Recent content by andrewkirk

Roughly: For the first five years we will not see the mirror. After five years we will see the mirror reflecting the image of ourselves ten years earlier. To be more precise, think about it like this. Say we can contact intelligent beings that live near that point 5 light years away, and we...

It means you first make a new polynomial ##g(x)## by substituting ##-x## into the formula for ##f(x)##. So for instance if ##f(x)=3x^2+2x+1## then ##g(x)=3(-x)^2+2(-x)+1=3x^2-2x+1##. Then you use the Upper Bound Theorem to try to find an upper bound for roots of ##g(x)##. If you find ##c## as...

Unfortunately that observation doesn't tell us anything about how the game selects racers. The actual selection will occur in a tiny fraction of a second and has nothing to do with the slow appearance of names on the screen. The slow appearance would be designed to simulate the feeling of real...

There can be no logical argument for that. What logical system could it use? It would have to be a logical system without modus ponens. Of course we could make up a logic system without MP but then we're just left asking what is the justification for the rules of deduction in that system? You...

The problem specification is complete except in one point: This is incomplete. It needs to specify how the randomness operates, and what the probabilities are. Here are two different examples, that will give different outcomes: 1. for a round there will be a defined set of valid allocations of...

The formula looks wrong. I think it should have ##t## where it says ##x##, and it should have ##-b## where it says ##b##. See here for the correct formula for the Richards curve (aka "generalised logistic curve"). It gives multiple versions. The above corresponds to the first version shown, with...

You assumed Mrs Holland could buy 88/21 cans, which is 4 4/21 cans. Nobody will sell her 4/21 of a can, so she has to buy 5 cans to have enough paint. She will have 17/21 of a can of paint left over.

Because the rule for partial differentiation is to calculate the derivative using the formula given and not looking through any variables that may themselves be functions of other variables. In your alternative approach you are treating t as a function of p, looking through that function...

Yes it's different, as each cut is a set of rational numbers, while we don't think of real numbers as being sets of other numbers. We think of them (or at least I do) as pin-point, atomic objects, not collections. The purpose of Dedekind's cut construction is to show that, from basic set theory...

Intensive properties are are often calculated by dividing one extensive property by another. So if the extensive properties have small fluctuations, so will the associated intensive ones. You can think of an intensive property of a large system as an average across the whole system of intensive...

You need to decide whether you are defining a vector or a vector field. A vector field has a curl. A single vector does not. The definition you've given above says it is a vector. Further, a vector field must be defined at every point in the space we are considering, so there is no role for...

You are correct. The question does not need to specify the initial velocity. It only needs to say that the sphere is given the same velocity. Given the height to which it rolls, we can infer the initial velocity, and we would infer a smaller velocity than 5 m/s. The question should not have...

The answer to 1 is "Yes". The answer to 2 is "nothing". It is a meaningless statement because the concept of 'bias' only applies to estimators. A correct statement would be "The standard deviation of a sample from a population is a downwards-biased estimator of the standard deviation of the...

We do know some things about the metric on ##Y##. By dint of ##Y## being a completion of ##X##, we know that ##d_Y(i(x1),i(x2))=d_X(x1,x2)##. So we only need to consider ##d_Y(y,i(x1))## and ##d_Y(y1,y2)## where ##y, y1,y2\in Y - i(X)##. To do that, we need to define ##f##. For ##w\in e(X)##...

To prove that, all you need to do is prove that: (1) 1 is an upper bound for all the ##f_n##. Can you show that, for any n, ##f_n\le 1##? Start by writing ##f_n## as an expression in only ##n##, rather than using "...." as above, which is vague and leads directly to confusion. (2) We can find...