studiot

Alfred, have you checked up on Latin Squares? They are not difficult to understand. Remember I said the question is like them, not exactly the same. Personally I don't think it is possible to develop a single mathematical expression that covers all possible cases with the front loading described. You would have to chop up the word count range and perhaps the % of abbreviations as well into segments and develop a separate formula for each.

Yes I was including Michel in the 'you guys'. (Would I ever exclude him?) The problem with those diagrams is that, whilst they are an excellent summary to someone familiar with this stuff, they contain a lot of information and there is no explanation as to where any of it comes from. Hence all the toing and froing about this in the posts. Thanks for the vote.

The (linear) wave equation in one dimension is given by a differential equation [math]\frac{{{\partial ^2}y}}{{\partial {t^2}}} = {v^2}\frac{{{\partial ^2}y}}{{\partial {x^2}}}[/math] With (one) commonly used solution [math]y = a\sin \frac{{2\pi }}{\lambda }\left( {x - vt} \right)[/math] As As you can see it connects space and time. The standing wave obeys the differential equation but has a solution [math]y = 2a\sin \left( {\frac{{2\pi x}}{\lambda }} \right)\cos 2\pi nt[/math] This separates space and time so for any time t you can plot the value of y over x. Edit Doing this with the wave equation is the beginnings of quantum theory by the way.

I honestly think that the diagrams you guys are bandying about are far too complicated for first understanding. In particular they run for way too many years. A few years are sufficient to establish the principles.

Yes he has miscopied. +1 mark please review your first post and correct it.

For a large word population I don't think it matters what distribution you assume; sampling every 5th word will divide it into uniform blocks, a bit like Latin Squares, and the result will tell you what the distribution is. This assumes all blocks are equally treated according to the distribution function. in the seeding of abbreviations. But this population is being compared to a second circumstance where the first bock or few blocks are treated differently from the rest. And the question was essentially which circumstance will lead to a better answer? The answer to this is that neither will always lead to a better answer. For very small word populations putting the abbreviations first is more likely to lead to a better answer. For very large word populations putting the abbreviations first is more likely to lead to a significantly worse answer, but will only be definitely worse for an infinite word population. The actual crossover point depends upon the actual populations.

I have to say that spacetime diagrams help some and hinder others. They are not necessary to understanding or working things out.

Yes the theorems variously attributed to Gauss, Green, Stokes are much used in Engineering and Physics to generate working equations. Green's other theorems are also used to help with some integration. They are also used in applied maths (numerical methods) to convert finite element to boundary element methods. Normally we do not used such a generally shaped volume of integration (It must be the mathematician in you that wants to be so general) because it is too difficult. The volume is generally a box, cylinder or sphere so we can use appropriate coordinates to simplify the calculation and is called the control volume. The control volume may be infinitesimal ie dx, dy, dz or it may be large but finite eg a reservoir. As to time independent functions you can integrate over the region in question potential functions are good. (Is R the best variable to use to describe your function ?) Scalar potentials - eg potential energy are much used in thermodynamics. Vector potentials can demonstrate static fields. So you need to expand your description of what you want to do. In general you either integrate a variable which is a function of position over some region to obtain the total amount of that variable (eg potential energy) in that region That is [math]\int {U\left( {{x_1},{x_2},{x_3}} \right)} d{x_1}d{x_2}d{x_3}[/math] Which gets you a sum total. But this is not integrating an equations, which you also mentioned. For this you (may) need Greens functions or Fourier transforms etc. For suitable equations that define functions this will get you another equation defining another function. A good example of this is integrating slopes, loads and moments on beams

+1

Sorry, Tim, I see what you mean. Try this. Actually, Tim, the frame of the clock(s) is of vital importance which are most certainly not present at all events in all frames, which they would be (could be) if they were present in all frames. I was not referring to Einsteins paper but the paintball games posts in this thread. In a way it is a shame that this thread had been split from another since it seems to have killed its progenitor. The twins offer a good opportunity to simply work through both time dilation and length contraction. In neither thread is anyone prepared to work properly through this. Furthermore the example I quoted above is the commencement of a simple analysis where the numbers drop out very easily. It also shows what happens to both clocks, only one of which (the rocket clock) is present at all important events. (Do you know what an 'event' is?) It is the sad fact that so many published 'explanations' of the twins tells less than half the story that IMHO confuses so many people. (I have to own to a bit of this myself in my very simple explanation of the asymmetry) I thought Mordred had already explained why you mistakenly think one clock 'ticks' faster and how you were misinterpreting spacetime diagrams, but obviously I was wrong since you persist in this view.

Actually, Tim, the frame of the clock(s) is of vital importance and most certainly not present at all events, which they would be if they were present in all frames. Michel, There is a big difference between length contraction and time dilation. An example for both of you. A rocket ship (observer A) leaves Earth (observer B) and take with it a clock and an one metre rule, marked off in 1cm graduations (0 through 100) and travelling at 0.8c. Both A and B will always 'see' the one hundred and one graduations on the ruler at all points in the journey. But they will differ in their assessment of the length of that ruler. As to the clocks, say A sends back a light flash every year, A will not see anything in the first three years, that is A's first flash will arrive at B when B's clock reads 3 years. The rocket clock is no longer in B's frame and B cannot measure on it.

Paste function not working. Clocks don't loose some ticks, they just never make them. A big source of confusion is identifying which frame a clock is in and not trying to take time differences between clocks in different frames.

Since you have visited the forum a couple of times since last posted, do I take it that you have lost interest in this thread of yours?

Was there something in my post you did not understand? It was a direct response to the OP assertion that the Earth has no (classical mechanics) connection to the plane. It was not an exhaustive analysis of all the forces acting, real or imaginary.

There is no simple anser to this, it is going to depend upon the sizes of the abbreviation and other word populations. Say there are 6 abbreviations and 30 words ( a short section). If you lump all the abbreviations together at the beginning you are guaranteed only 1 hit regardless of the subsequent number of words, but your accuracy will decrease as the number of words increases as you will never get another hit. If however the abbreviations are distributed it is possible to either get them all or miss them all depending upon the size of the rest of the word population. With 30 or greater as I offered you could get none or all six.

Robin, this question of yours has spawned at least one more thread and become very messy, which is why I originally avoided it. I did make a couple of posts containing questions designed to help you find the right view, But you have avoided answering them. When contemplating these questions it is very important to keep track of which clock is where and which system you are measuring in. A wrong placement here leads to embarrassing paradoxes and misunderstanding. In particular you should always find out which clock is at all the events - There will in general be only one of these. If you are interested I will work through the Twins using largely logic, although a little easy maths will be needed. The main thing to get the correct time differences, is to know which reading on which clocks can be used directly and which have to be transformed. Logic is need for this, not maths. But I will only do this as a discussion as I am not prepared to do all the work here. So In the Twins there are no clocks in the Earth system that are present at every event. Can you say which clock or clocks are at all the significant events in the the history? We will also be following this from swansont as it is a highly significant hint. +1

Please confirm the following with reference to my diagrams attached. Sorry for the quality of the hasty sketches. 1) Mark off an interval, ab, along the x axis shown by lines through a and b parallel to the y axis. Draw in y1(x) and y2(x) as the bounds in the xy plane. This defines a surface (area) in the xy plane. 2) Erect a z axis perpendicular to the xy plane showing the bounding area we are working in. I have shown this hatched. 3) Erect rectangular columns over this area from xy plane to the bonding surfaces given by z1(xy) and z2(xy) This defines the volume we are working in now for your function R The general wave equation is a connection between time and space which introduces an extra variable we have not catered for (time) To take time into account we have to account for the flux of R crossing the volume boundary as defined above, as well as the waves already within the volume. A more restrictive wave equation is the time independent equation of standing waves. If the waves are standing there is no flux across the boundaries so time may be discounted and a simple spatial volume integral employed.

The following 4 slides from the Cambridge University Physics teaching site may help you think about Imatfaal's comments as well as mine.

Forgive me but I can't see a c anywhere.

You are running ahead of yourself. Much of modern theory in this area revolves around the the difference between linear (euclidian) and affine spaces and transformations. This distinction is often not clearly drawn. Mordred mentioned rotations. There are also what are known as 'forbidden' rotations' we are important for non symmetrical (handed or chiral) objects. Operators (mathematically) are a particular way of expressing relationships (equations and so on) for convenience of mathematical processing.

Hmm, a difficult one we have thought about in our family. +1 Let us look back ways over history, at least over the time when society was rich enough supply youth with the time and resources for (I hate to use this word) leisure activity, on a widespread and general scale. In the late Victorian and post Victorian era, mechanical hobbies abounded. Meccano, Hornby and many others spring to mind. Model making was a hobby that carried on into adulthood and enormous effort was put into these. These hobbies also followed the development of technology. So steam gave way to IC engines and then to radio and then to amateur electronics, amateur computers (hardware) then software. In these and other hobbies there was a cycle of constructors and users. Photography is a really good example here. Some like to construct then get bored once they have built it, some really prefer to use and buy kit ready made. Amateur radio is a really good example. Not all these hobbies are scientific/technological. Fishing is the most popular sport in the UK. So why am I mentioning these? Well they have all displayed similar characteristics to those mentioned in the OP. Has society suffered or benefited from them? Many have also arrived, burgeoned and then declined. There used to be at least 5 amateur electronics magazines in the UK. All have now closed.

A niggle here. It is not dimensionally true to say you can subtract distance (or its square) from time (or its square). Time has dimension T The constant, c, has dimensions LT-1 When multiplied together the result has dimension, L

Interesting how Nature in the form of animal biology displays the reverse. Our muscles can only contract (pull), they cannot push. So muscular activity in our bodies is determined by the balance between two opposing muscles. Muscles work in pairs, pulling in opposite directions.

Sorry if I miscopied this but wonderful windows10 will again not let me copy and paste or use the quote function. Although his profile does not indicate much wtf posts as though he is or was a professional mathematician so it was good to see my words in post 3 echoed and much extended, in a particularly understandable and helpful way. +1 geordief, A really good place to look for this is the Cambridge University Text for SMP (school mathematics project) by E A Maxwell Geometry by Transformations This connects old and new style geometry and stirs in a little group and isometry theory rather well.

I believe my great great great great great grandfather's dog once mentioned something about angels and pinheads.