TITLE: Find the Number of Zeros of $14z^{100}-5e^z$ in the Unit Disc. What are their Multiplicities?
QUESTION [2 upvotes]: This is an old qual problem. I consider the function defined by $f(z)=14z^{100}-5e^z$ and apply Rouche's Theorem. Let $g(z)=14z^{100}$. Then for $z$ on the boundary of the unit disc, $\vert f(z)-g(z)\vert=\vert 5e^z\vert<14=\vert g(z)\vert$. Since $g$ has 100 zeros in the unit disc, counting multiplicities, $f$ also has 100 solutions in the disc. However, just because $g$ has a zero of order 100 doesn't mean $f$ does (WolframAlpha gives distinct solutions http://www.wolframalpha.com/input/?i=14z%5E100+%3D+5e%5Ez). How do I determine the multiplicities of the zeros. Wouldn't I have to find the location of each root? I am not sure if I have the correct machinery to tackle this part.

REPLY [5 votes]: If $a$ is a zero of $f$ then the multiplicity is how many derivatives of $f$ vanish also at $a$.
We have that $f'(z)=1400z^{99}-5e^z$.
Then $f(z)=f'(z)=0$ implies that $14z^{100}=1400z^{99}$. This can only happen if $z=0$ or $z=100$. From these the only inside the unit disc is $z=0$. But $z=0$ is not a solution of $f(z)=0$.
Since in the unit disc it never happens that $f$ and $f'$ vanish simultaneously then all the zeros there must be simple.


TITLE: Time until absorption markov chain with weights
QUESTION [1 upvotes]: I have a markov chain with one absorbing state. In each state i, ti seconds must elapse before moving to the next state. Starting at t0 what is the expected time to reach t3?
t0 = 5,
t1 = 10,
t2 = 15,
t3 = 20
My transition matrix looks like this:
\begin{bmatrix}3/4&1/4&0&0\\4/5&0&1/5&0\\6/7&0&0&1/7\\0&0&0&1\end{bmatrix}
I know that I need to solve this as a system of linear equations namely each equation dictated by:
$$h_i = \sum_{k=0}^Np_{ik}h_k+1$$
But I'm not sure how the "+ 1" or the time/weight affects it. Any help is appreciated.

REPLY [1 votes]: The +1 in the equation for state $i$ would be replaced by $t_i$, in which case this system of equations is exactly what you would get using first transition analysis. Intuitively, you can think of it as "expected hitting time from state i is equal to time spent in the state i + expected hitting time from the state you visit from state i (which is given by the summation term)"


TITLE: Determine if the graph is Hamiltonian using Dirac's theorem
QUESTION [0 upvotes]: I know that Dirac's theorem states “If $G$ is a graph with $n$ vertices, $n \geq 3$, each of degree at least $n/2$, then $G$ is Hamiltonian”, but how do I use this to prove that a graph with $99$ edges on $15$ vertices is Hamiltonian? Should I use a degree sum formula and compare that to the results of Dirac's theorem?

REPLY [0 votes]: Let $G=(V,E)$ be such that $|V|=15$ and $|E|=99$.

To show that $G$ is Hamiltonian it suffices to show $\deg(v)\ge 8$ for all $v\in V$.

Suppose instead that $\deg(v)\le 7$ for some $v\in V$.

Then for the graph $H=G-v$, the sum of the degree is at least
$$
2{\,\cdot\,}99-2{\,\cdot\,7}=184
$$
contradiction, since $H$ has $14$ vertices, so the sum of the degrees is at most
$14{\,\cdot\,}13=182$.


TITLE: Parametrization of a matrix drawn randomly from $SU(n)$ (using Haar measure)
QUESTION [1 upvotes]: I have been trying to find a (simple) parametrization of a random Unitary matrix, drawn from $SU(n)$, in terms of random variables.
A trivial example would be a matrix drawn from $U(1)$,
$$M = [e^{i\theta}]$$
where $\theta$ is a random variable uniformly drawn from $[0, 2\pi)$.
Any reference would be appreciated.

REPLY [1 votes]: $\mathrm{SU}(2)$ is simple:
$$U=\begin{bmatrix}
\cos\theta&e^{i\psi}\sin\theta\\
-e^{-i\psi}\sin\theta&\cos\theta
\end{bmatrix},\quad \theta\in(0,\pi/2), \psi\in(0,2\pi),$$
with Haar measure
$$\mu(d U)\propto\sin\theta\cos\theta d\theta d\psi.$$
For general $\mathrm{SU}(n)$ see this paper. The idea is to generalise the Euler parametrisation using the previous $2\times 2$ matrix as building blocks.


TITLE: Examples of calculating perverse sheaves on algebraic varieties with easy stratification.
QUESTION [9 upvotes]: This question is also asked in mathoverflow https://mathoverflow.net/questions/232589/examples-of-calculating-perverse-sheaves-on-algebraic-varieties-with-easy-strati
I have been learning intersection homology and perverse sheaves in the following way. I started by reading the first $7$ chapters of Kirwan and Woolf's book
http://www.amazon.com/Introduction-Intersection-Homology-Theory-Edition/dp/1584881844.
Then, I read chapter $8$ of the book http://www.math.columbia.edu/~scautis/dmodules/hottaetal.pdf which introduced the theory of perverse sheaves using the language of $t-$ structure. After reading the abstract construction of the category of perverse sheaves for an algebraic variety or analytic space, I hope to see examples of calculating the perverse sheaves for spaces with easy stratification, such as $\mathbb{CP}^n$. Could anyone please provide some interesting examples? Thanks!

REPLY [1 votes]: I remember reading this example somewhere but I cannot find the source now:
Consider $\mathbb{P}^1_{\mathbb{C}}$ the projective line and the stratification $\mathbb{A}^1, \{\infty\}$. Now what are the irreducible perverse sheaves here?
Well to get an irreducible perverse sheaf you must choose one of your strata and then a local system there. In this case, local systems both on the point and on the affine line are trivial (since the fundamental group is trivial), so you only have the two irreducible perverse sheaves $IC_{\mathbb{A}^1}, \delta_{\{\infty\}}$ where the latter is the scyscraper sheaf.
Now if we were in a category like the representations of finite groups, where objects decompose in irreducibles, you would have basically described the whole category. But there are indecomposable non-irreducible perverse sheaves, so a total description should also find them:
For starters, the obvious thing to do now is to play around with pushfowards. Indeed, consider the distinguished triangle, for $i: \{\infty \}\rightarrow \mathbb{P}^1$ and $j:\mathbb{A}^1\rightarrow  \mathbb{P}^1$ the obvious inclusions, and apply it to $j_* IC_{\mathbb{A}^1}.$
$$ 0\rightarrow j_{!}j^* j_* IC_{\mathbb{A}^1}\rightarrow j_* IC_{\mathbb{A}^1}\rightarrow i_*i^*j_* IC_{\mathbb{A}^1}\rightarrow 0$$
Notice that for the last term, one may naively think that $i^*j_* IC_{\mathbb{A}^1}=0$. This is not true if you remember more carefully the definition of the inverse image: You have to look at the neighborhoods around the image, which will be nonempty for every neighborhood, and indeed $i^*j_* IC_{\mathbb{A}^1}$ is the trivial sheaf. Then the pushfoward will give you $\delta_{\infty}.$
For the first term,$j^*j_*$ basically acts trivially, and then notice that $j_!$ will give you exactly the middle extension, so the first term is $IC_{\mathbb{P}^1}.$
So we get
$$ 0\rightarrow IC_{\mathbb{P}^1}\rightarrow j_* IC_{\mathbb{A}^1}\rightarrow \delta_{\infty}\rightarrow 0.$$
The middle sheaf is indecomposable (why?) so you found one of your indecomposable sheaves.
In a similar fashion, you can show $j_! IC_{\mathbb{A}^1}$ is also an indecomposable sheaf, and it satisfies the dual triangle.
Now notice that perverse sheaves have enough projectives. What is the projective cover of $\delta_{\infty}?$ If you try to identify it with a direct sum of the ones we have already found you will see that it is impossible. Indeed, this projective cover, let it be $P_{\infty},$ is also indecomposable, and these are all your indecomposable sheaves.
By the way, you may want to try playing around with $P_{\infty},$ it is a nice exercise.
Also it is nice to work out the Riemann-Hilbert correspondance in this case.


TITLE: Checking whether a point is on an elliptic curve
QUESTION [2 upvotes]: I have the point $(2,1)$ and I would like to check whether it is on my elliptic curve.
The elliptic curve is defined by $y^2 = x^3 + 3x + 1$ in $GF(7)$.
My solution to check whether or not the point is on the curve was to substitute the point into the equation:
$$ (1)^2 = (2)^3 + 3(1) + 1. $$
From inspection one may observe the left side does not equal the right side, therefore, the point is not on the curve.
However, I suspect this is incorrect I was just looking for some guidance. Thank you so much.

REPLY [2 votes]: You have a small error there. Given the curve equation in $\operatorname{GF}(7)$ check the point $P(2,1)$
$$y^2 = x^3 + 3x + 1$$
put $x= 2, y=1$
\begin{align}
(1)^2 &= (2)^3 + 3(\mathbf{\color{blue}{2}}) + 1 & & \pmod{7}\\
1 &= 8 + 6 + 1 & &\pmod{7}\\
1 &= 15 = 2\cdot7+1 & &\pmod{7}\\
1 &= 1 & &\pmod{7}\\
\end{align}
Therefore, the point is on the curve.
Your mistake was $(1)^2 = (2)^3 + 3(\mathbf{\color{red}{1}}) + 1 \pmod{7}$


TITLE: Area of a right trapezoid with bases $15$ and $8$, and height $9$
QUESTION [0 upvotes]: I can't figure out how they got that.
I got 1080, they got 103.5.
How???

REPLY [0 votes]: Hint : area of Trapezoid  is calculated by
$$ \frac{base_a + base_b}{2} \cdot  height   $$

REPLY [0 votes]: As has been explained in comments, your formula doesn't work. Instead, split it up into two shapes. You have two shapes, a $7\times 9$ right-angled triangle, and an $8\times 9$ rectangle. Calculate their areas and the area of the trapezium is their sum.

REPLY [0 votes]: Hint
Divide it into these polygons and determinate the areas separately. You might need to use the Pythagorean theorem.

If not, you might want to use the formula for the area of trapezoides:
$$\text{Area}=\frac{\text{base1}+\text{base2}}{2}·\text{height}$$


TITLE: What's $\sum_{k=0}^n\binom{n}{2k}$?
QUESTION [5 upvotes]: How do you calculate $\displaystyle \sum_{k=0}^n\binom{n}{2k}$? And doesn't the sum terminate when  2k exceeds n, so the upper bound should be less than n?
EDIT: I don't understand the negging. Have I violated a rule of conduct or something?

REPLY [3 votes]: $\newcommand{\angles}[1]{\left\langle\, #1 \,\right\rangle}
 \newcommand{\braces}[1]{\left\lbrace\, #1 \,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\, #1 \,\right\rbrack}
 \newcommand{\ceil}[1]{\,\left\lceil\, #1 \,\right\rceil\,}
 \newcommand{\dd}{{\rm d}}
 \newcommand{\ds}[1]{\displaystyle{#1}}
 \newcommand{\expo}[1]{\,{\rm e}^{#1}\,}
 \newcommand{\fermi}{\,{\rm f}}
 \newcommand{\floor}[1]{\,\left\lfloor #1 \right\rfloor\,}
 \newcommand{\half}{{1 \over 2}}
 \newcommand{\ic}{{\rm i}}
 \newcommand{\iff}{\Longleftrightarrow}
 \newcommand{\imp}{\Longrightarrow}
 \newcommand{\pars}[1]{\left(\, #1 \,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\pp}{{\cal P}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\vphantom{\large A}\,#2\,}\,}
 \newcommand{\sech}{\,{\rm sech}}
 \newcommand{\sgn}{\,{\rm sgn}}
 \newcommand{\totald}[3][]{\frac{{\rm d}^{#1} #2}{{\rm d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\, #1 \,\right\vert}$
$\ds{\sum_{k = 0}^{n}{n \choose 2k}:\ {\large ?}}$

$$
\mbox{Note that}\quad\sum_{k = 0}^{n}{n \choose 2k}
=\sum_{k = 0}^{\color{#c00000}{\large\infty}}{n \choose 2k}.
$$

$$
\mbox{We'll use the identity}\quad
{m \choose s}=\oint_{0\ <\ \verts{z}\ =\ a}
{\pars{1 + z}^{m} \over z^{s + 1}}\,{\dd z \over 2\pi\ic}\,,\quad s \in {\mathbb N}
$$

Then,
  \begin{align}
&\color{#66f}{\large\sum_{k = 0}^{n}{n \choose 2k}}
=\sum_{k = 0}^{\infty}\oint_{\verts{z}\ =\ a\ >\ 1}
{\pars{1 + z}^{n} \over z^{2k + 1}}\,{\dd z \over 2\pi\ic}
=\oint_{\verts{z}\ =\ a\ >\ 1}{\pars{1 + z}^{n} \over z}
\sum_{k = 0}^{\infty}\pars{1 \over z^{2}}^{k}\,{\dd z \over 2\pi\ic}
\\[3mm]&=\oint_{\verts{z}\ =\ a\ >\ 1}{\pars{1 + z}^{n} \over z}
{1 \over 1 - 1/z^{2}}\,{\dd z \over 2\pi\ic}
=\oint_{\verts{z}\ =\ a\ >\ 1}{\pars{1 + z}^{n}\,z \over \pars{z - 1}\pars{z + 1}}
\,{\dd z \over 2\pi\ic}
\\[3mm]&={\pars{1 + 1}^{n}\times 1 \over 1 + 1}=\color{#66f}{\large 2^{n - 1}}
\end{align}


TITLE: How to prove that the maximal solution of $x'(t) = e^{-g(x(t))^2}$ is global?
QUESTION [1 upvotes]: Let $g : \mathbb{R} \rightarrow \mathbb{R}$ be a function of class $C^{\infty}$. We consider the Cauchy's problem :
\begin{cases}
x'(t) = e^{-g(x(t))^2} \\
x(0)=0 &
\end{cases}
I have to show that the maximal solution $u$ of this system is defined on $\mathbb{R}$. But I don't know how to study this problem, because we have not enough informations on $g$... Someone could help me (and sorry for my bad english) ?

REPLY [3 votes]: Let
$$
G(x) := \int_0^x e^{g(s)^2}\, ds,
\qquad x\in\mathbb{R}.
$$
The function $G$ is of class $C^1(\mathbb{R})$ and $G'(x) = e^{g(x)^2} \geq 1$ for every $x$, so that $G$ is strictly increasing and $G(\mathbb{R}) = \mathbb{R}$.
In particular, its inverse function $G^{-1}$ is defined on all $\mathbb{R}$.
The solution of your Cauchy problem is then given by
$$
x(t) = G^{-1}(t)
$$
and it is defined for every $t\in\mathbb{R}$.
Quick answer: the r.h.s. is a $C^1(\mathbb{R})$ bounded function, hence solutions to Cauchy problems are global (i.e. defined on all $\mathbb{R}$).


TITLE: Describe $R=\mathbb{Z}[X]/(X^2-3,2X+4)$
QUESTION [5 upvotes]: I need to describe a ring $R=\mathbb{Z}[X]/(X^2-3,2X+4)$

I know that its element would be of the form $\{f(x)+(X^2-3,2X+4)|f(x)\in \mathbb{Z}[X]\}$
After that will i divide $f(x)$ by $X^2-3$ first and then by $2X+4$ to reduce the elements  in $R$?

REPLY [1 votes]: I have another answer.
$\mathbb Z[x]/(x^2-3,2x+4) = (\mathbb Z[x]/(x^2-3))/(x^2-3,2x+4)/(x^2-3)$(by third isomorphism theorem)
now define a map $\mathbb Z[x]$ to $\mathbb Z[\sqrt{3}], $
$F(f(x))=f(\sqrt{3})$( evaluated at root $3)$
then $F$ is surjective homomorphism.By 1st isomorphism theorem
$\mathbb Z[x]/(x^2-3)=\mathbb Z[\sqrt{3}]$
and $(x^2-3,2x+4)/(x^2-3)=(2x+4+(x^2-3))$ and when it undergoes $F$,resultant ideal will be $(2\sqrt{3}+4)$
so finally description will be $\mathbb Z[\sqrt{3}]/(2\sqrt{3}+4)$


TITLE: Is this Quadrilateral possible?
QUESTION [1 upvotes]: Is this figure possible to achieve?
$AD > BC$; $\angle BCD$ = $\angle BAD$ = $90^{\circ}$ .
And this is supposed to be a closed quadrilateral.
I'm confused as to whether this is viable - since $AD > BC$, and the angles are $90^{\circ}$, how would this figure close?

REPLY [2 votes]: The figure seems perfectly alright.
The confusion in your mind may be because you are considering the angle at D to be $90^{\circ}$. Imagine it to be an acute angle and then it would be fine.
Note that one of the hypotenuse BD is $\sqrt{34}$ by pythagorus theorem. And hence the remaining side is of length $4$.
Try to draw the figure with $D$ as an acute angle and your confusion would be clear.


TITLE: $f:G\rightarrow G'$ is a non trivial homomorphism, show that if $ord(G)$ is a prime then $f$ is onto.
QUESTION [0 upvotes]: Let $f:G\rightarrow G'$ be a non trivial homomorphism,
Prove that if $|G'|$ is prime, then $f$ is onto.
My Attempt:
As $f$ is homomorphism, $f(gg')=f(g)f(g')$ for all $g,g'\in G$.  If $|G'|$ is prime then what?

REPLY [4 votes]: Hint: The image of $f$ is a subgroup of $G'$. What size can a subgorup of $G'$ have?


TITLE: A college consists of $4$ professors and $8$ students. In how many ways can a party of four to be chosen so as to include at least one professor?
QUESTION [0 upvotes]: The Solution I have tried:
$$^8C_3 \cdot ^4C_1= 224$$
But answer is wrong.
The correct answers mentioned in mcqs are: $325,~425,~525,~625$

REPLY [4 votes]: The groups of four that can be formed are
$$
\binom{12}{4}
$$
but this number also includes groups with no professors. How many of these can be formed? Exactly
$$
\binom{8}{4}
$$
So the answer is
$$
\binom{12}{4}-\binom{8}{4}=425
$$


TITLE: Does convergence in $C(\Bbb R)$ imply convergence in $\mathcal D'(\Bbb R)$
QUESTION [0 upvotes]: Sequence of continuous functions $(f_k)$ which converge to $f$ in $C(\Bbb R)$. Does it imply that $f_k$ will converge to $f$ in $\mathcal D'(\Bbb R)$ as well? Or we need something more to make this assertion?
[Here $\mathcal D'(\Bbb R)$ is a linear continuous functional on $\mathcal D(\Bbb R)$ ; collection of all linear maps: from $\mathcal D(\Bbb R)$ to $\Bbb R$)
and we say $f\in$ $\mathcal D'(\Bbb R)$, is continuous in the sense that
for all $\phi_n\to \phi$ in $\mathcal D(\Bbb R)$ => $(f,\phi_n)\to(f,\phi)$ in $\mathcal D'(\Bbb R)$]

REPLY [2 votes]: It seems that the type of convergence considered is: $f_n\to f$ in $C(\mathbb{R})$ if for every fixed compact $K\subset \mathbb{R}$ we have that $f_n\to f$ uniformly in $K$. By using this definition we have that $f_n\to f$ as a distribution, i.e. if we define $T_n(\phi)=\int_\mathbb{R} f_n\phi$ and $T(\phi)=\int_\mathbb{R}f\phi$, $\phi\in \mathcal{D}(\mathbb{R})$ then $T_n\to T$ in $\mathcal{D}'(\mathbb{R})$
To prove the claim in the last paragraph, let $K=\overline{\operatorname{support}(\phi)}$, where by definition $K$ is a compact set. Hence, $f_n\to f$ uniformly in $K$, which implies that
$T_n(\phi)=\int_\mathbb{R}f_n\phi=\int_K f_n\phi\to \int_Kf\phi=\int_\mathbb{R} f\phi=T(\phi)$ for every $\phi\in \mathcal{D}(\mathbb{R})$


TITLE: Angle properties Ratio of Triangles
QUESTION [0 upvotes]: Given that $PX$ = 8.1 cm, $QX$ = 6.3 cm, $SX$ = 4.5 cm, $RX$ = 3.5 cm
Find ratio area of triangle $PQX$ : area of triangle $PSX$
What I've tried so far:
$PQX$:$SRX$ = 81:25
$QRX$:$PSX$ = 49:81
Took LCM
and got 3969:2025 = 49:25
The Answer Sheet states
7:5

REPLY [2 votes]: Well, in this case since for the both the triangle base is on the same line and point P is common so height h will be common
$ \frac {ar(PQX)}{ar(SRX)} = \frac {\frac {1}{2}h.QX} {\frac {1}{2}h.SX}$
So ratio of areas will simply be ratio of bases i.e. $\frac {QX}{SX}$=$\frac{6.3}{4.5}$=$\frac{7}{5}$


TITLE: Is $\Pr(X<Y|X<a)\geq \Pr(X<Y)$?
QUESTION [2 upvotes]: Given that $X$ and $Y$ are non-negative random variables, and $a$ is a non-negative constant, is $\Pr(X<Y|X<a)\geq \Pr(X<Y)$? I mean, is that $X<a$ gives useful informaiton on the guess of $X<Y$?
Thanks a lot in advance.

REPLY [4 votes]: Not necessarily. It holds of course if $X$ and $Y$ are independent.
But if they are not independent, anything can happen. Suppose for example that $Y=X^2$ holds and $a=1$ ...

In the independent case, note that
$$ \mathrm P(X<Y \mid X<a) ~\ge~ \mathrm P(Y\ge a\mid X<a) ~=~ \mathrm P(Y\ge a\mid X\ge a) ~\ge~ \mathrm P(X<Y \mid X\ge a) $$
where the middle equals sign is because $X$ and $Y$ are independent.
Since $\mathrm P(X<Y)$ is a weighted average of the two extremes of this inequality, it cannot be larger than the left-hand side.


TITLE: Definition of CW complex: weak topology vs quotient
QUESTION [4 upvotes]: In the definition of a CW complex we construct a topological space by gluing  $n$-cells to the $n-1$-skeleton, and then we give to it another topology (it already has one: the one given by the gluing operation which is a quotient topology) choosing a particular family of closed sets as fundamental covering (or equivalently the initial topology given by the  characteristic maps).
I imagine that the two topologies (the quotient and the weak one) are different (otherwise why call it "weak"? It would be enough to define the CW complex as a gluing of cells), but I would make good use of an example. Can anyone help me?
Thanks

REPLY [7 votes]: The two topologies are exactly the same, as you can easily prove by induction on $n$.  Indeed, supposing the two topologies are the same on the $(n-1)$-skeleton $X^{n-1}$, the quotient topology on $X^n$ has the property that a map out of it is continuous iff its restriction to $X^{n-1}$ is continuous and its composition with the characteristic map of each $n$-cell is continuous.  But by induction, the restriction to $X^{n-1}$ is continuous iff the composition with the characteristic map of each cell of dimension $<n$ is continuous.  This shows the quotient topology has the same universal property as the weak topology.
So there is no need to refer to the weak topology induced by the cells in defining a CW-complex, and some authors do not do so (for instance, Hatcher's Algebraic Topology just defines the topology as the quotient topology).  For infinite-dimensional CW-complexes, you can define the topology as the colimit topology with respect to the inclusions of the skeleta, where the skeleta are inductively given the quotient topology.
As for why the topology is called "weak", "weak topology" is sometimes used as a synonym for "colimit topology" (but beware in functional analysis it is instead used as a synonym for "limit topology"!).  In particular, for instance, the weak topology on an infinite CW-complex is typically much finer than other natural topologies that may exist.  For instance, a wedge of countably infinitely many circles can be given a topology by identifying it with the Hawaiian earring, a subset of $\mathbb{R}^2$, but the CW-complex topology is much finer than the Hawaiian earring topology.


TITLE: Existence of $f$ such that $(f \circ f)(x)=-x^3+1$
QUESTION [1 upvotes]: Is there any function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f$ has primitives on $\mathbb{R}$ and $$(f \circ f)(x)=-x^3+1$$for any $x$ real number?

I don't even know how to start solving this problem. Obviously, $f$ is bijective. if $f$ would have been differentiable one ideea would be to use $[F(f(x))]'=-x^3+1$ or something like this.

REPLY [4 votes]: Derivatives, defined over open intervals, by Darboux's theorem, are always Darboux functions. As you also pointed out, the function is bijective. Injective Darboux functions are always monotone (think about how you prove this for injective continuous functions, using the intermediate value theorem; it's the same proof). Hence, $f$ is either monotone increasing or decreasing.
But, in either case, one would expect $f \circ f$ to be increasing, and $x \mapsto 1 - x^3$ is (strictly) decreasing. This is a contradiction, hence no such $f$ can have a primitive.


TITLE: Does totally proper forcing imply countable distributivity?
QUESTION [3 upvotes]: For a suitable model $M$ for $Q$ and a condition $q \in Q$ we say that $q$ is $(M,Q)$-generic if whenever $r \leqslant q$, $D \in M$ dense, $D \subset Q$, $r$ is compatible with an element of $D \cap M$.
If $\lbrace p \in Q \cap M \colon q \leqslant p \rbrace$ is an $(M,Q)$-generic filter, then $q$ is called totally $(M,Q)$-generic.
$Q$ is totally proper if whenever $M$ is a suitable model for $Q$ and $q \in Q \cap M$, $q$ has a totally $(M,Q)$-generic extension.
A forcing notation $P$ is $\kappa$-distributive if the intersection of $\kappa$ open dense sets is open dense.

Now let $P$ be a totally proper forcing notation. Does it follow that $P$ is proper and countable distributive?
I know that the way back holds and want to know if it is equivalent.

REPLY [4 votes]: The answer is yes. Assume $Q$ is totally proper. It follows that $Q$ is proper. To see that it is countably distributive, let $\sigma$ be a name for an $\omega$-sequence of ordinals. Find a suitable $M$ with $\sigma\in M$. By total properness, we get a condition $q$ whose upward cone is an $M$-generic filter for $P$. In particular, $q$ decides every particular value of $\sigma(\check n)$. So $q$ forces that $\sigma$ is in the ground model. So the forcing cannot add any new countable sequences of ordinals, and so it is countably distributive.


TITLE: Prove that the convex hull of a set is the smallest convex set containing that set
QUESTION [7 upvotes]: How do you prove that the convex hull of A is the smallest convex set containing A?
edit: definition of a convex hull: Given a set A ⊆ ℝn the set of all convex combinations of points from A is called the convex hull of A.

REPLY [3 votes]: Young, when you wrote

How do you prove that the convex hull of A is the smallest set
  containing A?

You meant that convex hull of A is the minimal convex set containing A, right?
To show this, which part is your definition?
The linear-algebraic characterization?
You can see that any intersection of convex sets containing A is also a convex set containing A.


text:
\begin{document}
\pagenumbering{arabic}

\title{Spectral density of the non-central correlated Wishart ensembles}
\author{Vinayak}
\email{vinayaksps2003@gmail.com}
\affiliation{Instituto de Ciencias F\' isicas, Universidad Nacional Aut\' onoma de M\' exico, C.P. 62210 Cuernavaca, M\' exico}
\begin{abstract}

Wishart ensembles of random matrix theory have been useful in modeling positive definite matrices encountered in classical and quantum chaotic systems. We consider nonzero means for the entries of the constituting matrix $\mathbb{A}$ which defines the correlated Wishart matrix as $\mathbb{W}=\mathbb{AA}^{\dagger}$, and refer to the ensemble of such Wishart matrices as the non-central correlated Wishart ensemble (nc-CWE). We derive the Pastur self-consistent equation which describes the spectral density of nc-CWE at large matrix dimension.
\end{abstract}
\pacs{02.50.Sk, 05.45.Tp, 89.90.+n}

\maketitle
\renewcommand*\thesection{\Roman{section}}
\renewcommand*\thesubsection{\thesection.\Roman{subsection}}

\section{Introduction}\label{Intro}
Random matrix theory (RMT) is no longer a specialized topic but its applications in a vast domain of science and advent of naive techniques made it important  in active area of research not only in physics \cite{Mehta, Brody81, TGW:98} and mathematics \cite{Wilks,Muirhead} but also in various other scientific disciplines \cite{Potters:2005, Seba:03, Sismology, diverseAT, bio}. The Wishart model for the correlation matrices \cite{Wishart}, introduced long way back in 1928, is probably the origin of RMT. In recent research, this model has gained much attention incorporating various generalizations in trend purposefully to model positive definite matrices encountered in classical or quantum chaotic systems. For instance the Wishart model, which incorporates actual correlations \cite{marchenko, Silverstien, SenM, MousSimon, Burda:2005, Baik:2005, vp2010,guhr1, guhr2}, gives a better platform to understand the underlying correlations in quantitative finance \cite{Finance1, Finance2, Finance3, Finance4} and also for practical statistical signal processing applications, including synthetic aperture radar, extra-solar planet detection, and multi-antenna wireless communications \cite{Muller:Review}. Ensemble of such Wishart matrices are known the correlated Wishart ensemble (CWE). Similarly, some other generalizations like using fixed-trace Wishart matrices while modeling the density matrices in quantum entanglement problems \cite{Fixed-Tr}, or power-map deformation of Wishart matrices \cite{vrt:2013, VBPS:2014} in the context of short time series analysis of multivariate systems, have also been useful.

In a general sense the Wishart model may be defined as $\mathbb{W}=\mathbb{AA}^{\dagger}$ where $\mathbb{A}$ is of dimension $N\times T$. The matrix entries $A_{j\nu}$, for $1\le j\le N$ and $1\le \nu\le T$, are Gaussian variables with mean $\mu_{j\nu}$, variance $\sigma^{2}$ and with correlations, $\xi_{jk}$, between the $j$'th and $k$'th  rows of $\mathbb{A}$. In a usual set up where $\mu_{j\nu}=0$ and $\xi$ is diagonal with $1$, this model defines Wishart or Laguerre ensembles (WE) where a lot is known in terms of Laguerre polynomials for the eigenvalue statistics \cite{Laguerre, APSG}. If the off-diagonal terms of $\xi$ are not $0$, then the model defines the CWE. Using Dyson's classification of invariant ensembles \cite{Mehta, Potter:book}, the three invariant CWEs can be defined as the correlated Wishart orthogonal ensembles (CWOE), correlated Wishart unitary ensembles (CWUE) and correlated Wishart symplectic ensembles (CWSE). In this paper paper we consider rather a simple generalization for the all three invariant CWEs using $\mu_{j\nu}\ne 0$ which defines the non-central correlated Wishart ensembles (nc-CWE). Predating the Gaussian ensembles of RMT \cite{Mehta,Potter:book}, such non-central matrices were introduced in mathematical statistics to better the so-called {\it null hypothesis} supplied by CWE \cite{TWAnderson}. However, the non-central Wishart ensembles (nc-WE) have been revisited recently in the context of signal processing \cite{MIMO} and in mathematical statistics \cite{nc-WE1,nc-WE2,Forrester}. In physics, however, nc-CWEs have been used recently in the context of density matrices \cite{vmarko} remarking that the zero-mean condition is a priory not valid for the density matrices. It is also worth mentioning that the nonzero mean condition has also been studied for the Gaussian ensembles \cite{Gaussian, Forrester,ap81}.

For CWE, the spectral density is known in terms of a Pastur self-consistent equation \cite{marchenko,Silverstien, SenM, Burda:2005,vp2010} which is valid for large $N$ and $T$ with finite ratio $N/T=\kappa$. For $\xi=\mathbf{1}_{N}$, where $\mathbf{1}_{N}$ is $N\times N$ identity matrix, the Pastur equation yields the famous Mar\v cenko Pastur density for WE. For finite $N$ and $T$, CWE poses a serious difficulty and thus exact result is known only for the spectral density \cite{MousSimon, guhr1, guhr2} while the two-point spectral correlation is known only asymptotically \cite{vp2010} for large matrices. The Pastur equation, however, has never been investigated for nc-CWE and perhaps even for nc-WE. Our focus in this paper is to obtain the Pastur equation using the binary correlation method \cite{Brody81, ap81, vp2010, vin2013, vinLuis:2014} and investigate some important features like how do nonzero means and correlations affect the ensemble-averaged bulk density and the ensemble-averaged mean positions of the eigenvalues separated from the bulk. The latter has been important in RMT applications \cite{Bouchaud:2009}.

The paper is organized as follows. In the next section \ref{Prelim} we will describe the model and fix our notations. In Sec. \ref{LoopEq} we will derive the loop equation. In Sec. \ref{CWE} we re-derive the Pastur equation by solving the loop equation for the CWE case and discuss analytical results for the separated eigenvalues as derived in Ref. \cite{vp2010}. In the first part of Sec. \ref{nc-WE} we will specialize in deriving the Pastur equation for nc-WE. In the second part of the same section we will derive the ensemble-averaged mean position of the separated eigenvalues and discuss their universality with the CWE case. Finally in the third part, with Monte-Carlo simulations we will illustrate our analytical result for the bulk density. Similarly, in Sec. \ref{nc-CWE} we will generalize the method of Secs. \ref{CWE} and \ref{nc-WE} and derive the Pastur equation for nc-CWE in the first part. Next, we will discuss about the separated eigenvalues and the bulk density respectively in the second and third part. Finally, we summarize our work with discussions in Sec. \ref{conclusion}.

\section{Preliminaries}\label{Prelim}
The model we are interested in is defined as
\begin{equation}
\mathbb{W}=\mathbb{AA}^{\dagger}/T,
\end{equation}
where $\mathbb{A}$ is $N\times T$ and
\begin{equation}
\mathbb{A}=\xi^{1/2}\mathbf{A}+\mathsf{B},
\end{equation}
so that, $\overline{\mathbb{A}}=\mathsf{B}$ and $\overline{\mathbb{W}}=v_{1}^{2}\xi+\mathsf{BB}^{\dagger}/T$ where $\xi$ is the $N\times N$ positive definite fixed (nonrandom) matrix defining correlations between rows of $\mathbb{A}$ and $\mathsf{B}$ is the $N\times T$ fixed matrix which representing the ensemble averaged $\mathbb{A}$. Here we have used an {\it overbar} for the ensemble averaging. Indeed, $\mathbf{A}$ is the random matrix where the matrix entries $A_{jk}$ are real Gaussian variables with mean $0$ and variance $v_{1}^{2}$ for nc-CWOE where the Dyson index $\beta=1$. Similarly for the nc-CWUE, $\beta=2$ and we consider $A_{jk}=A^{(1)}_{jk}+\ii A^{(2)}_{jk}$ where $\mathsf{A}^{(1)}$ and $\mathsf{A}^{(2)}$ statistically equivalent but independent Gaussian matrices with mean $0$ and variance $v_{2}^{2}$. Finally, for the nc-CWSE symplectic ensembles $\beta=4$ and $\mathsf{A}$ is composed of $4$ statistically equivalent but independent Gaussian matrices with mean $0$ variance $v_{4}^{2}$ written in terms of $\mathbf{1}_{2}$ and two-dimensional matrix representative of quaternion units $\tau_{\gamma}$ where $\gamma=1,...,3$. Then $\mathbb{A}^{\dagger}$ is the transpose, Hermitian conjugate and dual of $\mathbb{A}$ respectively for $\beta=1,2,$ and $4$.The joint probability density of the matrix elements of $\mathbf{A}$ is given by the Gaussian probability measure,
\begin{equation}\label{MGaussian}
\mathcal{P}(\mathbf{A})\propto \exp\left[
-\text{Tr}\frac{\mathbf{A}\mathbf{A}^{\dagger}}{2v_{\beta}^{2}}
\right].
\end{equation}
Since variance supplies the scale for the statistics, we fix the scale as $v_{\beta}^{2}=\sigma^{2}\beta^{-1}$ \cite{MonFrench}. With out loss of generality we consider $T\ge N$.

We use the binary correlation method in order to obtain the ensemble-averaged spectral density, $\overline{\rho}_{\mathbb{W}}(\lambda)$. In this method it is convenient to deal with the Stieltjes transform or the resolvent of the density while the resolvent, $\overline{\mathbf{g}}_{\mathbb{W}}(z)$, is defined as
\begin{equation}
\overline{\mathbf{g}}_{\mathbb{W}}(z)=\overline{\langle \,\left(z\mathbf{1}_{N}-\mathbb{W}\right)^{-1}\,\rangle_{N}},
\end{equation}
where $z=\lambda\pm \ii \epsilon$ for positive infinitesimal $\epsilon$ and the angular brackets stand for the spectral averaging, e.g. $\langle \mathbf{H} \rangle_{K}= K^{-1}\, \text{tr}\, \mathbf{H}$ for $K\times K$ dimensional $\mathbf{H}$. Then $\overline{\rho}_{\mathbb{W}}(\lambda)$ can be determined via the relation
\begin{equation}\label{rhoG}
\overline{\rho}_{\mathbb{W}}(\lambda)=\lim_{\epsilon\to 0} \frac{\mp}{\pi} \Im \, \overline{\mathbf{g}}_{\mathbb{W}}(z).
\end{equation}
In order to calculate $\overline{\mathbf{g}}_{\mathbb{W}}(z)$ we use the moment expansion:
\begin{equation}\label{mom1}
\overline{\mathbf{g}}_{\mathbb{W}}(z)=\sum_{n=0}^{\infty}\frac{{\bf \overline{m}}_{n}}{z^{n+1}},
\end{equation}
where $\overline{{\bf m}}_{n}$ is the $n$'th moment of $\overline{\rho}_{\mathbb{W}}(\lambda)$ defined as
\begin{equation}\label{mom2}
\overline{{\bf m}}_{n}=\int d\lambda\, \lambda^{n}\overline{\rho}_{\mathbb{W}}(\lambda)=\overline{\langle \mathbb{W} ^{n}\rangle}_{N}.
\end{equation}

In principle, the problem is solved once we obtain a closed form of $\overline{\mathbf{g}}_{\mathbb{W}}(z)$. As in \cite{vp2010}, we could have started the moment expansion (\ref{mom1}, \ref{mom2}) to obtain $\overline{\mathbf{g}}_{\mathbb{W}}(z)$, but due to the additional term $\mathsf{B}$ the expansion results nontrivial combinations of $\mathsf{A}$ and $\mathsf{B}$. Further complications will arise in the ensemble averaging of this series with respect to the jpd (\ref{MGaussian}). We simplify the problem regarding the ensemble averaging first by using the trick of linearization \cite{Nowak-Nowak}. Following Ref. \cite{Nowak-Nowak}, we define
\begin{equation}
\mathsf{X}=\frac{1}{\sqrt{T}}\left[
\left(\begin{matrix}
\mathbf{0} & \mathsf{A}\\
\mathsf{A}^{\dagger} & \mathbf{0}
\end{matrix}\right)
+\left(
\begin{matrix}
\mathbf{0} & \mathsf{B}\\
\mathsf{B}^{\dagger} & \mathbf{0}
\end{matrix}
\right)
\right],
\end{equation}
where we replaced $\xi^{1/2}\mathbf{A}$ by $\mathsf{A}$. In the following, we use $(N+T)\times(N+T)$ matrices defined as
\begin{equation}
\tilde{\mathsf{A}}= \left(\begin{matrix}
\mathbf{0} & \mathsf{A}\\
\mathsf{A}^{\dagger} & \mathbf{0}
\end{matrix}\right), ~~~ \tilde{\mathsf{B}}=\left(
\begin{matrix}
\mathbf{0} & \mathsf{B}\\
\mathsf{B}^{\dagger} & \mathbf{0}
\end{matrix}
\right).
\end{equation}
Note that the eigenvalues of $\mathsf{X}^{2}$ coincides with those of $\mathbb{W}$ with a two-fold degeneracy for each. We define resolvent, $\overline{\mathbf{g}}_{\mathsf{X}}(u)$, for the spectral density $\overline{\rho}_{\mathsf{X}}(y)$ of $\mathsf{X}$, as
\begin{equation}\label{gx}
\overline{\mathbf{g}}_{\mathsf{X}}(u)=\overline{\langle (\mathsf{U}-\mathsf{X})^{-1} \rangle}_{N+T}, ~\text {where}~
\mathsf{U}=u\mathbf{1}_{N+T},
\end{equation}
and $u=y\pm\ii\epsilon$. In what follows, we calculate $\overline{\mathbf{g}}_{\mathbb{W}}(z)$ using $\overline{\mathbf{g}}_{\mathsf{X}}(u)$. For the first and so on, moments of $\overline{\rho}_{\mathsf{X}}$ are related with the moments of $\overline{\rho}_{\mathbb {W}}$ via
\begin{equation}\label{gwgu}
z \overline{\mathbf{g}}_{\mathbb{W}}(z)-1=\frac{N+T}{2N}\left(u(z)\overline{\mathbf{g}}_{\mathsf{X}}(u(z))-1\right),
\end{equation}
and $u^{2}=z$. Below, we will calculate the ensemble average of
\begin{equation}\label{GLX}
\mathsf{G}^{(\mathsf{X})}_{\mathsf{L}}(u)=\mathsf{L} \left(\mathsf{U-X}\right)^{-1},
\end{equation}
where
\begin{equation}
\mathsf{G}^{(\mathsf{X})}=\left(
\begin{matrix}
G_{11} & G_{12}\\
G_{21} & G_{22}
\end{matrix}
\right).
\end{equation}
$G_{jj}$ are square blocks, of dimensions $N\times N$ and $T\times T$ respectively for $j=1$ and $2$, and $G_{12}$ and $G_{21}$ are rectangular blocks of dimensions $N\times T$ and $T\times N$ respectively and $\mathsf{L}$ is an $(N+T)\times (N+T)$ arbitrary fixed matrix. $\mathsf{L}=\mathbf{1}_{N+T}$ gives $\mathsf{G}^{(\mathsf{X})}$ which on the spectral averaging yields $\overline{\langle\mathsf{G}^{(\mathsf{X})}}(u)\rangle_{N+T}=\overline{\mathbf{g}}_{\mathsf{X}}(u)$. Finally, we define the ratio
\begin{equation}
\kappa=N/T.
\end{equation}

\section{The loop equation}\label{LoopEq}
We notice that the large-$u$ expansion of $\mathsf{G}^{(\mathsf{X})}_{\mathsf{L}}(u)$ has non-trivial combinations of $\tilde{\mathsf{A}}$ and $\tilde{\mathsf{B}}$. Since $\tilde{\mathsf{B}}$ is a fixed matrix, we may use
\begin{equation}\label{KUB}
\mathsf{K}= \left(\mathsf{U}-\tilde{\mathsf{B}}\right)^{-1},
\end{equation}
and expand $\mathsf{G}^{(\mathsf{X})}_{\mathsf{L}}(u)$ the for small $\mathsf{K}$ (or equivalently for large $u$). It is worth mentioning that this trick has been used in the context of non-central Gaussian ensembles in Ref. \cite{ap81}. Then the large-$u$ expansion of Eq. (\ref{GLX}) can be written as
\begin{equation}
\mathsf{G}^{(\mathsf{X})}_{\mathsf{L}}(u)= \mathsf{LK}\sum_{n=0}^{\infty} \left(\tilde{\mathsf{A}}\mathsf{K}\right)^{n}.
\end{equation}
Since $A_{jk}$ are centered at $0$, the odd-$n$ terms of the above expansion are identically $0$ on the ensemble averaging. Thus the ensemble-averaged series reduces to
\begin{equation}\label{series}
\overline{\mathsf{G}^{(\mathsf{X})}_{\mathsf{L}}}(u)= \mathsf{LK} +\mathsf{LK}\overline{\tilde{\mathsf{A}}\mathsf{G}^{(\mathsf{X})}\tilde{\mathsf{A}}\mathsf{G}^{(\mathsf{X})}}.
\end{equation}
In order to perform the ensemble averaging for the remaining terms we use the jpd (\ref{MGaussian}) with $\mathsf{A}=\xi^{1/2}\mathbf{A}$ and derive the following exact identities, valid for arbitrary fixed $\Phi$ and $\Psi$,
\begin{eqnarray}
\label{Id-1}
\frac{1}{T} \overline{\mathsf{A}\Phi\mathsf{A}^{\dagger}\Psi}&=& \sigma^{2} \langle\Phi\rangle_{T} \xi \Psi,\\
\label{Id-2}
\frac{1}{T} \overline{\mathsf{A}^{\dagger}\Phi\mathsf{A}\Psi}&=& \sigma^{2} \langle\xi\Phi\rangle_{T} \Psi,\\
\label{Id-3}
\overline{\mathsf{A}\Phi\mathsf{A}\Psi}&=&\frac{(2-\beta)\sigma^{2}}{\beta}\Psi\tilde{\Phi},
\end{eqnarray}
where $\tilde{\Phi}= \Phi^{t}$, for $\beta=1$ where $\Phi^{t}$ is the transpose of $\Phi$, $\tilde{\Phi}= \Phi$ for $\beta=2$ and $\tilde{\Phi}= -\tau_{2}\Phi^{t}\tau_{2}$. As the identities suggest, we consider only the terms resulting form the binary associations of $\mathsf{A}$ with $\mathsf{A}^{\dagger}$ and avoid terms resulting from the binary associations of $\mathsf{A}$ with $\mathsf{A}$. With the help of these identities we calculate only the leading order terms of the series in Eq. (\ref{series}). We find
\begin{equation}\label{avSeries}
\overline{\mathsf{G}^{(\mathsf{X})}_{\mathsf{L}}}=\mathsf{LK}+\mathsf{LK}\Sigma \overline{\mathsf{G}^{(\mathsf{X})}},
\end{equation}
where the equality is valid only in the leading order and
\begin{equation}\label{Sigma}
\Sigma=\sigma^{2}\left(
\begin{matrix}
\xi\overline{\langle G_{22}\rangle}_{T} & \mathbf{0}\\
 \mathbf{0} & \kappa\overline{\langle\xi G_{11}\rangle}_{N}\mathbf{1}_{T}
\end{matrix}
\right).
\end{equation}
In the derivation of the above Eqs. (\ref{avSeries}, \ref{Sigma}) we have avoided binary associations across the traces as those also result terms of $O(N^{-1})$. Substituting now $\mathsf{L}\to \mathsf{L}(\mathbf{1}_{N}-\mathsf{K}\Sigma)^{-1}$ in Eq. (\ref{avSeries}), and then Eq. (\ref{KUB}), we finally get
\begin{equation}\label{GLSigma}
\overline{\mathsf{G}^{(\mathsf{X})}_{\mathsf{L}}}(u)=\mathsf{L}\left(\mathsf{U}-\tilde{\mathsf{B}}-\Sigma\right)^{-1}.
\end{equation}

In order to calculate the inverse of the matrix in the right-hand-side (r.h.s) of the Eq. (\ref{GLSigma}), we use the Schur decomposition. For instance, using $\mathsf{M}=\mathsf{U}-\tilde{\mathsf{B}}-\Sigma$, we may write
\begin{equation}\label{mat-inv}
\mathsf{M}^{-1}=\left(
\begin{matrix}
\mathbf{a} & \mathbf{b}\\
\mathbf{c} & \mathbf{d}
\end{matrix}
\right)^{-1}=
\left(
\begin{matrix}
\mathsf{S}^{-1} & -\mathsf{S}\mathbf{b}\mathbf{d}^{-1}\\
-\mathbf{d}^{-1}\mathbf{c}\mathsf{S}^{-1} & (\mathbf{d}-\mathbf{c}\mathbf{a}^{-1}\mathbf{b})^{-1}
\end{matrix}
\right),
\end{equation}
where $\mathsf{S}=\mathbf{a}-\mathbf{b}\mathbf{d}^{-1}\mathbf{c}$ and
\begin{eqnarray}
&&\mathbf{a}= u\mathbf{1}_{N}-\sigma^{2}\xi \overline{g}_{22}, ~~\mathbf{b}=-\frac{1}{\sqrt{T}}\mathsf{B},\nonumber\\
&&\mathbf{c}=-\frac{1}{\sqrt{T}}\mathsf{B}^{\dagger}, ~~\mathbf{d}=(u-\sigma^{2}\kappa\,\overline{g}_{11;\xi})\mathbf{1}_{T}.
\end{eqnarray}
We have used here more general spectral averaged quantities, defined as
\begin{equation}\label{Spec-Ave}
\overline{g}_{jj;\mathcal{L}}=\langle\mathcal{L} \overline{G}_{jj}\rangle_{K},
\end{equation}
with $\mathcal{L}$ as an arbitrary fixed matrix of the dimension $N\times N$ and $T\times T$ and $K$ is $N$ and $T$, respectively for $j=1$ and $2$. For example, the spectral-averaged quantity $\overline{g}_{22}$ is obtained by using $\mathcal{L}=\mathbf{1}_{T}$ in definition (\ref{Spec-Ave}) for the corresponding  upper diagonal-block matrix of the r.h.s. of Eq. (\ref{Sigma}). Similarly, for the lower diagonal-block matrix we have used $\mathcal{L}=\xi$: $\overline{g}_{11;\xi}=\langle\xi \overline{G}_{11}\rangle_{N}$.

Next, we use $\mathsf{L}=\mathbf{1}_{N+T}$ in Eq. (\ref{GLSigma}) and compute $\overline{\mathbf{g}}_{\mathsf{X}}(u)$ using Eq. (\ref{gx}). We get
\begin{equation}\label{gu-g11g22}
\overline{\mathbf{g}}_{\mathsf{X}}(u)=\langle(\overline{g}_{11}\mathbf{1}_{N}\oplus \overline{g}_{22}\mathbf{1}_{T})\rangle_{N+T},
\end{equation}
where $\oplus$ stands for the direct sum and
\begin{eqnarray}
\label{g11}
\overline{g}_{11}&=&
\llan

\frac{1}
{u\mathbf{1}_{N}-\sigma^{2}\xi\,\overline{g}_{22}-\frac{\zeta}{(u-\sigma^{2}\kappa\overline{g}_{11;\xi})}}

\rran_{N},\\
\label{g22}
\overline{g}_{22}&=&
\llan

\frac{1}
{(u-\sigma^{2}\kappa\overline{g}_{11;\xi})\mathbf{1}_{T}-\frac{1}{T}\mathsf{B}^{\dagger}\,(u-\sigma^{2}\xi\,\overline{g}_{22})^{-1}\mathsf{B}}

\rran_{T}.\nonumber\\
\end{eqnarray}
In the above equation we have introduced a positive definite matrix $\zeta=\mathsf{BB}^{\dagger}/T$. Note that $\overline{\mathbf{g}}_{\mathbb{W}}(z)$ can be obtained by calculating $\overline{\mathbf{g}}_{\mathsf{X}}(u)$ using the relation (\ref{gu-g11g22}) and then using the relation (\ref{gwgu}).

\section{Pastur Equation for CWE}\label{CWE}

For our model, $B_{jk}=0$ defines the CWE. The spectral density of CWE has been derived by several authors \cite{marchenko, Silverstien, SenM, Burda:2005, vp2010} using different techniques. As mentioned before, for large $N$ and $T$ with finite ratio $\kappa$, the spectral density is known in terms of the Pastur self-consistent equation. Below we give an alternative method to obtain the Pastur density for CWE by solving the loop equation (\ref{g11}, \ref{g22}).

We first note that in this case Eqs. (\ref{g11}, \ref{g22}) reduce to
\begin{equation}\label{g11g22-CWE}
\overline{g}_{11}(u)=\llan(u\mathbf{1}_{N}-\sigma^{2}\xi\, \overline{g}_{22})^{-1}\rran,~~\overline{g}_{22}(u)=(u-\sigma^{2}\kappa \overline{g}_{11;\xi})^{-1},
\end{equation}
where
\begin{equation}\label{g11xi-CWE}
\overline{g}_{11;\xi}(u)=\llan\xi\,(u\mathbf{1}_{N}-\sigma^{2}\xi\, \overline{g}_{22})^{-1}\rran
=\frac{u\overline{g}_{11}(u)-1}{\sigma^{2}\overline{g}_{22}(u)}.
\end{equation}
We may also write $\overline{g}_{22}$ as
\begin{equation}
u\overline{g}_{22}(u)=1+\sigma^{2}\kappa \overline{g}_{11;\xi}(u) \overline{g}_{22}(u).
\end{equation}
Using the second equality of Eq. (\ref{g11xi-CWE}) in the above equation we get
\begin{eqnarray}\label{g11g22}
\overline{g}_{22}
&=&
\frac{\kappa \,u\overline{g}_{11}+1-\kappa}{u}.
\end{eqnarray}
This is a very useful equation because not only it establishes a linear relation between $\overline{g}_{11}$ and $\overline{g}_{22}$ that we need to solve the loop equation but also when inserted in Eqs. (\ref{gwgu},\ref{gu-g11g22}) it leads to another useful identity:
\begin{equation}\label{gzgu}
z\overline{\mathbf{g}}_{\mathbb{W}}(z)=u(z) \overline{g}_{11}(u(z)).
\end{equation}
As it will shown ahead, the above two relations (\ref{g11g22}, \ref{gzgu}) are also valid for nc-WE and nc-CWE. Finally, we use these two relations in Eq. (\ref{g11g22-CWE}) with $z=u^{2}$ and obtain the Pastur density for CWE:
\begin{equation}\label{Pastur-CWE}
\overline{\mathbf{g}}_{\mathbb{W}}(z)=\llan\left[z\mathbf{1}_{N}-\sigma^{2}(1-\kappa+z\kappa\overline{\mathbf{g}}_{\mathbb{W}}(z))\xi\right]^{-1}\rran_{N}.
\end{equation}
As noted in Ref. \cite{vp2010} that this result is independent of the Dyson-index $\beta$ because of the scaling $v_{\beta}^{2}=\sigma^{2}/\beta$. The same holds true for the other Pastur equations we derive below.

For a nontrivial spectrum of $\xi$ the analytic solution is complicated. Thus, this equation has to be solved numerically. To this end an efficient algorithm is discussed in Ref. \cite{vp2010} where various cases of $\xi$ have been worked out. However, analytically we can solve the Pastur equation when it is quadratic. For instance, consider the $\xi_{jk}=\delta_{jk}$. For this choice the Pastur equation (\ref{Pastur-CWE}) yields the resolvent
\begin{equation}\label{GMP}
\overline{\mathbf{g}}_{\mathbb{W}}(z)=\frac{z-\sigma^{2}(1-\kappa)-\sqrt{(z-\sigma^{2}(1-\kappa))^{2}-4z\kappa\sigma^{2}}}{2\kappa z\sigma^{2}},
\end{equation}
where we have considered the negative sign so that $\overline{\mathbf{g}}_{\mathbb{W}}(z)$ behaves as $z^{-1}$ for large $z$. Next, the inverse transform (\ref{rhoG}) of this resolvent gives the famous Mar\v cenko Pastur density:
\begin{equation}\label{MPlaw}
\overline{\rho}_{\text{MP}}(\lambda)=\frac{\sqrt{(\lambda_{+}-\lambda)(\lambda-\lambda_{-})}}{2\pi \kappa\sigma^{2} \lambda}
\end{equation}
where $\lambda_{\pm}=\sigma^{2}(\sqrt{\kappa}\pm1)^{2}$.

It has been shown in Ref. \cite{vp2010} that Eq. (\ref{Pastur-CWE}) can also be solved for the equal-cross-correlation matrix model, viz. $\xi_{jk}=\delta_{jk}+(1-\delta_{jk})\, \mu_{0}^{2}$. Notice that in this case $\xi$ is diagonal plus a rank-$1$ matrix. Thus for $N\mu_{0}^{2}>\sqrt{\kappa}$ the spectral density we find is composed of a bulk and a separated eigenvalue provided $N\mu_{0}^{2}>\sqrt{\kappa}$:
\begin{equation}\label{den-eqcor}
\overline{\rho}_{\mathbb{W}}(\lambda)=\overline{\rho}_{0}(\lambda)+N^{-1}\delta(\lambda-\overline{\lambda}_{N}).
\end{equation}
The bulk density, $\overline{\rho}_{0}(\lambda)$, described by the Mar\v cenko Pastur law (\ref{MPlaw}) with a rescaled variance $\sigma^{2}(1-\mu_{0}^{2})$. The ensemble-averaged position of the separated eigenvalues, $\overline{\lambda}$, is given by
\begin{eqnarray}\label{Ebar-CWE}
\overline{\lambda}_{N}&=&\sigma^{2}\frac{[(N-1)\mu_{0}^{2}+1)][(N-\kappa)\mu_{0}^{2}+\kappa]}{N\mu_{0}^{2}}\nonumber\\
&\simeq& \sigma^{2}\frac{(N\mu_{0}^{2}+1)(N\mu_{0}^{2}+\kappa)}{N\mu_{0}^{2}}.
\end{eqnarray}
A simple generalization of the equal-cross-correlation matrix is a block diagonal matrix, which is again diagonal plus a finite-rank matrix. In this case the above result can be easily generalized for other separated eigenvalues. However, it has been shown in Ref. \cite{vp2010} that even for more complicated $\xi$ analytic result for the $k$'th separated eigenvalue $\overline{\lambda}_{k}$ can be written as
\begin{equation}\label{Ebark-CWE}
\overline{\lambda}_{k}=\sigma^{2}\lambda^{(\xi)}_{k}\left(
1-\kappa+\lambda^{(\xi)}_{k}\llan\,\mathbb{Q}_{k}(\lambda^{(\xi)}_{k}\,\mathbf{1}_{N}-\xi)^{-1}\rran_{N}
\right),
\end{equation}
where $\lambda^{(\xi)}_{k}$ is the $k$'th eigenvalue of $\xi$ and $\mathbb{Q}_{k}=\mathbf{1}_{N}-|k\rangle\,\langle k|$ is the projection operator to the $k$'th eigenstate $|k\rangle$ of $\xi$.

\section{Pastur equation for nc-WE}\label{nc-WE}
nc-WE is perhaps the simplest case next to WE or CWE. The nc-WEs have already been addressed in Ref. \cite{Forrester} using different methods. Since the Pastur equation has never been given explicitly, below we derive the Pastur equation for nc-WEs.

We begin with using $\xi=\mathbf{1}_{N}$ in  Eqs. (\ref{g11}, \ref{g22}) which results
\begin{eqnarray}\label{g11-NCW}
\overline{g}_{11}&=&\llan
\frac{1}
{(u-\sigma^{2}\overline{g}_{22})\mathbf{1}_{N}-\frac{\zeta}{u-\sigma^{2}\kappa\overline{g}_{11}}}
\rran_{N},
\\
\label{g22-NCW}
\overline{g}_{22}&=&
\llan
\frac{1}
{(u-\sigma^{2}\kappa\overline{g}_{11})\mathbf{1}_{T}-\frac{\eta}{u-\sigma^{2}\overline{g}_{22}}}
\rran_{T},
\end{eqnarray}
where in the second equality we have used $\mathsf{B}^{\dagger}\mathsf{B}/T=\eta$. Notice that except for the zeros, $\zeta$ and $\eta$ both have the spectrum. As mentioned above, Eqs. (\ref{g11g22}, \ref{gzgu}) also hold here. To show this we first write
\begin{eqnarray}\label{g11g22-ncWE}
\overline{g}_{22}
&=&
\frac{1}{T}\sum_{j=1}^{N}
\left[
\frac{u-\sigma^{2}\overline{g}_{22}}
{(u\mathbf{1}_{N}-\sigma^{2}\kappa\overline{g}_{11})(u-\sigma^{2}\overline{g}_{22})-\lambda^{(\zeta)}_{j}}
\right]
\nonumber\\
&+&
\frac{(1-\kappa)}{(u\mathbf{1}_{N}-\sigma^{2}\kappa\overline{g}_{11})}.
\end{eqnarray}
Next, we use Eq. (\ref{g11-NCW}) in the above equality and obtain (\ref{g11g22}) which consequently implies the relation (\ref{gzgu}). Finally, we use the relations (\ref{g11g22},\ref{gzgu}) and substitute $u^{2}=z$ to simplify the loop equation (\ref{g11-NCW}) into the self-consistent equation for $\overline{\mathbf{g}}_{\mathbb{W}}(z)$. This method yields the Pastur equation for nc-WE:
\begin{equation}\label{Pastur-NcW}
\overline{\mathbf{g}}_{\mathbb{W}}(z)=
\llan
\frac{1}{
[z-\sigma^{2}(1-\kappa+z\kappa\overline{\mathbf{g}}_{\mathbb{W}}(z))]\mathbf{1}_{N}-\frac{\zeta}{1-\sigma^{2}\kappa\,\overline{\mathbf{g}}_{\mathbb{W}}(z)}
}
\rran_{N}.
\end{equation}

If we set now $\zeta=0$, then we indeed get resolvent of the Mar\v cenko-Pastur density (\ref{GMP}). Otherwise, if we set $\sigma^{2}=0$ then it will give the resolvent corresponding the spectrum of $\zeta$. Like the Pastur equation for CWE, here as well, Eq. (\ref{Pastur-NcW}) depends on the spectrum of $\zeta$ and thus has be solved numerically when it has a non-trivial spectrum. Below we consider a rank-$1$ matrix $\mathsf{B}$ which closely related with the equal-cross correlation matrix model of the CWE. However, unlike CWE in this case the bulk density is not rescaled with variance but remains the same as for the WE (\ref{MPlaw}). Using the techniques of Refs. \cite{vp2010,ap81} we start with this simple choice to calculate the ensemble averaged position of the separated eigenvalues and generalize this result for the bulk density different from the Mar\v cenko Pastur density.

\subsection{Separation of Eigenvalues}
We begin with a simple choice for $\mathsf{B}$, viz.
\begin{equation}
B_{jk}=\mu.
\end{equation}
Then the only nonzero eigenvalue of $\zeta$, $\lambda^{(\zeta)}_{N}=N\mu^{2}$. In this case, from Eq. (\ref{Pastur-NcW}) we get
\begin{eqnarray}\label{gncw-FinRank}
&&\overline{\mathbf{g}}_{\mathbb{W}}(z)=\overline{\mathbf{g}^{(0)}}(z)\nonumber\\
&+&
\frac{N^{-1}}{
z-\sigma^{2}(1-\kappa+z\kappa\overline{\mathbf{g}}_{\mathbb{W}}(z))-\frac{\lambda^{(\zeta)}_{N}}{1-\sigma^{2}\kappa\,\overline{\mathbf{g}}_{\mathbb{W}}(z)}}.
\end{eqnarray}
Here we have used
\begin{eqnarray}\label{g0-ncw}
\overline{\mathbf{g}^{(0)}}(z)&=&
\Big\langle \mathbb{Q}_{N}

\Big[
z\mathbf{1}_{N}-\sigma^{2}(1-\kappa+z\kappa\overline{\mathbf{g}}_{\mathbb{W}}(z))\nonumber\\
&-&\frac{\zeta}{1-\sigma^{2}\kappa\,\overline{\mathbf{g}}_{\mathbb{W}}(z)}
\Big]^{-1}

\Big\rangle_{N},
\end{eqnarray}
where $\mathbb{Q}^{(\zeta)}_{k}=\mathbf{1}_{N}-|k\rangle\langle k|$ and $|k\rangle\langle k|$ is the projection operator for the eigenstate $|k\rangle$ corresponding to the eigenvalue $\lambda^{(\zeta)}_{k}$. Solving Eq. (\ref{gncw-FinRank}), while ignoring the second term, we retrieve the Mar\v cenko-Pastur result (\ref{GMP}) for the bulk density while it is understood that the bulk density is normalized to $1-1/N$. However, in the the above equation we do not drop the term containing $\zeta$ and treat this term as for a general $\zeta$. The ensemble-averaged mean position of the separated eigenvalues can be identified from the pole in the second term of Eq. (\ref{gncw-FinRank}) as
\begin{equation}\label{pole-ncw}
\overline{\lambda}_{N}=\sigma^{2}(1-\kappa+\overline{\lambda}_{N}\kappa\overline{\mathbf{g}^{(0)}}(\overline{\lambda}_{N}))+\frac{\lambda^{(\zeta)}_{N}}{1-\sigma^{2}\kappa\,\overline{\mathbf{g}^{(0)}}(\overline{\lambda}_{N})},
\end{equation}
where we have used $\overline{\mathbf{g}^{(0)}}$ instead of $\overline{\mathbf{g}}$ and ignored $O(N^{-1})$ terms. Using this in Eq. (\ref{g0-ncw}) we obtain
\begin{equation}\label{gEbar-ncw}
\overline{\mathbf{g}^{(0)}}(\overline{\lambda}_{N})=\frac{\Phi_{N}}{1+\sigma^{2}\kappa\Phi_{N}},
\end{equation}
where
\begin{equation}
\Phi_{N}=\llan \mathbb{Q}_{N}(\lambda^{(\zeta)}_{N}\mathbf{1}_{N}-\zeta)^{-1}\rran_{N}.
\end{equation}
Next, by using Eq. (\ref{gEbar-ncw}) in Eq. (\ref{pole-ncw}), we obtain $\overline{\lambda}_{N}$. Following Ref. \cite{vp2010}, we can also generalize this result for the $k$'th separated eigenvalue, $\overline{\lambda}_{k}$ as
\begin{equation}
\overline{\lambda}_{k}=(1+\sigma^{2}\kappa\Phi_{k})[\sigma^{2}(1-\kappa)+\lambda^{(\zeta)}_{k}(1+\sigma^{2}\kappa\Phi_{k})].
\end{equation}

The above result is of course different from that for the CWE (\ref{Ebark-CWE}). However, for the rank-one $\mathsf{B}$ this result gives
\begin{equation}\label{Sprt-NCW}
\overline{\lambda}_{N}=\frac{(N\mu^{2}+\sigma^{2})(N\mu^{2}+\sigma^{2}\kappa)}{N\mu^{2}},
\end{equation}
which is valid only if $N\mu^{2}>\sqrt{\kappa}\sigma^{2}$ otherwise the separated eigenvalue will be absorbed in the Mar\v cenko-Pastur-bulk. Interestingly, it also coincides with (\ref{Ebar-CWE}) for $\mu=\mu_{0}$ and $\sigma^{2}=1$. In Ref. \cite{vmarko}, this correspondence has been exploited without any analytical treatment for the nc-WE. There are the parameters chosen as $\mu=\sqrt{r/N}$ and $\sigma=(1-r)/\sqrt{N}$ in the n-WUE case and $\mu_{0}=\sqrt{r}$ and $\sigma=1/\sqrt{N}$ in the CWUE case. Indeed, for these parameters the two results (\ref{Ebar-CWE}) and (\ref{Sprt-NCW}) coincide in the leading order.

\subsection{The Bulk Density}

\begin{figure}
        \centering
               \includegraphics [width=0.5\textwidth]{Rank-N.eps}
                \caption{(Color online) Spectral density of the nc-WOE where $B_{j\nu}=\delta_{j\nu}\mu\sqrt{j}$ for $N=1024$, $T=2N$, $\sigma^{2}=1$, and $\mu=0,1,3$ and $10$. Solid lines in this figure represent the theory obtained from the numerical solution of \ref{Pastur-NcW} and open circles represent the histogram data obtained from the Monte-Carlo simulation of $\mathsf{C}$. The dashed line represent the Mar\v cenko-Pastur formula. In the inset (a) we show the density for $\mu=10$ while in (b) we show the density for $\lambda\le1$ with the same $\mu$.}
\label{Rank-N}
\end{figure}

It is important to point out that the Eq. (\ref{Pastur-NcW}) describes only the bulk density and not the density of the separated eigenvalues. Analytically it has been proved for CWUE that the density of the separated eigenvalues is described by a Gaussian distribution \cite{Baik:2005} and numerically the same is found to be valid for the nc-WUE case \cite{vmarko}. To obtain the bulk density for a non-trivial $\zeta$, Eq. (\ref{Pastur-NcW}) has to be solved numerically. We thus use the Newton's method described in Ref. \cite{vp2010} to solve Eq. (\ref{Pastur-NcW}). We consider
\begin{equation}
f(\overline{\mathbf{g}^{(n)}_{\mathbb{W}}}(z))-\overline{\mathbf{g}^{(n)}_{\mathbb{W}}}(z)=0,
\end{equation}
at given $z$ where $f(\overline{\mathbf{g}^{(n)}_{\mathbb{W}}}(z))$ is the r.h.s. of Eq. (\ref{Pastur-NcW}) for $\overline{\mathbf{g}^{(n)}_{\mathbb{W}}}(z)$, and $n$ represent the iteration-number starting from $0$ with an initial guess $\overline{\mathbf{g}^{(0)}_{\mathbb{W}}}(z)$.

To illustrate the result (\ref{Pastur-NcW}) we use $B_{j\nu}=\delta_{j\nu}\mu\sqrt{j}$, for $0\le j\le N$ and $\sigma^{2}=1$. In Fig. \ref{Rank-N} we compare our theory with the Monte-Carlo simulations for $N=1024$ dimensional matrices. In the main figure, we show results for $\mu=1$ and $3$ while for $\mu=0$ we plot only the Mar\v cenko Pastur density. As can be seen from this figure that the density tends to attain a uniform shape as $\mu$ is increased. This is closely predicted by the theory. In two insets, (a) and (b), we show result for $\mu=10$. As shown in (a) our theory gives reasonable account of the data through out the support for the density. In (b), we notice oscillations for $\lambda<1$ which is almost consistent with the theory.

\section{Pastur equation for nc-CWE}\label{nc-CWE}
Having specialized in CWE and nc-WE cases we now consider $\xi\ne\mathbf{1}_{N}$ and $\zeta\ne0$ in Eq. (\ref{g11},\ref{g22}). We first note that Eq. (\ref{g22}) can be written as
\begin{eqnarray}
\overline{g}_{22}&=&\frac{1-\kappa}{u-\sigma^{2}\kappa\overline{g}_{11;\xi}}+ \kappa
\Big
\langle
\Big[
(u-\sigma^{2}\kappa\overline{g}_{11;\xi})\mathbf{1}_{N}
\nonumber\\
&-&\zeta (u-\sigma^{2}\xi \overline{g}_{22})^{-1}
\Big
]^{-1}
\Big
\rangle_{N}.
\end{eqnarray}
Using this and Eq. (\ref{g11}) one finds the relation (\ref{g11g22}) and consequently the relation (\ref{gzgu}). Next, exploiting relations (\ref{g11g22}) and (\ref{gzgu}) with $u^{2}=z$ in Eq. (\ref{g11}) we obtain a coupled Pastur equation for nc-CWE:
\begin{equation}\label{Pastur-nc-CWE}
\mathbf{\overline{g}}_{\mathbb{W};L}(z)=
\llan L
\frac{1}
{z\mathbf{1}_{N}-\alpha_{1}(z,\mathbf{\overline{g}}_{\mathbb{W}}(z))\,\xi-\alpha_{2}(\mathbf{\overline{g}}_{\mathbb{W};\xi}(z))\zeta}
\rran_{N},
\end{equation}
where $L$ is an arbitrary $N\times N$ matrix and
\begin{eqnarray}\label{alphabeta}
\alpha_{1}(z,\mathbf{\overline{g}}_{\mathbb{W}}(z))&=&\sigma^{2}(1-\kappa+\kappa z\mathbf{\overline{g}}_{\mathbb{W}}(z)),\nonumber\\
\alpha_{2}(\mathbf{\overline{g}}_{\mathbb{W};\xi}(z))&=&[1-\sigma^{2}\kappa\mathbf{\overline{g}}_{\mathbb{W};\xi}(z)]^{-1}.
\end{eqnarray}
Choices $L=\mathbf{1}_{N}$ and $L=\xi$ yield respectively $\mathbf{\overline{g}}_{\mathbb{W}}(z)$ and $\mathbf{\overline{g}}_{\mathbb{W};\xi}(z)$ and thus complete the result. It is easy to see that results (\ref{Pastur-CWE}) and (\ref{Pastur-NcW}) are immediate from the result  (\ref{Pastur-nc-CWE}), for the choices $L=\mathbf{1}_{N}$ and $\zeta=0$, and $L=\mathbf{1}_{N}$ and $\xi=\mathbf{1}_{N}$, respectively in (\ref{Pastur-nc-CWE}).

\subsection{Separation of eigenvalues}
It is also important to note that in general Eq. (\ref{Pastur-nc-CWE}) can not be simplified to the eigenvalues of $\xi$ and $\zeta$ unless they commute with each other. Therefore, unlike the Pastur equation, it is difficult to extend the results (\ref{Ebar-CWE}, \ref{Sprt-NCW}) to the nc-CWE case.

We consider $\xi_{jk}=\delta_{jk}+(1-\delta_{jk})\mu^{2}_{0}$ and $B_{jk}=\mu\delta_{jk}$. Note that $\xi$ is a diagonal plus a rank-$1$ matrix and $\mathsf{B}$ is also a rank-$1$ matrix. In this case we can write Eq. (\ref{Pastur-nc-CWE}) as
\begin{equation}
\mathbf{\overline{g}}_{\mathbb{W}}(z)
=\overline{\mathbf{g}^{(0)}_{\mathbb{W}}}(z)+\frac{1}{N} \frac{1}{z-\overline{\lambda}_{N}}.
\end{equation}
Here, since $(N-1)$ eigenvalues of $\zeta$ are identically zero, we have
\begin{equation}
\overline{\mathbf{g}^{(0)}_{\mathbb{W}}}(z)=
\llan\mathbb{Q}_{N}
\frac{1}
{z\mathbf{1}_{N}-\alpha_{1}(z,\mathbf{\overline{g}_{W}}(z))\,\xi}
\rran_{N},
\end{equation}
where $\mathbb{Q}_{k}$ corresponds to the $k'$th eigenstates of $\xi$ and
\begin{equation}\label{eqLB}
\overline{\lambda}_{N}=
\lambda^{(\xi)}_{N}\,\alpha_{1}(\overline{\lambda}_{N},\mathbf{\overline{g}}_{\mathbb{W}}(\overline{\lambda}_{N}))+
\lambda^{(\zeta)}_{N}\,\alpha_{2}(\mathbf{\overline{g}}_{\mathbb{W};\xi}(\overline{\lambda}_{N})).
\end{equation}
Next, we write
\begin{eqnarray}\label{gWxi}
\overline{\mathbf{g}_{\mathbb{W};\xi}}(z)&=&\overline{\mathbf{g}_{\mathbb{W};\xi}^{(0)}}(z)+O(N^{-1}),\nonumber\\
\overline{\mathbf{g}_{\mathbb{W};\xi}^{(0)}}(z)
&=&
\llan\mathbb{Q}_{N}\,\xi
\frac{1}
{z\mathbf{1}_{N}-\alpha_{1}(z,\mathbf{\overline{g}}_{\mathbb{W}}(z))\xi}
\rran_{N}.
\end{eqnarray}

We notice a relation between $\alpha_{1}(z,\overline{\mathbf{g}^{(0)}_{\mathbb{W}}}(z))$ and $\alpha_{2}(z,\overline{\mathbf{g}^{(0)}_{\mathbb{W};\xi}}(z))$:
\begin{equation}\label{iden-alphabeta}
\alpha_{1}(z,\overline{\mathbf{g}^{(0)}_{\mathbb{W}}}(z))=\sigma^{2}\alpha_{2}(\overline{\mathbf{g}^{(0)}_{\mathbb{W};\xi}}(z)).
\end{equation}
To obtain the above result we write $\overline{\mathbf{g}^{(0)}_{\mathbb{W}}}(z)$ in terms of $\overline{\mathbf{g}^{(0)}_{\mathbb{W};\xi}}(z)$ and then use Eq. (\ref{alphabeta}). This relation simplifies Eq. (\ref{eqLB}) as
\begin{equation}\label{LB-beta}
\overline{\lambda}_{N}=\alpha_{2}(\overline{\mathbf{g}^{(0)}_{\mathbb{W};\xi}}(z))\, [\sigma^{2}\lambda^{(\xi)}_{N}+\lambda^{(\zeta)}_{N}].
\end{equation}
Further, using the above equation in Eq. (\ref{gWxi}), we find
\begin{equation}\label{gxi-beta}
\overline{\mathbf{g}_{\mathbb{W};\xi}^{(0)}}(\overline{\lambda}_{N})=[\alpha_{2}(\mathbf{\overline{g}}_{\mathbb{W};\xi}(\overline{\lambda}_{N}))]^{-1}\Psi_{N},
\end{equation}
where
\begin{equation}\label{Psi}
\Psi_{N}=\llan
\mathbb{Q}_{N}\, \xi
\frac{1}{\sigma^{2}(\lambda^{(\xi)}_{N}\mathbf{1}_{N}-\xi)+\lambda^{(\zeta)}_{N}\mathbf{1}_{N}}
\rran_{N}.
\end{equation}
Substituting Eq. (\ref{gxi-beta}) in the definition of $\alpha_{2}$, we find
\begin{equation}\label{beta-Psi}
\alpha_{2}(\mathbf{\overline{g}}_{\mathbb{W};\xi}(\overline{\lambda}_{N}))=1+\sigma^{2}\kappa \Psi_{N}.
\end{equation}
Finally, we use the above result in (\ref{LB-beta}) and obtain
\begin{equation}\label{Sprt-nc-CWE}
\overline{\lambda}_{N}=(1+\sigma^{2}\kappa \Psi_{N})(\sigma^{2}\lambda^{(\xi)}_{N}+\lambda^{(\zeta)}_{N}).
\end{equation}

This result can be generalized to the block-diagonal $\xi$ and $\zeta$ with dimensionally the same blocks where each block of $\xi$ is represented by equal-cross-correlation matrix while the corresponding $\zeta$ block is of the rank-$1$. For this setup one can generalize results (\ref{Psi}) and (\ref{Sprt-nc-CWE}) replacing the subscript $N$ by $k$ for the $k$'th separated eigenvalue.

Solving the above equation for equal-cross-correlation matrix $\xi$ and a rank-$1$ matrix $\zeta$ we obtain the ensemble averaged mean position for the separated eigenvalue as
\begin{equation}
\overline{\lambda}_{N}=\frac{(N\Delta^{2}+\sigma^{2})(N\Delta^{2}+\sigma^{2}\kappa)}{N\Delta^{2}},~\Delta^{2}=\mu_{0}^{2}\sigma^{2}+\mu^{2},
\end{equation}
where the above result is valid for $N\Delta^{2}>\sqrt{\kappa}$. This result is an interesting generalization of the corresponding results for the CWE and nc-WE where the bulk density is described by the Mar\v cenko Pastur density with a rescaled variance $\sigma^{2}(1-\mu_{0}^{2})$ as in Eq. (\ref{den-eqcor}).

\begin{figure}
        \centering
               \includegraphics [width=0.5\textwidth]{excr.eps}
                \caption{(Color online) The spectral density for the nc-CWOE where $\xi_{jk}=\delta_{jk}+(1-\delta_{jk})\mu_{0}^{(|j-k|)}$ and $\mathsf{B}_{j\nu}=\mu^{|j-\nu|}$ with  $\mu=0.5$, $\sigma^{2}=0.25$. In the main figure the density is shown on semi-log plot for $\mu_{0}=0.1,\,0.3$ and $0.5$ respective with {\it black, orange} and {\it blue} colors. Solids lines represent the theory (\ref{Pastur-nc-CWE}) for nc-CWE and dashes lines represent the corresponding $\mu=0$ cases. In the inset, theories for $\mu=0$ and $\mu=0.5$ are compared on log-log scale.}
\label{excr}
\end{figure}

\subsection{A non-trivial Example}
For non-trivial $\xi$ and $\zeta$ Eq. (\ref{Pastur-nc-CWE}) can be solved numerically. Thus one has to extend the algorithm for two equations of two variables, viz.
\begin{eqnarray}
f_{1}(\overline{\mathbf{g}^{(n)}_{\mathbb{W}}}(z), \overline{\mathbf{g}^{(n)}_{\mathbb{W};\xi}}(z))-\overline{\mathbf{g}^{(n)}_{\mathbb{W}}}(z)&=&0\nonumber\\
f_{2}((\overline{\mathbf{g}^{(n)}_{\mathbb{W}}}(z),\overline{\mathbf{g}^{(n)}_{\mathbb{W};\xi}}(z))-\overline{\mathbf{g}^{(n)}_{\mathbb{W};\xi}}(z)&=&0,
\end{eqnarray}
where $f_{1}(\overline{\mathbf{g}^{(n)}_{\mathbb{W}}}(z),\overline{\mathbf{g}^{(n)}_{\mathbb{W};\xi}}(z))$ and $f_{2}((\overline{\mathbf{g}^{(n)}_{\mathbb{W}}}(z),\overline{\mathbf{g}^{(n)}_{\mathbb{W};\xi}}(z))$ are the r.h.s of (\ref{Pastur-nc-CWE}) respectively with $L=\mathbf{1}_{N}$ and $L=\xi$. Next, we start with initial guesses $\overline{\mathbf{g}^{(0)}_{\mathbb{W}}}(z)$ and $\overline{\mathbf{g}^{(0)}_{\mathbb{W};\xi}}(z)$ for a given $z$ and use the Newton's method to obtain the solution in the machine precision.

To illustrate the result we solve Eq. (\ref{Pastur-nc-CWE}) for $\xi_{jk}=\delta_{jk}+(1-\delta_{jk})\mu_{0}^{(|j-k|)}$, where $\mathsf{B}_{j\nu}=\mu^{|j-\nu|}$ with $\mu=0.5$ and $\mu_{0}$ is varied as $\mu_{0}=0.1,\,0.3$ and $0.5$. Also we choose $\sigma^{2}=0.25$ and $N=512$ with $T=2N$. The result is shown in Fig. \ref{excr} where open circles represent the histogram data obtained from the Monte-Carlo simulation of $\mathsf{C}$ and solid lines are obtained from the numerical solution of the theory (\ref{Pastur-nc-CWE}) where we have considered $N=512$. As shown in the figure, the theory reasonably explains numerical results. In this figure we also compare theory for $\mu=0.5$ for the corresponding CWOE ($\mu=0$). As can be seen in this figure, the nonzero mean not only changes the density profile but also shifts non-trivially the spectrum.

\section{Summary and Discussions} \label{conclusion}
We have studied nc-CWE and obtained exact result for the spectral density at large matrix dimension. The derivation is formalized in two steps, viz. first we obtain the loop equation for $\mathsf{X}$, which eigenvalues are closely related with those of $\mathbb{W}$, and secondly we derive the Pastur equation for $\mathbb{W}$ from the loop equation. With this formalism we have derived the Pastur equation for CWE, nc-WE and for the nc-CWE. For all the three cases we have exploited a linear relation between the averaged quantities $u\overline{g}_{11}$ and $u\overline{g}_{22}$. We notice that in the first two cases the Pastur equation depends on the eigenvalues of positive definite symmetric matrices, $\xi$ and $\zeta=\mathsf{BB}^{\dagger}/T$. We have shown that in general, unlike CWE and nc-WE, the spectral density for nc-CWE does not depend simply on the spectra of $\xi$ and $\zeta$ rather more intricately on the matrices.

From the Pastur equation, we have worked out the ensemble-averaged mean position of the separated eigenvalues for the nc-WE. For CWE this has been worked out in Ref. \cite{Baik:2005, Forrester, vp2010}. Following Ref. \cite{vp2010}, we have given the result for a general $\zeta$ in the nc-WE case. In nc-CWE case the the Pastur equation is more complicated. However, we have been able to worked out the ensemble-averaged mean position of the separated eigenvalues for some especial cases of nc-CWE. As for the CWE and nc-CWE, for more general cases we have used the Newton's method to solve the Pastur equation numerically. We have supplemented our theoretical result with numerics with some non-trivial examples.

Finally, it would be interesting to extend this generalization for the Wishart model of nonsymmetric correlation matrices those dealt in Refs. \cite{vin2013, vinLuis:2014}. Another important extension of this work is related to short time series often encountered in the correlation analysis of multivariate complex systems. For short time series, $N>T$ resulting a correlation matrix which is singular with significantly many zero eigenvalues. In Ref. \cite{GuhrKaebler,GuhrShaefer} the power map method is proposed and used recently in \cite{VBPS:2014} as tool to get rid of this degeneracy. This method results an spectrum emerging from the zero eigenvalues when the exponent is very close to $1$. It has been shown in Ref. \cite{vrt:2013} that the so emerging spectrum is very sensitive to correlations and we believe that the study of the emerging spectra corresponding nc-CWE is very important.

\section{Acknowledgments}
The author is grateful to Thomas H. Seligman and Luis Benet for useful discussions. Financial support from CONACyT through Project No. 154586 and No. PAPIIT UNAM RR 113311 is acknowledged. The author was supported by DGAPA/UNAM as a postdoctoral fellow.

config: arxiv


TITLE: Combined Variation Formula Derivation
QUESTION [2 upvotes]: I'm teaching a pre calculus class and I'm running into this seemingly simple concept that I just can't derive.
So we just went over direct and inverse variation, and have arrived at an example where y is directly proportional to T AND indirectly proportional to V.  Here the book just threw it out there that the formula to use is $y = \frac{kT}{V}$ but I can't seem to understand where this formula came from.
I mean, we have that $y = k_1 T$ and $y = \frac{k_2}{V}$ but where did the $y = \frac{kT}{V}$ come from? I could only obtain such an equation by squaring $y: y^2 = k_1 T \frac{k_2}{V} = \frac{kT}{V}$ for $k = k_1 k_2$.
I feel like I'm missing something very fundamental here...... Thanks in advance!

REPLY [0 votes]: "Directly proportional to $T$" and "inversely proportional to $V$" both refer to cases where all other variables are held constant. Hence in the relation $y=k_1T$, $k_1$ contains an expression with $V$ held constant, and since $V$ is inversely proportional to $y$ we can write $k_1=\frac kV$, which yields the desired expression of $y=\frac{kT}V$. Likewise, we can write $k_2=kT$ for the other relation $y=\frac {k_2}V$ and obtain the same result.


text:
\section{Heurisitic derivation of the information theoretic phase transition}\label{sec:allOrNothing}

In this section we analyze the potential function in order to heuristically locate the information theoretic transition in the special case of the spiked Wigner model with Bernoulli prior $P_X={\rm Ber}(\rho)$. The main hypotheses behind this computation are $i)$ that the SNR $\lambda=\lambda(\rho)$ varies with $\rho$ as $\lambda=4\gamma |\ln \rho|/\rho$ with $\gamma >0$ and independent of $\rho$; that $ii)$ in this SNR regime the potential possesses only two minima $\{q^+,q^-\}$ that approach, as $\rho\to 0_+$, the boundary values $q^-=o(\rho/|\ln \rho|)$ and $q^+\to\rho$. For the Bernoulli prior the potential explicitly reads

\begin{align*}
	&i_n^{\rm pot}(q,\lambda,\rho)
	\nn
	&\equiv\frac{\lambda (q^2+\rho^2)}{4} -(1-\rho)\EE\ln\Big\{1-\rho+\rho e^{-\frac12\lambda q+\sqrt{\lambda q}Z}\Big\}-\rho\,\EE\ln\Big\{1-\rho+\rho e^{\frac12\lambda q+\sqrt{\lambda q}Z}\Big\}\,.
\end{align*}
We used that
\begin{align}
I(X;\sqrt{\gamma} X+Z)=-\EE \ln \int dP_X(x)e^{-\frac12\gamma x^2+\gamma Xx+\sqrt{\gamma}Zx}+\frac12\EE[X^2]\gamma\,.	\label{linkPsiI}
\end{align}
Let us compute this function around its assumed minima. Starting with $q^-=o(\rho/|\ln \rho|)$ (this means that this quantity goes to $0_+$ faster than $\rho/|\ln \rho|$ as $\rho$ vanishes) we obtain at leading order after a careful Taylor expansion in $\lambda q^- \to 0_+$ (the symbol $\approx$ means equality up to lower order terms as $\rho \to 0_+$)

\begin{align}
	i_n^{\rm pot}(q^-,\lambda,\rho)
	&\approx \frac{\lambda (q^-)^2}4+\frac{\lambda\rho^2}4-\frac{\rho(\lambda q^-)^2}8\approx\frac{\lambda\rho^2}4= \gamma \rho |\ln \rho|\,.	\label{solq-}
\end{align}

For the other minimum $q^+\to\rho$, because $\lambda q^+ \to +\infty$ the $Z$ contribution in the exponentials appearing in the potential can be dropped due to the precense of the square root. We obtain at leading order

\begin{align*}
	i_n^{\rm pot}(q^+,\lambda,\rho)&\approx 2\gamma\rho|\ln \rho|-\ln\{1+\rho^{1+2\gamma}\}-\rho \ln\{1+\rho^{1-2\gamma}\}\,.
\end{align*}
Here there are two cases to consider: $\gamma>1/2$ and $0<\gamma \le 1/2$. We start with $\gamma>1/2$. In this case the potential simplifies to
\begin{align*}
	i_n^{\rm pot}(q^+,\lambda,\rho)&\approx  \rho|\ln \rho|\,.
\end{align*}

Now for $0<\gamma\le 1/2$ we have

\begin{align*}
	i_n^{\rm pot}(q^+,\lambda,\rho)&\approx  2\gamma\rho|\ln \rho|\,.
\end{align*}

The information theoretic threshold $\lambda_c=\lambda_c(\rho)$ is defined as the first non-analiticy in the mutual information. In the present setting this corresponds to a discontinuity of the first derivative w.r.t. the SNR of the mutual information (and we therefore speak about a``first-order phase transition''). By the I-MMSE formula this threshold manifests itself as a discontinuity in the MMSE. In the high sparsity regime $\rho \to 0_+$ the transition is actually as sharp as it can be with a $0$--$1$ behavior. This translates, at the level of the potential, as the SNR threshold where its minimum is attained at $q^-$ just below and instead at $q^+$ just above. So we equate $\lim_{\rho\to 0_+}i_n^{\rm pot}(q^-,\lambda_c,\rho)=\lim_{\rho\to 0_+}i_n^{\rm pot}(q^+,\lambda_c,\rho)$ and solve for $\lambda_c$. This is only possible, under the constraint $\gamma >0$ independent of $\rho$, in the case $\gamma >1/2$ and gives $\gamma=1$ which is the claimed information theoretic threshold $\lambda_c(\rho)=4|\ln \rho|/\rho$. Repeating this analysis for the Bernoulli-Rademacher prior $P_X=(1-\rho)\delta_0+\frac12\rho(\delta_{-1}+\delta_1)$ leads the same threshold, which suggests that the transition is only related (for discrete priors) to the recovery of the support of the signal.

Another piece of information gained from this analysis is that around the transition the mutual information divided by $n$ is $\Theta(\rho |\ln \rho|)$. Therefore the proper normalization for the mutual information is $(n \rho|\ln \rho|)^{-1}I(\bX;\bW)$ for it to have a well defined non trivial limit in the regime $\rho\to 0_+$.

Finally for $\gamma \le 1$ the minimum of the potential is attained at $q^-$ and the rescaled mutual information $(n \rho|\ln \rho|)^{-1}I(\bX;\bW)$ equals $\gamma$ as seen from \eqref{solq-}. If instead $\gamma \ge 1$ the minimum is attained at $q^+$ and the mutual information instead saturates to $1$, so we get the asymptotic singular function $\gamma \mathbb{I}(\gamma \le 1)+\mathbb{I}(\gamma \geq 1)$.


TITLE: Finite complement topology over $\Bbb{R}$ is not second-countable under ZF?
QUESTION [1 upvotes]: Under $\mathsf{ZF+AC_\omega}$, the space $\Bbb{R}$ with finite complement topology is not second-countable, since a countable union of countable subsets of $\Bbb{R}$. However, such argument doesn't work if we does not assume any form of choice.
My question is, $\Bbb{R}$ with finite complement topology is not second-countable under ZF (Zermelo-Fraenkel set theory without choice), or it is consistently second-countable? I would appreciate your answer.

Proof of "$\Bbb{R}$ with finite complement topology is not second-countable" under choice: Let assume $\{V_n\}_{n<\omega}$ be a countable basis of the space. We know that $\Bbb{R}-\{x\}$ is open for each $x\in \Bbb{R}$ so for each $x$ there is $n<\omega$ s.t. $V_n\subset \Bbb{R}-\{x\}$ so $$\bigcap_{n<\omega} V_n\subseteq \bigcap_{x\in\Bbb{R}} \Bbb{R}-\{x\} = \varnothing.$$
However, $\bigcap_{n<\omega} V_n = \Bbb{R} - \bigcup_{n<\omega} (\Bbb{R}-V_n)$ and $(\Bbb{R}-V_n)$ is finite set. Since countable union of at most countable sets is at most countable, $\bigcup_{n<\omega} (\Bbb{R}-V_n)$ is countable so $\bigcap_{n<\omega} V_n$ is not empty, a contradiction.

REPLY [1 votes]: Here's the thing about $\Bbb R$. It's linearly ordered. And this means that finite subsets are well-ordered canonically by that linear ordering.
In particular, this means that the countable union of finite sets of real numbers is always countable. So the usual proof works, with the minor modification that now we need to appeal to the properties of $\Bbb R$ to claim that $\bigcup(\Bbb R-V_n)$ is countable.