studiot

Not sure why you need this for Chemical Engineering but good on you for wanting to know. As a ChemE student you will be familiar with differential equations in general and the separation of variables in particular. The good news is that because the potential energy of an electron in an atom is a function of distance from the nucleus only it is possible to separate the variables in the quantum differential wave equation (Schrodinger etc). This means that we can express the wavefunction as the product of three functions, each one only involving one of the three coordinates So we have in spherical coordinates [math]\psi = R\left( e \right)\Theta \left( \theta \right)\Phi \left( \phi \right)[/math] So of the four quantum numbers, n, l, m and s m is the magnetic quantum number and s the spin number and do not contribute to potential energy. For an S shell electron l = 0 and the angular part is therefore constant. So we are left with the radial part R(r) The probability of finding an S electron in a small element of volume, dv at a distance between r and (r + dr) is then [R(r)]2dv. Also the probability of finding the electron anywhere between between r and (r+dr) That is lying in a spherical shell of radius r and thickeness dr is given by replacing dv by the volume of the shell volume = 4πr2dr So we have The probability = 4π[R(r)]2 r2dr and the function 4π[R(r)]2 r2 is the radial probability distribution function you have plotted. Does this help?

Vitamin B12 injections sometimes come in dark brown ampoules in a box but the daily injection regime usually only lasts a week then tails of quickly to three monthly intervals. So keep them in the box until you want to use them, remove only one at a time. You could wrap the box in foil, rather than the ampoules if you really must but I rather doubt the manufacturer would offer something unsuitable. Keep the box in a cool/cold dark place such as a refrigerator. Storage If you need to store Neo-B12® Injection, keep it in the original pack until it is time for it to be given. ... Keep Neo-B12® Injection in a cool dry place, where the temperature stays below 25°C. Do not store this medicine or any other medicine in the bathroom or near a sink. ... Keep it where children cannot reach it. More items... Neo-B12 Injection - NPS MedicineWise

I suggest the problem here is that energy and momenta (in the plural) have to do with determinism, which was introduced many pages back. This is because both in classical and quantum theory, knowledge of all the momenta of a system at any instant will entirely 'determine' its subsequent trajectory in phase space. Note momenta has a special meaning in this case which includes the normal mass x velcoity type. Two things about this. Firstly I think the relationship between 'determinism' or fully determined and a scale of indeterminism is similar to the relationship between 'certainty' and a scale of uncertainty or probability. I often have to remind folks that a probability of 1 has more than one meaning. Secondly there is the question introducing of Chaos theory to the mix. Returning to free will Go back to my example of ice cream flavours. But this time you have a choice of vanilla or chocolate and you are only going to have one ice cream. A priori you have a choice and therefore can exercise the high level 'free will' as offered by Eise. But this changes, at the instant of choice, since you can no longer choose say vanilla, having ordered the chocolate. So a posteriori you do not have this free will. This is rather like the famous Monty hall problem, which alters the probability of things between a priori and a posteriori.

I don't agree that this constitutes either a necessary or a sufficient condition for emergence. I have already given an example where this is shown. A phenomenon may emerge due to configuration of the system.

English has a technical word for objects in the general sense that is nouns. But English is blessed because it distinguishes between certain types of noun. In particular it distinguishes 'abstract' nouns and 'concrete' nouns. Abstract nouns are objects that only exist in the mind. They are concepts. Concrete nouns exist in physical reality, they have physical substance. You can pick one up and weigh it, poke it and so on. So a cow is a concrete noun and a unicorn is an abstract noun. So swansont's physical is our attempt to create a physical version of a theoretical object which means 'as close as we can get'. I have tried to draw this distinction in my examples in the previous posts. Does this help? Secondly, by saying you understood my explanation of 'almost' I hope you understood that this means the process 'as close as we can get' in the physical world. I hope you understood the distinction between the single discontinuous point and the infinity of continuous ones. This distinction becomes very important in the branch of mathematics known as Analysis because there are types of discontinuity. One of these is known as a 'removable discontinuity' which property greatly assists analysis.

Good question +1, but since the statue toppling mania has spread to other countries it should be noted that targetting has now become indiscriminate and worse an excuse for attack and counterattack by one community on another. Arguments against the Bristol slave trader Colston have been widely publicised since his statue was dragged into the harbour. Less widely reported was the counterattack on a statue of a modern black writer and actor (Fagon) in that same city. Still in my region we have the entirely idiotical threat to the statue of Admiral Blake, the British Admiral noted for freeing slaves. So communities should be examining their consciences as to whether this has already gone too far.

Theology ??

Let us say that this was a good try at an idealisation, BUT Just as with a discharging capacitor that in theory takes forever to discharge fully, your photon would take forever to bounce back and fore whetween the mirrors a countable infinity of times.

Yes, that's a shorter way to put it. Physical can mean either that in some experiment we extrapolate along an asymptote as joigus has offered. My example from analytical chemistry of infinite dilution would be where you might do this, but there are many pratical situations where this might happen eg measuring the charge on a capacitor. My example from Thermodynamics is of the type mentioned by swansont where a heat sink or source is such that its temperature is not altered by the addition or extraction of some particular quantity of heat. That is how Carnot cycles are analysed.

An example of a function that is 'continuous almost everywhere would be [math]\left\{ {f\left( x \right):f\left( x \right) = 1,x \ne 0,f\left( x \right) = 0,x = 0;x \in R} \right\}[/math] Here the function is continuous for an infinity of points but discontinuous for one single point at the origin. Note that we can handle infinity and use infinity in Mathematics, but infinity is not 'almost infinity' it is infinity. For example the 'point or line at infinity' in projective geometry. This is example is different from your optics one since your table says both that the image does not exist and the image is at infinity. A better optical example would be a source (object) at infinity, which has meaning.

In mathematics the problem is the 'almost'. So no it makes no sense. In Physical Science then yes phrase may have value, for instance an 'almost infinite thermal reservoir' in Thermodynamics or 'almost infinite dilution' in Chemistry. In each physical case the phrase measn that the quantity concerned is so large compared to the change considered that the quantity is constant or unaffected by the change. This is just making use of one mathematical property of infinity that infinity plus or minus x is still infinity.

Hi Mordred, did you mean [math][math]{g_{\mu \nu }} = {\eta _{\mu \nu }} + {h_{\mu \nu }}[/math][/math] or [math]{g_{\mu \nu }} = {\eta _{\mu \nu }} + {\hbar _{\mu \nu }}[/math]

It means that it does not have one value but many depending upon which direction you look in. Note that permittivity mu is also a tensor which only has one value in an isotropic situation. Otherwise you have the same situation in an anisotropic situation. The B and H vectors then point in slightly different directions.

As I understand it, an inability to recognise 'social' clues like this is characteristic of these people. You have to be a special type of person to be calm enough not to take this as rudeness, not easy for me.

I totally agree, as did our lecturer, he was very persuasive that current classification systems are very wide of the mark. Interestingly his lectures were very well presented, although he spent quite a bit of time telling us that he was classified on the scale. So yes, I think these alternative ways of thinking can be a gift and a great one at that, not a disorder at all. My way of thinking is different again, and I'm not totally sure how it can be described in the context of free will. Edit (whoops wrong thread) As regards communication with such persons, I suggest everyone is different so each communication must be treated on its merits, difficult and frustrating though that may be.

Thank you for those replies, especially the second one. Thank you also joigus for your lipid/micelle example if an emergent phenomenon. I had never considered those associations that way before. Which brings me to my usual example of an emergent phenomenon, arch action. This only 'emerges' only when the last component voussoir is in place and not until. Or does it? Arching action occurs when there is only one component and can be likened to my earlier example to Eise of a perfect circle.

Thanks to all contributing to this excellent thread. I have been watching with interest hoping to learn. I am of the opinion that some of the members posting questions here are 'on the autistic spectrum' and that is the seat of the difficulty in communicating with them, not that they are 'cranks'. Just before Covid struck here, I attended an excellent series of WEA lectures (supported by the Wellcome foundation) entitled "The Challenge of Diagnosing Psychiatric Disorders" partly to see if there was a better way for me to reach these people. One point came out was that Asperger's was incorporated in the autism spectrim in the latest (2013) version of the classification bible - DSM-5 The lecturer passionate about the subject, also offers online courses.

I would like to examine this statement more closely (how's that for free will ?). I see what you mean, but cannot each higher layer regard evrything below it as 'the base' ? Is there such a thing as 'the base' or bottom layer or is the structure like an open set in MAthematics? After all we can reduce down from components to molecules to atoms to subsections to atomic particles to quarks to fields to...... Secondly there is the question about 'independence'. Does that imply you cannot substitute a different 'base' and still achieve the same effect?

Hi, MigL, the Natural numbers is the wrong choice of set for that example. Bounded sets and subsets (such as the natural numbers) of real numbers are characterised by two bounds An upper bound or maximum number and a lower bound or minimum number. Neither bound need be part of the set, but the maxima and minima are part of the set. In the case of the natural numbers, they have no upper bound and no maximum but they are 'bounded below' by any number (integer)less than 1. The set in your example is really two sets - The set of negative integers which is bounded above by any integer >= 0 and the set of the positive integers which is bounded below by any integer <= 0 Finally none of these sets can set up a circle as described by Marcus. That requires complex numbers in the complex plane, which has different boundary requirements or alternatively the cartesian product set R x R. But it was a good question though. +1

Isn't that just so clear ? +1

Not 'a' problem but 'the' problem. I don't follow your line of thinking here. I did not introduces the layers (though I have agreed the idea has merit) But I think you did.

You then have the problem of defining the level or layer at which free will becomes apparent. I am reminded of a description by Stafford Beer about another slippery concept - value. I don't have the exact quote but he wrote to the effect: Conside a pound of apples, which has value in itself. Give it to a first class chef and he will produce something of greater value from those apples. Give those same apples to some ham fisted idiot and he will reduce those apples to a slimy mess of no value at all.

Hello, Marcus, I see you are thinking outside the box again. Quite right, any complete curve/line divides the plane into two regions, either both infinite or one finite if the curve forms a loop.

@joigus and @eise It has been interesting following your discussion; you have both raised many good points. But I have to say I don't fully agree with either of you. I am not completely clear what either of you mean by reductionism or emergent phenomena. Eise's layer analysis contains some interesting approaches. Do you think an action (proposed, desired or real) could be free in one layer but not free in another ? But I think the drive towards a sort of 'reductionism' that reduces choice to the level of a Turing machine is oversimplicication. A Turing machine is not capable of allowing for or determining 'net forces', either internal or external, yet most activity in Nature is the result of some 'netting' effect. Eise, your computer program example seems to me to be similar to the question "Do perfect circles exist in Nature?" A brick encloses some physical space so you could define a perfect circle as passing through a particular set of elements of that brick, at some level or 'layer'. So is the circle like your program - Something you can perhaps isolate? So I return to my earlier analysis. Can free will be partial? Are there degrees of free will? Is free will itself a 'layered' phenomenon?