studiot

You are using it every day to protect yourself from harmful charged particles coming from space.

Yes it's true. Zap was talking about the existence or at least the presence (presumably on SF) of two other people. Isn't that what 'being here' means?

Doesn't a mirror do exactly this? Why do folks so often forget that acceleration is a vector and that change of direction is an acceleration? Too much Christmas pud?

Since I have already said that I agree with swansont, I would would expect extraordinary evidence to support a claim that the probability is vanishingly low.

Therefore the a priori probability (the usually quoted one) is exactly 1, as swansont told you many posts ago. This is evidence of what exactly? And how does it stack up against all the other data, you have not taken into account?

This is the wrong day for riddles sir.

I missed this before, but thanks for listening. What gravitational field would this be ? It seems you consider whatever causes a 'gravity well' to be somehow separate from any other gravitational field. How does this work?

This is easy to say and difficult to prove. Evidence please.

Butch you are allowing your thoughts to wander all over the place. Try focusing on the subject. Think about it like this. What is the size (mass, diameter, volume whatever) of the smallest orange? There must have been one somewhere on /earth and perhaps one is recorded in the Guinness book, like the height of the tallest man. Whatever it is is just that. It does not mean that oranges or men are quantised. For discrete objects like particles. So for discrete individual objects the smallest mass is the mass of the smallest object, whatever that is. However a field that has energy acts as though it has mass (albeit very small) from relativity theory, not quantum theory. And as far as we know there is no quantised limit to this. Merry Christmas.

Yet null methods of measurement have long been known to be be the most accurate.

I don't see how that matters. After the energy levels in say thorium are not inter multiples of those in say hydrogen or oxygen. Nor are the differences between them, which is why they have different spectral lines. Added to which the OP asks if there is a minimum, not is it quantised. Minima occur in continuous systems too. Happy Christmas.

Assuming you refer to 'rest mass', surely it must be since mass is a property of particles and energy and the smallest mass must therefore be given by the rest mass of the smallest indivisible particle or quantum of energy?

No I don't agree. On the basis that you consider a theorem that is a couple of hundred years old to be wrong, perhaps you don't agree with Pythagoras' Theorem, which is a couple of thousand years old? Consider these examples for a moment: Firstly What is the probability that someone will nuke New York in 2019? Note this has never happened so there is no prior data. Secondly if I give you all the race results at Kempton Park for the last century you will have lots of data. So what would the probability of 'upshot' winning the 4.15 this coming Saturday 5th January 2019 be? Both of these questions rquire Bayes methods to answer them. Firstly What is the probability that someone will nuke New York in 2019? Note this has never happened so there is no prior data. Secondly I give you all the race results at Kempton Park for the last century you will have lots of data. So what would the probability of upshot winning the 4.15 this coming Saturday 5th January 2019 be? Both of these questions rquire Bayes methods to answer them. Firstly What is the probability that someone will nuke New York in 2019? Note this has never happened so there is no prior data. Secondly I give you all the race results at Kempton Park for the last century you will have lots of data. So what would the probability of upshot winning the 4.15 this coming Saturday 5th January 2019 be? Both of these questions rquire Bayes methods to answer them. I said I would give more so: Remember that probabilities refer to a proposed future event. This is because, as I said there are three types of probability commonly known as 1) Prior (or a priori) probability 2) Empirical probability and 3) Intuitive or subjective probability. For a probability of X = exactly 1 these mean 1) X must always occur. 2) means that X has always occured in the past, but that it does not mean that X will (ever) occur again in the future. 3) Means that; based on additional information, not direct observation of previous trials, we make an informed guess that X will occur. How are we getting on?

Statistics and probability doesn't work like that. That is the purpose of Bayes theorem and Bayesian statistics.. This even allows you to calculate the probability of something that has never happened before.

All your posts include too much off topic material. You have yourself pointed out that you asked two specific questions. Comments should be largely limited to addressing (replies to) these. Something like this is good and enhances the flow of conversation (oils the wheels) but not the paragraphs you have added. Question 1 was not about probabilities it was about your responders (me in this case) so I told you that Mathematics was pretty important to me. My apologies that I said retired Mathematician. I should have said retired Applied Mathematician. Yes it answers my question and is the right way to carry on a discussion +1 Read again what I said because I didn't say there were three probabilities - How can they be since I also defined them to be equal to 1? What I said was ther are three different meaning to stating the probability is 1. This is only true of (applies to) the probability values 1 and zero, it is not true for other intermediate values of probability eg p = 0.75 By assigning a probability of 0.75 to some possible outcome, say X, you are automatically saying that there is at least one other possible outcome, say Y, (then with a probability of 0.25). But by assigning a probability of 1 you are saying there are no other possible outcomes. There is more but it is too late to go into more detail tonight so I will let you think about this first.

Hello HongKong and welcome. Perhaps these sketches will help? It is all very well to say perpendicular. But perpendicular to what? We talk about the normal to a curve or a surface and we mean at right angles to a tangent to that curve or surface. In the case of the circle in the first diagram the normal is the same line as the radius. We often talk about an outward normal as I have drawn. The radius then forms the inward normal. When we come to a plane or a straight line the normal still at right angles to the tangent but we often call the tangent a parallel.. If we have a wiggly line the normals at different point point in different directions as shown. They are not parallel to each other. Finally for the slope or wedge (very common in Physics) ther normals at any to any face or side are parallel to each other. Does this help? The last diagram shows also how to clearly distinguish between { horizontal & vertical} and {normal and tanget} as pairs.

Hmm debatable. I looked at this earlier but my impression was there was far too much off topic dross to be worth bothering with. However since you have posted this in applied Mathematics and I am a retired Mathematician I will offer answers to both your questions. 1) I can see no relaible connection between Mathematics and Religion. 2) Since we are where we are the probability of it happening is exactly 1. However for probabilities of 1 and zero there are three very and fundamentally different meanings of the phrase probability is exactly 1. Do you understand enough probability to know the implications of this and can you tell me which one of these three you are refering to ?

To both new members here

Quiet. Ask yourself the following questions. 1) What system does Newton's 2nd Law apply to or represent.? Answer it represent a single body or system expressed as a free body diagram. Note the crucial part. There is only one mass. 2) What system does Newton's Law of gravitation apply to or represent? Answer it represents two bodies that interact with each other. Note the crucial part. There are two masses. This is the first place your student is going wrong.

Linear algebra is as good a place as any, but you do need to know the Physics of what you are applying it to. Remember that modern mathematical geometry is algebraic in nature so higher algebra underlies the application. I don't know what level your Physics or Maths is at, only that it is presumably up to undergraduate standards? So read this extract first and then come back with some details. I have highlighted the relevant section. It boils down to the connection between the triangle inequality and algebra, as applied to (quantum) Physics. The article is about Emmy Noether, of Noether's Theorem fame, which theorem you will need. Some good books to approach this from a mathematical point of view are An older book Mathematical Foundations of Quantum Mechanics G W Mackey Benjamin A Modern Book The Mathematical Principles of Quantum Mechanics D F Lawden There is a Dover version of this. Drawing together the maths background is a mammouth task (but very worthwhile) and will entail a great deal of background reading.

Then you are just making trouble for reasons best known to yourself so there is no point my having further discussion with you.

I'm glad you are not teaching me Physics. What was that you said about two equal masses a fixed distance apart. Google is your friend. We also had a long thread about it here recently.

Oh come now, I'm sure you know better than that. setting a = 0 is nothing more than noting that the masses are not undergoing accelerating movement towards each other. This means that the net accelerating force F = 0. It does not mean that either m1 or m2 are zero. If that were the case then the bodies would exert zero gravitational effect on each other, as with photons. For massive bodies it may mean they are fixed or held as with Cavendish's balls. But even though they are not moving towards each other massive objects still experience a gravitational force in Newtonian mechanics. The bottom line is that the original diagram is incomplete since it does not explain how the masses came to be in the position shown. Any force due to gravity must be additional to any force that either drew them there or holds them there must be taken into account when calculating F. And the offered analysis does not do this.

So explain to me the value of m given by quiet's equations 2, when a = 0 ? (Since the plan is to substitute this value or its square into equation 3) Or explain to me how you can validly divide by zero in Mathematics? Here is the version I'm looking at in post#1. I'm still hoping quiet will see why you cannot substitute for m in any Physics or Mathematical system

I am not saying the same thing. The way to explain it to your students is simple. F = ma is an equation of dynamics, and only meant for non equilibrium systems Newton's law of gravitation is not. So that force is only exerted if the object is accelerating. The force of gravity is exerted whether the objects are in motion or not. Physically you can't combine the equations in the way you have, although algebraically you can, the result makes no sense. Furthermore the 'force' of gravity is normally put in the frame of the larger object (eg the Earth) You are trying to work in some external frame. Any teacher should be able to guide his students in the correct direction about this.