# Search results

1. ### Proving Completeness of a function space

Hi , is the following correct ? (an outline of the proof ) Given an arbitrary Cauchy sequence (f_{n}) we have that \forall \epsilon > 0, \exists n_{\epsilon} \leq m < n \, s.t \sup_{0 \leq x < \infty} \frac{|f_{n}(x) - f_{m}(x)|}{x^{2} + 1} < \epsilon g_{n}(x) = f_{n}(x)/(x^{2} + 1)...
2. ### Proving Completeness of a function space

Let F = \left\{f : [0, \infty) \rightarrow R, norm(f) = \sup_{x \in [0,\infty)} \frac{|f(x)|}{x^{2} + 1} < \infty\right\} Is F complete , under the given norm ? My approach was to look at the pointwise limit of an arbitrary Cauchy sequence, but I am not able to prove that it converges in...
3. ### Stationary distribution

Hi, A Markov chain is aperiodic if all the states are in one class (as periodicity is a class property and the chain itself is called aperiodic in your case) and starting from state i, there is a non-zero probability of transition to state i (this is of course given by your definition of d(i))...
4. ### Question about an integral.

Hi, if f(t) >= 0 for t >=0, then using (g(x/t) + 1) would be a better idea ? Now if f(t) = f^{+}(t) - f^{-}(t) , where f^{+}(t) and f^{-}(t) denotes max(f(t),0) and max(-f(t),0) respectively then one can use g(x/t) + 1 for f+(t) and g(x/t) - 1 for f-(t) to obtain a bound.
5. ### Doubt about Implicit differentiation

Sorry but what is "i" ? I did not understand what this means. Why is it not needed if x is a function ? And could you please point me to some easy references on calculus for functionals ? Thank you !
6. ### Doubt about Implicit differentiation

Sorry about the confusion. Yes, x is an implicit function of \lamba , but the idea is that x(\lambda) is a family of functions paramterized by \lambda
7. ### Doubt about Implicit differentiation

Hi all, I was reading a paper in which implicit differentiation was used as follows x \in R, \lambda \in R Given G(x,\lambda) = 0 \frac{\partial G(x,\lambda)}{\partial x} \frac{\partial x}{\partial \lambda} + \frac{\partial G(x,\lambda)}{\partial \lambda} = 0 My doubt is...
8. ### Use of method of undetermined coefficients

Use of "method of undetermined coefficients" Hi all, Suppose I have a equation f(z+1) - f(z) = z^{1/2} , \forall z \geq 0 eq (1) then is it possible to solve this equation by the method of undetermined coefficients ? It is usually seen in textbooks that the forcing function is taken to be...