# Tagged Questions

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### Are there logarithm functions for arbitrary rings?

The logarithm function for $\mathbb{R}$ satisfies $\log xy = \log x + \log y$ whenever both $\log x$ and $\log y$ are defined. Are their conditions for a ring $R$ which guarantee the existence of a ...
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### convexity of the product of two entropy-like functions

Consider the functions $T_p(q)= \sum_i q_i^p$, where p>1 and q is a finite-dimensional vector satisfying $\sum_i q_i = 1, q_i >0$ (ie, a probability mass function). In information-theoretic terms, ...
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### Why is the negative entropy Lipschitz with respect to the $1$-norm (Over)?

Let $\left\|x \right\| = \sum_{i=1}^{i=n}\left|x^i\right|$ and $d\left(x\right)=\sum_{i=1}^{i=n}x^i\ln x^i$ where $x\in R^n$ and $\sum_{i=1}^{i=n}x^i=1$ How to prove: For all $x, x'$, \left| ...
A conditional entropy can be expressed in the following way, $H_{V_t}(V_s) = -\sum_{s,t}p(s,t)\log{p_t(s)} = -\sum_{s,t}p(s,t)\log{\frac{p(s,t)}{\sum_{s'}{p(s',t)}}}$ $s$ and $t$ are defined ...
### Upper bound for $-t \log t$
While reading Csiszár & Körner's "Information Theory: Coding Theorems for Discrete Memoryless Systems", I came across the following argument: Since $f(t) \triangleq -t\log t$ is concave and ...