Rk/fixes

wiki/acknowledgements.md

@@ -19,7 +19,7 @@ permalink: /acknowledgements/
 ## Support
 
   - Supported by [Centre for Theoretical Physics, Polish Academy of
-    Sciences](http://www.cft.edu.pl/en/).
+    Sciences](http://www.cft.edu.pl).
 
   - Hosted and maitained by [Institute of Theoretical and Applied
     Informatics, Polish Academy of Sciences](http://www.iitis.pl/en/).

wiki/index.md

@@ -39,12 +39,12 @@ is forming an image of $\mathbb{R}P^1 \times \mathbb{R}P^1$.
 
 ## Page created by
 * [Łukasz Pawela](https://www.iitis.pl/en/person/lpawela)
-* [Piotr Gawron](https://pgawron.github.io)
+* [Piotr Gawron](https://piotrgawron.eu/)
 * [Jarosław Adam Miszczak](https://www.iitis.pl/en/person/jmiszczak)
 * [Zbigniew Puchała](https://www.iitis.pl/en/person/zpuchala)
 * [Karol Życzkowski](http://chaos.if.uj.edu.pl/~karol/)
 * [Paulina Lewandowska](https://www.iitis.pl/en/node/2654)
-* [Ryszard Kukulski](https://iitis.pl/en/node/2619)
+* [Ryszard Kukulski](https://scholar.google.pl/citations?user=Lict0ugAAAAJ&hl=en)
 
 
 ## Support

wiki/numerical-range.md

@@ -14,9 +14,9 @@ range* (also called *field of values* {% cite murnaghan1932field %}) as a
 subset of the complex plane:
 
 $$
-W(A)=\\{ z \in \mathbb{C}: z=\bra{\psi}
+W(A)=\{ z \in \mathbb{C}: z=\bra{\psi}
 A\ket{\psi}, \\ \ket{\psi} \in {\mathbb{C}}^d, \\
-\braket{\psi}{\psi}=1\\}.
+\braket{\psi}{\psi}=1 \}.
 $$
 
 Following a common convention we denote here

wiki/numerical-range/examples/2x2.md

@@ -9,7 +9,7 @@ permalink: /numerical-range/examples/2x2/
 
   - For a Jordan matrix $J=\begin{pmatrix}0&1\\\0&0\end{pmatrix}$ the
     numerical range forms a circular disk of radius one half, $W(J) =
-    D(0,)$.
+    D(0, 0.5)$.
 
 | Example of a circular numerical range.                               |
 | --- |

wiki/numerical-range/examples/4x4.md

@@ -7,8 +7,7 @@ permalink: /numerical-range/examples/4x4/
 ---
 ### Example 1
 
-A generic matrix $M=\begin{pmatrix} 1 & 1 & 1 & 1\ 0 & \ii & 1 & 1\ 0
-& 0 & -1 & 1\ 0 & 0 & 0 & \ii \end{pmatrix}$ has an oval–like numerical
+A generic matrix $M=\begin{pmatrix}1&1&1&1\\\0&\ii&1&1\\\0& 0 & -1 & 1 \\\ 0 & 0 & 0 & \ii \end{pmatrix}$ has an oval–like numerical
 range $W (M)$. 
 
 |                                                                            |
@@ -17,8 +16,8 @@ range $W (M)$.
 
 ### Example 2
 
-The matrix $M=\begin{pmatrix} 1 & 1 & 1 & 1\ 0 & \ii & 1 & 1\ 0 & 0 & -1
-& 1\ 0 & 0 & 0 & \ii \end{pmatrix}$ has a numerical range $W (M)$ with
+The matrix $M=\begin{pmatrix} 1 & 1 & 1 & 1 \\\ 0 & \ii & 1 & 1 \\\ 0 & 0 & -1
+& 1 \\\ 0 & 0 & 0 & \ii \end{pmatrix}$ has a numerical range $W (M)$ with
 one flat part of the boundary $\partial W$. 
 
 |                                                                            |
@@ -27,8 +26,8 @@ one flat part of the boundary $\partial W$.
 
 ### Example 3
 
-The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 1\ 0 & \ii & 0 & 1\ 0 & 0 & -1
-& 0\ 0 & 0 & 0 & 1 \end{pmatrix}$ has a numerical range $W (M)$ with two
+The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 1 \\\ 0 & \ii & 0 & 1 \\\ 0 & 0 & -1
+& 0 \\\ 0 & 0 & 0 & 1 \end{pmatrix}$ has a numerical range $W (M)$ with two
 flat parts of of the boundary $\partial W$. 
 
 |                                                                            |
@@ -37,8 +36,8 @@ flat parts of of the boundary $\partial W$.
 
 ### Example 4
 
-The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 1\ 0 & \ii & 1 & 0\ 0 & 0 & -1
-& 0\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ with
+The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 1 \\\ 0 & \ii & 1 & 0 \\\ 0 & 0 & -1
+& 0 \\\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ with
 two parallel flat parts of of the boundary $\partial W$. 
 
 |                                                                            |
@@ -47,8 +46,8 @@ two parallel flat parts of of the boundary $\partial W$.
 
 ### Example 5
 
-The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 1\ 0 & \ii & 0 & 0\ 0 & 0 & -1
-& 0\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ with
+The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 1 \\\ 0 & \ii & 0 & 0 \\\ 0 & 0 & -1
+& 0 \\\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ with
 three flat parts of $\partial W$ connected with corners and one
 oval–like part. 
 
@@ -58,8 +57,8 @@ oval–like part.
 
 ### Example 6
 
-The matrix $M=\begin{pmatrix} \ii & 0 & -1 & 0\ 0 & 0 & -1 & 0\ 1 & 1
-& 1-\ii & 0\ 0 & 0 & 0 & 1 \end{pmatrix}$ has a numerical range $W (M)$
+The matrix $M=\begin{pmatrix} \ii & 0 & -1 & 0 \\\ 0 & 0 & -1 & 0 \\\ 1 & 1
+& 1-\ii & 0 \\\ 0 & 0 & 0 & 1 \end{pmatrix}$ has a numerical range $W (M)$
 with three flat parts of $\partial W$ with only one corner and two
 oval–like parts. 
 
@@ -69,8 +68,8 @@ oval–like parts.
 
 ### Example 7
 
-The matrix $M=\begin{pmatrix} 1 & 0 & 1 & 0\ 0 & \ii & 0 & 1\ 0 & 0 & -1
-& 0\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ with
+The matrix $M=\begin{pmatrix} 1 & 0 & 1 & 0 \\\ 0 & \ii & 0 & 1 \\\ 0 & 0 & -1
+& 0 \\\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ with
 four flat parts of $\partial W$. 
 
 |                                                                            |
@@ -79,8 +78,8 @@ four flat parts of $\partial W$.
 
 ### Example 8
 
-The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \ii & 0 & 1\ 0 & 0 & -1
-& 0\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ pair
+The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 0 \\\ 0 & \ii & 0 & 1 \\\ 0 & 0 & -1
+& 0 \\\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ pair
 of flat parts of $\partial W$ connected with a corner connected with two
 oval–like parts. 
 
@@ -90,8 +89,8 @@ oval–like parts.
 
 ### Example 9
 
-The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 0\ 0 & \ii & 0 & 0\ 0 & 0 & -1
-& 0\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ equal
+The matrix $M=\begin{pmatrix} 1 & 0 & 0 & 0 \\\ 0 & \ii & 0 & 0 \\\ 0 & 0 & -1
+& 0 \\\ 0 & 0 & 0 & -\ii \end{pmatrix}$ has a numerical range $W (M)$ equal
 to the convex hull of eigenvalues. 
 
 |                                                                            |

wiki/numerical-range/examples/doubly-stochastic.md

@@ -12,14 +12,22 @@ permalink: /numerical-range/examples/numerical-range-of-doubly-stochastic-matric
 A doubly stochastic matrix $A \in \mathbb{R}^{n \times n }$ is a matrix
 for which the entries are non-negative while the row and column sums are
 all equal to one $\sum_{j=1}^{n} a_{ij} = 1 \text{ for } i=1,\ldots,n$
-and $\sum_{i=1}^{n} a_{ij} = 1 \text{ for } j=1,\ldots,n$
+and $\sum_{i=1}^{n} a_{ij} = 1 \text{ for } j=1,\ldots,n$.
 
 ### Theorem
 
-Let A be $4\times 4$ doubly stochastic matrix. Then, $ W(A)$consists of
-elliptical arcs and line segments if and only if$$ := (A) - 1 +  $is an
-eigenvalue of A (multiple, if $$\mu =1$$). If, in addition$ A - 1 - 3\>0
-, (A -1 - 3)^2 - (A^A) + 1 +2  + ^2 \> 0 \$\$\$\$
+Let A be $4\times 4$ doubly stochastic matrix. Then, $ W(A)$ consists of
+elliptical arcs and line segments if and only if 
+
+$$
+\mu := \tr(A) - 1 + \frac{\tr(A^3) - \tr(A^\top A^2)}{\tr(A^\top A) - \tr(A^2)}
+$$
+
+is an eigenvalue of $A$ (multiple, if $\mu =1$). If, in addition 
+
+$$
+\tr(A) - 1 - 3 \mu >0, (\tr(A) -1 - 3 \mu)^2 - \tr(A^\top A) + 1 +2 \det(A) / \mu + \mu^2 > 0,
+$$
 
 then $W(A)$ also has corner points at $\mu$ and 1, and thus four flat
 portions on the boundary. Otherwise, 1 is the only corner point of
@@ -29,8 +37,8 @@ elliptical arc.
 ### Example 1
 
 Consider the doubly stochastic matrix : $A =
-\begin{pmatrix} 0&1/3&1/4&5/12\ 1/3&0&1/2&1/6\ 1/4&9/32&1/4&1/6\ 5/12&37/96&0&19/96
-\end{pmatrix}$
+\begin{pmatrix} 0&1/3&1/4&5/12\\\ 1/3&0&1/2&1/6\\\ 1/4&9/32&1/4&1/6\\\ 5/12&37/96&0&19/96
+\end{pmatrix}$.
 
 Using above theorem, we compute $\mu = -1/3.$ By calculating the
 characteristic polynomial and computin the conditions from Theorem we
@@ -47,8 +55,8 @@ Fig. 1 we give $W(A)$, and the horizontal ellipse it contains, $W(A_1)$.
 ### Example 2
 
 Consider the doubly stochastic matrix : $A =
-\begin{pmatrix} 0&1/3&1/4&5/12\ 1/3&0&1/2&1/6\ 1/4&1/8&1/4&3/8\ 5/12&13/24&0&1/24
-\end{pmatrix}$ By coincidence, we again compute $\mu = -1/3$, and though
+\begin{pmatrix} 0&1/3&1/4&5/12\\\ 1/3&0&1/2&1/6\\\ 1/4&1/8&1/4&3/8\\\ 5/12&13/24&0&1/24
+\end{pmatrix}$. By coincidence, we again compute $\mu = -1/3$, and though
 the characteristic polynomial again has $\mu$ as a root. The formulas in
 inequalities on Theorem evaluate to $7/24$ and $59/576$ respectively, so
 the number of flat portions is still the same. However, Fig. 2 shows

wiki/numerical-range/examples/ginibre.md

@@ -11,8 +11,7 @@ The figures below depict numerical range for large random matrices drawn
 from different ensembles. Gray areas denote numerical ranges and red
 dots denote spectra of matrices.
 
-Matrices are normalized so that for every matrix $A$
-$\Tr(AA^\dagger)=\dim(A)$.
+Matrices are normalized so that for every matrix $A$, $\Tr(AA^\dagger)=\dim(A)$.
 
 In the figures below
 
@@ -29,7 +28,7 @@ largest modulus.
 Complex Ginibre matrices $G_d$ of dimention $d$ with entries $\xi_{ij}$,
 where $\mathbb{E} \|\xi _{ij}\|^2 =1/d$. As we mention in the
 introduction, by the circular law, the spectrum of $G_N$ is
-asymptotically contained in the unit disk. Note $ \\|G\_d\\| \_{}^2=d$.
+asymptotically contained in the unit disk. Note $\mathbb{E} \\|G\_d\\| \_{}^2=d$.
 
 Upper triangular random matrices $T_d$ of dimension $d$ with entries
 $T_{ij}=\xi_{ij}$ for $i \<j$ and $T_{ij}=0$ elsewhere, where
@@ -42,8 +41,8 @@ denote complex eigenvalues of $G_d$. In order to ensure the uniqueness
 of the probability distribution on diagonal matrices, we assume that it
 is invariant under conjugation by permutations. Note that integrating
 over the Girko circular law one gets the average squared eigenvalue of
-the complex Ginibre matrix, $||<sup>2=*{0}^1 2x^3 dx=1/2$. Thus,$
-\\|D\_d\\| *{}</sup>2=d/2$.
+the complex Ginibre matrix, $ \int \_{0}^{1} 2x^3 dx=1/2$. Thus, $
+\mathbb{E} \\|D\_d\\|^2=d/2$.
 
 Diagonal unitary matrices $U_d$ of order $d$ with entries $U_{kl}=\exp(i
 \phi_k) \delta_{kl}$, where $\phi_k$ are independent uniformly

wiki/numerical-range/generalizations.md

@@ -42,5 +42,7 @@ permalink: /numerical-range/generalizations/
     `B`](/numerical-range/generalizations/numerical-range-of-a-with-respect-to-b)
   - [Maximal numerical
     range](/numerical-range/generalizations/maximal-numerical-range)
+  - [Joint numerical
+    range](/numerical-range/generalizations/joint-numerical-range)
   - [Essential numerical
     range](/numerical-range/generalizations/essential-numerical-range)

wiki/numerical-range/generalizations/c-numerical-range.md

@@ -83,7 +83,7 @@ we denote all $p$-Schatten-class operators defined by
 \mathcal{B}^p(\mathcal{X}, \mathcal{Y}) \coloneqq \\{  C \in \mathcal{K}(\mathcal{X}, \mathcal{Y}) | \sum_{n=1}^\infty s_n(C) ^p < \infty \\}
 \end{equation}
 
-for $p \[1,\infty )$ whereas the Schatten-$p$-norm of matrix $A$
+for $p \in \[1,\infty )$ whereas the Schatten-$p$-norm of matrix $A$
 $$
 \| A \|_p \coloneqq \left( \sum_{n=1}^\infty s_n(A)^p \right)^{1/p},
 $$

wiki/numerical-range/generalizations/essential-numerical-range.md

@@ -24,9 +24,9 @@ the essential numerical range of $T$ by $W_e(T ) = \\{\lambda \in
 
 ### Properties
 
-For any $z \in \mathbb{C}$ and let $\sigma_e(T) = \\{ \lambda \in
+For any $z \in \mathbb{C}$ and $\sigma_e(T) = \\{ \lambda \in
 \mathrm{C} : \exists (x_n)_{n \in \mathbb{N}} \subset \mathcal{D}(T )
-\\,\\,\\, \text{with} \\,\\,\\, \|\|xn\|\| = 1, (x_n) \rightarrow 0,
+\\,\\,\\, \text{with} \\,\\,\\, \|\|x\_n\|\| = 1, (x_n) \rightarrow 0,
 \|\|(T- \lambda)x_n \|\| \rightarrow 0 \\}$ then we have
 {% cite bogli2020essential %}
 

wiki/numerical-range/generalizations/numerical-range-of-a-with-respect-to-b/examples.md

@@ -9,8 +9,8 @@ permalink: /numerical-range/generalizations/numerical-range-of-a-with-respect-to
 
 ### Example 1
 
-Diagonal matrix $A=\begin{pmatrix} 1 & 0 & 0 & 0\ 0 & 1+2\ii & 0 & 0\ 0
-& 0 & 3\ii & 0\ 0 & 0 & 0 & 0 \end{pmatrix}$ with respect to matrix $B
+Diagonal matrix $A=\begin{pmatrix} 1 & 0 & 0 & 0\\\ 0 & 1+2\ii & 0 & 0\\\ 0
+& 0 & 3\ii & 0\\\ 0 & 0 & 0 & 0 \end{pmatrix}$ with respect to matrix $B
 =\1_4$.
 
 #### Trace norm
@@ -23,15 +23,17 @@ Diagonal matrix $A=\begin{pmatrix} 1 & 0 & 0 & 0\ 0 & 1+2\ii & 0 & 0\ 0
 
 #### Infinity norm
 
-![]({{ "/assets/numerical-range/generalizations/nr_diagonal_id_inf.gif" | relative_url }}) Under this
+![]({{ "/assets/numerical-range/generalizations/nr_diagonal_id_inf.gif" | relative_url }}) 
+
+Under this
 assumption, the numerical range $w_{\|\|\cdot\|\|_\infty}(A;\1_4)=
 W(A)$.
 
 ### Example 2
 
-Matrix $A=\begin{pmatrix} 6+\ii & 0.4 & 0 & -0.1\ 0.1 & 1-3\ii & -0.3\ii
-& 0\ 0 & 0 & 0.5 & 0 \end{pmatrix}$ with respect to matrix
-$B=\begin{pmatrix} 1.2 & 0 & 0 & 0\ 0 & \ii & 0 & 0\ 0 & 0 & -1 & 0
+Matrix $A=\begin{pmatrix} 6+\ii & 0.4 & 0 & -0.1\\\ 0.1 & 1-3\ii & -0.3\ii
+& 0\\\ 0 & 0 & 0.5 & 0 \end{pmatrix}$ with respect to matrix
+$B=\begin{pmatrix} 1.2 & 0 & 0 & 0\\\ 0 & \ii & 0 & 0\\\ 0 & 0 & -1 & 0
 \end{pmatrix}$
 
 #### Trace norm

wiki/numerical-range/generalizations/numerical-range-of-a-with-respect-to-b/higher-rank-numerical-range-for-rectangular.md

@@ -17,10 +17,14 @@ and [numerical range of rectangular
 matrices]({{ "/numerical-range/generalizations/numerical-range-of-a-with-respect-to-b" | relative_url }}).
 
 Let $M,N$ be positive integers, that satisfy $M \geq N$. For every $k
-\in \\{1,\ldots,N\\}$ define the following set: $\mathcal{X}= \left\\{
+\in \\{1,\ldots,N\\}$ define the following set: 
+
+$$
+\mathcal{X} = \left\\{
 (X,Y): X \mbox{ is } N \times (N-k+1) \mbox{ isometry matrix },
 Y=\left\[ \begin{array}{c\|c} X & 0\ \hline 0 & U \end{array} \right\],U
-\in \mathcal{U}_{M-N} \right\\}.$
+\in \mathcal{U}_{M-N} \right\\}.
+$$
 
 ## Definition
 
@@ -60,20 +64,20 @@ induced matrix norm and for every unit vector $z$ and for all $(X,Y) \in
 
 ## Example 1
 
-Diagonal matrix $A=\begin{pmatrix} 1 & 0 & 0 & 0\ 0 & 1+2\ii & 0 & 0\ 0
-& 0 & 3\ii & 0\ 0 & 0 & 0 & 0 \end{pmatrix}$ with respect to matrix $B
-=\1_4$ with $\|\|\cdot\|\| = \|\|\cdot\|\|_\infty$. Then,
-$\Lambda_{k,\|\|\cdot\|\|}(A;B) = \Lambda_k(A)$ is $k$--numerical range
+Diagonal matrix $A=\begin{pmatrix} 1 & 0 & 0 & 0\\\ 0 & 1+2\ii & 0 & 0\\\ 0
+& 0 & 3\ii & 0\\\ 0 & 0 & 0 & 0 \end{pmatrix}$ with respect to matrix $B
+=\1\_{4}$ with $\|\| \\cdot \|\| = \|\| \cdot \|\|_\infty$. Then, 
+$ \Lambda\_{k,\|\|\cdot\|\|}(A;B) = \Lambda\_k(A)$ is $k$--numerical range
 of $A$. In this case, $k=2$.
 
 ![]({{ "/assets/numerical-range/generalizations/2k_nr_diagonal.png" | relative_url }})
 
 ## Example 2
 
-Matrix $A=\begin{pmatrix} 6+\ii & 0.4 & 0 & -0.1\ 0.1 & 1-3\ii & -0.3\ii
-& 0\ 0 & 0 & 0.5 & 0 \end{pmatrix}$ with respect to matrix
-$B=\begin{pmatrix} 1.2 & 0 & 0 & 0\ 0 & \ii & 0 & 0\ 0 & 0 & -1 & 0
-\end{pmatrix}$ with $|||| = ||||\_$and$k=1$. Then,
+Matrix $A=\begin{pmatrix} 6+\ii & 0.4 & 0 & -0.1\\\ 0.1 & 1-3\ii & -0.3\ii
+& 0\\\ 0 & 0 & 0.5 & 0 \end{pmatrix}$ with respect to matrix
+$B=\begin{pmatrix} 1.2 & 0 & 0 & 0\\\ 0 & \ii & 0 & 0\\\ 0 & 0 & -1 & 0
+\end{pmatrix}$ with $\|\| \cdot \|\| = \|\| \cdot \|\|\_2$ and $k=1$. Then,
 $\Lambda_{1,\|\|\cdot\|\|}(A;B) = w_{\\\| \cdot \\\|}(A; B)$.
 ![]({{ "/assets/numerical-range/generalizations/1k_nr_rectangular.png" | relative_url }})
 

wiki/numerical-range/generalizations/p-k-numerical-range.md

@@ -78,7 +78,7 @@ when:
 ## Theorem for Hermitian matrices
 
 Let $A$ will be Hermitian matrix of dimenasion $n$ and let $pk \le n$. The
-set $\Lambda_{p,k}(A)  \emptyset $ if and only if $\lambda_{jk}(A) \geq
+set $\Lambda_{p,k}(A)  \neq \emptyset $ if and only if $\lambda_{jk}(A) \geq
 \lambda_{n-(p-j+1)k+1}(A) \quad \text{for } j=1,\ldots,p.$
 
 Furthermore, for a given matrix $B \in M_n$ we can obtain

wiki/numerical-range/generalizations/p-th-matricial-range.md

@@ -35,7 +35,7 @@ properties of $W(p:A)$.
 Let $A \in M_n$ will be any matrix of dimension $n$, then in general
 case the set $W(p:A)$ is non-convex {% cite thompson1987research %}. The
 following theorems (see {% cite li1991k %}) give us the conditions to
-matrix $A$ and its eigenvalues so as to the set $W(p:A) will be convex.
+matrix $A$ and its eigenvalues so as to the set $W(p:A)$ will be convex.
 
 ### Theorem
 

wiki/numerical-range/generalizations/q-numerical-range.md

@@ -28,7 +28,7 @@ Properties of $W_q(A)$ of a matrix $A$ of dimension $d \times d$
 ## Example
 
 The numerical aproximation of $W_{13/14}(A)$, where $A =
-\begin{pmatrix} 0&1&1/2\ 0&0&1\ 0&0&0 \end{pmatrix}.$
+\begin{pmatrix} 0&1&1/2\\\ 0&0&1\\\ 0&0&0 \end{pmatrix}.$
 
 ![]({{ "/assets/numerical-range/qnr1.png" | relative_url }})
 

wiki/numerical-range/generalizations/restricted-numerical-range.md

@@ -9,7 +9,7 @@ permalink: /numerical-range/generalizations/restricted-numerical-range/
 
 ## Definition
 
-Restricted numerical range of an $d \times d$ matrix $A$ is defined as:
+Restricted numerical range of an $d \times d$ matrix $A$ is defined as
 $W_\mathrm{R}(A) = \left\\{ \mathrm{Tr}A\rho, \\; \rho \in
 \Omega_\mathrm{R} \subset \Omega \right\\}$. The symbol
 $\Omega_\mathrm{R}$ denotes any arbitrary subset of set $\Omega$ of all