studiot

That's what I was saying. +1

Yes there were 'dogooders' in every time and era. I think more land was distributed to ex soldiers than anyone else though, following the Punic Wars, especially the second one.

Apart from being physically impossible, that's too harsh. You learned things in the past, lots of things. When in the present or the future you come across something relevent to what you learned in the past you need to recall it. I am / about to face a particular situation What did I, my father/uncle/brother/colleague do in that situation I witnessed it for the first time ? The key is of course it has to be relevent going forward, not some morbid fascination with the past.

I agree so +1 to iNow. I would make a comment on this idea as follows Perhaps this is an inherited survival trait ? But as in so many things in Nature it is also a question of balance. (Do you remember that record ?) And the balance between all our traits can so easily get out of kilter. There are many 'Talking Therapies' from CBT to 'The Turnaround Principle'. You may not have heard of this latter one but it involves in re-examining the belief by considering the opposite "If I am worried about X - What if The oppiste of X is true ? "

Thank you for sharing this. +1 This tie up between the extreme northern peoples reminds me of this epic novel, wwhich I found to be a really good read. https://en.wikipedia.org/wiki/Last_of_the_Breed

Just a thought (well 2 actually) sologuitar has posted a series of similarly worded titles and these are in the homework thread. I wonder if this has triggered some additional anti bot /anti spam software here. I have seen some other maths sites swamped by what are clearly repeat posts.

Well it would be helpful if you has indicated that your expansion comments were not in opposition to what I meant but additional to it. I was responding to your assertion ( which I agreed with) that for practical operational reasons individuals were (and still are) called or elected to represent a whole bunch of people, usually geographically defined. I am simply pointing out that historically many (most?) of these individuals actually only represented their own personal interests and noboby else's.

I understand the situations you are referring to but they are not what I mean. It's very difficult to explain if you just keep simply telling me I'm wrong, when you are not even understanding what I am saying. Baronial landowners didn't vote for or against "Husbands voted against their own wives, daughters and mothers." Such matters didn't even come into their consideration.

So far as i can see all these check out. But you should not use use to check work you are going to submit for marks. Here is a simple check you can run for yourself. Say you are asked to simplify expression A / expression B and end up with expression C / expression D You want to know if they are equal ie you have not made a mistake. [math]\frac{A}{B}? = \frac{C}{D}[/math] CROSS MULTIPLY The results should be identical if you are correct. a simple example should show what I mean To test if [math]\frac{{{x^2}}}{{{x^3}}} = \frac{1}{x}[/math] CROSS MULTIPLY [math]{x^2}*x = {x^3}*1[/math]

No, read it again more carefully. The land (and later industrial) Barons represented noone but themselves individually.

I wish that were true, but the mental state of far too many is that even in good or the best of times they still dwell on past bad times and even imagine that these are still there or are afraid that they will return. Phsychiatry has yet to come up with satisfactory measures in this case, though positiveism helps some.

I'm sorry to tell you that you have completely missed my point. Self interest is exactly that. A simple example the Member (of Parliament) for say Oxford was a local Landowner. He represented his large landholdings, ie himself, although officially he 'represented' all those who live on and around his land.

Not really. but there is some confusion in this thread. I see that you are looking for values of x that make the fraction undefined ie division by zero. If x is a real number that is correct. But it is confusing at least and incorrect at worst to say x = 0 works. For the reason you have already given putting x = 0 leads to the fraction being undefined. So x is not equal to 0. We are all agreed on this. You are also correct to note that there is no real number p such that (p2 +1) = 0 So the correct form of reasoning goes that x cannot be equal to p, whatever p might be, since it is not a real number. Thus the only restriction on numbers that you cannot choose for x or must exclude to be x = 0

https://www.bbc.co.uk/news/business-66656369 Robots are trained to help revive coral reefs

My understanding of all past 'democracies', all the way back to the Magna Carta and beyond to SPQR, was that the voters only represented their own self interest, they did not represent anybody else (ie those who were not allowed to vote).

Yes it cannot be changed, but is it really 'gone'. I thought that was the whole point of this thread. Surely Dan was feeling the influence of past events and wished to find ways to mitigate this? @DanGonzal Welcome this is a topic that affects everybody to some degree or other. +1

Nothing at all. I'm sorry, it was me not you. Preparing latex formulae is difficult for me these days as I have to do it on my old computer,save it, and then transfer it to a newer computer when I can get on one. Whilst I was doing this for probably your first of the 'bunch' you must have added more so I only quickly posted in this one which was the wrong thread. Thus I didn't understand why the formula under discussion had 'changed'. I deleted my post but had to put something in its place. Sorry if this caused more confusion. But hey, never mind, I see we have new mathematically minded member so welcome @e jane aran FYI please note the site 5-posts-in- the- first- 24-hours limit. This is a security measure that greatly reduce the rubbish the overworked moderators have to deal with. After that time expires you will be unlimited. Thank you for the help you offered. +1

Edit ignore this I now see you have a bunch of these

Indeed so. +1

Welcome. I see from this question you are studying Henderson Hasselbalch so you must already have quite a bit of simpler stuff under your belt. You also seem to indicate that you are chemistry as subsidiary to some bio or medical course. In which case this book will help greatly. The relevant section here I don't have more time this morning, but I see there are other chemists about. Also I recommend you ask a moderator to move this to chemistry since we can work through it there. Homework help is more restrictive and your question is not homework.

I think those disciplines that activeley use tensors have a head start on those which could do but usually avoid them. I wish I had had a copy of this book when I first met tensors. Geologists make extensive use of certain tensors so one can get an immediate handle on the subject via rock mechanics and, to a lesser extent, soil mechanics. The build up from one dimension is echoed in the explanation of GR given by Baez https://math.ucr.edu/home/baez/einstein/einstein.pdf and also by Koleki https://www.grc.nasa.gov/www/k-12/Numbers/Math/documents/Tensors_TM2002211716.pdf Essentially this view runs that you have a coordinate system (which is therefore geometric) and a bunch of (physical) quantities of interest at some or all points spanned by the coordinate system. This setup is organised into an array of coordinate points, an array of quantities and an array of coefficients connecting the first two arrays. The coeffeciant array being the tensor array. Its coefficients may be constants or expressions which depend upon the coordinates. When one starts some form of continuum mechanics it is usual to first work in one dimension, then two, then three and so on. This offers the opportunity to go back to a lower dimension to see what is happening, although there is always the possibility of loosing some aspect that only occurs in higher dimensions . An example of such loss is the loss of the difference between contra and co variance in one dimension. Some other books with useful visula models are from Fleisch and Bickley

I think this subject has links to your other current question about tensors so the two threads should be read together. Consequently also the references I make to in each have material relevent to the other thread. Here is a good maths text about the requirements of dimension theory, for a variety of spaces, The index of this book is particularly unusual as it contains potted definitions for many important terms. A second book I am going to place here, although it also contains good material about Hilbert Spaces it it more relevant to spaces in general than visualising tensors.

Welcome Nicely put. All good stuff. +1 Hope to see more in future.

That is not an accurate interpretation of my OP. My perception of infinite would not have an end or a final point. In what way is that not accurate ? Your very first line introuces doubts about ends of infinity. So I tried to deal with them one at a time To which I replied about the starting point Here I tried to point you at something I believe you already know. Namely that there are infinite counting numbers and you can start from any of them. Later posts deal with the end point because Infinity is so strange that you can have an infinity between a definite start point and a definite end point; or you can have a definite start point but no end point or you can have a definite end point but no start point or you can have neither end point nor start point. Knowing this and being able to pick the right case is surely key to understanding the issue you then ask about? That is not an accurate interpretation of my OP. My perception of infinite would not have an end or a final point. In what way is that not accureate ? Your very first line introuces doubts about ends of infinity. So I tried to deal with them one at a time To which I replied about the starting point Here I tried to point you at something I believe you already know. Namely that there are infinite counting numbers and you can start from any of them. Later posts deal with the end point because Infinity is so strange that you can have an infinity between a definite start point and a definite end point; or you can have a definite start point but no end point or you can have a definite end point but no start point or you can have neither end point nor start point. Knowing this and being able to pick the right case is surely key to understanding the issue you then ask about?

🙂 Hopefully we we wprk it out for BB in the end I admire your steadfastness so far.