Recent content by Jaggis

Just to avoid further confusion: of course they can't be possible simultaneously. So, if the process is isothermal, it can't be isobaric - and vice versa. However, I didn't initially claim that only one of the processes is possible. They are both possible depeding on the heat that flows into...

But there is constant external pressure due to at least the weight of the piston and atmospheric pressure, though? That is, unless we're doing the experiment in space or something. This pressure is external, not necessarily the same as the pressure of the gas when it is heated. I never said...

I shall now quote a book "Physics for Scientists and Engineers" by Douglas C. Giancoli, 4th edition (2009), page 508 (ISBN-13: 9780131495081). The text in bold is just as at appears in the original text. "Let us assume that the gas is enclosed in a container fitted with a movable piston, Fig...

So, the pressure must be lower in the isothermal case. But what if we just do an experiment and see the volume growing and the weight being lifted? If we know the mass of the weight, the area of the piston and measure how much the weight is being lifted, how can we then know whether the process...

For example, in this video it is described how in an isothermal expansion or compression how heat and work balance each other out in an isothermal process so that the temperature of the gas stays constant. So, when would this apply if the situation I described is isobaric? PS. Sorry for...

We have a piston with ideal gas in it and a weight. The weight is placed on the piston. The gas is heated externally and the gas expands. Will the expansion be isobaric or isothermal? One argument would be: the expansion will be isobaric because the weight is providing constant pressure. The...

Ah! It is as simple as that. Yes, convergence would have to hold for all t \in [-1,1] so it is enough to find an individual t that won't satisfy it.

Does the sequence \{f_n\}=\{\cos{(2nt)}\} converge or diverge in Banach space C(-1,1) endowed with the sup-norm ||f||_{\infty} = \text{sup}_{t\in (-1,1)}|f(t)| ? At first glance my intuition is that this sequence should diverge because cosine is a period function. But how to really prove...

Hi, I have trouble with the following problem: Gaussian random variable is defined as follows \phi(t) = P(G \leq t)= 1/\sqrt{2\pi} \int^{t}_{-\infty} exp(-x^2/2)dx. Calculate the expected value E(exp(G^2\lambda/2)). Hint: Because \phi is a cumulative distribution function, \phi(+\infty) =...

OK, thanks for your help.

OK. I see your point. I think. Then I suppose that we want to use the Neumann series for φ. Because we demand that φ is an isomorphism, we need || φ-I || < 1. As in || φ-I || = || T+KT - I || <1. Now it remains to show that this holds if ||K|| <1? But can we choose K freely? The way I...

Oh, yes. You are right. My bad. Because I + K has a bounded inverse that can be represented as an infinite sum (according to the Neumann series theorem) ∑||K||^k, we need ||K|| <1. If φ(K) = T + TK, would the solution have something to do with the following reasoning: T + TK = T(I + K)...

Hi, I have some trouble with the following problem: Let E be a Banach space. Let A ∈ L(E), the space of linear operators from E. Show that the linear operator φ: L(E) → L(E) with φ(T) = T + AT is an isomorphism if ||A|| < 1. So the idea here is to use the Neumann series but I can't really...

This solution would indeed make sense if the condition has to hold for all i∈ℕ. Actually, how I initially read the problem was "for all a∈A and i must be an integer" but now I realize that it probably was meant "for all a∈A and for all i = 1,2,...". If the solution really is that A is empty...

This is nitpicking, but as I understand it, " for all a∈A" means "for all elements a that belong to A", which already states that A has elements. To leave an opening for the possibility of an empty set, it should say: "if a∈A (, condition (1) holds)". To say that all a∈∅ have a certain (albeit...