text:
After a Grim Year for LGBT Rights, the Way Forward
Written by adminon April 16, 2018
Ukraine took significant steps toward a less cumbersome and abusive legal gender-recognition procedure, and in a groundbreaking Botswanan case, a transgender man won a seven-year battle for legal gender recognition. While the lawsuit was rejected on jurisdictional grounds, the judge minced no words in denouncing Lively's "crackpot bigotry" that aided "a vicious and frightening campaign of repression against LGBTI people in Uganda."
Indonesia's Constitutional Court deliberated on, but rejected on procedural grounds, a proposal to criminalize all sex outside of marriage, including same-sex conduct. It is in middle-ground countries — those that are neither fully accepting, nor overtly hostile — where the most promising developments are taking place, and where courts are playing a crucial role in defending and advancing the rights of LGBT people. Since 2014, the Egyptian government of President Abdel Fattah al-Sisi has systematically arrested gay and bisexual men and transgender women, with several hundred imprisoned for same-sex conduct. In September 2017, the police conducted a new wave of arrests after revelers at the acclaimed Mashrou' Leila concert displayed a rainbow flag. Nonetheless, Indonesia accepted two recommendations during its Universal Periodic Review (UPR) before the UN Human Rights Council pertaining to LGBT people: one on the protection of LGBT human rights defenders, and another on prioritizing equality and nondiscrimination, including in relation to LGBT people. Dating apps have provided an indispensable means for LGBT people to connect, form community, and find avenues for dating and sex. But when infiltrated, they are also effective tools for blackmailers and extortionists, and for authorities to pursue LGBT people. In South Korea, for example, where homosexual conduct is not permitted in the military, army investigators seized suspected gay soldiers' smart phones and scanned dating apps to entrap gay soldiers. Transgender women found themselves hounded by vigilantes. In the United States, a legal settlement was reached in a case brought by adoptive parents involving medically unnecessary cosmetic surgery on an intersex infant. The Israeli government initially opposed, then accepted joint parent adoption for same-sex couples. Businesses in Singapore stepped up to the podium to support the annual Pink Dot festival after the government forbade sponsorship by multinational corporations. Despite these directives, Tanzania repeatedly threatened LGBT human rights defenders, along with other groups working on controversial issues, and raided meetings and workshops. In Africa, the Tanzanian government closed down drop-in centers providing HIV services for key populations, contending that the centers "promote homosexuality." When activists and lawyers held a meeting in October to discuss a legal challenge to policies that limit the right to health, police arrested and detained them without charge. In China, a court ruled against a public hospital that had forced a gay man into so-called conversion therapy, which is widespread in both state and private medical settings, although homosexuality is not officially regarded as a mental disorder. The legislature passed a bill that replaced marriage with domestic partnerships for same-sex couples, undoing a previous Supreme Court ruling on equal marriage. In April, news broke of a wide-scale purge against gay and bisexual men in Chechnya. Security and police officials rounded up presumably gay men and tortured them in informal detention facilities. The United States has sanctioned the Chechen leader, Ramzan Kadyrov, and another Chechen official. Despite efforts by anti-gay marriage activists, Finland, Germany, Malta, and Australia embraced marriage equality, while constitutional courts in Taiwan and Austria ruled that current laws defining marriage as "between a man and a woman" were unconstitutional, instructing parliament to revise the law within a set timeframe. But the best approach to defending and advancing rights is not through a blueprint but by tailoring strategy to context.
State-sponsored homophobia
Political homophobia remains a potent symbol for autocratic leaders and a convenient football for competing ideologies. President Donald Trump issued a memorandum barring transgender people from serving in the military, although this has been blocked by courts and contradicted in practice. An executive order opened the way for religious exemptions, and poses a threat to the rights of women and LGBT people. The US government withdrew protection for transgender students under Title IX in February, and in October issued a directive stating that transgender people are not protected from employment discrimination under Title VII, reversing a 2014 Justice Department position. ¤
Graeme Reid is director of the lesbian, gay, bisexual, and transgender rights program at Human Rights Watch. Despite a surge in violent attacks on transgender women in Pakistan, activists there have pushed legal recognition and social services inclusion with human rights bodies and regional and national legislatures. Yet this cultural revival takes place against a backdrop of uncertainty in the United States with a rollback of rights for LGBT people domestically and a wavering of support internationally. This tension played itself out in a highly publicized, frosty standoff between Vice President Mike Pence and the ice skater Adam Rippon during the Winter Olympics in Seoul. The African Commission on Human and Peoples' Rights has increasingly championed the rights of LGBT people and those who defend them. When Canadian Prime Minister Justin Trudeau, in an emotional address to a cheering parliament, apologized for past injury to gender and sexual minorities, he sent a signal to the world that exclusion is not necessarily a permanent state of being and that governments can make amends for past discriminatory policies. The ability to form community, to associate, organize, and assemble are of great importance to LGBT groups around the world and yet these fundamental rights are often denied. Legislatures in the US states of South Dakota, Alabama, and Texas curtailed adoption rights for same-sex couples. In a strongly worded judgment that could have broad impact, a US district court ruled in a case brought against the anti-gay Christian evangelist Scott Lively for fomenting homophobia in Uganda. The principles will provide guidance for governments that are willing to reconsider their stance on LGBT rights. But the price is high — it means hiding fundamental aspects of the self, which takes its toll, both personally and socially. At least 79 survivors or people otherwise affected by the purge fled Chechnya with the assistance of Russia's leading LGBT support group and were eventually resettled in Canada and several European Union countries. ¤
An overview of LGBT rights by country, courtesy of Human Rights Watch. The UN's first independent expert on violence and discrimination based on sexual orientation and gender identity, Vitit Muntarbhorn, presented a report to the UN General Assembly in September that outlined his vision of the steps that need to be taken to combat violence and discrimination: decriminalization of same-sex relations, effective anti-discrimination measures, legal recognition of gender identity, de-stigmatization, sociocultural inclusion, and promotion of education and empathy.
School environment and sex education
Schools can be hostile environments for LGBT youth who struggle with rejection and an absence of positive affirmation at a time when they are most vulnerable.
Abuses in medical settings
Paradoxically, because ideas about sexuality and gender identity are deeply embedded in medical science, the helping professions can be a source of trauma and abuse for LGBT people. Japan revised its national bullying prevention policy to explicitly include LGBT students, but failed to update its national educational curriculum, which still ignores LGBT issues. No doubt, as the visibility of LGBT people grows, and rights are achieved in some parts of the world, there has been a strong response from groups that stand against LGBT rights and that are better organized, funded, and more sophisticated than in the past. In the United States, 26 transgender people murdered in what appear to be hate crimes were commemorated on November 20, Transgender Remembrance Day. Queer Eye and RuPaul's Drag Race garner attention on the small screen. The news from the British overseas territory Bermuda was less positive. In contrast, in Brazil, a judge overruled a long-standing decision by the Federal Council of Psychology to ban licensed psychologists from engaging in sexual orientation change efforts. The closet provides a measure of protection from social condemnation, legal prohibition, or worse — violent attack. National and international medical authorities have been at the forefront of calls to ban forced anal examinations to determine if individuals have engaged in anal sex. The investigation has made little progress, but the crimes have not gone unheeded. I'm not buying it," Rippon said in response to a reported overture for a meeting. There has also been a spike in political homophobia, as unpopular minorities bear the brunt of majoritarian rhetoric and authoritarian rule. It is also a dangerous precursor to discrimination and violence against LGBT people.
Freedom of expression, association, and assembly
For LGBT people around the world, invisibility has been a double-edged sword. Ugandan police raided and shut down the Queer Kampala International Film Festival, as well as a week of scheduled activities for annual Pride Week, while an Egyptian media regulatory body declared a media blackout on positive reporting on homosexuality. While activists in India protested the shortcomings in a proposed law seeking to protect the rights of transgender people, Nepal took concrete steps toward a trans-inclusive civil service. In Turkey, the governor of Ankara imposed an indefinite ban on all public LGBT events in the province. In contrast, Bulgarian police stepped in to protect Sofia Pride from threatened disruptions by an ultra-nationalist, virulently homophobic group.
Moving forward in perilous times
For those who have been at the receiving end of state-sponsored crackdowns, there is cold comfort in seeing the stories of global progress that are incremental but significant. Despite an international outcry, two men convicted of having consensual sex in Aceh were subjected to 85 lashes with a cane in public under Sharia (Islamic law) applicable in that region. In regions of the world experiencing violent crackdowns, the activists focus on security, and evading the dragnet. A writ petition filed against a 2013 judgment that upheld the constitutionality of the sodomy law will be heard by the Supreme Court this year. At its 60th ordinary session in May, the commission emphasized the need to protect all human rights defenders, including those working for the protection of LGBT rights, and to ensure freedom of assembly rights for LGBT groups. And yet, looking beyond these egregious abuses, there was also remarkable progress. Positive court rulings in several jurisdictions enhanced protections for LGBT people, while several countries gave legal recognition to same-sex relationships and transgender identities. The United Nations General Assembly received a report on violence and discrimination against LGBT people. In two separate cases, the Justice Department filed amicus briefs in efforts to curtail LGBT protections against workplace and public accommodations. "You mean Mike Pence, the same Mike Pence that funded gay conversion therapy? The 2016 launch of the Equal Rights Coalition, the first intergovernmental network to advance the rights of LGBT people, signifies a growing willingness of the almost 40 member states to play a more proactive international role. Vocal political support from governments, including the United States, has helped LGBT groups internationally and is part of the reason why organizations have been more visible and vocal. Annual survey results in Britain and France revealed significant increases in bias-motivated attacks against people based on sexual orientation or gender identity. It will also send a signal that these human rights issues are not being taken seriously, paving the way for increased discrimination and violence against LGBT people. In the United States, activists would do well to defend ground at risk by sweeping religious exemptions, and holding the government to account for its human rights record internationally. APRIL 16, 2018
IN THE UNITED STATES, 2018 heralds a bumper year for queer culture: while award-winning Call Me by Your Name is enjoying widespread public accolade and generous critical acclaim, classics including Torch Song Trilogy, The Boys in the Band, and Angels in America are lined up on Broadway. In some countries, such as Indonesia, the immediate goal is to prevent sweeping discriminatory laws from being passed. The most disturbing trend was the scale and frequency of arbitrary arrests, state-sponsored discrimination, and violence against LGBT people. The brutal consequence of intensified homophobia orchestrated or facilitated by agents of the state was evident in Azerbaijan, Russia's Chechen Republic (Chechnya), Egypt, Indonesia, Tajikistan, and Tanzania.
Legal and policy developments
In a political environment in which the rights of LGBT people remain an unpopular cause, courts can play a positive role in affirming fundamental rights. If that support wanes, LGBT groups will be more vulnerable and exposed. In a significant milestone, a group of experts issued the updated Yogyakarta + 10 principles, which provide guidelines for the interpretation and application of international human rights law regarding sexual orientation and gender identity. The European Court of Human Rights condemned Russia's "gay propaganda" law in a decision that Russia is obligated to abide by, even though three months after the ruling a Russian court found an activist, Evdokia Romanova, guilty under the same law. In the United States, Utah repealed its "no promo homo" law, which curtailed discussions of homosexuality in schools, but seven other US states retain similar laws. The settlement augured well for intersex activists in the United States who have been campaigning against such surgery for decades to little avail. Meanwhile the Philippines issued a gender-inclusive policy in schools that offers express protection on grounds of sexual orientation and gender identity, but effective implementation is lacking. During the year, Indonesian authorities conducted raids against LGBT people, exploiting an existing anti-pornography law to apprehend at least 200, and bring charges against gay men arrested in a hotel, as well as night clubs and saunas. The United States withheld a portion of military aid to Egypt, due to stated concerns about human rights abuses, specifically a law that places onerous restrictions on nongovernmental organizations. This is a road map that activists around the world can rally around. Of course, governments are not the only sources of violence against LGBT people. Looking back over a tumultuous year, for lesbian, gay, bisexual, and transgender (LGBT) people in many parts of the world, 2017 was grim by any standard. In the United States, discriminatory bills that sought to restrict access to facilities for transgender students were withdrawn in South Dakota and died without coming to a vote in Texas. The Obama administration explicitly included sexual orientation and gender identity as elements of a broader human rights agenda guiding foreign policy. In India, hope was rekindled by a Supreme Court judgment that ruled privacy a fundamental right and made express reference to section 377 of the penal code, a vestige of colonial rule that outlaws same-sex conduct. The authorities mined the men's social media accounts and contacts for the names of other men, who were also rounded up. Detainees were released to elder male relatives in public rituals of humiliation, and their captors encouraged families to commit so-called "honor killings."
Sustained international pressure compelled the Kremlin to press Chechen authorities to suspend the purge and to open a federal inquest. For activists, it was a year of responding to a seemingly endless cycle of unfolding crises. The examinations have no forensic value or medical basis, and can amount to a form of torture or sexual assault, yet are still practiced in several countries. Men across the Netherlands took to holding hands in public in a demonstration of solidarity with a gay couple who were victims of a violent assault. Yet, by and large, Egypt's allies focused on investment, migration, and counterterrorism, and ignored Egypt's abysmal human rights record. But courts can also be an instrument of repression. The American Medical Association board of trustees, two pediatrics organizations, and three former US Surgeons General called for an end to medically unnecessary surgeries on intersex children. Meanwhile, there were some positive developments, as courts in South Korea and Mozambique stepped in to facilitate the registration of LGBT groups.
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\section{Introduction}
The Euler and Betti numbers of moduli spaces of stable sheaves on
complex surfaces have received much attention in the past, both in
mathematics and physics. Computation of the generating functions
of these numbers is notoriously difficult; a generic result is only
known for rank 1 sheaves \cite{Gottsche:1990}. Yoshioka
\cite{Yoshioka:1994, Yoshioka:1995} has computed the generating
functions of the Betti numbers for rank 2 sheaves on a ruled
surface using wall-crossing. The generating functions for rank
2 sheaves on the projective plane $\mathbb{P}^2$ follow from these by
the blow-up formula \cite{Yoshioka:1994, Yoshioka:1995}. Some
qualitative differences appear for rank 3, in which case only three Poincar\'e
polynomials are known \cite{Yoshioka:1996}. Using a different
approach, namely toric geometry, Refs. \cite{Klyachko:1991,
  Kool:2009, weist:2009} have computed generating functions
of the Euler numbers for sheaves of rank 2 and 3 on $\mathbb{P}^2$.

The connection between sheaves and $\mathcal{N}=4$ supersymmetric gauge theory relates the
Euler number of the moduli space to the supersymmetric index (or
BPS-invariant) \cite{Vafa:1994tf}. Another consequence of this connection is that the
$SL(2,\mathbb{Z})$ electric-magnetic duality group of gauge theory implies modular properties for the
generating functions of the Euler numbers \cite{Vafa:1994tf}, which was verified among
others for sheaves on $\mathbb{P}^2$ with rank 1 and 2 \cite{Gottsche:1990, Yoshioka:1994}.
Ref. \cite{Bringmann:2010sd} identified the modular properties for the
Betti numbers, which correspond to refined BPS-invariants
\cite{Dimofte:2009bv}. Although modularity has
been proven useful for computations for rational surfaces \cite{Minahan:1998vr,
  Yoshioka:1998ti, Gottsche:1998}, mathematical justifications of
physical expectations, in particular a so-called holomorphic anomaly
\cite{Vafa:1994tf, Minahan:1998vr}, have been limited to rank $\leq$2 \cite{Gottsche:1990, Yoshioka:1994,
  Yoshioka:1998ti} since generating functions for $r>2$ were not known.

This motivated the present work, which computes the generating functions of the
Betti numbers for stable sheaves with rank 3 on the rationally ruled surface
$\mathbb{\tilde P}^2$ and on $\mathbb{P}^2$. Using the results from \cite{Manschot:2009ia, Manschot:2010xp, Bringmann:2010sd},
the generating functions take a particularly compact form in terms of
modular functions and indefinite theta functions. The latter are
convergent sums over a subset of an indefinite lattice
\cite{Gottsche:1996}. The lattice for rank 2 has
signature $(1,1)$ and the corresponding functions are well-studied in the literature
\cite{Gottsche:1996, Zwegers:2000}. Interestingly, the lattice for rank 3 has signature $(2,2)$
and the corresponding function is of a novel form. A detailed discussion
of the modular properties of this function will appear in a future article \cite{Bringmann:2011}.

The computations in this article rely on wall-crossing and the blow-up
formula, analogously to the computations by Yoshioka for rank
2, and can be extended to rank $>3$ if desired. To arrive at the generating functions, the (semi-primitive)
wall-crossing formula of \cite{Denef:2007vg, Dimofte:2009bv} is
applied to determine the change of the Betti numbers across a wall of marginal stability.
This wall-crossing formula is derived in physics both for
field theory \cite{Gaiotto:2008cd} and for supersymmetric black holes
in supergravity \cite{Denef:2007vg, Andriyash:2010qv}, which correspond to D$p$-branes (or sheaves) supported on $p$-cycles of a
Calabi-Yau threefold. The validity of the wall-crossing
formula for this article can be motivated by viewing the surface as a (rigid) divisor of a
Calabi-Yau threefold, and it is further confirmed by agreement of the
generating functions with older results in the mathematical literature \cite{Yoshioka:1996,
  Kool:2009, weist:2009}.  It is well-known that this wall-crossing
formula is equivalent with the mathematical wall-crossing formulas derived for Donaldson-Thomas
invariants \cite{Joyce:2008, Kontsevich:2008}. In fact, also
mathematical arguments exist that these formulas are
 applicable for invariants of moduli spaces of sheaves on $\tilde \mathbb{P}^2$ since
$K^{-1}_{\tilde \mathbb{P}^2}$ is numerically effective \cite{Joyce:2004, Kontsevich:2010px}.

Wall-crossing for sheaves (or D4-branes) supported on divisors in
Calabi-Yau threefolds \cite{Manschot:2009ia} is another
motivation for this paper. Considering the sheaves on surfaces without
the embedding into a Calabi-Yau simplifies the system and in this way
helps to understand the Calabi-Yau case. On the other hand supergravity gives a useful
complementary viewpoint, and suggests for example that
the generating function for BPS-invariants can be computed  using enumeration of so-called flow trees
  \cite{Denef:2000nb}. This approach was taken in
  \cite{Manschot:2010xp}, and the present article provides an
  illustration and confirmation of this technique. Subsection
  \ref{subsec:flowtrees} gives a brief introduction to flow trees, however
  the discussion in this article is mostly phrased in terms of sheaves and characteristic classes,
  because the notion of a moduli space is most rigorously defined in
  this context.
\\
\newline
The outline of the paper is as follows. Section \ref{sec:sheaves}
reviews the necessary properties of sheaves, including wall-crossing
and blow-up formulas. Subsection \ref{subsec:flowtrees} gives a brief
introduction to flow trees. Section \ref{sec:euler} computes the
Euler numbers of the moduli spaces for rank 2 and 3, followed by the computation of the
Betti numbers in Section \ref{sec:betti}. Appendix \ref{app:modfunctions}
lists various modular functions, which appear in the generating functions
of the Euler and Betti numbers.

\section*{Acknowledgements}
\noindent I would like to thank Emanuel Diaconescu, Lothar G\"ottsche and
Maxim Kontsevich for helpful and inspiring discussions, and IHES for hospitality.
This work is  partially supported  by ANR grant
BLAN06-3-137168.

\section{Sheaves}
\label{sec:sheaves}

\subsection{Sheaves and stability}
The Chern character of a sheaf $F$ on a surface $S$ is given by
ch$(F)=r(F)+c_1(F)+\frac{1}{2}c_1(F)^2-c_2(F)$ in terms of the
rank  $r(F)$ and its Chern classes $c_1(F)$ and $c_2(F)$. It is
convenient to parametrize a sheaf by ch$(F)$ since it is additive:
$\mathrm{ch}(F\oplus G) =\mathrm{ch}(F)+\mathrm{ch}(G)$. Define
$\Gamma:=(r,\mathrm{ch}_1,\mathrm{ch}_2)$. Other frequently occuring quantities are the determinant
$\Delta(F)=\frac{1}{r(F)}(c_2(F)-\frac{r(F)-1}{2r(F)}c_1(F)^2)$, and
$\mu(F)=c_1(F)/r(F)\in H^2(S,\mathbb{Q})$.

Let $0\subset F_1 \subset F_2\subset \dots \subset F_s=F$ be a
filtration of the sheaf $F$. The quotients are denoted
by $E_i=F_i/F_{i-1}$ with $\Gamma_i=\Gamma(E_i)$.

\begin{lemma}
With the above notation, the discriminant $\Delta(F)$ is given by
\begin{equation}
\Delta(F)=\sum_{i=1}^s\frac{r(E_i)}{r(F)}\Delta(E_i)-\frac{1}{2r(F)}\sum_{i=2}^s
\frac{r(F_{i-1})\,r(F_i)}{r(E_i)} \left(\mu(F_{i-1})-\mu(F_i) \right)^2.\non
\end{equation}
\end{lemma}

\begin{proof}
Consider first the filtration for $s=2$: $0\subset F_1\subset
F_2=F$, such that $\Gamma(F)=\Gamma(F_1)+\Gamma(E_2)$. Application of the definitions
and some straightforward algebra lead to:
$$
\Delta(F)=\frac{r(F_1)}{r(F)}\Delta(F_1)+\frac{r(E_2)}{r(F)}\Delta(E_2)-\frac{r(F_1)r(E_2)}{2r(F)^2}(\mu(F_1)-\mu(E_2))^2.
$$
Applying this equation iteratively on $F_1$ leads to the
lemma.\footnote{\, Note
that this is different from Ref. \cite{Yoshioka:1996} (Lemma 2.2).}
\end{proof}

The notion of a moduli space for sheaves is only well defined after
the introduction of a stability condition. To this end let $C(S)\in
H^2(S,\mathbb{Z})$ be the ample cone of $S$.

\begin{definition}
Given a choice $J\in C(S)$, a sheaf $F$ is called $\mu$-stable if for every subsheaf $F'$,
$\mu(F')\cdot J <\mu(F)\cdot J$, and $\mu$-semi-stable if for every subsheaf $F'$,
$\mu(F')\cdot J \leq\mu(F)\cdot J$. A wall of marginal stability $W$ is a (codimension
1) subspace of $C(S)$, such that $(\mu(F')-\mu(F))\cdot J=0$, but
$(\mu(F')-\mu(F))\cdot J\neq 0$ away from $W$.
\end{definition}

Let $S$ be a K\"ahler surface, whose intersection pairing on
$H^2(S,\mathbb{Z})$ has signature $(1,b_2-1)$.  Since at a wall,
$(\mu_2-\mu_1)\cdot J=0$ for $J$ ample,
$(\mu_2-\mu_1)^2<0$. Therefore, the set of filtrations for $F$, with
$\Delta_i\geq 0$ is finite.

\subsection{Invariants and wall-crossing}
Ref. \cite{Vafa:1994tf} shows that the BPS-invariant of $\mathcal{N}=4$ gauge
theory on $S$ equals the Euler number (up to a sign) of a suitable compactification of the
instanton moduli space, i.e. the Gieseker-Maruyama moduli space
$\mathcal{M}_J(\Gamma)$ of semi-stable sheaves on $S$ (with respect
to the ample class $J$). The topological classes $\Gamma$ of
the sheaf are determined by the topological properties of the instanton.
The complex dimension of $\mathcal{M}_J(\Gamma)$ is given by:
\begin{equation}
\dim_{\mathbb{C}} \mathcal{M}_J(\Gamma)=2r^2\Delta-r^2\chi(\mathcal{O}_S)+1.\non
\end{equation}
The BPS-invariant $\Omega(\Gamma;J)$ corresponds to the Euler number of $\mathcal{M}_{J}(\Gamma)$
\begin{equation}
\Omega(\Gamma;J)=(-1)^{\dim_\mathbb{C}\mathcal{M}_J(\Gamma)}\chi(\mathcal{M}_J(\Gamma)),\non
\end{equation}
if the moduli space is smooth and compact. The mathematical rigorous
definition of the BPS-invariant is more involved if these
conditions are not satisfied \cite{Yoshioka:1995}.
The rational invariants \cite{Nakajima:2007, Joyce:2008, Kontsevich:2008}
\begin{equation}
\label{eq:ratinv}
\bar \Omega(\Gamma;J)=\sum_{m|\Gamma} \frac{\Omega(\Gamma/m;J)}{m^2}
\end{equation}
are also particularly useful for our purposes
\cite{Manschot:2010xp}.

To state the changes $\Delta\Omega(\Gamma;J_\mathcal{C}\to J_{\mathcal{C}'})$ across walls of marginal
stability, we define the following quantities:
\begin{equation}
\label{eq:intprod}
\left<\Gamma_1,\Gamma_2\right>=r_1r_2(\mu_2-\mu_1)\cdot K_S , \qquad \mathcal{I}(\Gamma_1,\Gamma_2;J)=r_1r_2(\mu_2-\mu_1)\cdot J.
\end{equation}
These definitions follow quite naturally from formulas in physics \cite{Diaconescu:2007bf, Manschot:2009ia,
  Manschot:2010xp}.

The change $\Delta\Omega(\Gamma_1+\Gamma_2;J_\mathcal{C}\to J_{\mathcal{C}'})$, for $\Gamma_1$ and $\Gamma_2$ primitive,
is \cite{Yoshioka:1996,Denef:2007vg}:
\begin{eqnarray}
\label{eq:deltaprimitive}
\Delta\Omega(\Gamma_1+\Gamma_2; J_\mathcal{C}\to
J_{\mathcal{C}'})&=&\textstyle{\frac{1}{2}} \left( \mathrm{sgn}(\mathcal{I}(\Gamma_1,\Gamma_2;J_{\mathcal{C}'}))- \mathrm{sgn}(\mathcal{I}(\Gamma_1,\Gamma_2;J_\mathcal{C})) \right)\\
&&\times\, (-1)^{\left<\Gamma_1,\Gamma_2\right>}\left<\Gamma_1,\Gamma_2\right>
\Omega(\Gamma_1;J_{W_\mathcal{C}})\,\Omega(\Gamma_2;J_{W_\mathcal{C}}),\non
\end{eqnarray}
with
$$
\mathrm{sgn}(x)=\left\{ \begin{array}{cl} 1, & x>0, \\ 0, & x=0, \\ -1, & x<0.\end{array} \right.
$$
The subscript ${W_\mathcal{C}}$ in $J_{W_\mathcal{C}}$ refers to a point in $\mathcal{C}$
which is sufficiently close to the wall $W$, such that no wall is
crossed for the constituents between the wall and $J_{W_\mathcal{C}}$. The wall
is independent of $c_2$, and therefore a sum over $c_2$ appears in the
next section.

For the computation of the invariants for rank 3, one also needs the
semi-primitive wall-crossing formula \cite{Denef:2007vg}:
\begin{eqnarray}
\label{eq:semiprim}
\Delta\Omega(2\Gamma_1+\Gamma_2;J_\mathcal{C}\to
J_{\mathcal{C}'})&=&\textstyle{\frac{1}{2}} \left( \mathrm{sgn}(\mathcal{I}(\Gamma_1,\Gamma_2;J_{\mathcal{C}'}))- \mathrm{sgn}(\mathcal{I}(\Gamma_1,\Gamma_2;J_\mathcal{C})) \right)\non\\
&&\times \left\{-2\left<\Gamma_1, \Gamma_2 \right>
\Omega(2\Gamma_1;J_{W_\mathcal{C}})\,\Omega(\Gamma_2;J_{W_\mathcal{C}}) \right.\non\\
&&\quad +(-1)^{\left<\Gamma_1,\Gamma_2\right>}\left<\Gamma_1,\Gamma_2\right>\Omega(\Gamma_1;J_{W_\mathcal{C}})\,\Omega(\Gamma_1+\Gamma_2; J_{W_\mathcal{C}}) \\
&&\quad \left. +\textstyle{\frac{1}{2}} \left<\Gamma_1, \Gamma_2\right>\Omega(\Gamma_2;J_{W_\mathcal{C}})\,\Omega(\Gamma_1;J_{W_\mathcal{C}})\,(\left<\Gamma_1, \Gamma_2\right>\Omega(\Gamma_1; J_{W_\mathcal{C}})-1)\right\}.\non
\end{eqnarray}

Define the generating function for $\bar \Omega(\Gamma;J)$:
\begin{equation}
\label{eq:genfunction}
h_{r,c_1}(\tau;S,J):=\sum_{c_2} \bar \Omega(\Gamma;J)\,q^{r\Delta-\frac{r\chi(S)}{24}}.
\end{equation}
Twisting a sheaf by a line bundle gives an isomorphism of moduli
spaces, which implies $h_{r,c_1+\mathbf{k}}=h_{r,c_1}$ if
$\mathbf{k}\in H_2(S,r\mathbb{Z})$. It is therefore sufficient to
determine $h_{r,c_1}$ only for $c_1 \mod r$. Explicit computation of $h_{r,c_1}(\tau;S,J)$ is typically complicated. A generic
result exists just for $r=1$ \cite{Gottsche:1990}:
\begin{equation}
\label{eq:genr=1}
h_{1,c_1}(\tau;S)=\frac{1}{\eta(\tau)^{\chi(S)}},
\end{equation}
with $\eta(\tau)$ defined in Eq. (\ref{eq:etatheta}). The dependence
on $J$ could be omitted here, since the moduli space of rank 1 sheaves
does not depend on a choice of ample class.

The next proposition gives
the universal relation between generating functions for $S$ and its
blow-up $\tilde S$. This appeared first for $r=2$ in
\cite{Yoshioka:1994, Vafa:1994tf} and for general $r$ in
\cite{Yoshioka:1996}. Proofs are given in \cite{Li:1999} for
$r=2$, and \cite{Gottsche:1998} for general $r$.
\begin{proposition}
\label{prop:blowup}
Let $S$ be a smooth projective surface and $\phi: \tilde S \to S$ the
blow-up at a non-singular point, with $C_1$ the exceptional
divisor of $\phi$. Let $J\in C(S)$, $r$, and $c_1$ such that
$\gcd(r,c_1\cdot J)=1$. The generating functions $h_{r,c_1}(\tau;S,J)$ and
$h_{r,c_1}(\tau;\tilde S,J)$ are then related by the ``blow-up
formula'':
\begin{equation}
h_{r,\phi^* c_1-kC_1}(\tau;\tilde S, \phi^* J)=B_{r,k}(\tau)\, h_{r,c_1}(\tau;S,J),\non
\end{equation}
with
\begin{equation}
B_{r,k}(\tau)=\frac{(-1)^{(r-1)k}}{\eta(\tau)^r}\sum_{\sum_{i=1}^ra_i=0 \atop a_i \in \mathbb{Z}+\frac{k}{r}} q^{-\sum_{i<j}a_ia_j}.\non
\end{equation}
\end{proposition}
The factor $(-1)^{(r-1)k}$ is a consequence of the relative sign between the
BPS-invariant and the Euler number. The two relevant cases for this article are $r=2,3$:
\begin{equation}
B_{2,k}(\tau)=(-1)^k\frac{\sum_{n\in \mathbb{Z}+k/2} q^{n^2}}{\eta(\tau)^2},\qquad B_{3,k}(\tau)=\frac{\sum_{m,n \in \mathbb{Z}+k/3} q^{m^2+n^2+mn}}{\eta(\tau)^3}.
\end{equation}

\subsection{Flow trees}
\label{subsec:flowtrees}
This subsection gives a brief introduction to flow trees, since the
computations in the next sections are inspired by it. More information
can be found in Refs. \cite{Denef:2000nb, Denef:2007vg}. See Ref.
\cite{Manschot:2010xp} for a discussion which is more adapted to the present context.

Flow trees appear in the analysis of D-brane bound
states. D-branes are equivalent to coherent sheaves in the ``infinite
volume limit''. A flow tree is an embedding of a rooted tree $T$ in $C(S)$ (or
more generally, the moduli space), which satisfies a number of
``stability'' conditions. The tree can be parametrized by a nested list,
e.g. $((\Gamma_1,\Gamma_2),\Gamma_3)$, and represents a decomposition
of the total charge $\Gamma=\sum_{i}\Gamma_i$. The change of $J$ along the edges of the tree,
is determined by the supergravity equations of motion. The endpoints of the
flow tree represent ``elementary'' constituents which do not decay in
$C(S)$, for example rank 1 sheaves. Generically, only in a special
chamber in $C(S)$, the chamber with the attractor point, the total moduli space corresponds to the moduli space
of these elementary constituents.

The existence of a tree as a flow tree is determined by ``stability conditions'' at its vertices. The class
$J$ lies at a wall for the two merging trees if it is a trivalent vertex. For example, the stability of the subtree  $(\Gamma_1,\Gamma_2)$ in
$((\Gamma_1,\Gamma_2),\Gamma_3)$, is determined at a (specific) point of the wall for $\Gamma_1+\Gamma_2$ and $\Gamma_3$. If
all conditions are satisfied the tree does exist as a flow tree. The attractor flow conjecture states that the ``BPS Hilbert
space'' is partitioned by flow trees \cite{Andriyash:2010qv,
  Denef:2000nb}. This implies that the BPS-index can be computed in
principle by enumerating flow trees, once the BPS-indices of the
endpoints are known \cite{Denef:2007vg}.

One of the advantages of flow trees is that they give an algorithmic
procedure to test for the stability of a composite object at a given
point in moduli space. A simplifying feature is that they do not distinguish between subobjects and quotients, in
contrast to the stratification of the set of sheaves using
(Harder-Narasimhan) filtrations.

Small changes are necessary to utilize flow trees in the present
context, since the manifold is a surface instead of a 3-dimensional
Calabi-Yau threefold. One difference is the choice of the boundary of $C(\mathbb{\tilde P}^2)$ as reference
point in the moduli space, instead of the attractor point. For
$\mathbb{\tilde P}^2$ one does not need to
solve for $J$ along the edges, not even for the flow trees with
3 centers, since a wall in $C(\mathbb{\tilde P}^2)$ is only a single
point (projectively). With these observations, it is not difficult to
realize that the generating functions for the (refined) BPS-invariants
in Sections \ref{sec:euler} and  \ref{sec:betti} can be obtained
either using wall-crossing or enumeration of flow trees.

\section{Euler numbers}
\label{sec:euler}

This section computes the generating function of Euler numbers of the
moduli spaces of semi-stable sheaves of rank 2 and 3 on $\mathbb{\tilde P}^2$ and
$\mathbb{P}^2$. First, some rudiments of ruled surfaces are reviewed.

\subsection{Some properties of ruled surfaces}
A ruled surface is a surface $\Sigma$ together with a surjective
morphism $\pi: \Sigma\to C$ to a curve $C$, such that the fibre $\Sigma_y$
is isomorphic to $\mathbb{P}^1$ for every point $y\in C$. Let $f$ be the
fibre of $\pi$, then
$H_2(\Sigma,\mathbb{Z})=\mathbb{Z}C\oplus\mathbb{Z}f$, with
intersection numbers $C^2=-e$, $f^2=0$ and $C\cdot f=1$. The canonical class is
$K_\Sigma=-2C+(2g-2-e)f$. The holomorphic Euler characteristic
$\chi(\mathcal{O}_\Sigma)$ is for a ruled surface $1-g$. An ample
divisor is parametrized as $J_{m,n}=m(C+ef)+nf$ with $m,n\geq 1$.

The blow-up $\phi: \mathbb{\tilde P}^2\to \mathbb{P}^2$ of the
projective plane $\mathbb{P}^2$ at a point is equal to the rationally
ruled surface with $(g,e)=(0,1)$. The exceptional divisor of $\phi$ is $C$, and
the hyperplane $H$ of $\mathbb{P}^2$ equals $\phi^*(C+f)$. The remainder of
this article restricts to the case $(g,e)=(0,1)$, although many
results are easily generalized to generic $(g,e)$.

\subsection{Rank 2}
Our aim is to compute the generating function
$h_{2,c_1}(\tau;\mathbb{\tilde P}^2,J)$ \footnote{\, Since almost all generating
  series in this section are for $\mathbb{\tilde P}^2$, it is omitted
  from the arguments of $h_{r,c_1}$ in the following.}
defined by Eq. (\ref{eq:genfunction}). To learn about the set of
stable sheaves on $\mathbb{\tilde P}^2$ for $J\in C(S)$, it is
useful to first consider the restriction of the sheaves on $\mathbb{\tilde
  P}^2$ to $f$. Namely the restriction $E_{|f}$ is stable if and only if $E$ is $\mu$-stable for
$J=J_{0,1}$ and in the adjacent chamber \cite{Huybrechts:1996}. However, since every bundle of rank $\geq 2$ on
$\mathbb{P}^1$ is a sum of line bundles \cite{Grothendieck:1957},
there are no stable bundles with $r\geq 2$ on $\mathbb{P}^1$. Therefore $\Omega(\Gamma;J_{0,1})=0$ for
$\Gamma=(r(F),-C-\alpha f,\mathrm{ch}_2)$ with $r(F)\geq 2$ and
$\alpha=0,1$. The computation for $c_1(E)=-\alpha f$ is more complicated, and is dealt with in the end of this subsection.

To determine $h_{2,-C-\alpha f}(\tau;J_{m,n})$, one can either change
the polarization from $J_{0,1}$ to $J_{m,n}$ (see Figure \ref{fig:mspace})
and keep track of $\Omega(\Gamma;J)$ across the walls, or enumerate the flow trees for $J_{m,n}$.
\begin{figure}[h!]
\centering
\includegraphics[totalheight=7cm]{mspace}
\caption{The ample cone of $\mathbb{\tilde P}^2$, together with the
  three walls for $\Gamma=(2,-C-f,2)$, namely for $(a,b)=(1,0)$,
  $(2,0)$, $(3,0)$.}
\label{fig:mspace}
\end{figure}
The only possible
filtrations are $0\subset F_1\subset F$, with $r_i=1$. Therefore the
primitive wall-crossing formula (\ref{eq:deltaprimitive})
suffices, and the $\Omega(\Gamma_i)$ do not depend on the moduli. In
the following, $c_1(E_2)$ is parametrised by $bC-af$. As is customary for
flow trees, the constituents are treated symmetrically, such that $a,b$ run over $\mathbb{Z}$. Since $r_1=r_2$, this sum needs to be
multiplied by $\textstyle{\frac{1}{2}}$. The walls are then at $\frac{m}{n}=\frac{2b+1}{2a-\alpha}$, with $m,n\geq 0$.
See Figure \ref{fig:mspace} for the walls for $\Delta(F)=\frac{9}{4}$, $r(F)=2$.

The various quantities appearing in
$h_{2,c_1}(\tau;J_{m,n})$ become in terms of $a$ and $b$:
\begin{eqnarray}
&\Delta(F)&=\textstyle{\frac{1}{2}} \Delta_1+\textstyle{\frac{1}{2}} \Delta_2+\frac{1}{8} (2b+1)^2+\frac{1}{4}(2b+1)(2a-\alpha),\non\\
&\left<\Gamma_1,\Gamma_2\right>&=-(2b+1)+2(2a-\alpha),\\
&\mathcal{I}(\Gamma_1,\Gamma_2;J_{m,n})&=(2b+1)n-(2a-\alpha)m.\non
\end{eqnarray}
It is now straightforward to construct the generating function using (\ref{eq:deltaprimitive}):
\begin{eqnarray}
\label{eq:genfunction2}
h_{2,-C-\alpha f}(\tau;J_{m,n})&=&-\frac{1}{2}\frac{1}{\eta(\tau)^8}\sum_{a,b\in\mathbb{Z}} \textstyle{\frac{1}{2}}
(\,\mathrm{sgn}((2b+1)n-(2a-\alpha)m)-\mathrm{sgn}(2b+1)\,)\\
&&\times\, (-(2b+1)+2(2a-\alpha))\,q^{\frac{1}{4} (2b+1)^2+\frac{1}{2}(2b+1)(2a-\alpha)}.\non
\end{eqnarray}
The $-$-sign in front is due to
$(-1)^{\left<\Gamma_1,\Gamma_2\right>}$, the $\textstyle{\frac{1}{2}}$ appears because
$r_1=r_2$, and $\eta(\tau)^{-8}$ arises from the sum over $\Delta_i$ and
(\ref{eq:genr=1}). Ref. \cite{Bringmann:2010sd} proved that for
$J_{m,n}=J_{1,0}$, the generating functions are
\begin{eqnarray}
\label{eq:genseriesr21}
&&h_{2,-C-f}(\tau;J_{1,0})=3B_{2,0}(\tau) \mathfrak{h}_{1}(\tau)/\eta(\tau)^6,\\
&&h_{2,-C}(\tau;J_{1,0}) = 3B_{2,1}(\tau)
\mathfrak{h}_{0}(\tau)/\eta(\tau)^6, \non
\end{eqnarray}
where $\mathfrak{h}_{i}(\tau)$ are generating functions of the class
numbers (\ref{eq:genclassj}). The half-integer coefficients for
$c_1(F)=-C$ arise because $J_{1,0}$ is a wall for $F$. Application of Proposition \ref{prop:blowup} gives the known generating
functions for $\mathbb{P}^2$ \cite{Vafa:1994tf}. This gives incidentally also
the correct result for $c_1(F)=0$, even though $\gcd(r(F),c_1(F)\cdot J)\neq 1 $.

To compute $h_{3,-C-f}(\tau;J_{m,n})$, one needs explicit expressions
for $h_{2,-\alpha f}(\tau;J_{m,n})$, $\alpha=0,1$. Fortunately, it is not necessary to deal with the singularities
in the moduli space explicitly. One can either apply modular transformations or blow-down and blow-up again for $J_{m,n}=J_{1,0}$ and then
apply the wall-crossing formula. One
finds in both cases for $J_{1,0}$:
\begin{eqnarray}
\label{eq:genseriesr22}
&& h_{2,-f}(\tau;J_{1,0})= -3B_{2,1}(\tau)
\mathfrak{h}_{1}(\tau)/\eta(\tau)^6,\non\\
&& h_{2,0}(\tau;J_{1,0})=-3B_{2,0}(\tau)
\mathfrak{h}_{0}(\tau)/\eta(\tau)^6.\non
\end{eqnarray}
The Fourier coefficients $\bar \Omega(\Gamma; J_{1,0})$ of $h_{2,0}(\tau;J_{1,0})$ are not integers,
since $\Gamma$ might be divisible by 2. One finds for the generating
function of $\Omega(\Gamma;J_{1,0})$  using (\ref{eq:ratinv}):
\begin{equation}
-3B_{2,0}(\tau) \mathfrak{h}_{0}(\tau)/\eta(\tau)^6-1/4\eta(2\tau)^4.\non
\end{equation}
The wall-crossing formula provides now the generating functions for
generic $J\in C(\mathbb{\tilde P}^2)$:
\begin{equation}
\label{eq:genseriesr2}
h_{2,\beta C-\alpha f}(\tau;J_{m,n})=h_{2,\beta C-\alpha
  f}(\tau;J_{1,0})+\Delta h_{2,\beta C-\alpha f}(\tau;J_{m,n}),
\end{equation}
with
\begin{eqnarray}
\label{eq:Deltah}
&&\Delta h_{2,\beta C-\alpha f}(\tau;J_{m,n})=\non\\
&&\qquad (-)^{\beta}\frac{1}{2}\frac{1}{\eta(\tau)^8}\sum_{a,b\in\mathbb{Z}} \textstyle{\frac{1}{2}}
(\,\mathrm{sgn}(-(2a-\alpha )\,)-\mathrm{sgn}((2b-\beta )n-(2a-\alpha)m)\,)\\
&&\qquad \times\, (-(2b-\beta)+2(2a-\alpha))\,q^{\frac{1}{4} (2b-\beta)^2+\frac{1}{2}(2b-\beta)(2a-\alpha)}.\non
\end{eqnarray}

\subsection{Rank 3}
Using the results of the previous subsection, the Euler numbers of the
moduli space of stable sheaves with $\Gamma(F)=(3,-C-f,\mathrm{ch}_2)$ can  be computed. This
computation has to deal with two additional complications:
\begin{itemize}
\item[-] semi-primitive wall-crossing is possible for sheaves with $\Gamma(F)=2\Gamma_1+\Gamma_2$,
\item[-] the BPS-invariants of a constituent with $r=2$ do themselves
  depend on the moduli, and need to be determined sufficiently close to the appropriate wall.
\end{itemize}

Since no stable sheaves do exist for $c_1(F)=-C-f$, all sheaves
are composed of 2 constituents with rank $r_1=1$ and $r_2=2$, or 3 constituents
with rank $r_i=1$, $i=1,2,3$. Therefore the formulas of Ref. \cite{Manschot:2010xp} for
the enumeration of flow trees with 3 centers are applicable. There it was
explained that the semi-primitive wall-crossing formula for
$2\Gamma_1+\Gamma_2$ simplifies, if 1) the invariants are
evaluated at a point on the wall $J_W$ instead of $J_{W_\mathcal{C}}$, and 2)  it is written in terms of the rational invariant $\bar \Omega(\Gamma,
J_W)$. With these substitutions, one finds that Eq. (\ref{eq:semiprim}) is equal to:
\begin{eqnarray}
\Delta\Omega(2\Gamma_1+\Gamma_2;J_\mathcal{C} \to J_{\mathcal{C}'} )&=&\textstyle{\frac{1}{2}} \left(
  \mathrm{sgn}(\mathcal{I}(\Gamma_1,\Gamma_2;J_{\mathcal{C}'}))-
  \mathrm{sgn}(\mathcal{I}(\Gamma_1,\Gamma_2;J_\mathcal{C})) \right)\left<\Gamma_1,\Gamma_2\right>\non\\
&&\times \left( -2\, \bar \Omega(2\Gamma
_1;J_W)\,\Omega(\Gamma_2)+(-1)^{\left<\Gamma_1,\Gamma_2\right>}\,\Omega(\Gamma_1)\,\Omega(\Gamma_1+\Gamma_2;J_W)\right).
\end{eqnarray}
One observes that the extra terms due to semi-primitive wall-crossing are naturally
included into the terms for primitive wall-crossing.

The Euler numbers can now be obtained by simply implementing the
formulas. Choose again $c_1(E_2)=bC-af$. Then, the walls are at
\begin{equation}
\frac{m}{n}=\frac{3b+2}{3a-2}, \qquad m,n \geq 0.\non
\end{equation}
For the generating function follows:
\begin{eqnarray}
\label{eq:genf2}
h_{3,-C-f}(\tau;J_{m,n})&=&\frac{1}{\eta(\tau)^{4}}\sum_{a,b\in
  \mathbb{Z}}\textstyle{\frac{1}{2}} (\, \mathrm{sgn}((3b+2)n-(3a-2)m)-\mathrm{sgn}(3b+2)
\,)\non \\
&&\times (-1)^b\,(-(3b+2)+2(3a-2))\,q^{\frac{1}{12}(3b+2)^2+\frac{1}{6}(3b+2)(3a-2)}\\
&& \times h_{2,bC-af}(\tau;J_{|3b+2|,|3a-2|}).\non
\end{eqnarray}
Expansion of the first coefficients gives for $J_{m,n}=J_{1,0}$:
\begin{equation}
h_{3,-C-f}(\tau;J_{1,0})=q^{-\frac{5}{6}}\left(3\,q^2
  + 69\,q^{3}+792\,q^4+6345\,q^5+\dots\right).
\end{equation}
One finds with Proposition \ref{prop:blowup}:
\begin{equation}
\label{eq:euler3p2}
h_{3,-H}(\tau;\mathbb{P}^2)=\frac{h_{3,-C-f}(\tau;J_{1,0})}{B_{3,0}(\tau)},
\end{equation}
which is also equal to $h_{3,H}(\tau;\mathbb{P}^2)$.
This result agrees with the coefficients given by Corollary 4.10 of
Ref. \cite{weist:2009},\footnote{\, Note that the result of
  Ref. \cite{weist:2009} differs from (\ref{eq:euler3p2}) by
  $\eta(\tau)^{-9}$, since that article considers vector bundles
  instead of sheaves.} and Corollary 4.9 of Ref. \cite{Kool:2009}.

\section{Betti numbers}
\label{sec:betti}
This section computes the Betti numbers of the moduli spaces of
stable sheaves with $\Gamma(F)=(3,-C-f,\mathrm{ch}_2)$ using
wall-crossing for refined (or motivic) invariants
$\Omega(\Gamma,w;J)$. To define these invariants, let
$p(X,s)=\sum_{i=0}^{2\dim_\mathbb{C}(X)}b_is^i$, with $b_i$
the Betti numbers $b_i=\dim H^2(X,\mathbb{Z})$, be the Poincar\'e
polynomial of a compact complex manifold $X$. Then I define the refined
invariant in terms of the Betti numbers by:
\begin{equation}
\Omega(\Gamma,w;J):=\frac{w^{-\dim_\mathbb{C}\mathcal{M}_J(\Gamma)}}{w-w^{-1}}\, p(\mathcal{M}_J(\Gamma),w).\non
\end{equation}
The primitive wall-crossing formula reads for $\Omega(\Gamma,w;J)$ \cite{Yoshioka:1996}:
\begin{eqnarray}
\Delta \Omega(\Gamma,w;J_\mathcal{C}\to J_{\mathcal{C}'})&=&-\textstyle{\frac{1}{2}} \left( \mathrm{sgn}(\mathcal{I}(\Gamma_1,\Gamma_2;J_{\mathcal{C}'}))- \mathrm{sgn}(\mathcal{I}(\Gamma_1,\Gamma_2;J_\mathcal{C})) \right)\non\\
&&\times\left( w^{\left<\Gamma_1,\Gamma_2\right>}- w^{-\left<\Gamma_1,\Gamma_2\right>}\right)\Omega(\Gamma_1,w;J)\,\Omega(\Gamma_2,w;J).\non
\end{eqnarray}

Using the semi-primitive wall-crossing formula for refined invariants
\cite{Dimofte:2009bv}, it becomes clear that the analogue of $\bar
\Omega(\Gamma;J)$ for refined invariants is:
\begin{equation}
\label{eq:refw}
\bar \Omega(\Gamma,w;J)=\sum_{m|\Gamma} (-1)^{m+1}\frac{\Omega(\Gamma/m,w^m;J)}{m}.\non
\end{equation}
The generating function is naturally defined by:
\begin{equation}
h_{r,c_1}(z,\tau;S,J)=\sum_{c_2} \bar \Omega(\Gamma,w;J)\, q^{r\Delta(F)-\frac{r\chi(S)}{24}}.
\end{equation}
Note that the power of the denominator in (\ref{eq:refw}) is 1 whereas it was 2 in
Eq. (\ref{eq:ratinv}). This leads to an interesting product formula when an additional sum over the rank is
performed. The generalization of Proposition \ref{prop:blowup} gives \cite{Yoshioka:1996}:
\begin{equation}
\label{eq:blowupw}
B_{2,k}(z,\tau)=\frac{\sum_{n\in \mathbb{Z}+k/2}
  q^{n^2}w^{2n}}{\eta(\tau)^2},\qquad B_{3,k}(z,\tau)=\frac{\sum_{m,n\in \mathbb{Z}+k/3}q^{m^2+n^2+mn}w^{4m+2n}}{\eta(\tau)^3}.
\end{equation}
The generating function of refined invariants for $\mathbb{\tilde P}^2$ and $r=1$ is:
\begin{equation}
h_{1,c_1}(z,\tau)=\frac{i}{\theta_1(2z,\tau)\,\eta(\tau)}.\non
\end{equation}

Now the computation is completely analogous to Section
\ref{sec:euler}. The generalization of Eq. (\ref{eq:genfunction2}) is:
\begin{eqnarray}
\label{eq:genf2w}
h_{2,-C-\alpha f}(z,\tau;J_{m,n})&=&\textstyle{\frac{1}{2}} \frac{1}{\theta_1(2z,\tau)^2\,\eta(\tau)^2}\sum_{a,b\in
  \mathbb{Z}} \textstyle{\frac{1}{2}} (\,\mathrm{sgn}((2b+1)n-(2a-\alpha)m)-\mathrm{sgn}(2b+1)\,)\non\\
&&\times \left(w^{- (2b+1)+2(2a-\alpha)}-w^{
    (2b+1)-2(2a-\alpha)} \right)\,q^{\frac{1}{4}
  (2b+1)^2+\frac{1}{2}(2b+1)(2a-\alpha)}.
\end{eqnarray}
This gives for $J_{m,n}=J_{1,0}$:
\begin{equation}
\begin{array}{ll}
h_{2,-C-f}(z,\tau;J_{1,0}) & = -B_{2,0}(z,\tau)\, g_{1}(z,\tau)/\theta_1(2z,\tau)^2, \non\\
h_{2,-C}(z,\tau;J_{1,0}) & = -B_{2,1}(z,\tau)\,g_0(z,\tau)/\theta_1(2z,\tau)^2, \\
h_{2,-f}(z,\tau;J_{1,0}) & = -B_{2,1}(z,\tau) \, g_{1}(z,\tau) / \theta_1(2z,\tau)^2, \non \\
h_{2,0}(z,\tau;J_{1,0}) & = -B_{2,0}(z,\tau) \,g_0(z,\tau)/\theta_1(2z,\tau)^2.\non
\end{array}
\end{equation}
The invariants for generic $J_{m,n}$ are obtained using the
generalization of $\Delta h_{2,\beta C-\alpha a}(\tau; J_{m,n})$
(\ref{eq:Deltah}). This gives for rank 3:
\begin{eqnarray}
\label{eq:h3Cf}
  h_{3,-C-f}(z,\tau;J_{m,n})&=&-\frac{i}{\theta_1(2z,\tau)\,\eta(\tau)}\sum_{a,b\in \mathbb{Z}}\textstyle{\frac{1}{2}} (\, \mathrm{sgn}((3b+2)n-(3a-2)m)-\mathrm{sgn}(3b+2)
\,)\non \\
&&\times \left(w^{- (3b+2)+2(3a-2)}-w^{ (3b+2)-2(3a-2)} \right)\,q^{\frac{1}{12}(3b+2)^2+\frac{1}{6}(3b+2)(3a-2)}\\
&& \times\, h_{2,bC-af}(z,\tau;J_{|3b+2|,|3a-2|}).\non
\end{eqnarray}

With Eq. (\ref{eq:blowupw}) for rank 3, the final result for $\mathbb{P}^2$ is:
\begin{equation}
\label{eq:r3w}
h_{3,-H}(z,\tau;\mathbb{P}^2)=\frac{h_{3,-C-f}(z,\tau;J_{1,0})}{B_{3,0}(z,\tau)}.
\end{equation}
The Betti numbers for $2\leq c_2\leq 6$ are presented in Table
\ref{tab:betti}. The first three lines agree with the three
Poincar\'e polynomials presented by Yoshioka \cite{Yoshioka:1996}.

\begin{table}[h!]
\begin{tabular}{lrrrrrrrrrrrrrrr}
$c_2$ & $b_0$ & $b_2$ & $b_4$ & $b_6$ & $b_8$ & $b_{10}$ & $b_{12}$ & $b_{14}$
& $b_{16}$ & $b_{18}$ & $b_{20}$ & $b_{22}$ & $b_{24}$ & $b_{26}$ & $\chi$ \\
\hline
2 & 1 & 1 & & & & & & & & & & & & & 3 \\
3 & 1 & 2 & 5 & 8 & 10 & & & & & & & & & & 42  \\
4 & 1 & 2 & 6 & 12 & 24  & 38 & 54 & 59 & &&&&& & 333\\
5 & 1 & 2 & 6 & 13 & 28 & 52 & 94 &  149 & 217 & 273 & 298 & && & 1968\\
6 & 1 & 2 & 6 & 13 & 29 & 56 & 108 & 189 & 322 & 505 & 744 & 992 &
1200 & 1275 & 9609 \\
&
\end{tabular}
\caption{The Betti numbers $b_n$ (with $n\leq
  \dim_\mathbb{C} \mathcal{M}$) and the Euler number $\chi$ of the moduli spaces of stable sheaves
  on $\mathbb{P}^2$ with $r=3$, $c_1=-H$, and $2\leq c_2\leq 6$.}
\label{tab:betti}
\end{table}

Note that Eq. (\ref{eq:r3w}) is rather compact and
expressed in terms of modular functions. Electric-magnetic duality suggests that
$h_{3,-H}(z,\tau;\mathbb{P}^2)$ exhibits modular transformation properties. Indeed, one
observes a convergent sum over a subset of an indefinite lattice of signature
$(2,2)$, when one substitutes the explicit expression for $h_{2,bC-af}(z,\tau;J_{|3b+2|,|3a-2|})$ in Eq. (\ref{eq:h3Cf}). Similar sums over lattices of signature $(n,1)$
appeared earlier in the literature for rank 2 sheaves \cite{Gottsche:1996,
  Gottsche:1998}, which can also be seen from Eq. (\ref{eq:genf2w}). A detailed discussion
of the modular properties of $h_{3,-H}(z,\tau;\mathbb{P}^2)$ and the
computation of $h_{3,0}(z,\tau;\mathbb{P}^2)$ will appear in a future
article \cite{Bringmann:2011}.

redpajama_set_name: RedPajamaArXiv


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Using excellent analytical skills, this person analyzes data and campaign results with an ability to observe patterns or anomalies in information. Specific job responsibilities include development, management and implementation of effective integrated social media campaigns, presentation of recommended strategies and campaign evaluation. Regular monitoring of campaign implementation and in-the-moment optimization of campaign tactics based on analysis, as well as data reconciliation and reporting. Strong ability to manage and optimize campaign performance to meet client goals. Performs ongoing market and industry research to maintain knowledge of trends, social media channels, audience/consumer behavior and competitor research.
An independent and critical thinker with a collaborative team spirit, the Social Media Specialist possesses the ability to visualize the big picture and is passionate and articulate in crafting compelling recommendations to present to clients. This person has a strong desire to be part of a team, which includes individuals with diverse backgrounds and expertise. A member of our team at A. Bright Idea asks the right questions, of themselves, the team and our clients, shares insights across teams, communicates constantly (early and often!) with clients and colleagues, leads and engages with a positive attitude and shares the spotlight for a job well done. A true "Bright Light" for solving communication challenges, a Social Media Specialist at A. Bright Idea thrives in our fun, fast-paced, high volume, quick-on-your-feet, respectful-of-all-ideas environment.


This worked for me! .
Same issue. qcow2 file was 20+gb, so I removed it. Upon re-running my 2 (small!) containers, it ballooned back to 8gb and then stabilized. I upgraded to rc3 today and it then jumped to 14gb, where it's now sitting.
Resize the image after it was created won't work. You can check the result by running a df -h on any container: docker run --rm alpine df -h. You can see the space left remains the same (about 60G).
In order to increase the image storage.
Create a new "template" image with desired size: qemu-img create -f qcow2 ~/data.qcow2 120G where the 120G is my new size.
Is there a maximum size for docker volumes?
Increase docker container disk space on OS X?
I deleted the file as described. However, I did an export of the images I wanted to retain before doing so. Then i imported the images after the delete. Worked great.
Same thing happened on one of my Linux boxes. So not strictly Mac related.
I have the same issue as above mentioned, tried thousands of things over the internet, no success!, as it turns out I changed my docker storage driver from devicemapper to aufs and everything now works like a charm.
I found that qemu-img was not already available on my machine. I was able to obtain it by brew install qemu.
This worked like a charm for me. Thanks!
But this command DID help!
i delete this file and lose all my images !
Tried using --quiet with same results.
Ah! Thank you! docker system prune -a worked wonderfully for me.
Previously when I got the out of space error, I would spend a lot of time finding and deleting "dangling" and "exited" processes, rebuilding all my images and restarting docker. But it seems with system prune -a you SHOULD NOT stop any processes or reboot docker — that apparently would cause the images to be cleaned up.
Then `y` to delete lots of containers, concluding (in my case) with: `Total reclaimed space: 23.26 GB`.
This fixed me thank you very much.


text:
Oblivious Insomniac
Pop culture musings from the perspective of an autistic, queer, trans woman with a love of all things sci-fi.
'Sympathy for the Devil' – The Suicide Squad
Posted byAmber Poppitt September 30, 2021 November 14, 2021 Posted inDC, Film
The Suicide Squad (2021) is pretty much what you'd expect a Suicide Squad movie directed by James Gunn to be like. What I mean is, it's fairly ruddy decent. A brass-necked, uncensored, somewhat sentimental and occasionally contemplative tale about a villainous batch of twisted misfits unwillingly taking on the role of superhero for the day. Following on from his dazzling success over at Disney with Guardians of the Galaxy Vol.1 and 2 (2014/2017), it is little wonder Warner Bros decided to snap him up when he became temporarily unemployed. Guardians successfully took a group of morally dubious misfits and turned them into lovable stars of their own show. Fans often commented on how Gunn had technically already made a successful Suicide Squad flick all the way back in 2014, when the Guardians franchise found itself regularly and aptly compared to DC's then-developing supervillain project. Therefore it was something of a no-brainer for execs to invite him aboard when opportunity arose.
Successfully achieving the task of framing its antagonistic protagonists as likeable and relatable is no easy task, one that could quite easily trip over itself if not executed well. An attempt to reinvent monsters as saviours could become redundant or toxic in the hands of a less talented filmmaker. Fortunately, this isn't the case here. The Suicide Squad doesn't tumble into the usual trappings of glamorising or justifying the actions of awful people . The film refuses to shy away from that fact that the characters driving its narrative have all done unspeakable things throughout their lives. They'll even continue to do a select number of unspeakable things even as the movie progresses. The film doesn't automatically sympathise with them from the word go. Neither does it directly attempt to absolve them of their crimes. Gunn's approach isn't to explicitly side with its lead characters. Instead he take's his applying context and logic to the thieves, murderous and psychopaths occupying the screen. They don't simply become good guys by doing good things, t hey become unethical protagonists who are products of a deeply unethical world. They are bad people at the lower end of a chain of bad people, captured in a vicious and uncaring cycle.
The reframing of antagonists as protagonists doesn't take place straight away, but gradually unfolds as the narrative progresses. Similarly to the way in which Alien (1979) doesn't reveal who the lead star is until the third act, the leads of The Suicide Squad don't earn their main-character statuses until a fair portion of their co-stars have been killed off. When the movie first opens, we are introduced to a vast array of nefarious folk stretching across the spectrum of sin. Their crimes vary wildly; from a man who was exposed to an inter-dimensional virus by his mother as a child, to a kid-slaughtering beast shaped like a weasel. With little time, the film goes about wiping a majority of these characters from the face the earth; usually in a dizzyingly brutal manner. Gradually, the cast is whittled down to a select few. From the fog of viscera, a band of leads emerge. All of them are still criminals who were locked away in a bid to protect the societies they belonged to, yet when the time comes for them to take control of the plot, Gunn has spent much of the film applying as much context to their characters as possible.
Let's take Robert Dubois' Bloodsport (Idris Elba) as our first case in point. Gunn's depiction retcons the characters' white supremacist background from the source material and reinvents him as a British mercenary serving time for shooting Superman with a Kryptonite bullet. This version of Dubois' prime reason for earning a supervillain badge leans entirely down to his attack on Clarke Kent. On the surface, attempting to kill the Man of Steel is certainly not a decent act to commit. Not only is the person you're aiming to injure capable of bursting Dubois like a balloon, he's also perceived as a beacon of hope and justice across the globe. Shooting superman is a surefire way to soil one's CV. Then again, we have no idea what context in which this particular battle was set in. What motivated Dubois to take such a risky shot in the first place? I ask this because we've seen Superman do some pretty terrible things over the past couple of years. That shiny, boy wonder image from the Richard Dinner movies isn't the most dominant image held of Clark Kent in today's culture. He's snapped someone's neck, caused unspeakable amounts of collateral damage while tussling with General Zod in Metropolis, and has even attempt to kill the Justice League in both cuts of their own movie. Who knows what skullduggery was kicking off to motivate Dubois to fire a Krypto bullet at him. Furthermore, we've even seen the Dark Knight himself attempt to commit the very same crime of trying to murder supes with Kryptonite weaponry app the way back in Batman V Superman: Dawn of Justice (2016). If we're to buy the idea that Bruce Wayne can get away with this kind of act, surely it's no stretch of the imagination to consider Bloodsport to be not the worst supervillain out there. I think Gunn was aware of the current perception of Superman when writing this plot point into this movie, allowing him to set up a villain who's actions can simultaneously be considered reprehensible and potentially justifiable simultaneously.
Next up there's Harley Quinn, who's a fairly awful person among other things, but for the most part, is depicted her as a surviver in some form of recovery after being groomed by that gangster/Joker hybrid we saw her knocking about with several years back. Although she's not going to win any awards for most well-behaved student anytime soon, The Suicide Squad distances her enough from her past to make her more sympathetic than before. When she kills, and boy does she kill, she does so to people who are either torturing her, or are assisting in the torture being inflicted upon her. She even outright shoots a man dead after he expressed a willingness to inflict harm upon children, which she expresses deep distress at. As with Dubois, Quinn's brutality is somewhat muffled by the types of people she's choosing to harm.
Another who inevitably survives this movie is King Shark (Sylvester Stallone). If you were to watch this movie on mute without any subtitles, you'd be forgiven for thinking Shark King was one of the most grotesque characters in this feature. Much of the promotional footage that dropped leading to its release showed this character devouring fully conscious men and tearing them to shreds. All that viscera and agony surely couldn't be caused by a cuddly bundle of joy. Except King Shark's depiction here is vastly different to previous iterations of the character, so much so, there are times in which he comes across as somewhat adorable. He's portrayed as an extremely dangerous toddler incapable of comprehending the pain and suffering his actions cause. There appears to be so little going on in King Shark's brain, it's almost impossible to hold him accountable for his actions. He's a big, dumb, powerful beast who's been locked away solely for being a big, dumb, powerful beast. His actions aren't driven by any form motive or malice. He's more or less an R-rated King Kong. He may cause devastation, but he wouldn't have done so had the nasty humans not put him there in the first place!
The most prominent example of the film's surviving line up is Cleo Cazo's Ratcatcher 2 (Daniela Melchior), who perhaps serves as the movies emotional core. Cazo is the daughter of the decased Ratcatcher (Taika Waititi). Cazo was raised homeless, and her father suffered from a heroin addiction which would inevitably take his life. After escaping to America, Cazo attempts to survive by pursuing a criminal career, for which she quickly results in her incarceration. Cazo's experience of rejection and imprisonment by the society she was born into led her has led her to where she is today. She is very much a product of a society that has failed and misunderstood her before long before she was even born. In a rather lovely flashback during the film's final act, a young Cazo asks her father why he chose to control rats in the first place. Her father responds by proclaiming "rats are the lowliest and most despised of all creatures, my love. But they have purpose. So do we all".
Ratcatcher's response to young Cazo's question cuts deep into the core of The Suicide Squad's underlying themes of rejection and exploitation. Chewed up and spat back out by a society now using them as disposable weapons, the survivors of this mission may have been cast into the rouges gallery of their universe, yet for the most part, they are beings damaged and imprisoned by an unforgiving society which doesn't think twice about using them as expendable commodities. While the surviving line-up who make the cut from supervillain to protagonist aren't the most delightful of individuals, there's an element of sympathy to them, warming us enough to them in time for the final battle. It's a trick Gunn utilised in Guardians too, albeit through a more a family-based model than a societal one. It's a solid way of making a concept like this function without too much it feeling too mean-spirited or ham fisted.
In spite of all his work to get us to side with these characters, however, Gunn also likes to remind us that we're still watching a film populated with villains. In order to maintain the premise of the movie, a balancing act must be committed. On the one hand, we need to like these characters on some level, yet we must also acknowledge that they aren't your traditional goodies. The Suicide Squad achieves this usually in the form of humour, which is the best way of allowing Gunn to have his cake and eat it. Want to depict the people you're trying to get everyone to root for as devious supervillains? Serve up said reminders in the form of a gruesome joke.
Perhaps one of the best examples of this comes early on in the movie, in which audiences are treated to a lengthy sequence where our "heroes" brutally slaughter a group of perceived enemies. The violence grows more creative and grotesque as the minutes progress. The punchline of the joke comes when we learn of the colossal misunderstanding that's occurred. The bad guys Bloodshot and the gang were mutilating weren't bad guys at all, but members of a gorilla faction also trying to overthrow the same regime that they are. It's an uncomfortably hilarious scene in which audiences are both reminded of the sorts of people we're hanging out with here, whilst also reminding them how messed up the whole concept of a The Suicide Squad movie really is.
All of this is great stuff, and confirms Gunn's talent at making these sorts of movies. Yet the film is far from perfect, however, as there is something of an identity crisis at its core which does cause it some issues at times.
First things first, I have no intention to bad mouth Suicide Squad (2016) in this post. It's far too common to describe the 2021 version as "better than the last" when asked to offer an opinion. I'll admit, I have also been guilty of this on several occasions. I'm now starting to feel a little bad at the idea of constantly throwing shade at a film that's been repeatedly dragged through the mud for over half a decade now. I therefore have no more desire to kick a feature that's already been pummelled to a pulp at every opportune. Nevertheless, because of this features' frankly bizarre and confuzzled relationship with its predecessor, we will need to examine the pair side-by-side to some extent. Which leads me onto my next point.
After the critical failure of David Ayer's Suicide Squad back in 2016, topped off with the divisive fan discourse pulling the Snyderverse vision apart, Warner Bros panicked and made a colossal U-turn with regards to it's proposed extended universe. This resulted in various DC projects branching away from the interconnect concept and becoming standalone features instead. 2019's Joker is perhaps the most noteworthy example of this to date, functioning as a film that aggressively adapts part of the Batman source material to make a feature as isolated as its main lead is from society. Though controversial, the film was a collosal hit, and since then, DC opted to stop trying to mimic the Marvel formula, focusing on developing a series of solitary projects instead.
Except this vision hasn't been the most clear cut of decisions. Follow up features such as Wonder Woman (2017), Aquaman (2018), Shazam (2018), Birds of Prey (2020), and Wonder Woman 84 (2020) have all kind of taken place in the Snyderverse. At least we're led to believe this is the case, what with all the returning characters and references alluding to a shared space. Only their lack of direct references and sketchy continuity to preceding movies implies they are only connected to an extent. To make matters even more convoluted, 2021 saw the inasne and entertainingly ambitious Zack Snyder's Justice League release on HBO max, which further tried to allude to a wider interconnect vision that will never come to pass. Obviously Zack Snyder's Justice League was intended as an expensive side-project for fans. Still, it's existence further tried.to act as a hub for a redundant timeline that's still kind of ongoing, albeit via a series of projects that partly distance themselves from the timelines they claim to belong to. If Wonder Woman 84 (2021) takes place in its own self-contained universe, yet is also follow on from a film that referenced and followed on from Batman v Superman, then how do all these films fit together? Also, if Zack Snyders' Justice League does take place in a totally different timeline, why does it feature more references and connections to the Suicide Squad, Wonder Woman 84 and Batman V Superman than the supposed "canon" Justice League (2017)?While all of these semi-connected/semi-not-connect plots may have some "logic" applied to them after the upcoming Flashpoint movie gets made, at the moment, we're left unsure as to what is a sequel or spin off, and what isn't. While Warner Bros. aren't the only guilty party in this respect (Sony and Marvel currently have their own continuity gaffs going on with Venom [2019] and Morbius [2022], but that's for another post), a sizable chunk of their DC projects can sometimes feel as though they are trying to both be apart of something bigger while simultaneously trying to be standalones.
The Suicide Squad is another such film that suffers from this problem. Bringing James Gunn aboard was a sound creative and financial decision on Warner Bros behalf. After the negativity following on from its predecessor's release, the studio wanted to do some damage control in order to save the property from slipping into financial turmoil. Their plan was to chuck out Ayer's vision – or at least the studio's initial vision – and bring in a filmmaker with a successful track record at making these sorts of movies. Fast forward a few years, and you have James Gunn's The Suicide Squad; a fresh vision with a new story and direction for the sub-franchise. Except despite being a reboot, it's also a sort of a sequel too. It doesn't directly reference any of the events from the first film, but it also brings in a bunch of cast members from the first movie. There are few hints suggesting Ayers' Suicide Squad happened in this timeline – such as returning characters acknowledging one another as old companions – yet all the emotional scarring or character development from the previous is practically non-existent.
This kind of sequel/not sequel approach can at times distract from the bigger picture. It wants to be a reboot constructed by someone with a vision separate from the original, but at the same time it occasionally forgets this and shifts back to being a sequel. This wouldn't be a problem if the film positioned itself as a sequel headed in a new direction, but the fact that both the story and the filmmakers refuse to acknowledge this in either direction creates an ambiguity that gives the feature something of an identity crisis. It disregards its predecessor while looting its corpse at the same time.
Perhaps this sequel ambiguity is more of a pet peeve on my behalf than a genuine problem with the film itself. Nevertheless, I can't help but feel that The Suicide Squad sometimes suffer from the same franchise insecurity a number of its neighbouring projects share.
In any case, the film as a whole holds out really well. James Gunn is evidently in his element with this one, and the creative freedom offered to him helps to provide a thrilling new vision for the Suicide Squad series to bask in. Funny, entertaining, heartfelt and a film that has genuinely has something to say. Here's to hoping Warner Bros keep Gunn on their pay grade. If we can get more DC flicks out of this guy, then Marvel will have to seriously consider upping their game.
Published by Amber Poppitt
I'm an autistic, trans writer from the UK with dreams of someday becoming a professional screenwriter. I also happen to be a huge film/TV/novel enthusiast with an undying obsession toward Doctor Who.	View more posts
'The Second Coming' – What Russell T Davies' Return means for the Future of Doctor Who
'The Birth of a Perished Vision' – Zack Snyder's Justice League
Oblivious Insomniac, Blog at WordPress.com.

redpajama_set_name: RedPajamaCommonCrawl


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As US scrambles to fill highly skilled jobs, Germany's vocational education system offers ideas
Lukas Graf travelled coast-to-coast in the US discussing apprenticeships and higher education.
Since the 1990s, various administrations – both Republican and Democratic – have launched programmes to expand the pool of highly skilled labour in the US. President Donald Trump's Executive Order in 2017 to expand and reform apprenticeships followed suit - amid ever greater demand for workers with advanced training. At 3.7% unemployment in October 2018, the US job market was so tight that the Federal Reserve warned the gap in highly skilled labour in some industries could restrain growth.
This crisis has been long in the making. In contrast to Germany, the US has no uniform, nationally mandated system of training for vocations. At the same time, the cultural bias toward 'college for all' in the last half century has deterred young people from pursuing highly skilled jobs acquired through occupational training - and it has saddled many of them with debt. Now the US is looking at ways to make such jobs more attractive.
Many hope that Germany's much-revered apprenticeship system, long held up as a model for other countries, could provide some answers. The Obama administration, too, created a fund for apprenticeships in 2015. "This may be one of the few bipartisan issues left at the moment," not to mention a refreshing sign of transatlantic cooperation, says Lukas Graf, Assistant Professor of Educational Governance at the Hertie School, after a trip to the US last summer to present his research on skill formation and labour markets in the US and Germany.
But he term "apprenticeship" has become something of a catch-all phrase for a work-based system of training highly skilled workers. In fact, next to traditional apprenticeships, Germany has another system combining on-the-job training and higher education that may be a better fit for the US, Graf says. So-called dual-study programmes, a hybrid of paid work in firms and post-secondary education, are on the rise in Germany where they are breaking down traditional boundaries between higher education and vocational training.
Germany's 'learn and earn' dual-study concept may be more compatible with the US, where there is an established community college-based system of vocational education, such as that offered by Central Piedmont Community College, a leader in occupational training, which Graf visited in Charlotte, North Carolina.
Germany's apprenticeships are traditionally work-based programmes beginning at the secondary-school (high school) level. German high-school pupils traditionally choose around age 16 whether to pursue a vocational track or continue on a university-bound track - as opposed to the US, where vocational education often begins after graduation from high school. In Germany, nationally recognised qualifications are designed jointly by stakeholders: business associations, chambers of commerce and unions in a particular industry, and are mandated by federal law. Around half of all young people born in the same year currently choose to enter an apprenticeship in one of the 330 certified vocations. At the end of 2016, there were 1.32 million apprentices in Germany, with around 20 percent of all companies in Germany participating in the training system.
In US, stakeholders debate the best path
While the Obama efforts had already increased funding for vocational education in the US, the Trump administration wants apprenticeships that are structured and supported at the industry level. Somewhat different from previous proposals, this would give more authority to employers.
"This strategy has sparked a lot of debate," Graf says. "And those involved are seeking knowledge from countries like Germany with a long experience in this area."
Rather than talking to policy-makers "inside the beltway" in Washington, Graf was invited by the American Council on Germany, which is the American sister organisation of the Atlantik Brücke, and the German-American Chamber of Commerce, to spend time with experts working at ground level across the US. At meetings with stakeholders, practitioners, people in education and training, and deans of community colleges in San Francisco, Atlanta, and Charlotte, North Carolina, Graf presented the German model and discussed its compatibilities – and incompatibilities – with that of the US.
"These were people from different sectors who don't always have the opportunity to exchange their experiences: representatives from high schools, community colleges, liberal arts colleges, employers and local politicians," Graf said. "Everywhere I heard the same thing – 'it's great we are all finally talking to each other'."
The reasons for the rise of dual-study programmes in Germany, as opposed to traditional apprenticeships, are common to both countries: besides skills that meet stringent occupational standards, future workers will likely need the kinds of innovative capacity that that students get from a university education, Graf told his audiences in the US. In turn, this speaks in favour of programmes that combine practical skills from the workplace and the thinking and theory of the seminar room.
In the US, "some work-based higher education programmes resemble the German dual-study programmes, including higher-end apprenticeship programmes offered by American community colleges as well as a range of co-op programmes," Graf said in a 2017 paper published in the American Political Science Association Journal, PS, together with Justin J.W. Powell of the University of Luxembourg. In addition, the lack of comprehensive national occupational standards, weak cooperation between business associations and unions and decentralized relationships between firms across industry may make dual-study more attractive than traditional apprenticeships in the US.
What can they learn from each other?
Although the German model of apprenticeships has a long established history, this, too is undergoing changes to adapt to new work demands and changes in education – and policymakers in Germany even look to the US for inspiration. While in the US, employers want more practical skills among academic graduates, in Germany there is a demand for more academic skills for those in trained occupations, Graf points out.
"Actually, we have a lot to learn from each other," he says. "The US can inspire Germany to create more flexible links between academic education and vocational training as well as to promote co-op programs in which students rotate between employers for training."
Some argue that old-school German apprenticeship training is too focused on how industry worked in the past, and not connected enough to the new world of technology and innovation. "The German economy needs more creative capacities in emerging sectors – but how do you link the old skilled world to the new world that requires innovation?" Graf asks.
The dual-study system is not without pitfalls, however. Some in Germany argue it will lead to an academisation of areas whose traditional strength is in hands-on skilled workers. And it may also give too much power to employers – a criticism of the Trump apprenticeships as well.
"Apprenticeships in Germany presuppose collective governance, so in transferring this model to higher education, stakeholders need to be organised," Graf says. In the dual-study system, the key partners tend to be employers and educators, rather than the traditional social partners of business associations and employee representatives.
This throws up a host of issues. "So, for example, when unions are involved, they usually aim to standardise wages, and as a result it is more difficult for companies to poach employees after investment in training," Graf points out. "Can apprenticeships be successful in a liberal economic model, like the US and not just in places where a system of collective bargaining between employers and labour has a long tradition, like Germany?"
On the other hand, such cooperation creates synergies between employers and educators to work together to reduce the skills mismatch in a technologically changing economy – both in Germany and the US. But "if private actors in for-profit companies, with specific skill demands, are engaged in education as policy makers, we need to make sure that students get a broad skill-set that benefits society at large in the future." Graf says.
The question policymakers in both countries need to consider is: "Whose interests prevail in these set-ups – is it the employers or the higher education institutions? That is, are the higher education organisations in a position to put constraints on employers, for example to enhance collective action or provide collective goods like transferable skills?"
More about Lukas Graf
Lukas Graf , Assistant Professor of Educational Governance
PhD Workshop brings together the next generation of higher education scholars
Social, labour and education policy
German education is better than its reputation - study


Of course, the biggest draw in New York for your dog is Central Park, which is an urban oasis with miles of paths, wide open meadows and plenty of things to do with your dog. Pets are welcome in Central Park as long as they and their owners follow the Park's rules and regulations. Specifically, dogs must be leashed at all times except in designated off-leash areas. There are dog drinking fountains located around the Park. Try to keep Rover from digging in the flowerbeds or playing (or drinking) from fountains and bodies of water. There are 23 designated dog friendly areas in Central Park. There are a handful of areas of the park that are off limits to dogs, most notably the ball fields, playgrounds, sand volleyball court, lilac walk, reservoir running track and sheep meadow. Central Park Paws is an organization that holds events specifically for dogs and their owners year-round in Central Park. Check out our many off-leash dog parks with maps and all! Everything's listed in our New York City Travel Guide.
When it comes to pet friendly New York City hotels, Manhattan is paws-itively amazing! Vacation Pet Friendly has the largest listings of the impressive list of pet friendly hotels as well as various information and weight or breed restrictions. If you think traveling to New York City with your dog is going to shut you out of the best hotels – you could not be more wrong. Hotels like the London NYC, all of the W hotels, the Loews Regency, the SoHo and TriBeCa Grand Hotels, the Waldorf Towers and the Carlyle take pets. Book these hotels and more through Vacation Pet Friendly and unlock special deals for our pet owners! Start your search here!


The latest meeting details are posted to the DDA Yosemite National Park meeting page as they come in.
The 2000 DDA meeting will be held at Yosemite National Park, California, April 9 to 12. The first scientific session begins on Sunday the 9th at 2 pm. There will be a welcoming reception on the evening of the 9th and a banquet the evening of the 11th. The DDA Yosemite meeting page contains links to information about Yosemite National Park, including travel information.
More to come — keep an eye on the Yosemite meeting page.
Lodging deadline: We have reserved a block of rooms at the Yosemite Lodge, but that block goes away after March 9. To get a room in the DDA block, you must make reservations by then. For reservations information, please see the DDA Yosemite room reservations page. Of specific note: you MUST use the registration form supplied by the Yosemite Concession Services and available on the DDA Yosemite room reservations page. Room reservation forms can be either mailed or faxed to Yosemite Concession Services.
It occurs to me to suggest that you add a note to the Newsletter encouraging attendees to share rooms with others. Our room block is not all that large, the rooms are pricey (the flood is the culprit here), and the attractions of Yosemite are its great outdoors (NOT the four walls of your room).
Airports of choice include SF and Fresno. Bakersfield is worth mentioning, even though Fresno is closer to the Park. Of course, Los Angeles is always in the equation, particularly for members linking up with our southern California attendees for the final carpool to the Park.
There are four entrances to the Park; three from the West and one from the East (via Lee Vining). The entrance from the East will be closed due to snow at the time of our meeting. The western entrances each have their advantages. The northernmost entrance (via Rte. 120) is best for attendees arriving via San Francisco or other points north. The southernmost entrance (via Rte. 41) is similarly best for attendees such as myself arriving from points in southern California. The central entrance (via Rte. 140) provides access through El Portal, where the Cedar Lodge is located. Cedar Lodge provides comfortable accommodations closest to but outside of the Park. For our attendees slow to make room reservations, Cedar Lodge may be the best option after our room block fills up.
A map, of moderate adequacy, is available of the Yosemite environs at the Yosemite.com link on our website. In all honesty, if you are unfamiliar with the necessary details of the roads in California, I think you should be advised to get a detailed road map of the state from your travel agent.
Click on the "I want to use the Web form" button.
Fill in the number of authors and the number of collaborations/teams.
Click on the "Create an abstract form for me!" button.
There is no separate charge for submitting an abstract. The abstract fee is included in the meeting registration fee.
Detailed instructions for electronic abstract submission are shown below.
The meeting registration form is available at http://dda.harvard.edu/meetings/yosemite2000/reg.html.
Instructions, fees, etc. are listed on the form. Please print it out and mail it along with your check to the address specified near the top of the form.
The DDA is again providing up to two $400 stipends each on a competitive basis to students attending the 2000 DDA meeting in Yosemite National Park. (See http://dda.harvard.edu/ddastip00.html for information about the stipend program.) The applications for the 2000 meeting are due by February 1, so act quickly! The applicants will be notified by e-mail, and the stipend will be presented to the recipient(s) at the meeting.
You should be receiving the 2000 Election Ballot and candidate statements the week of February 28 (in accordance with the DDA bylaws). When you receive your ballot, please mark it and return it promptly in a signed envelope. Results will be announced via email a few days before the meeting, as well as at the meeting.
The DDA is moving towards allowing members to cast their ballots via electronic means (email, web, FAX), in addition to the traditional snail mail. As with all things formal, the process takes time. Currently, the DDA Committee is hashing out the exact wording of the proposed changes. In accordance with the DDA bylaws, these changes will be submitted to the AAS Council at the next Council meeting in July. After that, the changes (as well as pertinent comments from the AAS Council and the DDA Committee) will be distributed to the DDA membership. The amendments will then be discussed at the 2001 DDA Business meeting, followed by formal ballots sent to the entire membership. Assuming the amendments pass, then the electronic voting options will be available for the 2002 DDA elections.
The U.S. Naval Observatory is making available a CD-ROM version of USNO/AE98, the high precision ephemerides of fifteen of the largest asteroids. These are the asteroid ephemerides which will appear in the Astronomical Almanac beginning with the edition for the year 2000. USNO/AE98 includes the asteroids 1 Ceres, 2 Pallas, 3 Juno, 4 Vesta, 6 Hebe, 7 Iris, 8 Flora, 9 Metis, 10 Hygiea, 15 Eunomia, 16 Psyche, 52 Europa, 65 Cybele, 511 Davida, and 704 Interamnia. USNO/AE98 is based on the JPL DE405 planetary ephemerides and covers the period 1799 November 16 (JD 2378450.5) through 2100 February 1 (JD 2488100.5). The CD-ROM also includes software to compress the ephemerides and interpolate the ephemerides for any arbitrary date within the range of USNO/AE98.
The CD-ROM is ISO 9660 compliant and will work on all systems with ISO 9660 compliant CD-ROM drives.
For more information on USNO/AE98, including the new mass determinations for Ceres, Pallas, and Vesta, see Hilton, J.L. 1999, 'U.S. Naval Observatory Ephemerides of the Largest Asteroids,' Astronomical Journal, 117, pp. 1077-1086. (Click here for ADS Abstract.) For a copy of the CD-ROM contact either James Hilton or The Astronomical Applications Department, U.S. Naval Observatory, 3450 Massachusetts Ave. NW, Washington, DC 20392, or send an email to jhilton@aa.usno.navy.mil.
The AAS is again handling abstracts for our DDA meeting. The AAS web page to do this has not been set up yet, so the information below is only a highly-likely guess, based on last year's instructions. When the AAS page is ready (the first week of February), I'll immediately send out a notice with further instruction via email. You cannot submit an abstract until the AAS form is ready after the first of February.
When you go to the above link, press the "I want to use the Web form" button, then select "DDA 2000", fill in the number of authors and number of collaborations, and press the "Create an abstract form for me!" button.
If you have any questions about electronic submission while you are preparing your abstract, please contact the AAS Executive Office for assistance. You can call the office at 202-328-2010, or send email to abs-help@aas.org.
Two acknowledgments are sent. The first is a copy of the submission sent within minutes to the submitter's browser. The second is sent to the first/presenting author's email address, it confirms acceptance and includes a running identification number.
If you do not receive these confirmations, DO NOT RESUBMIT THE ABSTRACT. Please contact the Executive Office by calling 202-328-2010 or by sending email to abs-help@aas.org.
If you notice an error in your abstract after submission, send details of the correction to abs-help@aas.org and the correction will be made for you. We cannot guarantee that corrections received after the abstract deadline will be processed.
NOTE: If you do not have access to the Web, the AAS Executive Office will make arrangements to include your abstract. Please contact them for assistance, 202-328-2010, or send email to abs-help@aas.org.


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Q: Convert HTML to properly formatted attributedString I need to convert HTML data that consists of
..
,
..
 and  elements into the attributedString with a proper formatting. I want to assign
 to UIFontTextStyleHeadline1 and
 to UIFontTextStyleBody and store image links. I need the output to be attributedString with heading and body elements only and I will handle the images separately.
So far, I have this code:
NSMutableAttributedString *content = [[NSMutableAttributedString alloc]
         initWithData:[[post objectForKey:@"content"]
    dataUsingEncoding:NSUTF8StringEncoding]
              options:@{NSDocumentTypeDocumentAttribute: NSHTMLTextDocumentType,
                   NSCharacterEncodingDocumentAttribute: [NSNumber numberWithInt:NSUTF8StringEncoding]}
   documentAttributes:nil error:nil];

which outputs to something like this:
Heading
{
    NSColor = "UIDeviceRGBColorSpace 0 0 0 1";
    NSFont = "<UICTFont: 0xd47bc00> font-family: \"TimesNewRomanPS-BoldMT\"; font-weight: bold; font-style: normal; font-size: 18.00pt";
    NSKern = 0;
    NSParagraphStyle = "Alignment 4, LineSpacing 0, ParagraphSpacing 14.94, ParagraphSpacingBefore 0, HeadIndent 0, TailIndent 0, FirstLineHeadIndent 0, LineHeight 0/0, LineHeightMultiple 0, LineBreakMode 0, Tabs (\n), DefaultTabInterval 36, Blocks (null), Lists (null), BaseWritingDirection 0, HyphenationFactor 0, TighteningFactor 0, HeaderLevel 2";
    NSStrokeColor = "UIDeviceRGBColorSpace 0 0 0 1";
    NSStrokeWidth = 0;
}{
    NSAttachment = "<NSTextAttachment: 0xd486590>";
    NSColor = "UIDeviceRGBColorSpace 0 0 0.933333 1";
    NSFont = "<UICTFont: 0xd47cdb0> font-family: \"Times New Roman\"; font-weight: normal; font-style: normal; font-size: 12.00pt";
    NSKern = 0;
    NSLink = "http://www.placeholder.com/image.jpg";
    NSParagraphStyle = "Alignment 4, LineSpacing 0, ParagraphSpacing 12, ParagraphSpacingBefore 0, HeadIndent 0, TailIndent 0, FirstLineHeadIndent 0, LineHeight 0/0, LineHeightMultiple 0, LineBreakMode 0, Tabs (\n), DefaultTabInterval 36, Blocks (null), Lists (null), BaseWritingDirection 0, HyphenationFactor 0, TighteningFactor 0, HeaderLevel 0";
    NSStrokeColor = "UIDeviceRGBColorSpace 0 0 0.933333 1";
    NSStrokeWidth = 0;
}
Body text, body text, body text. Body text, body text, body text.
{
    NSColor = "UIDeviceRGBColorSpace 0 0 0 1";
    NSFont = "<UICTFont: 0xd47cdb0> font-family: \"Times New Roman\"; font-weight: normal; font-style: normal; font-size: 12.00pt";
    NSKern = 0;
    NSParagraphStyle = "Alignment 4, LineSpacing 0, ParagraphSpacing 12, ParagraphSpacingBefore 0, HeadIndent 0, TailIndent 0, FirstLineHeadIndent 0, LineHeight 0/0, LineHeightMultiple 0, LineBreakMode 0, Tabs (\n), DefaultTabInterval 36, Blocks (null), Lists (null), BaseWritingDirection 0, HyphenationFactor 0, TighteningFactor 0, HeaderLevel 0";
    NSStrokeColor = "UIDeviceRGBColorSpace 0 0 0 1";
    NSStrokeWidth = 0;
}

I am new to attributedString and seek for an efficient way to convert these attributes into the standard fonts mentioned above. Thank you.

A: If somebody would seek something similar I am on the end using TFHpple librabry to separate images from text elements in HTML data and then I change format attributes of the attributedString like this:
NSString *contentString = [self parseHTMLdata:bodyString];

NSMutableAttributedString *content = [[NSMutableAttributedString alloc] initWithData:[contentString dataUsingEncoding:NSUTF8StringEncoding] options:@{NSDocumentTypeDocumentAttribute: NSHTMLTextDocumentType, NSCharacterEncodingDocumentAttribute: [NSNumber numberWithInt:NSUTF8StringEncoding]} documentAttributes:nil error:nil];

// prepare new format
NSRange effectiveRange = NSMakeRange(0, 0);

NSDictionary *attributes;

while (NSMaxRange(effectiveRange) < [content length]) {

attributes = [content attributesAtIndex:NSMaxRange(effectiveRange) effectiveRange:&effectiveRange];

    UIFont *font = [attributes objectForKey:@"NSFont"];

    if (font.pointSize == 18.0f) {

        [content addAttribute:NSFontAttributeName value:self.headlineFont range:effectiveRange];

    } else {

        [content addAttribute:NSFontAttributeName value:self.bodyFont range:effectiveRange];
    }
}

And the hpple part:
- (NSString *)parseHTMLdata:(NSString *)content
{
    NSData *data = [content dataUsingEncoding:NSUTF8StringEncoding];

    TFHpple *parser = [[TFHpple alloc] initWithHTMLData:data];

    NSString *xpathQueryString = @"//body";

    NSArray *elements = [[[parser searchWithXPathQuery:xpathQueryString] firstObject] children];

    NSMutableString *textContent = [[NSMutableString alloc] init];

    for (TFHppleElement *element in elements) {

        if ([[element tagName] isEqualToString:@"h2"] || [[element tagName] isEqualToString:@"p"]) {

            if ([[[element firstChild] tagName] isEqualToString:@"a"]) {

                // image element, just save it in array
            } else {

                // pure h2 or p element
                [textContent appendString:[element raw]];
            }
        }
    }

    return textContent;
}

Checking the font size in attributes may seem fragile, if it would cause some problems I can dig deeper into paragraph style which holds the heading/body tags.

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