon equation have negative energy solutions like Dirac one or are they (the solutions) somehow different? if they are the same, why the problem of negative energy is always referred to Dirac equation and not to KG?


In my complex analysis course I'm supposed to compute all entire functions with the given requirement for the Real part. How do I work on this? I hav no clue how to work this out. Can someone please give me a hint?


MPM combines elements of both the finite element method (FEM) and the particle-based methods. It is particularly suitable for problems involving large deformations or fluid-solid interaction. Would you apply it (after proper editing) to wave propagation problems, where there is no mass transportation, but only energy transportation?


How does protons collide and produce heavier particles if light particles can not decay into heavier particles ? considering particles can not decay into heavier particles irrespective of their energy. How can we see heavier particles at CERN if only the protons are colliding ?


I am doing a course on rotational movement of solid rigid objects and a professor of the MIT symbolizes the axis around which the object rotates with a symbol that is a letter with a vertical line on top of it and another one below it. Is there a simple way to do that in a line of LaTeX?


"In physics, gravity is a fundamental interaction which causes mutual attraction between all things that have mass" "Light or visible light is electromagnetic radiation that has no mass" My question is how can gravity swallow light ?


What do you think is a suitable word to use instead of the informal "stuff" here, and is "appropriate" used correctly in this context? I will give you the number of the appropriate team that will be able to help you with questions with regard to administrative or financial [stuff].


In English, when speaking about going to the center of the town, it's a matter of going to the "downtown" so, my question is about the origins of the connotation of some "descent" (going down) - moreover, I noticed that the same phenomenon is present in many other languages...


"As we look on on the conflict" I want to use the phrase "look on" but am not sure how to incorporate it into the sentence. Would be "as we look on at the conflict"?


Is there an adjective of intensity (or similar term related to force) that ends in "-al"? like: space > spatial, time > temporal, etc. Intensity is meant as the physical attribute/dimension of some physical phenomenon. Sample sentence: "... the high (...adjetive of intensity needed here...), temporal [adjective of time], and spatial [adjective of space] resolving powers of macroscopic observations require that ... "


I am new to topic " Algebraic Grid Generation". I want to find a simple example where we solve the host equation, let us say the heat equation, numerically in the computational domain then transform it back to the physical domain. All the references I found discuss the general steps without providing a single practical example


What does "which" refer to in the following sentence? The final jurisdiction over what an intellectual is resides in culture, which becomes almost coterminous with a singular public opinion.(https://dictionary.cambridge.org/ja/dictionary/english/coterminous) I think it refers to an intellectual, but I'm not sure.


Like in adiabatic process there is closed system but in reality we know there is no such thing as closed system there will be still exchange of heat so it will not work adiabatically means it is only applicable for ideal gas not real gas?


I would like to know if it is possible to use the Concrete font together with the standard LaTeX mathematical font. If it is possible, could someone please tell me how to do this? My thanks in advance!


You encounter them periodically in movies, and, in certain places, in real life. I asked people who live in a region positively studded with castles, and all they could come up with was "the peeping slidy thingy in the entrance door." Which is amusing but doesn't answer the question. What's it called? And the slide - does it have a separate term?


A friend said something to their dog and it was meant for me. I told her that I am pretty sure that is passive aggressive, but don't think I am correct. What is this called? I attempted wording it several different ways, but continually received links that were not applicable.


Such as the natural linewidth is defined to be angular frequency, while the absolute frequency of laser is frequency. By far I haven't found a good way not to learn it by roting. Would anyone be willing to share with me your thoughts if you have a better way?


i am confused about a result in paper related to reflectivity from etched hole in silicon. The reflectance oscillation varies with surface roughness but i am not getting similar oscillation while simulating in fdtd lumerical. the model paper is based on fresnel equation.


It is possible to get quadrature squeezed light where on one quadrature the uncertainty is less than the of a coherent state while in the other one, orthogonal to it, it is larger. Is it possible, by doing an appropriate angle transformation, to go from a quadrature squeezed state to a photon-number squeezed state (sub-Poissonian distribution)?


I have a feeling that the word "ingredient" implies that it was intended to be there by human, while catechin is not an additive, but was naturally in the tea leaves. Giving another example, is polyphenol a healthy ingredient of wine, a composition, or a compound of wine?


I believe that electric current is a kind of force (even when we consider it to be opposite to the flow of electrons). If it is so then what is the nature of this force ? And also tell me if there are energy carriers in that constitute this force.


I've learned that the Hamiltonian Operator corresponds to the total energy of the system when applied to a general wave function. After applying and obtaining the measurement (energy), the wave function turns into the eigenfunction. Does the wave function before the operator is applied have to be an eigenfunction of the operator, or can the operator be applied to any wave function?


An interior point of an equilateral triangle ABC is M. The projections of M on the sides of the triangle respectively: points E, D, F. Let G be the centroid of the triangle EDF. Prove that G is the midpoint of OM if O is the circumcenter of triangle ABC.


I remember author of some textbook alluding to Zassenhaus and Knuth's bound of the zeros of a complex polynomial.Unfortunately after even after days of search I could not find some reference or source for it.I would be highly obliged if somebody could provide some source


Assuming that there is no form of resistance in the primary circuit, and that the secondary circuit contains resistance, will current in the primary circuit be infinite? Will resistance in the secondary circuit provide resistance in the primary circuit? Thanks in advance.


I came across a problem where a piston and gas arrangement was present and it was heated, so if the piston is free to move we call it isobaric. But is it only true if it is quasi static (reversible process) or also for a normal irreversible process, and if not will that process be some polytropic process.


This is from a test: What on earth have you done to your hair? Oh. I've had it cut, do you like it? I am wondering if the comma after "cut" is justified. Would a period, dash, or semicolon be better in its place?


Regarding linearly polarized electric fields that are produced by a dipole antenna and electric fields from a current carrying wire, are the equipotential surface the same as the magnetic fields? Because both are perpendicular to the electric fields.


I am writing a document using overleaf and spellcheker underlines all words with red, it looks like this: How can I fix it? By some reason it did't underline all words in other documents that I wrote.


I know that random forcing does not add unbounded reals. Is a Cohen real always unbounded? I can proof that the product of two random forcings adds a Cohen real. How can I show that the two step iteration of random forcings does not add cohen real?


Is the equation T=M/F where T is residence time, M is quantity and F is flow rate equivalent to W=L/lambda where W is waiting time, L is queue length and lambda is flow rate? It seems to me that the derivation of either equation would be equivalent to the derivation of the other.


I read a previous question "superconductor levitating in earth's magnetic field?" it seemed like the impact of gravity would make this unikely unless the superconductor was super light. Is it conceivable to combine the properties of aerogels and superconductors to create light material that repels the earth's field? basically is the synthesis even possible? and if so what might it require?


Programmer here. I sometimes catch myself saying that something is a Tautology False because it always evaluates to FALSE or OFF. I know Tautology means always True in discrete math but I use it to describe something that doesn't change. It's always constant in being False or Null, etc. Am I being illogical.


By default the hyperlink in the list of acronyms is in the page number. Is there a way to change this such that by clicking on both the acronym and page number it takes you to its first occurrence? Kind of like using linktoc=all in for the table of contents.


when I look up into the night sky I see planets, not twinkling like the other stars, but still illuminated. However, when the Hubble or James Webb telescopes approach planets we see the local features instead of an illuminated sphere in space. why is that?


I am looking for the very first published experiment in a relevant journal with all the following features: Michelson or Mach-Zender light interferometry is demonstrated The arms of the interferometer are optical fibers The light used is a slow sequence of single quanta (aka single-photons) generated with on-demand methods (e.g. Quantum dots). Does anyone know the reference of this article?


A red light and blue light enter a rectangular glass block normal to its surface at the same time. Strictly speaking, after passing through the block, which pulse exits first? Should I consider their paths inside glass slab to be same as the lights are entering normal to the surface?


I am building an online LaTeX editor and now face issues with slow compilation for documents of increased size. Everytime the user changes the document, the compiler has to recompile the whole document. Is it possible to only compile the changed part and speed up compile performance that way?


I know a and l, and that the dotted lines are parallel, how can i determine the angle alpha in terms of a and l? I've been working on it non-stop for the past two days and I can't find a solution, although my system seems fully constrained. Figure


The magnetic field is pointing upwards and uniform. Part C: Since the magnetic field does not do any work, the initial velocity of the bar should remain constant through time. But in the solution of the problem, the velocity is a function of time. Can anyone please explain this inconsistency?


I would like to use something that looks like the following table in my document, but I have no idea of how to make this using TikZ or other packages. Using a matrix leaves too much empty space above in my tries. The cells should be editable, so I can show the parsing steps.


Please could someone help explain to me the choice of delta here. I am unsure how the definition of uniform continuity is being applied. I guess it is something to do with when x and u fall in different intervals J or I? Thanks for your help!


While i'm doing the math homework, I find something very strange. I am confused by a textbook's answer. The Question The textbook's answer So, Does differentiation change the units of measurement in mathematical equations?


Is there a word for "a diverse amount of knowledge"? I would be used in a sentence like this: Trying to give a diverse amount of knowledge covering many fields and genres of text.


Electrons have a magnetic dipole moment. If a permanent magnet was held against one end of a copper rod, the delocalised electrons close to the magnet would be attracted and concentrate around the magnetic pole. This should result in the creation of a weak electret because of the uneven electron distribution.


As in, is there a manual for how to get good typography using the LaTeX typesetting system that is similar to Matthew Butterick's web-based book? MB's Practical Typography only shows examples for web, MS Word and Pages which is why I'm asking. That website is linked here: https://practicaltypography.com/


Which statement is correct? The change adds more info to the changelog about the previous commits on May xx, xxxx. Or: The change adds more info about the previous commits on May xx, xxxx to the changelog.


The problem I have is that the line around which the circle must be rotated cuts the circle in two parts. Any hint or help with what to do with it will be useful. Thanks.


I just don't understand how the voltage is adjusted to suspend the droplet. What do we use to adjust the voltage? Because, normally, even when electric and gravitational forces are balanced, the charge will moves with uniform velocity towards the oppositely charged direction.


I see a lot of books of Nonequilibrium QFT and they study the thermodynamics on quatum realms, but what is the difference about that and Quantum Thermodynamics? They study the same thing only with different approach?


Needs to prove that integral can be reduced to another form with specific substitution. Not sure how it can be proved. I think that t=sin(phi) on second stage but I can't understand how to start.


I want to see and compare all the beamer outer themes. Something like this. Is this available somewhere? I know I can generate examples myself, but I prefer to use something already available.


In the below sentence, The mission of the entity may include goals other than maximizing profit. is it implied that "maximizing profit" is a goal of the entity OR is it referring to only non-profit-maximizing goals?


I am student of mathematics and I have started to learn algebraic topology . I want to learn visual things . Can you please suggest books that can help me to increase visualization ?


Can Sagnac effect be observed in coiled, rotating electrical cables conducting electric signals, in a similar way to that observed in coiled, rotating optical fibers? And if so then who demonstrated it first in electric cables?


Consider a gas. Now, we already know the particle's duration of a collision is very small and it immediately bounces away from there. Can these be thought of as topological holes in one's phase space (since the particle barely spends anytime there)? What is the volume of these holes and do they obey extensivity?


Some people say that when the electrons are excited beyond the Mott gap, the electrons will just hop to the neighbor ions and stop there so will not contribute to electrical conductivity. Is this true?


One to the power of infinity seems to be undefined. Is the infinite power tower of one also undefined? EDIT What about one to the power of the cardinality of an infinite set? Does that do away with the issue of the (alleged) undefinedness?


Can we say the ability of computers to make decisions is based on a quantum property? I refer to the PN junction in semiconductors, is the phenomena related to PN junction a quantum property? Why I asked that, I wanted to know if the finest element able to make a decision is necessarily at quantum level. In example can we make a choice mechanically?


I know in an isosceles triangle, the height, median and angular bisector to the base are coincidental by SSS or other congruent theorems, but I searched all over the world and can't find whether there's a name for this property.


If object A's momentum is decreased, will it produce less force on object B upon collision, in comparison to a situation where A object had greater momentum? If this is true, does it follow that the velocity of an object is proportional to the amount of force it can produce on another object?


In this Penrose diagram for a charged/Rotating black hole, once your in the new universe you can continue onward to an infinite sequence of black holes/white holes. So I'm asking, where do those black holes and white holes come from?


When water is absorbed by soil for example, it gets dark. But when water is absorbed by tissue paper, and kept closer to another object, that object can be seen pretty clearly through the tissue paper. Isn't it possible that soil becomes translucent and the tissue paper becomes darker in colour?


I have seen that the electric field of a dipole antenna detaches and propagates. For a section of the detached electric field, will the antenna experience recoil if the detached field moves a charged object?


Ball is free to move in hollow tube, if you rotate tube ball will move outward. What force push ball outward in inertial frame and how ball trajectory looks? (I know that centrifugal force don't exist in inertial frame, but without this force I can't explain what force move ball outward...)


Ligo works by destructively interfering light from a laser using an interferometer. Its said that no light enters the photo detector when all mirrors are the same distance. However,based on my research,the light would just create an interference pattern and never completely destructively interfere. How exactly does ligo do this? Can I get it done on a smaller scale?


I am new in the community. I am reading the book "Linear models in statistics" of Rencher & Schaalje. Can someone argue if that's correct? How can you express "some simple sum of squares as quadratic forms in y"? picture from book "Linear models in statistics"


if a matrix A is m by n , and m>n , in this case we have one solution? i don't understand how more rows of a matrix can still have one solution


Hoping for some assistance with my possessive conundrum. If you were going on a trip for leaders (not run by leaders/organised by leaders), would it be: a Leaders Trip or Leaders' Trip? Thank you!


I am writing lyrics to a song, and the first two lines I have written read as follows: The spiral begins as these thoughts are imbued, "I'm hideous sin and I have no value." The idea I want to convey is the negative thoughts that saturate or permeate a person causes a downward spiral that can lead to despair.


The existence of Magnetic monopoles has not been proven or disproven. There are equartions for Charge black holes, There are equations for Rotating black holes, but are there equations for Magnetically charged black holes?


We say "I had to leave," but not "I'd to leave." Why? This is also unlike other auxiliary usage of the verb "had" (e.g. in past perfect tense), so the difference is more nuanced than the had being an auxiliary verb or not; why is this not contracted like other auxiliary usage of the verb?


Examples: I can throw a good throw. I want to run a long run today. Scoop me a scoop, please. She gave me drinks to drink. I really like these simple sorts of sentences, but I've never seen any specific term for them.


If I am taking takeout home in a plastic bag, is it better to remove all air from the bag when sealing it so no heat is lost to the surrounding air or keep air in the bag acting effectively like insulation? Let's assume the air is room temp/colder when I seal the bag.


The Theorem on the Textbook Is there anyone who can explain the theorem to me. I find it hard to understand. Thanks! By the way, what's the formal name of this theorem, I found nothing online.


Do you have interesting books presenting abelian variety (over an arbitrary field k) using the scheme point of view? Most of the lectures I know use the point of view presented in the first chapter of Hartshorne (so without scheme). I'm looking for something which presents the modern aspects/questions of those objects.


How to find the lateral surface of elliptical cone? Equation of lateral surface is Lateral surface equation but how to find the value of E in this equation? Here is an example. Example


Why does a rotating body, like a rod or a spring, exhibit elongation instead of compression even though every part of it experiences "centri-seeking" force? Is it plausible to understand this in an inertial frame of reference, rather than the conventional pseudo refrence frame?


While it's not too difficult to derive Snell's Law for acoustics e.g. from Huygen's principle like for light, I find surprisingly little resources on an equivalent to Fresnel formulas linking the reflected and refracted amplitudes, mostly limited to perpendicular incidence and sound intensity in dependence of acoustical impedance, but what are the formulas for e.g. pressure or density continuity at arbitrary angles of incidence?


If A and B are two normal matrices with the same eigenvectors, but different eigenvalues, where, A = UVU' and B = UWU', which operations with A and B (sum, multiplication, inverse, etc,.) will change the eigenvectors of the resultant matrix and which will not?


I typed a work using wx Maxima. I am really short on time. Is there a way I can convert wxmx/wxm file to LaTeX or PDF? Pls answers should be very basic (i.e a toddler can understand). Also this work includes long equations and matrices but no diagrams or tables, just plain text.Thanks.


We know that an eigenvalue decomposition of a matrix is to find those eigenvectors that are just scale for some coefficients. But what about Jordan matrix decomposition? I just learn how it is calculated, but I am not sure whether it has geometric intuition.


I am creating a list of special characters used for delimitation in my work. "-","_"," " are delimiters But the latex code is wrapping on " " "-","_"," " are delimiters How do I treat " " as a word that shouldn't be wrapped?


Fractal based measures help quantify the self-similar geometrical properties of set (in this case a time-series signal). It mostly represents the global characteristics of geometry of the signal. But with time series as data, we tend to look for local characteristics where Multi-Fractal Analysis is adopted. If my understanding is right about Multi-Fractal Analysis, what are Multi-Fractal Analysis methods available and can be adopted?


According to Bohr's hypothesis electrons can exist only at certain special distance from the nucleus only on certain particular orbits which is determined by Planck's constant, how does reach to this conclusion and what's the experiment behind it?


I know that composition of dl isn't correct generally, But I read in my text book that we can do that with logarithms and exponential with some conditions So what is these conditions? I don't find them in my text book


Do evaporating black holes emit gravitons? I know that hawking radiation consists of photons, and that for very small black holes it can also consists of sun atomic particles, but what about gravitons? Those are supposedly massless particles, can an astrophysical black hole emit those?


Is there a specific term for a question, such as Are you asleep? and Can you hear me?, where the binary is Response/No Response rather than Yes/No? I feel like there are other aspects of this concept that orbit this specific question, so any related discussion at all is appreciated.


So the way to measure electron spin is to split the beam with a magnet to obtain two beams. Essentially what gets measured is magnetic moment, that is later interpreted as the direction of spin multiplied by Bohr magneton. Is there any "direct" measurement of electron spin that ignores magnetic moment?


I know the radius of the Earth and the radius of the orbit (r). I need to find the length of arc (d) along the surface of the Earth given the graphical information presented. Thanks for any help you can provide!


In an RL series circuit where the current rate of change is positive, there is a voltage drop across the inductor, if voltage drop means an energy loss, then in what form the energy is lost in the inductor?


Assume that N points are distributed uniformly randomly within the unit square, and call these points anchor points. What is the mean, mean minimum, and mean maximum distance of a point randomly chosen within the square that is not an anchor point to its closest anchor point?


I want the 'Article Title' to not appear in the header of every page in my springer-nature latex document. May I know what to comment out / replace in the sn-jnl.cls file? Thanks.


I have a task where two equal dipoles are separated by a non-grounded, perfectly conducting thin plate. The dipoles are oriented along the plane direction and can be parallel or antiparallel oriented. I wonder what the force is between the plate and the dipoles and between the two dipoles and whether one can reach perfect force cancellation. Thank you very much! Alexey


Why are the so-called radio-transparent dielectric caps covering parabolic antennas based on a metal frame, which, in theory, will work like a Faraday grid, and if it does not absorb the entire signal, at least significantly attenuate it and create interference due to secondary radiation? Look inside Look outside


I have question about matrix derivation, there are two matrix derivation in png, the first is simple one that i wrote, the second matrix is my main question. I want to know is this true calculation? https://i.stack.imgur.com/yLRNY.png


As eg from Wikipedia: https://en.wikipedia.org/wiki/Notions_(sewing) The term is chiefly in American English... was also formerly used in the phrase "Yankee notions", meaning American products I'm wondering if anyone can provide a more detailed explanation of how this word came to have its sewing-related meaning.


He claimed to find the atmosphere vital and creatively invigorating. Perhaps there was something to this: he had taken his languishing novel out of the trunk for the first time in nearly a year. It by Stephen King Could you guys tell what he meant by saying that the novel got taken out of the trunk? Is this some kind of idiom?


Is it possible to create the following effect in latex? The core is that it have a distinct color for each word instead of using a shading, which can mess the text up. It's fine if this is done using javascript and svg. Credits for Celeste for this image...... I'm trying to port wavedash.ppt to latex using dvisvgm, but I'm kinda stuck.


What would be the correct form of this phrase? Atom forge or Atomic forge I know 'atomic' is an objective and 'atom' being a noun is correct when used with forge because you are forging something. But 'atomic' sounds right and I've seen that used before. Just want some clarification.


I have been learning more about sets lately and have stumbled upon notation (or I guess a language) such as this: I would like to know the name of this notation and perhaps some resources to learn it.


Is there a reference out there that only focus on (different)rank of matrices(with all kind of entries: real, complex, integers) and connects then further to ranks of tensors and further with the ranks multi-linear operators. Thank you in advance.