graphs – Representation of connected components in the $O(|E|)$ time/space variant of Karger’s algorithm

I’m trying to understand the various optimizations given in the original 1992 paper on Karger’s algorithm. Specifically, looking at section “3.1 Unweighted Graphs”, I don’t understand what data structure is used to represent the variant of the algorithm that works in $O(|E|)$ time and space (the other variant, which uses union-find and takes $O(|V|)$ space and $O(|E| log |E|)$ time, makes sense).

Specifically: how would you represent the edges so that it is possible to

  1. track the connected components induced by a sequence of $m$ edges in a random order in $O(m)$ time?
  2. contract a sequence of $m/2$ edges in a random order in $O(m)$ time?

I must be missing something obvious, but I couldn’t find any implementation for this variant (everyone seems mostly interested in implementing the union-find variant of the algorithm).

analytic number theory – Proving Integral representation of $zeta(n)$

Currently studying analytic number theory and was baffled by the following integral representation of $zeta(n)$

$$zeta(n) = dfrac{2}{1 – 2^{1 – n}}sinleft(dfrac{pi n}{2}right)int_0^infty dfrac{x^{-2n}}{pi x}(1 – pi x^2csc(pi x^2))dx$$

However there’s no proof for this provided in the text.

Any help or hints would be highly appreciated.

Thanks.

ra.rings and algebras – Representation theory terminology question

For a paper I’m writing, I need a term for a representation-theoretic concept that I’m sure someone has thought of before, so I thought I’d ask here rather than just make something up.

Let $G$ be a group and $R$ be a commutative ring. Consider an $R(G)$-module $V$. For any ideal $I$ of $R$, we have the submodule
$$I V = {text{$c cdot v$ $|$ $c in I$ and $v in V$}}.$$
What is the term for $R(G)$-modules $V$ such that all submodules are of this form? The ones I’m interested in have the additional property that if you ignore the $G$-action, then they are free $R$-modules (though not finitely generated!), but I doubt this matters for this question.

For an easy example, if $R = mathbb{Z}$ and $G = text{GL}(n,mathbb{mathbb{Z}})$, then $mathbb{Z}^n$ has this property.

If $R$ is a field, then this reduces the the usual notion of an irreducible representation, so I think of this as a version of irreducibility. But looking through my ring-theory books, I can’t find it anywhere.

group theory – How do I find a non-trivial 2D representation of $Z_2 subset $ SL(2, C)?

I am trying to find a non-trivial two dimensional representation of $Z_2 subset SL(2, C)$, but I am a little stuck.

Here is what I did so far:

For M $in$ SL(2, $Z_2$):
begin{bmatrix}a&b\c&dend{bmatrix}
with det M = ad – bc = 1 and a, b, c, d $in$ {0, 1}.

I also found that the order of SL(2, Z$_2$) is 6.

So putting the pieces together, I think that this means that I am looking for a representation of six 2 x 2 matrices that each have a determinant of +1, where my matrix elements are either 0 or 1.

I found the following 3 matrices that satisfy these conditions
begin{bmatrix}1&0\0&1end{bmatrix} begin{bmatrix}1&1\0&1end{bmatrix} begin{bmatrix}1&0\1&1end{bmatrix}

This is where I am stuck: I am unable to find 3 more matrices, only consisting of 0 and 1, that satisfy the condition that my determinant is 1. How do I proceed?

exponentiation – Integral representation of $f(x)=0^x$

Recently I had an argument with LuboŇ° Motl on Quora, where he had argued that $0^0$ should be left undefined in computer algebra systems, because $x^y$ has no limit at $(0,0)$ and $0^x=0$ at all $x>0$.

So, I decided to represent the function $0^x$ in integral form. Once heaving a universal representation of the function one would be able to see what its value at $x=0$ is (as well as around that point).

For that purpose, I armed myself with divergent integrals, Laplace transforms, etc, etc.

So, assuming that

$$0^{-n}=frac{W^{n+1}-w^{n+1}}{(n+1)!}$$

where

$$w^n=B_n(0)+nint_0^infty B_{n-1}(x)dx$$

and

$$W^n=B_n(1)+nint_0^infty B_{n-1}(x+1)dx$$

I obtained the following formula:

$$0^x=frac{B_{1-x}(1)-B_{1-x}+int_0^{infty } (x-1) x t^{-x-1} , dt}{Gamma (2-x)}=frac{(x-1) zeta (x,1)-(x-1) zeta (x,0)+int_0^{infty } (x-1) x t^{-x-1} , dt}{Gamma (2-x)}$$

The formula with Bernoulli polynomials works well in Mathematica at positive and negative integer $x$, returning expected divergent integrals, but the formula with Hurwitz Zeta fails at positive integers, returning expression with the symbol ComplexInfinity.

The both formulas fail at non-integer $x$. So, I wonder, whether it is possible to re-write these formulas in such a way that I would always obtain a convergent or divergent integral?

I tried to apply the integral representations of Hurwitz Zeta function from Wikipedia, but those return wrong results outside of their range of validity (unlike the built-in functions).

real analysis – Integral representation in the case of two variable function

Let $fin C^1(mathbb{R},mathbb{R})$, we have the follwoing integral representation : for all $a,bin mathbb{R}$ :
$$f(b)-f(a)=int_a^b f(t)dt.$$
Now if we consider $gin C^1(mathbb{R}^2,mathbb{R})$ do we have similar result for the mapping $g(f(.),.)$ ?

$$g(f(b),b)-g(f(a),a)= int_a^b f'(t)partial_x g(f(t),t)+partial_y g(f(t),t) dt$$
This is my suggestion.

rt.representation theory – Realizability of a real representation using real cyclotomic coefficients

Let $G$ be a finite group and $rho: G rightarrow GL(d,mathbb{C})$ an irreducible representation with Frobenius-Schur indicator $frac{1}{|G|}sum_{gin G} operatorname{tr} rho(g^2) = 1$. Thus $rho$ is a real representation.

A theorem by Brauer states that every irreducible representation over $mathbb{C}$ can be written using coefficients taken from $mathbb{Q}(zeta_n)$ where $zeta_n=exp(frac{2pi i}{n})$ where $n$ is at most the exponent of the $G$.

We also know that any real representation can be written with coefficients in $mathbb{R}$

Can we satisfy both?

For example, the irreducible representations for the dihedral groups can satisfy both conditions (real coefficients using cosines and sines of fractions of $pi$).

I can solve the problem over the algebraic integers by computing an antilinear equivariant involution $F$ such that $F(x) = J overline{x}$ for a suitable matrix $J$, and such that $rho(g) F(x) = F(rho(g) x)$, which implies $rho(g) J = J overline{rho(g)}$. Then I know that $J overline{J} = alpha mathbb{1}$ for a positive scalar $alpha$, and I set $J’ = J/ sqrt{alpha}$, so that $J’$ represents the complex conjugation. Now, $J$ can be found with cyclotomic coefficients (one iterates over a basis of the space of matrices with ${0,1}$ coefficients, and averages over the group). But the square root operation is not necessarily closed over the cyclotomics, this is where I am currently stuck.

Representation of Lie algebra to Lie group

Let $G$ be a semisimple simply connected Lie group with Lie algebra $mathfrak g$ and let $H$ be a subgroup of $G$ with Lie algebra $mathfrak h$. Suppose that I have a finite dimensional highest weight representation $V$ of the Lie algebra $mathfrak g$ and I restrict this representation to $mathfrak h$. Now is the restricted representation is a representation of $H$ as well ?

representation theory – Each Weyl group orbit in the character lattice of $V$ contains exactly one dominant weight

Let $V = mathbb{C}^3 otimes mathbb{C}^3$ be a representation of $G = SL_3(mathbb{C})$.
The weights of this representation is the set of $varepsilon_i + varepsilon_j$ for $i, j = 1, 2, 3$, where $varepsilon_i$ takes $text{diag}(h_1, h_2, h_3) in mathfrak{h}$ to $h_i$.

The Weyl group is $W = { 1, s_1, s_2, s_1s_2, s_2s_1, s_1s_2s_1 }$.
For a simple root $alpha_i$, the coroot $h_i$ is simply the matrix $E_{ii} – E_{i+1, i+1}$.
Then the pairing $$ langle varepsilon_j, h_i rangle alpha_i = begin{cases} -alpha_i & text{ if } i = j, \ alpha_i & text{ if } i = j -1, \ 0 & text{ else }. end{cases}$$
By the defining equation of root reflections, $$ s_i (beta) = beta – langle beta, h_i rangle alpha_i$$ for $beta in mathfrak{h}^*$, we have $$ s_i(varepsilon_j) = begin{cases} varepsilon_j-alpha_i & text{ if } i = j, \ varepsilon_j + alpha_i & text{ if } i = j -1, \ varepsilon_j & text{ else }. end{cases}$$
Using this last part, the $W$-orbit of the weight $varepsilon_1 + varepsilon_2$ is the set ${ varepsilon_1 + varepsilon_2, varepsilon_1 + varepsilon_3, varepsilon_2 + varepsilon_3 }$.

Dominant weights are non-negative integral linear combinations of the fundamental weights, $mvarpi_1 + n varpi_2$.
Expanding this out in terms of the $varepsilon_i$, I get $$ m varpi_1 + n varpi_2 = frac{1}{3} left( 2(m+n) varepsilon_1 + (2n-m) varepsilon_2 – (m+n) varepsilon_3 right).$$
Equating coefficients shows that none of the set of weights ${ varepsilon_1 + varepsilon_2, varepsilon_1 + varepsilon_3, varepsilon_2 + varepsilon_3 }$ are dominant weights, contradicting what I am supposed to show.
What have I done wrong?