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I cannot answer for florida foundations problems since you have provided no details. However it is worth pointing out that there is a huge difference between the chemistry of lime-carbonate 'concrete' and the modern portland cement alumino-silicate based concrete, that the Romans had no knowledge of. One major difference that is known is that the carbonate concrete never stops hardening, but this proces is slow, taking centuries to millenia. So I am not suprised to learn that the Romans developed accelerators. I see from the article that the example is of unreinforced concrete. That too is not suprising. Carbonate concrete pH is inappropriate for protecting iron and steel against corrosion, unlike portland concrete. This is posted in Engineering so it is unclear whether this is a scientific announcement about carbonate concrete or a question about failures in Florida ? Good discussion base either way. +1 A couple of years back I posted an article (SCIAM if I remember correctly) about the Roman stuff.

I'm pretty sure your analysis is far to simplistic. Further the DSM5 is well recognised as following decades behind science progress. It is really more of a legal classification than a scientific one. The simplest counterexample I can think of lies in a program I saw a couple of years before Covid about folks with unusual medical conditions. Several patients had sensory cross wiring so they say tastes, heard colours, and so on. Wierd but I have no reason to doubt the program producers. As regards the balance between the different ways people think, I understand the most common is to think in words and sentences and actually subvocalise their thoughts. Many of these have trouble visualising pictures - and then you get the distinction between visualising still and moving images. I tend to think in terms of concepts and pictures rather than words and sentences. I see the entire 'text' of the concept at once, not sequentially. This give me trouble when I try to set things down in words. They just come tumbling out and I can't get them down fast enough. How about your own experiences ?

Gentlemen can you please give others the time to go through the 9 pages of maths posted before closing this ? I am been rather busy with other things this w/e but would comment further as follows. The NS equation or equations is a single 3D vector PDE or 3 separate scalar ODEs. That is not enough by itself to solve since it only gives 3 equations for the 5 variables involved. To solve it we require to introduce two further equations. One way is to use the continuity equation and a gas law equation of the form density = a function of pressure and temperature. This I think on first reading is Seth's method. It may be possible to reduce the number of variables by specifying spherical symmetry. But a trumpet symmetry is decidedly non symmetiric. Much of the sound energy is focused. I do believe that @MigL is a NS equation specialist and would welcome his comments as well as whatever @Mordred comes up with.

To add to this excellent summary, +1, It is important to distinguish between shells and orbitals. Shells are groupings in space of several orbitals. The orbitals in a given shell are so shaped as to not intefere with each other in the space occupied by a given shell Note these are not solid shapes but surfaces. Imagine them as the skins of balloons, like the party balloons we twist together to form dogs etc.

My apologies. I put my comma in the wrong place which changes the meaning rather drastically. I said No reasoning, as in logic, does not work in equations. Equations are a mathematical concept and I can try to explain them more fully in a separate thread if you are interested ? Whereas I should have said No, reasoning as in logic, does not work in equations. In other words I was responding to your question "right ?" (meaning is my statement correct?) by saying No it is not correct. Reasoning has a particular form of expression and equations offer another different form of expression. When logic (which is part of philosophy) uses the word reasoning it means that that three distinct things are involved. Logic is about 'statements' and the 'connections' between them and something called truth values for these statements. It's layout in simple terms runs as follows Statement 1 (called the predicate) is connected to Statement 2 (called the subsequent) by a 'connective'. An equation is a particular form of a single mathematical statement called a 'relation'. No problem this community has many members ready to help those who really want it. By the way I have a couple of decades on you so don't be too hard on yourself. I'm sure there is plenty of life left in the old dog yet.

Good Morning Ben. Congratulations on your excellent English. I assume you are enduring the same as me, a miserable wet morning following a miserable wet night. I would like to make one comment and enter a plea. Can I suggest you are introducing too many topics into a single thread which makes my poor head spin - I don't know what it does to yours ? One topic per thread please. OK so you have asked about equations, though I don't see much maths in your posts. Equations have a particular significance in maths No reasoning, as in logic, does not work in equations. Equations are a mathematical concept and I can try to explain them more fully in a separate thread if you are interested ?

You said it yourself. By pressing her button. Transponders of one sort or another have been around for decades. They were certainly small enough by the mid 1970s to meet your description.

Well so far I've got as far as printing the pdf out and having a swift butchers. It's really good that you have declared (hopefully all) your variables at outset. +1 It's also interesting to see a chemical engineer using the ideal gas equation rather than z or w factors. Anyway I'm not sure which 2 equations you want to solve for, presumably the characteristic equations at the end 5.06 and 5.05 or 5.07 ?

fair comment +1

So what is a line spectrum from Sol and Sirius if it is not a comparison to two clocks in different widely separated gravity wells ?

What about the gravity well conditions I offered you ?

Please read more carefully the quote you made from my post. I did not suggest that DM can form atoms, or any other particles. I said explicitly we don't know if.......

I don't have this book to hand at the moment and I can't rememember if he covers wheels explicitly, but Steven Vogel does an excellent job investigating the reasons how and why nature and man achieve identical mechanical goals in the sifferent ways they do. Fo example no muscle can generate a push force by itself.

I really don't understand what's special about this proposed experiment Since the early days, it has been a tenet of astrospectroscopy that the pattern of spectral lines (including shifts thereto) and aberations are determined by phenomena described by GR. So light arising in galaxies (and therefore gravity wells) far far away mr skywalker, are calibrated using GR.

I still think your question is rather limited for no obvious reason. Here is a Wikipedia discussion on the wider subject. https://simple.wikipedia.org/wiki/Rolling_and_wheels_in_the_natural_world

I don't follow the significance of limiting this to organs? What about the tortoise ? What even do you mean by a wheel ? Here is an image of the famous falkirk wheel.

Obviously not in respect of whatever conditions are being applied. That is true of all models. We don't expect elastic analysis to match reality beyond the 'elastic limit'.

Actually nature uses rotation in some suprising ways. https://www.google.co.uk/search?hl=en-GB&source=hp&biw=&bih=&q=rotating+microbes&iflsig=AJiK0e8AAAAAY7g0_8kobF67VuYcPl8mgs7idJ3VWaj_&gbv=2&oq=rotating+microbes&gs_l=heirloom-hp.3..0i546l2.11096.32186.0.32384.19.16.1.2.2.0.176.2198.0j16.16.0....0...1ac.1.34.heirloom-hp..0.19.2308.MRvTaHdMUtY https://en.wikipedia.org/wiki/Gömböc good point +1

Of course not. All you have to do is out dt = 0 @t=0 into the equations to see that there must still be an apparent atttraction felt between massive bodies. It must still give measurably verifiable results in the case of static situations.

I would recommend against using the letter [math]v[/math] for clarity since not only does it come out different in the standard font used on this site but it also can represent velocity, as can [math]u[/math], and so can easily be confused with 'parameters'. Addition of velocities is decidedly non linear in SR.

To put it mathematically If LT represents a Lorentz transformation is represents a transformation of something. That something is one of the four coordinates, x,y,z,t. It is easy to show that aLT (x) = LT (ax) ; where a is some coefficient. and the same for the other three coordinates. As joigus noted, LT does not act directly on v, the relative velocity. which is a condition that forms part of the definition of linear in mathematics.

I was rather hoping you might read my post and say to yourself: "That's interesting I wonder why he said that and made the distinction between the formula and the transformation. Perhaps there is something I can learn here." And then ask what a professional mathematician meant when he said For instance the sine function is nonlinear, but the fourier series and transform is a linear application of it. This is a linear polynomial [math]a{x^5} + b{x^4} + c{x^3} + d{x^2} + ex[/math] although only the last term is specified by a linear formula.

The transformations are linear. The formulae involved are not. Edit since composing this I see a non productive exchange of red points.

You are stating british custom. I asked about french custom. My point is that there is no intrinsic correctness or right of this. It is a matter of custom and practice.

Answering the actual question When 1mol of calcium carbonate decomposes into 1mol of calcium oxide and 1mol of carbon dioxide, 177.8kJ of heat is absorbed.