studiot

The point is that in order to draw a continuous line your pen must pass through irrational points to get from one rational to the next. Equally for (physics) fields to follow the equations of physics, the field lines must pass through irrational points to get from one rational point to another and in order to make a completely closed line or shell or surface or whatever a the closure must block the flux from escaping through irrational points on that closure.

No. Does 4 implies 2 have any meaning? Yet 4 is related to 2 and in this case 2 is related to 4 since they are both even. Further this relation can be transferred so if 4 is related to 6 and 2 is related to 4 then 2 is related to 6. The three statements a R a is called reflexive relation a R b implies b R a is called symmetric relation If a R b and b R c then implies a R c this is called a transitive relation. Relations may satisfy none, one or two or three of these conditions If the last three are satisfied then the relation is called an equivalence relation and often given the symbol a ~ b Equality is an equivalence relation, but > is not since a > b does not imply b > a.

It is worth noting that there is some disparity in drawing the distinction between a relation and a function and considerable disparity in the notation employed. So it is always wise to check the definitions and notations used in any particular work. Here is simple explanation of how I like to think of relations Given a set A we often like show that a 'relation' exists between certain pairs of the elements of A. Such a relation can be expressed as a property statement that is true for some pairs of elements but false for other pairs. A good notation that does not clash with other uses (such as the ~ symbols which may be reserved for a special type of relation) is For all a, b in A a R b denotes that b is related to a c notR d denotes that d is not related to c A relation property may take many forms. For instance a is related to b if they have the same parity so all even numbers and all odd numbers are parity related. (This is a relation that is not connected to a function) Another relation that is not is function is the equation of a circle in a plane. That is because there are two values of y for every value of x and functions are defined as being single valued. This is the principle distinction between a function and a relation.

What do you mean by this ?

Hippasus already gave you one. Remember the ancient greeks were primarily geometers, not algebraicists. But not only geometers but constructive geometers. Note that most of Euclid is couched in terms like Proposition XXX To construct ........ So imagine their consternation when they discovered that they could not measure the length of that diagonal, although they could undoubtedly construct it.

jesus math is old. Was not aware of this fact. Thanks for this. To be fair what he showed was "There is no set of all sets" ie no Capo di Capo", no "One ring to rule them all. It is perfectly OK to posit a simpler universal set which is the complement of some set plus that set itself. I agree with the others that I saw no sense in your use of an arrow though. Nor do I understand your definition of a relation, which can be thought of as a partition of a set.

Of course it would. The diagonal of a unit square would be different for instance. I'm not sure if the triangle inequality or Schwartz inequality which underlie quantum theory would also be different.

Don't see why you need to get all fancied up; Hilbert space ? What is wrong with good old fashioned Hippasus ' square root of 2 ? Or Archimedes and his Pi ?

So how am I meant to take this ? You don't know anything about neighbourhoods but you don't want to consider them ? Why not ? Why is this not inconsiderate of others who are trying to help you ? As far as I can see you are having trouble reconciling the concept of infinity and infinite processes with your half completed half remembered past studies. Further you keep intorducing infinity and things which are not numbers or even necessarily to do with numbers. As soon as you start talking about integrals you have left the realm of numbers - real or otherwise - and entered the realm of functions. Functions are not numbers and numbers are not functions. In fact the Riemann integral is a mapping from the space of functions to the space of numbers (please note the correct use of the word space here) Not only that it is a linear mapping called a functional. Be that as it may here is one way in which neighbourhoods help avoid infinity. There are different types of neighbourhood defined. A simple neighbourhood of a general point p which includes the point p itself and a reduced neighbourhod of p which does not include the point p itself. By using the latter we may avoid infinity if we define a reduced neighbourhood of p as ∞ as being {x: x > c} where C is some chosen number. Do you understand this notation ? It means the set of all numbers greater than some C. The implications of this are huge. It means that it does nor matter whether there is an infinity or not - we only have to 'approach it' or 'tend to it'. Equally useful mileage can be obtained from the reduced neighbourhood of a real number point p, since it must include any 'nearest real number' - if such a thing exists. But we can still proceesd with out analysis if it does not. Also, and back to integrations and functions, we can use neighbourhoods directly with functions. We cannot so use the other derivations without modification. What's not to like ? Here is a link to a good article about intervals and neighbouthoods. Note they use 'a' for the point, p where i have use 'p'. https://math.fel.cvut.cz/en/mt/txtb/1/txe3ba1a.htm

Just type create recovery usb into the search box nxt to the windows button in the bottom left and follow the instructions that open up. You can see the screens you get in my links. Windows will work out how big the usb stick needs to be, it will probably be in the range of 8Gb to 16GB. Let Windows make the stick and put it away safely. Once a month or two or so update it and changes including udates and apps will be automatically included. When in trouble The stick will boot the pc and offer recover, repair, restore and all the other words in your list.

Don't forget that from W10 windows creates a required hidden partition on your hard drive, partly as a security measure. Simply copying the win partition will no longer work. Also many machines these days have machine specific and drive specific signature codes that must be correct.

OK so you are a dreamer shooting the breeze. Nothing wrong with that since you are asking for advice. Firstly gravity. Earth's gravity depends upon the mass of the Earth, not its orbit. So unless the meteor combined with Earth (as Phi said) and it was a mighty big one gravity would not have altered. BUT The idea of something altering is not so far off. You also mentioned the dinosaurs. Yes they were big. So were the plants in those times. W know from paleoclimatic data that the average temperatures were higher and there was much more of both oxygen and carbon dioxide about in those days. Here is a quote from professor Beerling (Sheffield) in The Emerald Planet - Oxford University Press So don't give up on having ideas, you may be right next time.

More or less yes, but you need both it is not a question of versus. Also sometimes fundamental principles are enough. Most times they are not. As for example in the phrase 'statically determinate' In two dimensions there are three fundamental equations available. After that you must introuduce others somehow if they are not enough - which is mostly true in real life.

You clearly don't want to understand since I offered you an alternative treatment using neighbourhoods and you ignored it totally. The beauty of neighbourhoods is that it does not depend upon whether or not there is a smallest real number or a nearest real number or what happens at infinity. And yet neigherbourhood analysis arrives at the same results as the other approaches you have been arguing about.

Happily I prepared this list of tips for someone like you just recently. (W10 is much like W11) I have emphasised the answer to your question in the list but the others may be of use. Windows 11 Create Local account https://support.microsoft.com/en-us/windows/create-a-local-user-or-administrator-account-in-windows-20de74e0-ac7f-3502-a866-32915af2a34d https://pureinfotech.com/create-local-account-windows-11/ Turn off one drive https://support.microsoft.com/en-us/office/turn-off-disable-or-uninstall-onedrive-f32a17ce-3336-40fe-9c38-6efb09f944b0 Create recovery stick https://support.microsoft.com/en-us/windows/create-a-recovery-drive-abb4691b-5324-6d4a-8766-73fab304c246 Save webpages https://support.microsoft.com/en-us/office/save-a-document-as-a-webpage-d1cf1c33-e854-4f9d-bd51-93f94da0f4e3 Skype some versions of w11 come already loaded some you have to get (free) from Msoft. So type 'get skype' into the search box and follow the instructions Recovery Options https://support.microsoft.com/en-us/windows/recovery-options-in-windows-31ce2444-7de3-818c-d626-e3b5a3024da5 https://www.online-tech-tips.com/computer-tips/how-to-boot-to-windows-11-10-system-recovery-options/ Wordpad Type wordpad into the search box at bottom left. This will allow you top open, write and save word docs Initial setup internet bypass https://www.wintips.org/how-to-bypass-internet-requirement-in-windows-11-22h2/

So it is good that you are interested enough to look. However I also said more about them than in your quote of my words. Note I was bringing in your comment on geometry. I was not disagreeing with it. Since I thought you and others might be interested The equations of constitution are the controlling statements or principles, preferably in the form of equations, in physics, chemistry or the science of interest, that are being invoked or brought into play. For example Newtons Laws of motion. Bernoullis's equation in fluid mechanics. The Law of mass action in chemistry. The electrochenical series in chemistry The differential equations of displacement of a loaded beam. Hopefully you get the idea. But these equations are applied to real systems they have no meaning in isolation. And real systems introduce constraints of their own. Many of these constraints are geometric ones, others may be conservation laws. So we often pair the continuity equation (conservation of mass) with Bernoulli conservation of charge with calculations on chemical reactions. Such equations are called the equations of compatibility. But as I said they need not be actual equations for instance we have the beam displacements at the supports must = zero to pair with the beam differential equation for which there are many mathematical solutions, and we must pick a solutions that satisfies the compatibility condition. This particular example is a simplified version of the basis of quantum mechanics, wher we must pick integer solurions to fit some system geometry. Are you beginning to get the idea ?

And ?

You may have crackpot tendencies, we all some in us I suspect, but you do try to reconcile your worries with conventional thinking. Crackpots tend to want to override everyone else. I can offer counterexamples for both of these claims. The Chinese observing and recording the supernova in the Crab Nebula in 1054 were just in the right place at the right time. They were not speculating any observation of the sort. The discovery of penecillin was a pure accident, no cnjecture, bold or otherwise was involved. In relation to your original comment that cause this thread. Have you ever heard of the equations of constitution and compatibility ? This is a very convenient way to divide up the maths. Compatibility is very often represented by some geometrical constraint.

I agree IMHO the 'machinery' is the interaction of physical properties. It should be pointed out and remembered that there are nearly always alternative methods, often using different physical properties to analyse or solve a particular phenomenon. It is very comforting when these different methods arrive at the same outcome. Even more comforting when that common outcome concurs with observation.

We all do it. I find a glass of Laphroaig helps at such moments.

My sincere apologies. I was rushing it. St Petersburg is in the Moscow time zone, which is GMT +3, although as you say it is sufficiently west to be actually 1 hour behind Notice the line in the BBC introduction Perhaps it always has been, I have not been able to determine this but in those days they used 'solar time', but for a whole area so it is reasonable to assume they followed the then capital.

They were well aware of the variation of time (of noon) with longitude by that date. I hour for every 15o longitude. Huygens published the first statement of what we now now as the astronomers and surveyors equation of time in 1665. I made my comments becasue I was suprised at the choice of St Petersburg, especially as it was a late established city and not made capital of Russia until 1712. But we had just fought a war with Russia. Of course the second reference was to the famous story about the international date line, published in 1872. Also at that time as noted there were many 'prime meridians' @Genady It is the meriadians, not the parallels that determine time. https://en.wikipedia.org/wiki/Prime_meridian#List_of_historic_prime_meridians_on_Earth The Russian one at that time was the Pulkova Meridian https://en.wikipedia.org/wiki/Pulkovo_meridian There was also the Paris rival to London https://www.solosophie.com/paris-meridian/

I find it interesting that knowledge of timezones was required in 1859, which was just after the Crimean War and just before the Jules Verne book "around the world in 80 days."

I wonder what temperatures you are working at ? The old DeaconProcess operated at 450oC. Here are some facts and figures about hypo.