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I am a professor of Mathematics at the University of Pisa, Italy.
22h

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Solution to a system of linear equations containing some inequalities
There is, of course, a lot of software, but this is simple enough to be done by hand. Assuming the first two equations being independent, solve them parametrically and plug $x_j=b_j t+c_j$ into the last two inequalities, obtaining $t$ in a solution interval (possibly empty). 
1d

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Norm of swapped power series in the unit disk
If you have an interesting motivation, this is maybe worth another question. 
1d

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Norm of swapped power series in the unit disk
E.g. $f$ with all $a_k$ real and positive are OK, as they reach the norm at $z=1$. 
1d

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Norm of swapped power series in the unit disk
A question: what can be said about a function $f\in H^\infty(D)$ whose norm is invariant by exchange $a_1\leftrightarrow a_k$ for any $k\ge 1$? 
1d

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Norm of swapped power series in the unit disk
That's what I was also thinking; though passing to the limit from permutations with compact support, to all permutations, seems delicate to me, as I only see local uniform convergence. 
1d

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Continuity in banach space for nonlinear maps
I think it is safer to have the family of balls locally finite too, and not only disjoint. Or also, to have the support of $f_n$ into the ball of radius $\epsilon/2$. Otherwise the resulting glueing function may fail to be continuous. For instance, in $\ell_\infty$ the unit balls centered at $(1+1/n)e_n$ are disjoint and well separated from each other, but any nbd of $0$ meets almost all of them, which allows the glued $f$ to be possibly discontinuous. 
2d

reviewed  Leave Open independent subset problems 
2d

reviewed  Leave Open Von Dyck Theorem 
2d

reviewed  Close About diagonal entries of the graph Laplacian 
2d

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Constructivity of zeros demanded by topological degree
In fact I 'm not sure I understand how this algorithm works. Case (A) may occur for a degree $1$ map, right? 
Nov 20 
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A question about the duality principle
In other words, an $n\times m$ matrix $K$ and its transpose have the same operator norms w.r.to the Euclidean norms on $\mathbb{R}^n$ and $\mathbb{R}^m$, that is $\K \_{2,2}=\K^T\_{2,2}$. A unit vector $f$ maximizes $\Kf\_2$ iff it is an eigenvector of the positive symmetric matrix $K^TK$ w.r.to its maximum eigenvalue , aka maximum singular value of $K$, squared. ( I'm not quite sure why you are then dealing with $K^T$ and not $K$, and I guess you mean $g$ to be normalized). 
Nov 15 
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Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that
Actually there is a lot of material online. Chapter 3 of these notes: perso.univrennes1.fr/michel.coste/polyens/SAG.pdf 
Nov 15 
answered  Can any bounded area defined by polynomial inequality in $\mathbb{R}^n$ be partitioned into simply connected finite area such that 
Nov 15 
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Orthogonal projection
OK, by invertible operator I mean a linear homeomorphism $ A:H\to H$ , thus more than just $0\in\rho(A)$ . So yes, the answer was somehow misleading. 
Nov 15 
revised 
Orthogonal projection
added 363 characters in body 
Nov 14 
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Orthogonal projection
But I understand $0\in\rho(G_0)$ as: $G_0$ being an invertible operator; therefore: $0\notin\sigma(G_0)$ and $\sigma(G_0)$ bounded... Why we don't need the latter? 
Nov 14 
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Orthogonal projection
Yes, but I preferred to write it independently from the assumption "compact resolvent", to make a precise statement. Yes, "compact resolvent" implies 0 is automatically isolated, and $\sigma(G)$ unbounded. 
Nov 14 
revised 
Orthogonal projection
added 138 characters in body 
Nov 14 
answered  Orthogonal projection 
Nov 14 
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Does this condition imply a polynomial is a product of linear factors
Also, what if $H(\lambda):=\prod_{j=1}^\lambda (j^2+1)$, for $\lambda\in\mathbb{Z}_+$ ? 