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Currently i include png/jpg files with graphicx package compiling with lualatex, no problem, in particular no error message because of bounding box not found. This question is about historic ways to compile because I don't want to break old code. First question: Is it true that when compiling into pdf with any of the m... | 0 |
The following pop science video discusses the experimental difficulty of measuring "quanta" of gravity. It seems to me that a significant argument involves how it would be difficult to investigate fluctuations of gravity on the plank scale, requiring detection methods that have densities that would collapse as a black ... | 0 |
In the diagram, OA is perpendicular to PQ. O is the centre of the circle. AC and EP meet at B on the circle and QC and AE meet at D on the circle. The goal is to prove that AP=AQ. I had tried to prove this by drawing a line through E parallel to PQ and I had proved some of the triangles were similar. However, this is t... | 0 |
Recently, I tried to understand the effect of Quantum levitation. I read that there is some well-known fact: Relative distance between superconductor and magnet must be preserved, because of flux pinning mostly occurred in type II superconductor. My rough understanding about this is, quantisation of (penetrated) magnet... | 0 |
I've noticed this quite often from Disney sound tracks. Lyrics are in a certain rhyming scheme - one line happens and the next line approaches the end of the line, but the last word subverts the rhyme. I have looked at this question and feel it's related, but not the same effect. Example: "In Summer" from Olaf in Froze... | 0 |
I am a first year student of applied mathematics. I currently find myself always trying to explain any phenomenon with mathematical models- the few ones (models) that I encountered till now. I always try to fit phenomenas to models like a data scientist does when he fits data to models. Recently I found this book: Robe... | 0 |
The boundary of a simple closed curve has each point on it, touching two adjacent points one on each side. Can the entire boundary be represented as a series of circles of the same diameter "d" where each circle is touching the adjacent circle at exactly one point only without overlap in areas between the two circles. ... | 0 |
I'm working on an algorithm to allow a vehicle to reverse and then line up and stop on a given spot facing a specific direction. The best approach I've been able to come up with involves two internally tangent circles with the larger extending from the point of tangency of the vehicle. Vehicle starts at point A and is ... | 0 |
A few days ago, I came to an amazing intuition while using GeoGebra, and I could not prove whether it was previously discovered or is it new, please mention a source if it already exists... The sum of the lengths of the tangents of two circles touching a conic section, each of which touches it at two points, and starti... | 0 |
I've read some description of certain house design as below: Living and dining rooms are towards the front with kitchens to the rear, with bedrooms either off the hallway or upstairs if it's a two-storey terrace. and here's the corresponding house plan: the definition of "off" in google: as I understand, and which can ... | 0 |
Suppose we have a closed circuit composed of a time constant voltage battery, a resistor and superconducting wires connecting them in series. In the equilibrium state, both the free charge the electric field in the interior of the superconducting wire should vanish, lest the free charge accelerates and there be no equi... | 0 |
why potential at any point in a uniform electric field is not same i.e. why potential difference between any two points in the uniform electric field is not zero? According to me if i bring a test charge from infinity to a point A in uniform electric field [assume the field is due to a positive charge] then i will have... | 0 |
In the Hertz experiment, when light falls on the anode, electrons are emitted and they move towards the cathode and thus flow through the circuit. However, when reverse voltage is applied and the polarity of the electrodes is reversed, electrons being emitted from the anode (formerly, a cathode) only come out in the ai... | 0 |
In fluids, in the state of equilibrium, there can only appear internal normal forces and these forces always have a tendency to compress the bounded volume in the fluid. Cited - Physics Galaxy ( Ashish Arora ) Would it be apt to say that in context with fluids, if I drop a block, the pressure exerted by the fluid on th... | 0 |
I want to recreate a plot where the area that is not visible from a certain point on earth (latitude and longitude) trough a full year. An example of this plots for example can be: Where in white is marked the zone a telescope is not seeing over all year. I searched in internet but i can not find anything related on ho... | 0 |
Some examples needing an explanation, thank you for the help. "I came up with (an) idea, we can do..." We should use "an", shouldn't we? But why, if we have a specific idea explained further? What about "He gave me a/the idea to admit to the university" then? "The conclusion is ..." Why should we use "the", when it's t... | 0 |
Consider for example the open sentence, x lives in India where 'x' is a variable, which can be substituted by various constants to make declarative sentences. For instance, Harshit lives in India Carol lives in India Zaid lives in India where 'x' has been replaced by constants 'Harshit', 'Carol' and 'Zaid', respectivel... | 0 |
In the most known incarnations of string theory, we have to compactify the needed, additional dimensions that are then taken to be a.) periodic and b.) "small" in some sense (most of the time their size/radius is taken to be on the order of the Planck length). Some models do propose so-called large extra dimensions but... | 0 |
Find the perspective sequence that maps collinear points A,B,C,D to D,C,B,A. Attempt: If we need to find a sequence of three perspectives that (A,B,C)->(A,C,B), where A, B, C are collinear, then first we mark p is a line through A, B, C, let S be some point that does not lie on p, let q be a line through A and S, let p... | 0 |
I would (like many before me) like to take my lecture notes using LaTeX. As writing pure LaTeX-code is very time consuming and not nearly as fast as handwriting, I can only really consider switching to LaTeX if I can use snippets and anything else that makes notetaking faster. I have discovered this post on how to get ... | 0 |
I am confused whether I should use "a" or "the" when I refer to an object given/identified by some other object. For example: The package is now in [a / the] deposit box with the number you received by SMS. For me this is the deposit box I previously stored the package in, but it seems to me, that for others (including... | 0 |
Take a primordial black hole and measure the Hawking radiation over a large amount of time by gamma-ray detectors, as well as a Large neutrino detector. Using theoretical calculations about the composition of Hawking radiation combined with the detected neutrino flux you can determine the neutrino luminosity. Combine t... | 0 |
Many problems in classical mechanics, classical and quantum field theory among others require the study of systems of partial differential equations where questions like existence and uniqueness cannot be answered with the typical counting-equations/variables arguments - questions which have physical relevance and that... | 0 |
I'm looking at a few English sentences, and I've realized that many verbs can be used as nouns in their infinitive form. For example. I am going on a run and in the present tense I am running. There are many more verbs that this can be done to. Are you going to take the jump? Why did you do three pitches? You know he w... | 0 |
I'm writing longer mathematical texts (lecture notes) regularly, and I don't know where else to ask this question. I regularly find myself enunciating (in the typographical sense) every statement I make. This means that I will put any even vaguely mathematical statement, not only theorems, corollaries, definitions, etc... | 0 |
I have written a long book in latex memoir. I now want to feed its content to a machine translator into another language (say, German). I will use the automated translation for a first pass for a better human translation. Translators do not understand latex. There are a couple of methods to go about feeding them. I cou... | 0 |
I am trying to learn about neural networks. I was reading the paper An Efficient Graph Convolutional Network Technique for the Travelling Salesman Problem which uses graph neural networks such that given a graph we get a hamiltonian cycle which hopefully is nearly optimal. However, they use as ground truth the concorde... | 0 |
i have taken two course in complex analysis, one variable and the other in several variable and the begining of complex manifolds, my understanding in one variable go well but in several variable our prof use two books, one of krantz and the other of Hoermander, i think due to my lack in the theory of integration(multi... | 0 |
I use LaTeX for a lot of my classwork (worksheets, graphics, etc). One thing I have not transitioned is my exam writing. An example page of an exam might look like this. This is just a portion, doesn't have the header and such, I'm comfortable working with those. I've not moved to LaTeX because of my need to be placing... | 0 |
I am posting the specific problem I am dealing with for reference and then explain the part I am confused with. My Understanding and approach to the problem I identified that in the ground frame the disc will have some inertia and thus simply remain stationary as the plank below slides away.When viewing from the plank'... | 0 |
This question is born out of the riddle where there are two characters, one that always lies and one that always tells the truth, and you want them to tell you which of two doors is the "good" or "bad" door. You can ask either character, "what would the other character say is the good door?" And the lier will answer wi... | 0 |
Is there a possible slight difference between protons and electrons behaviour while they are accelerated at speeds that cause relativistic changes to their mass. Why am I asking that? Because for a proton we know that it is made of several components which of them are the three quarks. If they have some speed inside th... | 0 |
An electromagnetic wave is usually illustrated with two sine wave shaped fields, one for the electric field and one for the magnetic field. As in the picture below: But as how I understand magnetism, it consists of two poles, north and south (alternatively described with field lines, with a direction, where north and s... | 0 |
Lill's method is a visual way to find the real roots of a polynomial equation. It involves drawing a straight line with lengths equal to the coefficients of the polynomial. The solution can be found by using a slope of a right triangle. Lill's method - Wikipedia The values of a polynomial function can be "defined" as a... | 0 |
I need to retrieve Roll and Pitch (but not Yaw) rotation values from rotation matrix (or euler angles, or quaternion, input can be different, fortunately, it can be converted from one form to another). [Info table] X points Left. Y points Up. Z points Forward. Rotation around local X axis is Pitch. Rotation around loca... | 0 |
When fluids are subject to external perturbations or put in certain geometries, they always show arguably the most fascinating structures : Starting from the eddies and vortices in pipe flows at large Reynold numbers, vortices of flows past an obstacle, or the vortices in fluids confined between two rotation cylenders.... | 0 |
Lets imagine you have a universe where the average matter density is extremely close the the boundary between one that will expand forever and one that will eventually re-collapse. Now lets imagine that there are very large scale fluctuations in density at the Gpc/Tpc scale and above. Could this result in differences i... | 0 |
There is a good discussion here (Referring to past times with "hence") that concludes that hence can't be used like "ago" to refer to a past event, not even to describe how much time has passed "since" that event. However, what if using the past perfect and then referring to time forward from that point, Could "hence" ... | 0 |
There is a closed continuous loop, and you have a stick of a random fixed positive length,so that You can slide the stick on the loop. Take the stick's midpoint. It's track will form another closed continuous loop. Take another stick of another random fixed positive length. Slide it on the new loop again. It's midpoint... | 0 |
Imagine a ball with diameter "a" at rest and another observer at rest at a distance of 'l' w.r.t to the ball in this rest frame.Say the ball suddenly starts moving at velocity v towards the observer .(Assume it accelerates instantaneously).What does the observer measure the distance between them?What does the observer ... | 0 |
Maybe this belongs on "Linguistics" but since it's about an English word, I suppose they'd only send me here. Over time, some words gather more and more meanings to themselves. Ironically, these words are often small like, "by". Other words seem to shed meanings, or rather, they devolve meanings to new words. In the Ne... | 0 |
In my current understanding, I have been told that the spin of quantum particles is simply intrinsic to them. That, particles are simply right-handed or left-handed... just because they are. To me, that seems then that we should then treat handedness like it is just as distinguishing of a property as mass, or charge, o... | 0 |
At the end my document, I get a new page that contains, Temporary page! LATEX was unable to guess the total number of pages correctly. As there was some unprocessed data that should have been added to the final page this extra page has been added to receive it. If you rerun the document (without altering it) this surpl... | 0 |
As we know, in EM wave, with the oscilation charge (for simplicity, charge was moving at constant speed, then accelerated and then continued moving with constant speed again), kinks are produced. The funny thing is as the kink moves outward, its length actually increase(the joint vector between the new field line and o... | 0 |
I want to simulate typewriter style typesetted math articles. There are a few major requirements: All characters in all environments be tt style and of the same size and width (including super and sub scripts in math mode). In math mode, sub and super scripts should be one character higher than the base. All emphasizin... | 0 |
We know that when the wave model of light was being used in physics there was Thomson Scattering Model to explain the interaction between a free electron and a light. The Thomson Scattering was explaining the force that acts on an electron when a light hits it, with the force that the electrical field component of that... | 0 |
I am trying to have better understanding of localized wave functions. Apparently free particle de Broglie waves are NOT normalizable and act as delocalized functions which was the original rationale behind their use for explanation of electrons in double slit experiment. so far so good ! But then we use Fourier transfo... | 0 |
I want a tool to easily make geometry diagrams. For example, say I wish to overlay two circles, and then fill their overlap with the color red, whilst coloring other areas with other colors? Let's say I want to make a triangle with curved sides and a flat bottom side; that could be done in no time with the right tool. ... | 0 |
Much of physics education focuses on equations of known possible reactions, how to balance them, how to predict the result if certain elements happen to react with each other. What I don't see a lot of is an explanation of what chemical reactions are likely to happen in a complex system. Assume I was building a physica... | 0 |
I am studying the Newton's Law's of Motion, and really confused about one thing. I know that bodies of very small mass get attracted to our earth due to a force it exerts, due to its huge mass. So that, according to the Newton gravitational equation, a given small mass on earth will be attract by a certain force and de... | 0 |
This question is perhaps more on the use of certain parameters in BibTEX than on the use of TEX/LaTEX itself: nevertheless I think it could be of some help to other persons who, like myself, need to redact large bibliographies. In order to list information on the existence of a review in the ZBmath database on a biblio... | 0 |
We know from molecular spectroscopy that incoming light on a molecule can change a molecule's rotational, vibrational and electronic energy levels. If the incoming light is, on the far-infrared and microwave region the molecule gets rotational energy.(microwave spectra) on the near-infrared region the molecule gets bot... | 0 |
For a given n dimensional chaotic system such as a chaotic attractor or really a time dependent chaotic dynamic system does there exist a higher dimensional representation where the behavior is in fact deterministic? This question comes from a data science problem in multivariate time series forecasting. Most real worl... | 0 |
While playing with GeoGebra a few days ago I came up with an initial property of my Cardioid curve, I heard this curve is well studied so I'm not sure on my guess if it's already discovered or new, adding references would be appreciated You have a Cardioid and its circle of perpendicular tangents The red point is an ar... | 0 |
I tried lots of research, but still couldn't wrap my head around the difference between free and mobile electrons. Free electrons, from wikipediaLink(I mean free particle bullet point, because other bullet points are not really free electron - as an example valence electron is not free, but mobile) are electrons that d... | 0 |
For some extended context, this is how Heads Up Displays (HUDs) and Reflector Sights work: However, as far as a I know, a lens will take collimated light and focus it to a very small point called the focal point. And since the reverse is also true, only the focal point can be collimated and "focused to infinity" to cre... | 0 |
If we bring a separable dipole (say a single ammonium chloride molecule) from infinity to the center of a capacitor, once the dipole stabilizes and the rotational energy is lost to electromagnetic radiation, the net work done is negative. If the capacitor plates be perforated and the field be strong enough to separate ... | 0 |
Recently, I play a game of Turing completeness where I utilize various gate circuits such as NAND, AND, and NOT to construct a circuit that satisfies the given truth table. I didn't learn digital circuits or relevant content. When I play this game, I want to know whether there are any theory to deal it. For example, gi... | 0 |
A parabola is supposed to be a conic section, obtained by slicing through a cone with a plane, like this from Wikipedia: But if you do this, what you get is just part of an elipse, since if the cone and the plane were bigger, eventually you'd end up with an elipse. And a section of an elipse is not a parabola - there i... | 0 |
I've attached an image that describes the light-rays refracted through a thin lens. P' represents the first real image that is created when light is refracted through the first convex mirror. But how does this translate to the second image point P'' through refraction? I understand the convex mirror formula and how if ... | 0 |
The reason we can see the color of objects is light rays undergo scattering in the subsurface of a material, and in their walk in the material lose some wavelengths, and finally exit at a random direction (link) However, aside from this, you see white highlights on an object from the light source. These can be diffuse ... | 0 |
(Edit: This was closed because it was marked as "opinion based" and I was asked to edit the question so that it could be answered by facts and citations. @Fumblefingers gave citations, thank you. On re-reading several times, I think yours is the correct interpretation, rather than assuming the existence of some actual ... | 0 |
I am aware that in the theory of classical infinite sums , one can not generally interchange the order of a double sum or do other infinite sum manipulations. However, these infinite sum manipulations can be valid if absolute convergence is available. I am working on a problem , where I would like to manipulate a doubl... | 0 |
After verifying that it was a real word, I just posted a chapter named "Processioning". I have begun to wonder what kind of thing that is. As far as I know there has never been a verb, "to procession", and "to process" is something else entirely. Yet, "they were processioning through town" makes perfect sense. I am thi... | 0 |
Suppose there is a ridiculously large bridge, fixed at either end (light seconds long at least). The bridge is constantly under the influence of gravity. If the ends are severed simultaneously, the whole bridge will fall. I assume simultaneity isn't a problem as the bridge is all in one reference frame? Will an observe... | 0 |
Usually in a Cartesian form of derivation in math and physics, I have seen that a particular formula is derived for simplicity by taking concerned points say in the first quadrant. Few examples are the derivation of mirror formula, lens formula, lens maker formula, distance between two points. Later, they extend the fo... | 0 |
I am an undergraduated student from a non-native English speaking country. In my country, when doing math, which in many contexts that we will be proving something like a theorem, a proposition. What matter is I feel like we don't have a variety of words while doing math. Everything we used in our assignment are just a... | 0 |
We are extracting transformation matrices for different joints from ARKit. These are relative to the parent joint. Since the positions are not very accurate, we have developed a machine learning model that corrects the positions based on ground truth (Vicon data). We only use position data in the model, so now the rota... | 0 |
Have an excel financial model, that pretty much is a time series of assets and liabilities. I would like to know what the optimal liability structure / mix is, subject to multiple balance sheet constraints. My objective function is to minimise funding cost. I tried to optimise using Excel Data Solver, but encountered t... | 0 |
In Meissner effect we say that below critical temperature there is an expulsion of external field lines or field lines do not penetrate inside the specimen. But here is what I am not able to understand, if the sample creates a mystical force which do not allow field line to pass then it can be imagined as if there is a... | 0 |
I've been told numerous times in high school that "first and second person pronouns in academic writing" should be avoided. This supposed wisdom is echoed in various style manuals as well. However, if you look at any authoritative text, like The Stanford Encyclopedia of Philosophy A simple search of "we" will yield cou... | 0 |
Background: ZFC in classical first-order logic. To make a definition, we first need to proof something does exist. So, this question is actually about proving something exists based on a recursion. I know that when you want to prove the existence of a certain function with the domain of ordinal numbers, you can use the... | 0 |
Good day, everybody. I program a simple computer game and there is a "dumbbel" inside of it, which should using some kind of rocket engine. The engine is fixed on one side of the dumbbel and creates a constant force F, which is ALWAYS orthogonal to the dumbbel. Since the dumbbel is ideal, the both sides have only mass ... | 0 |
For all the words I know, I can't seem to find the right word for this strange emotion. Some words I have tried come up short in one way or another. Confusion lacks the underlying hatred, annoyance ignores the triggering bafflement, skepticism is to forward to properly describe the abhorrent serenity caused by this mix... | 0 |
The normal proof goes as follows: if H(f) is a program that takes the source code of any program f and return whether it halts, we define G to be if(H(G)){ loop forever } else { halt } Which is a contradiction if you run H on G. My problem is the unjustified assertion that a program like G even exists, as presumably it... | 0 |
Suppose I have some process that I believe to be the result of a differential equation. It would be reasonable to regard this as the path of a point particle in some notional space. Suppose further that I know the exact shape thaat the path follows, and it is one with a simple description, such as a conic section or a ... | 0 |
In ceveral textbooks it is said that an ideal LC circuit does not lose its energy, it just bounces from the capacitor to the inductor coil back and forth. On the other hand, any system where charges are going to be accelerated radiate energy (that's should always happen when involved charges are going to be proper acce... | 0 |
Everywhere it is written that electrons flow through through wire of an eletric circuit, and the reason given is that the battery maintains high potential at one end, and low at another. But, if you think about it from first principle electricity& magnetism, then one would realize by intuition that the provided reason ... | 0 |
Central idea of Michelson and Morley experiment was that if single speed of light predicted is speed of light relative to ether, then as the earth moves through the ether, light travelling along the direction of motion must travel at a different speed relative to the earth and light moving at right angles to the motion... | 0 |
I don't know if this site is a suitable place to look for examples, but I have a lot of ideas that I would like to find a good example of, and I try to organize these things in a later book. I hope I get to a place where I can discuss my thoughts so that I find people taking these questions seriously, and I think they ... | 0 |
Urban Dictionary includes one explanation of "play" : Anything to do with sexual relations: fooling around, making out, oral sex or having intercourse. However, most of the formal-language dictionaries don't include this explanation, only interpreting the word as "to tease" which can describe something childlike such a... | 0 |
Any introduction of relative simultaneity usually is like this : Alice on a train car shines two beams of light towards both ends where two clock are. She sees both clocks see the light at the same time. Bob on the station looks at the same event and says the trailing clock was hit before the leading clock. I have no d... | 0 |
This question has come up with me in two different Final Fantasy games, where you want or need to collect every memory or skill through random selection where not all memories/skills are available until late in the (main) game (not in the random selection "game"). Collecting a memory/skill does not remove it from the s... | 0 |
I'm trying create a new single-item selection system that has either identical or close to the identical probability of an existing multi-item selection system. The current system has multiple "tables", each containing multiple items. Both a table and each item inside of that table have a selection probability. That is... | 0 |
In the book To Have or to Be Erich Fromm claims using "have" in English increased due to the rise of the market economy and Protestantism. Where one is alone in the market, with their personal relationship to God, rather than being part of a feudal economy and church where they are taken care of. How well is another co... | 0 |
When I try to solve a hard problem, I have no idea on what to do before I write anything on the paper: apparently, only after I spent hours on a particular problem I find the key idea by just a coincidence. This makes me feel that training myself by solving tons of hard problems on integration or geometry etc... is a w... | 0 |
The main statement is: the EM coupling constant is energy dependent, and the fine structure constant (FSS) is the low energy limit. This means that the flow of the coupling constant vanishes when the energy does not suffice any more to create the lightest charged particles, electrons and positrons (there only remain th... | 0 |
The problem arose of choosing a direction for studying and working within the framework of this topic: neural networks as universal approximators for solutions of partial differential equations. At the moment, as a rule, neural networks are used to obtain more accurate numerical approximations, stable, compared to clas... | 0 |
I have proven this question but somehow I feel like maybe I have done something wrong or I have been reckless somewhere would you please help me to verify my answer If S is a set with least upper bound and greatest lower bound property assume X and Y are nonempty subset of S such that every element of X <= every elemen... | 0 |
It is said that if one would attach a load physically to the rotating disc from center to rim no current would flow because both in the load wires and the disc itself currents would be generated in the same directions, everything would cancel out, right? But I don't understand why this is a problem? there is a simple w... | 0 |
Like 'pretty ugly'. The most used meaning of 'pretty' is contradictory to 'ugly', and it could feasibly be used to mean something like (if it were an oxymoron), 'someone who may be considered ugly, but is actually kinda pretty'. But of course it actually means, considerably ugly. Is there a term to describe such pairin... | 0 |
This is a question from an interested amateur. Math welcome (as I or another may investigate it some day) but don't expect me to understand it in your answer. Consider a star emitting light. In order to talk about it simply, think of a point light source, and ignore spatial perturbations (for the sake of phrasing the q... | 0 |
I play football. However, I dislike going and sometimes I just dont want to play it. But whenever someone tells me that i do not have to do it or that I can stop when I want, I put all the work into it. One reason I can assume is that my father played ball and I am the last son, he and my mother want me to play but I h... | 0 |
As we know, In a circuit, simple or complex, electric fields created by surface charges move electrons which creates current which creates magnetic field which can be coupled to other lines and induce voltage if it is changing by time, which also creates current and this current also creates magnetic field and voltage ... | 0 |
Currently the proven theory is the quantum field theory. This theory defines fields in "all spacetime" and particles are disturbances in these fields. These particles are punctual and interact through virtual particles. But string theory is not clear to me. This defines the strings (bosonic or fermionic) as the smalles... | 0 |
I am wondering how heat loss happens in sound waves through air. First of all, for wave energy to be transformed into heat, absorption should happen by a medium. But why wont absorption implicitly translate to faster vibrations which would state that even though wave energy was translated into heat vis absorption, that... | 0 |
We call G the formula that in the formal axiomatic system employed by Godel to formalise arithmetic has the meaning of: "I am not provable in this system". We know that this formula G is true in that axiomatic system since we can exclude that it is false. But why can we say that it is also true in arithmetic itself? Af... | 0 |
In security, we speak about different strategies of protection. Mainly, there are three to consider. First kind, usually called shell protection corresponds to the act of preventing the intrusion before the access. For instance, a door, fence, password on a website or silver galvanization of the utensils. That's usuall... | 0 |
I have seen videos that interview accent experts, like on why Americans do bad English accents (a reason given is a lack of exposure). But are there something like references that one could point to to rate accents? I would think at least one should survey native speakers to get a rating (I met a Kiwi who said he would... | 0 |
As oppose to a cylinder in two pieces rotating against each other, as in O'Neill's cylinder, what if the cylinder were all one piece and an external thrust, like rockets, set the cylinder in motion, rotating it, would it rotate indefinitely? Would inhabitants of the cylinder experience artificial gravity indefinitely, ... | 0 |
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