studiot

Glad to be of service.

To quote from the preface of the book Many of the important calculations in applied maths (engineering and physics) use functional analysis because engineers and scientists want numbers and statistics. Functionals are maps from diverse domains to some field, usually the real numbers. A good example is the definite integral, another is the time integral of the lagrangian (also called the action or the action integral). There are also many examples of everyday functionals in statistics, typically the mean and variance of a distribution is a functional. There is also the 'energy functional'. Indeed you can find functionals pretty well anywhere and everywhere you apply mathematics.

From much much battered copy of Spice (my second copy. ( It was so good I had the first one stolen from me). Sulphur is UK spelling, Sulfur is US. The americans have simplified quite few technical words. I'm sorry we didn't 'learn the table' , as some schools. Cambridge board never set any exam questions about the table itself in my day, only about the information it contained. The point being that you have to know a lot of the information for the table to assume real significance. The standard channel shaped long table was developed by about the 1920s. After this a single 'hanging' lanthanide line was introduced, followed by the double line in 1940. There are lots more in-between tables for those interested.

You mean this one ? Yes Professor K has written some excellent applied maths books, I remember preferring his book to the course set book at first year university, back in the 1960s.

It is a common misconception that there is one 'periodic table' that can be presented in several different forms. But in fact the situation is more complicated than this Yes indeed it does depend upon 'which table' you look at, so let us look at how this situation has arisen. So what is the periodic table and what is a period ? Early chemists discovered that the 'indivisible elements' could be listed in small groups with similar properies. The main distinguishing property they could measure back then was atomic weight, so they listed the elements by AW. Mendeleyev was the first to notice that it was more than just collections, he noticed a regularity in the occurence in the more interesting (to the chemists) regularity in those lists, even though there were some anomalies in place on the list. This is the table he published in 1869. The' periods' go across the table, left to right. And the groups (Gruppe) are tabulated vertically. You can see he was a bit mixed up about group 4. But the real question was and still is "Which properties to you use to generate the categories ?" The problem being that different categories will include or exclude different elements, although listing in order of atomic number moved several misplaced elements into a more coherent pattern. This straightforward listing led to the form you are probably thinking of known as the 'long form' This form has the advantage that atomic number indexes every element and can be presented in a reasonably compact form. But Chemistry is about a whole lot more than atomic weight and atomic number - that is really for physicists. Chemistry is about acids and alkalies, metals and non metals, chemical reactions and most particularly electrons and their role. Considering electrons we arrive at 4 groups and 7 periods, which tells us where all the electrons are This information can also be displayed in other ways and I am guessing that you have looked at one of these. Finally a very modern version of the long table showing a few extra properties eg metal/non metal.

Since it is too difficult for idiots like me to understand why a missing electron is a wave of any sort I would be grateful if geniuses like yourself would explain in simple words that I can understand.

Thank you for your reply and the negative point. Since I am guessing that English is not your first language I can see why you have failed to understand my opening post and subsequent replies. Yes I agree that if you are correct that there is no duality (in reality) none of my examples represent duality. But then you failed to understand that they were all examples for discussion, to stimulate the idea that there may be more than one type of duality. You also failed to take into account that I said it depends upon circumstance. So it is true that concrete is very very weak in tension, but very very strong in compression. So it is weak in some circumstances, but strong in others. That is why we use reinforced concrete. You would certainly fail a chemistry exam if you claim that aluminium cannot act as either an alkali or an acid depending upon circumstance (ph in this case) That is how you can get the substances aluminium sulphate and calcium aluminate. That is reality, but is it duality ? I find your statement that a hole (in electronics) is a wave quite interesting. Especially as a hole is the absence of something, because something is missing. So what is this wave made of ? No duality in reality ? Children in school use a pair of compasses to plot or draw a circle as a trajectory of points a fixed distance from a centre. But in some schools they also construct a circle by 'curve stitching'. That is they plot those points by stretching threads along tangents to that same circle. In fact they are laying the groundwork to one day understand dual spaces in mathematics.

Thank you for your reply, but I really don't see what it has to do with duality. Membrane potentials occur in Biology yes, but no dualism is involved. Two state ( on or off) systems occur widely, but this is entirely different from dualism. A good example would be binary logic, the two states are 1 and zero. In electronics a common implementation would be +5volts representating 1 and 0volts representing zero. This is called positive logic. But it is possible to do it the other way round, with the 5 volts representing zero and 0volts representing 1 This is called negative logic. Such a situation is indeed dualism. There are many such in Mathematics. As a matter of interest membrane potentials are really part of Chemisty, not Physics. A good treatment of their calculation is given in Physical Chemistry for Biochemists Price et al Oxford University Press

Thank you for your interest. Would you explain further, perhaps with some examples ?

Point taken, thanks. Spacetime it is. +1

I suppose this can be cast as a linear programming model, though I am used to a more direct physical solution as the parallel operation of dissimilar power sources is an important real world situation Is this homework/coursework ? Assuming this is homework, a hint for (a) Any linear programming problem requires the introduction of a constraint equation to bound the region where the constituent equations hold as there are not enough constituent equations by themselves (more unknowns than equations).

The OP states that this is about algebra, relational geometry and some other concepts are introduced. So let us take stock of the consequences this implies. algebra, geomtery and relations fall within the province of Mathematics. A relation is defined as a set of members each comprised of two elements, one element from each of two sets. The two sets may be actual copies of the same set so the relation is then between elements of one set. This is the case for geometric relations (relational geometry), where the elements are points of a geometric manifold. the process of creating/defining sets can be carrier further by taking the pairing set and another set to establish further relations. a metric is such a relation. But the OP has disbarred the use of such a relation so we are left with simpler geometry to work with. Less onerous geometric relations are still possible but we must look to other physical justification to provide the necessary rules. I'm sorry but the expression energy = space just will not cut it. However I do wonder if some process akin to the derivation of the Madelung Constant might suffice. Such a relation could yied a non metric, yet numeric (ie algebraic), statement of the relation between any two points of the geometric manifold. But no meaning can be attached to the statements "The distance between point a and point b is" or "point c is further from point a than point b is", since there is no metric.

You picked up my first point pretty well, that communications diffuculties impedes accurate assessment of abilities. I would like to add to this that different folks, both non autistic and presumably autistic can think in different ways. In particular some people can think in the abstract, others find this difficult to do. Some think in pictures, some in words and some in ideas or concepts.

I believe that one complicating factor is that most if not all autistic people experience difficulty in communicating with others, both autistic and non autistic. This difficulty affects both directions of communication.

You are not addressing my concerns. It doesn't matter how long or short your wires are, or what their cross sectional area is. Nor does it matter how long it takes to pass a given number of units of charge . All that matters is that each unit liberates one ion with one unit of charge. so counting the number of ions liberated equals the number of unit charges passing. That gives the total charge. No forces, vacuums, permeabilities, permitivities, etc are involved. Counting over a particular time period connects this as a curent to an existing dimensions (time) Avogadro's number connects it to another existing dimension (mass) Connection to length appears if you change from current to current density. But you cannot derive a basic unit of charge from these connections.

So how do you regard the status of the Faraday constant, F ? F = NAe = 96485 coulombs / mol where NA is Avogardo's number and e is the charge on the electron

Well I have a problem with this definition of the coulomb, which is surely independent of mass. Faraday showed that the amount (ie number) of ions liberated in an electrolysis is proportional to the strength of current flowing and the time for whixh it flows. From this we may draw the connclusion that the number of ions liberated is proportional to the amount od electricity which has passed throught the elecrrolyte since the current is the rate of flow of charge. But each different ion has a different mass hence the proportionality is independent of the mass liberated. Nowadays we use current density as easier to tie in with EM theory as the base electric unit, but you cannot do without one. Here are some comparison tables and definitions from older systems and Si. Not this was SI before 'count' or number was admitted as a valid dimension.

Alternatively from the guy that likes proving mathematical theorems by physical means.

Exactly so' It is very common to confuse subsets and elements (members). To make it more confusing a set may also be divided up in other ways - one such is a 'partition'.

This is a very roundabout way to say Creat a set that only contains those elements that are in both A and in B. - Which has another name. create a set which is the union of the set of all those elements that are in A but not in B with the set of all those elements which are in B but not in A. I note you website uses both the term elements and members. Have you lost interest in your previous set theory thread or were you just not going to answer me ?

You can't prove a definition. And there are many alternative axiomatic systems of set theory, but none are complete because of the underlying tensions identified by Godel and others.

Why not ? Thing is a perfectly acceptable general noun. I also think that if you wish to continue this side discussion it would be better done in another thread, so as not to confuse a beginner in set theory. In my view the standard introductory approach leaves much to be desired as it avoids introducing concepts that make set theory useful in so many other disciplines.

Yes my wording was a bit slack. But the set of every thing is most definitely an infinite set. So if you restrict things to those without a square root, that set does not have a square root and must be a member of itself. That is exactly why Russell introduced 'type' theory. also originally called category theory, but category theory now refers to a different subject.

The set of everything that is not a square root ?

Who are we ? Cantor's original word in German was indeed Elemente (plural). German, like English received the word element via Latin from Greek, and the word has has many different meanings through the millenia. Workers after Cantor discovered philosophical difficulties underlying basic set theory, which is possibly why Russell used the translation 'members' to distinguish aggregates (the English word for set at that time) which sets which contain other sets as members and sets which do not. Whether you chose to use the word elements or members, the important point is to distinguish between subsets, which are neither elements nor members, and the elements/members themselves of the set. Subsets are entites that are formed from the elements/members but are not generally members, which is why HbWhi5F has counted too many 'elements'. That way the definitions are self consistent.