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I can't seem to find any sort of concrete definition anywhere... it always seems a bit hand wavy :( In particular I want to know what is an isomorphism between two banach algebras? Thanks.

Homework Statement f is a continuous function on [0,1] M is the set of continuous functions on [0,1] which are 0 at 1 ... i.e. for all m in M, m(1) = 0 I want to know if it's true that |f(1)| = \inf\{\sup\{|f(t)+m(t)|: t \in [0,1]\} : m \in M\} The Attempt at a Solution So...

I don't follow, could you spell it out a bit more? So cos^2(x) = 1-x^2+\frac{1}{3}x^4-... Now what? Thanks

Homework Statement How can I calculate it for \frac{1}{1+cos^2(x)} by using the fact that \frac{1}{1+x^2} = 1 - x^2 + x^4 - ...? Homework Equations Given in the problem. The Attempt at a Solution I tried letting u = cos(x), then \frac{1}{1+cos^2(x)} = \frac{1}{1+u^2} = 1...

Thanks for your help guys... Yes Hurkyl I'm just doing functions on functions. dx I'm a little confused - if g_{*}f=g\circ f as Hurkly says then we'd have f mapping A to B, then g mapping B to C ... wouldn't that be OK? Perhaps I'm missing something serious here :S ... could you give me a...

"push-forward" ? Can someone help me understand what this is, in as simple terms as possible? If I have a function f: A\rightarrow B and another one g:B\rightarrow C I know the "pullback" f^{*}g: A\rightarrow C is f^{*}g = g\circ f (correct?) But what about the push forward f_{*}g? What...

This isn't homework per se... It's a question from a book I'm self-studying from. If f is continuous on [a,b] and differentiable at a point c \in [a,b], show that, for some pair m,n \in \mathbb{N}, \left | \frac{f(x)-f(c)}{x-c}\right | \leq n whenever 0 \leq |x-c| \leq \frac{1}{m}...

I'm considering taking honours next year in statistics with a project in financial mathematics. I'm really looking forward to taking more classes in this area, but I have a couple of worries I'd like to air... 1. How smart do you really have to be? I think I'm above the curve, I can get high...

Yeah, I can see it's not periodic and hence must be apeiodic, but what's going on with that definition? My understanding of it is that there has to be a special (fixed) value of n where you can go from anyone state to all the others, including back to that state... but that doesn't seem to hold...

anyone?

Homework Statement Transition matrix is 0 0 1 0 0 1 (1/3) (2/3) 0 "argue that this chain is aperiodic" Homework Equations definition of aperiodicity - there must exist a time n such that there is a non-zero probability of going from state i to state j for all i & j The...

OK, so Vout = 86/(860 x 0.06^2 x pi) = 8.84 m/s then... work (out) per unit mass = pressure rise/density + 0.5(Vout^2 - Vin^2) = 306.31 J/kg Output Power = mass flow rate x work per unit mass = 26342.94 W efficiency = 26342.94/31500 = 0.84 Is that right? The answer my lecturer...

it's one my lecturer picked out specifically from a fifth edition textbook... so I'd assume it's OK.. Thanks rainman, I think I've made some progress with it.. still a little stuck though... from the volume rate of 0.1 m^3/s, I get mass flow rate = 86 kg/s using m' = pv' Also...

i left out the oil's density.. don't know if that makes a difference to your query though? word for word the question reads: An oil pump is drawing 35kW of electric power while pumping oil with a density of 860kg/m^3 at a rate of 0.1m^3/s. The inlet and outlet diameters of the pipe are 8cm...

Hi guys, could someone help me on this? I'm stuck :( *A motor draws 35kW electricity, it operates at 90% efficiency. The motor is connected to an oil pump which has an inlet diameter of 8cm & an outlet diameter of 12cm. pressure rise in the oil pump is 400kPa. oil enters the pump at 0.1m^3/s...