Question: Title: $x^x(\ln(x)+1)=b$ Body: I want to solve for $x$ and have tried solving it myself and used many algebraic calculators including wolfram alpha, and I could not get any answer from anything. If you can solve it please show me how, and if it is impossible please explain why to me. Th...
I've been reading an article on Clarke critical values of subanalytic Lipschitz functions. There I've come across the following definition(s) of subdifferential: $$f: U \to \mathbb{R}^n, \ \ \ \ \emptyset \neq U \subset \mathbb{R}^n$$ $f$ is locally Lipshitz continuous Let $$x \in U:$$ The Fre...
In the book [1], Clarke et al. define the generalized gradient for a Lipschitz function $f:\mathbb{R}^n\to\mathbb{R}$ as follows. 8.1. Theorem (Generalized Gradient Formula). Let $x\in\mathbb{R}^n$, and let $f:\mathbb{R}^n\to\mathbb{R}$ be Lipschitz near $x$. Let $\Omega$ be any subset of zer...
According to various websites, for some function $f:X\to R$ we can define a map $$ D(x,v):= \lim_{y\to x} \sup_{t\searrow0} \frac{f(y+tv)-f(y)}{t} $$ and the Clarke Subdifferential (Def 1) is $$ \partial f(x) := \{v \in X^* : D(x,v) \geq v\}. $$ The Clarke Subdifferential generalizes the gra...
The Clarke subdifferential of $f(x)=|x|$ at $x=0$ is a set $[-1, 1]$, and $\{sign(x)\}$ otherwise, just like the ordinary subdifferential because of convex. Now for \begin{equation} g(x)= \begin{cases} |x|/2-x^2/4, \text { if }|x|\leq 1,\\ 1/4, \text { otherwise}, \end{cases} \end{equa...
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