- Hardy's Inequality: Problems $3.14$ and $3.15$ in Rudin's RCAIn Problem $3.14$, we prove (a) Hardy's inequality, (b) the condition for equality, and I shall talk about (c), (d) below. Problem $3.15$ is the discrete case of Hardy's inequality. I have asked three related questions in a single post itself, since all of them are related to Hardy's inequality, and none should be too involved. There are some existing posts on MSE related to these topics, so I…
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- What does holonomy measure?I have difficulty understanding conceptually what holonomy measures. it can return a phase shift of the vector transported parallel along the connection. If there is no phase shift, it means that the connection is flat, and if there is no phase shift, then it should indicate that the space is curved. But I have found examples where a flat space can have non-trivial holonomy, for example a cone has…
- Finding the bounds for $|e^z - 1|$ on unit circle.The sharp upper bound is relatively easy to find: $$|e^z - 1| = \left|\sum_{n = 1}^\infty \frac{z^n}{n!} \right| \leq \sum_{n = 1}^\infty \frac{|z|^n}{n!} = e^{|z|} - 1 = e - 1$$ and it is attained at $z = 1$. I am wondering if there is a simple way to obtain a positive lower bound. I am suspecting a sharp lower bound is $(1 - 1/e)$ but I cannot prove it. I was told that $(3 - e)$ is a positive…
- Generalization of the identity $2(m^2+n^2)=(m+n)^2+(m-n)^2$ for cubic caseI am very exited to see the following beautiful identity: $2(m^2+n^2)=(m+n)^2+(m-n)^2$ I wonder if I can generalize it for cubic case like the expression as follows: $a(n_{1}^3+n_{2}^3+n_{3}^3)=(\pm n_1\pm n_2 \pm n_3)^3+(\pm n_1\pm n_2 \pm n_3)^3+\cdot \cdot \cdot+(\pm n_1\pm n_2 \pm n_3)^3$ Where $a$ is some constant and $\pm$ means you are free to choose any sign. I tried it by caculating…