studiot

Well we know that 'practical mathematics' developed in early civilisations in Asia, the Middle East and independently the Americas. By practical maths I mean measurement for agriculture and building and taxation and even some more sophisticated measurements like river gauging in some places. Also developed were mathematical systems for timekeeping. We know this from written records where maths appears about the same time as other forms of writing. What is interesting is that, even in those early days, there were a few 'theoretical mathematicians' who explored further into the maths they had. Again we know this from their writing about the the solution of quadratic equations. The ancient Greeko-Romano ciivilisations were the first to try to work more systematically through theoretical material, particularly in geometry. Not much more was achieved in the theoretical (pure) maths until the middle of the 17 hundreds when there was a burgeoning of theoretical maths which carried on into 18 hundreds and even the beginning of the 19 hundreds. At this point many it became apparent that many theoretical concepts playe a much more important and fundamental role than was realised, for instance in group theory which started off as a purely theoretical concept, like greek geometry, but led to important relevations in basic physics. Both pure and applied maths have continued to develop apace since that time, each offereng the other insights . As a result the 19hundreds have seen a much more coherent overall structure for the subject. About proofs and axioms etc. It is debatable whether there is such a thing as scientific proof since the concept is at odds with the basic scientific tenet that it only takes one observed exception to disprove a rule, making observation King. Proof in the general everday sense is for lawyers. Mathematical proof is different in that no observation is required . Maths proof is really a demonstration that the proposal does not contradict any of the premises or axioms. It does not examine whether the axioms are 'correct'. Here are a few pages about this in relation to group theory.

A trawl round the net shows various support frames, nearly all have underpurlins, including your pictures. I did find some glorified deck chairs and some interesting sloping plasic tubs weighted with bags of earth or stone. These latter raise an interesting point as you seem to be thinking of buying only panels and assembling your own support. Have you considered climate conditions ? The panels are like sails or fence panels and are subject to significant variable wind forces. The variability means that the ground reactions are contantly being worked back and fore, which is how fences blow over as the posts works loose in the ground. This will also continually flex the panels themselves. A final thought. Panels are rated for 20 - 25 years life, but I wonder what suppliers / manufacturers would say about the guarantee if they are not mounted on the recommended supports. In the UK many suppliers resort to the " Not installed in accordance with recommendations/instructions." to wriggle out of early failure.

I have no idea what this means. Nor does this make any sense to me. You do realise that there are physical quantities in the material world that cannot be represented by a single number ?

Sure, post what it is you want to discuss. I would also be grateful if you would take note of my comment (now made several times) that there is a lot of Mathematics that is not about numbers. Maths is not a party trick to get you an answer. Unifying is actually quite a reasonable statement, probably more than you know, but the price of this is that there are explicit conditions to be met.

Interesting - substitution, also called replacement. Of course that is the function of the klein 4-group I was illustrating. But I'm not sure I would call group theory the most basic.

I see you have been back a few times now. Have you given up on this ?

There is not enough information about your proposal to properly comment. Surely the panel frame provides the support and specifies mounting methods ? I think what you might need for an array would be what roofers call an underpurlin. Certainly you would need to support the frame edges at these points, John's idea, though interesting, is not really suitable as these supports are designed to be as near as possible the same horizontal level, quite unlike yours.

I don't understand this connection to 'high frequency' All changes in electromagnetism and gravity move at the same speed - the speed of light. This is regardlees of any frequency components of that frequency.

Clearly you are both mistaken since not all maths relates to numerical operations. I have been trying to hint at this fact with the pictures I have been posting. Today most humans can count. And most, if they tell you anything, will say that maths is founded on set theory. But go back to the stone age or even further and you will find a more basic operation that of creating a tally. Archaeologists have found bone, stick and stone tally sticks. Set theory relies on an operation that is essentially the same as tallying, only is is called putting into one to one correspondence. What is counting anyway ? Interestingly the Austalian Aboriginals have (had) a number system that went - one, two, many. You don't need to actually count if you tally or put whatever you are counting into one to one corresponcence with a list of numbers. But before you can do that you need a yet more basic operation. You need to to be able to distinguish A from B, Here is a little imaginary tale from the land before time. A leader or chieftan (who could actually count modest numbers) wanted to get an idea of the size of the wild goat herd they were stalking. So he gave a lad a stick and a sharp flint and sent him to tally the heard (the boy could not count) telling hime to cut a mark on the stick for every different goat he saw, and to be sure not to tally the same goat twice (so the boy needed to be able to distinguish goat A from goat B). On another occasion he sent the boy to tally a band of enemies that he was worried about. Here are tales from more modern times "Come in number 5, your time is up" said the guy at the fairground. So what is number 5 ? Horse number 5 came in first place in the derby. Horse number 1 came in third. What happens if you add first to third ? or what is the arithmetic of ordinal numbers ? What happens if you add number 5 to number 1 or what is the arithmetic of numbers as labels ? So you have cardinal numbers (that you can add together) Ordinal number that you can't add together Numbers used as labels that you also can't add together and finally you have counting numbers - the natural or counting numbers do not include zero as if you count zero objects, you haven't counted anything. But be careful about saying you can add them together because sometimes it doesn't work. All of the above can be used to establish a basic set theory, which in turn can establish counting numbers from nothing at all (I said this before but you didn't spot it) Once that is done it can be extended to theories of infinities which allow us to work with the concepts. I think that is enough to be going on with but I would very much like to learn of KJW's response to my question.

Of course, this is a discussion site. And you normally offer thoughts of value, so I for one welcome them.

Thank you for offering this excellent example to enable us to take this thread deeper into the base of Mathematics.. So let us see what it can tell us before I ask a pertinent question. A general 'string' has what I will loosely call 'ends' for the moment. You mention an 'infinite string' Well the string ...111111 has one definite end as it starts (on the right) with 1, using one meaning of infinity, that of unending, it goes on forever, adding 1s on the left. So we say this particular string which we describe as infinite has only one end. Firstly are you aware of the ellipsis? That is the three dots both KJW and I have used ? It simply means that there is more but we haven't attempted to write it all. Useful shorthand. Since you want to look into the philosophy of things What, in your opinion, is the most basic operation or activity in Mathematics ? You did not take me up when I suggested that there are at least four different types of number in maths; are you not interested ? Once we have established the basics, we can move on to proof and its meaning because you have that incorrect in both maths and science - those two disciplines have significant significant differences in meaning.

Let's start here shall we. My dog is more intelligent than any AI yet constructed. It is not (perhaps yet) possible to train an AI to sniff out a whole range of substances that dogs can easily and naturally. So by running to an AI summary you are missing out on the important points being made on examples that are specifically designed to follow your desire to only work with mthas realting to the material world around us. MyQM comment has direct realw orld consequences, it is not some esoteric maths. The maths models the real world, not the other way round. Having said that it is suprising just how often the real world and the theoretical mathematical one follow the same path and at the sme time comforting. Thank you for telling me your understanding of the infinite. Religeous considerations have no place in mathematics,, and your appear to control and limit your progress in this matter. If you wish to pursue religion then I will have no further interest in this thread. This is despite my know;edge that a reasonable value for Pi is clearly stated in the Old Testament (and presumably the Torah) although not quite as good as the value in the Rhind Papyrus. Yes you have got an inkling of the idea here, although zero is indeed a number and the basis not all other numbers, rather than one. But you are hampered because I think you do not understand what a number (itself) is. In fact there are at least four types of object we call 'number'. And they are all as different as chalk and cheese. Further they obey different rules. Do you wish to learn more or do you wish to preach at me again ?

Jayant Narlikar was one of the co founders of the Steady State alternative to the big bang theory of cosmology. Although currently out of fashion, the big ban v steady state saga is reminiscent of the wave v particular theories of light which also see sawed back and fore in popularity. BBC NewsJayant Narlikar: Remembering the Indian scientist who cha...Jayant Narlikar who died at age 86, was dedicated to science education and shaped a generation of Indian researchers.

I put a lot of effort and that last post and , in genera, my posts link to each other to build up concepts. So it would be helpful to both of us if you read them through several times as I see you have either missed important parts of the content or have some misconceptions. So thank you for the response but please answer the questions, they are all designed to point to particular points of importance. I know the scottish dancing attachemnt was showey eye candy, but it was only meant to lead to another point. Actually if you look at the history of mathematics you will find that in all the early civilisations geomtry prereceded arithmetic and indeed progress was held back in most for thousands of years because of unwieldy symbolisations. That is why it is important to answer my introductory question about 1 and 2 and oneness and twoness. There is much more to discuss about symbols, did you notice I had introduced the word 'labels' ? Proper consideration would carry forward the i j k discussion where they are labels or indexes, and their relation to the square root of minus 1 is coincidental. This is because, as I said earlier their arithmetic is not your normal arithmetic where A time B gives the same result as B times A. Instead in their arithmetic A time B does not necessarily give the same result as B times A. This is vitally important in the real world because it is the reason for the Heisenberg Uncertainty Principle in Quantum Mechanics. More simply the use of plus and minus as directions in electrical circuit theory as explains the difficulty that arises because the directions for current and voltage are necessarily opposite. The rotating arm is a real world mechical object. There is no graphic plot needed for this. You cannot relate rotation angle to a point (such as 0). I did not suggest that. You must relate it to a line. Back to scottish dancing. That was from a chapter entiltled an infinity of curious arithmetics. An infinity. OK so what do you think infinity means, that you do not like it ? Here are some simple arithmetics presented as tables. The first is concentional algebraic substitution of the effect of replacing x by (-x). The second is just a similar letter substitutionT The third is purely symbolic using geometric shapes. But they are all the same

OK so let us start to offer some reasonable justification and explanation. First question. What do you think ' - ' (or +) or '1' or 2 or 12 are or mean in mathematics ? Many folks with insufficient background and even some who should know better fail to fully appreciate that these are symbols. They represent a mathematical concept so it can be manipulated and discussed. So 1 represents oneness - whatever that is - it is not oneness itself 2 represents twoness - whatever that is - it is not twoness itself and so on. + and - are even more tricky because not only are they just symbols, they do multiple duty, representing different mathematical concepts. Writing +90 or -90 may mean different things in different contexts. Yes they can represent the operation add 90 or subtract 90. But I am going to use the meaning 'signed numbers' also called positive and negative numbers in ordinary arithmetic. This is similar to positive and negative in electricity. But it represents direction not an arithmetical operation. So let us step into the real world and consider a rotating arm, exactly one unit long, fixed at one end to a pivot. as in Fig1 Since the arm is free to rotate in either direction I will label one direction positive or + and the other direction negative or - Then let us add two axes and set the pivot to the origin and align it along the first axis as in fig 2 Now let us swing the arm through exactly 90o in the negative direction. Call this operation multiplication by some as yet unknown number 'r' as inFig3 It can be seen that the free end of the arm now aligns with the -1 on the second axis Now swing the arm another 90o in the engative direction and note the free end now aligns with -1 on the first axis. So considering only the free end of the arm its starts of at position +1 and after being multiplied twice by 'r' it ends up at position -1 (actually I have called the operation multiplication but it is really something more subtle called composition in the arithmetic of 'operations') Algebraically we can write r2 (1) = -1 r2 = -1 r = √-1 So here is an example of the construction and meaning of 'i' in the real world operating on real material objects alone. But the arithmetic is not the arithmetic of ordinary numbers, it is the arithmetic of operations which is similar to, but not the same as that of ordinary numbers. and i is not a symbol for an ordinary number. But it is not imaginary either, that was a really unfortunate name from history. Here are two pages from the second chapter (called an infinity of arithmetics) of an interesting book on modern maths It is fascinating how many different operations in the real material world follow this pattern. This example is about the mathematics of Scottish Dancing.

If you have two or more generators of any type in parallel there will be a difference in their voltage output. This will result in a reverse current circulating in the parallel loop. Whether this is significant will depend upon the nature of the generators and their voltage difference. I have posted the relevant mathematics on this site at least twice now in other threads.

Congratulations, you have just discovered my point that there is no closed formula for this activity. It is a process. Philosophically there are two complementary processes. Analysis and Synthesis. It is many times simpler to measure something that is actually there (Analysis) than to create a desired something that is not yet there (Synthesis). Much Engineering is composed of the latter.

So you can put numbers into that formula, take a tape measure or whatever, and walk out onto the hill and point to a spot ? How ?

From the numbers and positions of the diodes I guess that the solar cells are grouped and the diodes are set to protect a group rather than individual cells. Are they bypass diodes or are they set to prevent fully illuminated cells form trying to pass current through shaded (and therefore lower voltage output) cells. Some better circuit diagrams would help.

I really don't know what that long thread is about because you seem to be preaching not just listening and discussing. In particular here is a real world engineering problem that does not satisfy your specification. You are to dig out part of a steeply sloping hillside and build a heavy reinforced concrfete box reservoir in the space. In order to support the new weight several lines of prebored piles are to be installed raking at 1 : 4 along the front or outer floor of the reservoir. Regardless of where each borehole starts on the hillside, the pilecap has to end up in the right place just at the underside of the reservoir floor to correctly distribute the load to the floor reinforcement. What closed formula can your AI pal offer to locate the drill point for each pile ?

There you are a much better coherent post and folks are already starting to off help and explanations. +1 to KJW. Thank you too for that post +1 also for encouragement. I think I now have a pretty good idea of where you are coming from. Now please don't misunderstand this next bit it is not a personal criticism, nor is it a criticism at all. It is an acknowledgement that folks are different and think differently,. I think perhaps that when you come across something that interests you, your mind suggests all sorts of loosly associated matters and that you started here by listing a great many of them for help. Now that I realise that this thread could become really productive. So well done for looking into i j k and finding the Miller indices. We can use this to look at the idea of indices as labels next post and start to work through your worries (hopefully others will chip in more and more) including you difficulties with infinity and electrical calculations with imaginary numbers.

Let me add to what swansont and markus have said, by asking Do you know what a functional is ? Why do you say that T is the kinetic energy density, not the kinetic energy ? never mind what "would be behind the tear or break" what does your system 'strain' into or what is behind that strain in your words ? In simple terms do you know what strain is ?

Thank you for your response. Actually there is a whole lot more to i, j k than this, which has dfinite solid connection to things like crystals and mineralogy and other applications. I wonder if you are hankering after this sort of approach, that used to be called practical mathematics. Here is a page from such a book an I wonder if you would post your reaction to it because if I am correct in my guess then you would enjoy this approach. What do you think ? The book is a british book published by Odhams called Mathematics in Theory and Practice

I should check your facts with the natives before posting next time. The relevant current body is Natural England and the relevant term is ancient woodland. It is however true, as exchemist says, that human occupation of the British Isles has led to a steady deforestation over thousands of years. Requirements for building, wars, farming clearance, sightlines (the Romans did a lot of this) and more have all played their part. These have been only partly offset by deliberate (re)planting, which has occurred only sporadically, for example the New Forest, by order of King Henry VIII.

Interesting use of emergent. Thank you. +1