5:27 AM
A new tag was created and added to 12 questions.
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Given $D\subset \mathbb{R}^n$. $f:D\longrightarrow\mathbb{R}^n$ is said to be a monotone operator if $$\left\langle f(x)-f(y),x-y\right\rangle\ge 0,\text{ for all }x,y\in D.$$ I wonder if this hypothesis is true about monotone operators. Given $D\subset \mathbb{R}^n$ and a differentiable operat...

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I found the following theorem: Given $H$ Hilbert space and a monotone operator $A\colon H\rightarrow H$, then A is maximal monotone if and only if $\operatorname{Range}(A+I)=H$. Note that: $A$ monotone (multivalued) means that $\forall u,v \in H$ and $\forall f\in Au, g \in Av$, then $(u-v,f-g)... 0 I trying to prove the following assestment Every linear monotone operator on$L^2 (0, \infty)$is bounded Any ideas? Thank you 2 I'm looking for an example of a (multi-valued) maximal monotone operator$A$mapping a Banach space$X$into its dual$X^*$such that the domain$D(A)=\{x\in X: Ax\neq\emptyset\}$is not convex. Preferably, the example should be simple (maybe with$X=\mathbb{R}^2$). Thanks a lot in advance for s... 2 I'm reading Barbus & Precupanu's 'Convexity and optimization in Banach spaces'. The authors define what I think is the resolvent for an operator$A: X \to X^*$(or subset$A \subset X \times X^*$) where$X^*$is the dual space: Let$ X,X^*$be reflexive and strictly convex and$A$maximal mon... 3 Suppose$X$is a real Reflexive Banach space. Let$A:X\rightarrow X^{\star}$be a Pseudo Monotone operator, i.e. if$u_{n}\rightharpoonup u$and$\limsup\langle Au_{n},u_{n}-u\rangle\leq 0$, then $$\langle Au,u-w\rangle\leq\liminf\langle Au_{n},u_{n}-w\rangle,\ \forall\ w\in X$$ where$\righthar...

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I'm learning monotone operator these days and can't figure out the meaning of the symbol. If you know the symbol please tell me. Thanks for your time!

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I am looking for a general definition of monotone condition for a function $G: \mathbb{R}^m \to \mathbb{R}^m$, and since I did not find a unique definition of monotone condition for multivariable function (see Monotonicity of function of two variables), I was wondering if the definition of monoto...

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In the paper Ernest K. Yyu, Stephen Noyd - A Primer on Monotone Operator Methods - Survey, the authors frame Iterative Refinement (of an approximate solution to a linear system $Ax=b$) in the context of monotone operators, Resolvents, and Cayley operators. They require $A+A^T \succeq 0$ to ensure...

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Definition: Let $H$ a Hilbert space. An unbounded linear operator $A: D(A) \subseteq H \to H$ is said to be monotone if it satisfies $$\forall v \in D(A),\ (Av, v) \ge 0$$ It is called maximal monotone if, in addition, $R(I + A) = H$, i.e., $\forall f \in H,\ \exists u \in D(A) \tex... 3 Let$H$be a real Hilbert space with inner product$\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm$\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let$A, B : H \rightarrow H$be monotone operators, that is (for both$A$and$B$) $$\langle A x - ... 2 Consider a convex optimization problem.$$\min_{u\in\Re^k} f(u)$$s.t.$g_i(u)\leq0,\ i=1,\ldots,m$Let$F(x)=F(u,\lambda)=(f'(u)+\sum_{i=1}^m\lambda_ig_i'(u),-g_1(u),\ldots,-g_m(u)):\Re^n\rightarrow\Re^n$, ($n=k+m$)$G=\{x=(u,\lambda)\in\Re^n:\lambda\geq0\}$.$f$and$g_i\$ are all convex. ...

The above queries did not find any older occurrences. (After the next update of SEDE, this instance should be inluded there.)

5 hours later…
10:53 AM
Mar 13 at 9:07, by Martin Sleziak
Maybe there should be on meta. (Or some other reasonable tag name for this topic.) There are questions such as: Privilege page on reduced ads doesn't show on this site, even though there are now advertisements or What does the “see reduced ads” privilege mean?
Mar 13 at 9:07, by Martin Sleziak
And I have recently asked this one: Are there ads on Mathematics Stack Exchange?

11:14 AM
I created the tag on meta.
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I find the ads about non-mathematical things annoying. I tried to click on one to report it but nothing worked. Is there a way a user of stack exchange can get rid of such ads? Some seem to appear for a few days and then go away, others then appear a few days later. Thanks for any information. Ad...