I joined Oxford Polytechnic in August 1988 and worked in the School of Education (as it then was) for 9 years, gradually increasing my hours as a course secretary, latterly as PA to Head of School, as my children got older. In 1997 I became Executive Assistant to the then new Vice-Chancellor, Professor Graham Upton and stayed with him till he retired in 2007, when I moved to a new position, Corporate Relations Manager, looking after our Honorary Graduates, the University Court, the Chancellor and other good friends of the University.
I "retired" in 2010 and was asked to join the Brookes Society Committee. I am a great advocate of Brookes and what it does, both for our students and for the local community, and I wanted to continue my involvement with the University, as well as helping to keep ex-members of staff in touch with each other and offering them the opportunity to remain involved, should they wish to do so.
Joining Oxford College of Technology as assistant librarian in January 1969, I became the Subject Librarian for architecture and planning in 1972, with responsibility for book acquisitions, enquiry work and instructing students in information resources in those subjects, which also encompassed construction and surveying.
In 1974-5 I was released on sabbatical in order to gain a Master's degree in Information Science at Loughborough University, and was subsequently able to develop areas of professional interest as Oxford Polytechnic grew in size and numbers. Projects I was involved in included founding ARCLIB, the UK architecture librarians' group, through which I developed useful overseas links with similar groups in France, Spain, Italy, Scandinavia and Germany.
Along with the University of Hertfordshire's Department of Engineering, I developed an early web resource for urban design, later spun out by Oxford Brookes as its first commercial company in 1998. Leaving the Library staff I became CEO of RUDI, the Resource for Urban Design Information.
I later chose to become self-employed and worked on various research projects for government and associated bodies, including some with Brookes' Schools of Planning and Architecture. I was always happy working at Oxford Brookes/Polytechnic/College of Technology, where I made many friends and since retiring in 2008 I have enjoyed meeting up with former colleagues at Brookes Society events. In joining the Committee I hope to continue the good work of those colleagues who originally set the Society up.
I joined the Business School in 2001 where I worked in the Undergraduate Office preparing examination papers and supporting the School's collaborations in Poland, India, Hungary, Germany, South Africa and Hong Kong. In October 2002 I became PA to the Dean of the School, a varied role which I enjoyed tremendously. I 'looked after' four Deans until my early retirement in July 2016. All four Deans worked towards moving the School to the heart of the University at Headington which happened finally in 2017.
Brookes Society has proved to be a good way to retain a link with the University and keep in touch with colleagues and friends.
I retired from the Publishing department at Brookes at the end of 2015, but returned to teach a couple of MA courses through 2016.
I started there in 1994, first teaching only part time and then as a half-time permanent Senior Lecturer, becoming full time the following year. From 2005 to 2010 I was the Publishing Field Chair (a now vanished role from Dave Scurry days, essentially the undergraduate course leader) and also did a three-year stint as an academic conduct officer. In the department I specialised in editorial subjects and academic publishing, as well as journals and scholarly communication, having come from editorial and training roles in academic publishing before I joined Brookes.
My own academic background is in Philosophy, as an undergrad at Exeter University and postgrad at UCL, though originally, a hundred years earlier, I had started out as an Engineering student in aerospace. I was also the Assistant Editor of the Political Quarterly for some fifteen years, retiring from that role in 2016.
I am a Chartered Accountant of some 44 years experience having been with Price Waterhouse Coopers from leaving University - being a student of theirs until I qualified and progressed with them to Partnership level. In 1988, I came to Oxford with PwC to integrate 12 companies into one and afterwards was sent off to the American Mid West to do the same for a world-wide pharmaceutical business that needed financial guidance.
I joined Oxford Brookes University in 2002 as Assistant Director of Finance at the time the university first needed outside commercial views to enable it to return to profitability.
I am married to Jan and we luckily have two children - Rachael and Oliver - who over the last 6 years have presented us with 4 grandchildren - 3 boys and 1 girl - all of whom keep us busy as grandparents!


text:
Life of Muḥammad: (18) The Raid on B. Qurayẓa and Mass Beheadings of the Jews
"According to what al-Zuhrī told me, at the time of the noon prayers Gabriel [Jibrīl] came to the apostle wearing an embroidered turban and riding on a mule with a saddle covered with a piece of brocade. He asked the apostle if he had abandoned fighting, and when he said that he had[,] he said that the angels had not yet laid aside their arms[,] and that he had just come from pursuing the enemy. "God [Allāh] commands you, Muhammad, to go to B. Qurayẓa. I am about to go to them and shake their stronghold."
Thus, Muḥammad is profiled as exhausted from war after the siege, but Jibrīl comes as a warring angel, commanding Muḥammad to attack the B. Qurayẓa. They have been successfully driven from the siege, and too, may be tired of any prospect of war at this juncture. As 'Alī approaches their fort, he hears "insulting language used of the apostle," and shortly thereafter, as Muḥammad approaches, he answers, calling to them, " 'You brothers of monkeys, has God [Allāh] disgraced you and brought his vengeance upon you?' They replied, 'O Abū'l-Qāsim, you are not a barbarous person.' "
A siege of twenty-five nights ensues as "Gabriel [Jibrīl] [ ] was sent to B. Qurayẓa to shake their castles and strike terror into their hearts" and "until they were sore pressed and God [Allāh] cast terror into their hearts." Ka'b b. Asad, their chief, knowing he has broken treaty with Muḥammad at the battle of the Trench, knows the B. Qurayẓa are desperate. So he presents three awful possibilities to his fellow Jews:
" 'O Jews, you can see what has happened to you; I offer you three alternatives. Take which you please. (i) We will follow this man and accept him as true, for by God [Allāh] it has become plain to you that he is a prophet who has been sent and that it is he that you find mentioned in your scripture; and then your lives, your property, your women and children will be saved.' They said, 'We will never abandon the laws of the Torah and never change it for another.' He said, 'Then if you won't accept this suggestion (ii) let us kill our wives and children and send men with their swords drawn to Muhammad and his companions leaving no encumbrances behind us, until God [Allāh] decides between us and Muhammad. If we perish, we perish, and we shall not leave children behind us to cause us anxiety. If we conquer we can acquire other wives and children.' They said, 'Should we kill these poor creatures? What would be the good of life when they are dead?'
"He said, 'Then if you will not accept this suggestion (iii) tonight is the eve of the sabbath and it may well be that Muhammad and his companions will feel secure from us then, so come down, perhaps we can take Muhammad and his companions by surprise.' They said, 'Are we to profane our sabbath and do on the sabbath what those before us of whom you well know did and were turned into apes?' He answered, 'Not a single man among you from the day of your birth has ever passed a night resolved to do what he knows ought to be done.' "
The three options are these:
Submit to Islām under the threat of death;
Kill innocent people in a desperate attempt to fight Muḥammad; or
Break the Sabbath.
Ka'b b. Asad behaves as one who may well believe Muḥammad fulfills the Tanakh, but again and if so, no scripture texts are cited. Or he sees no alternative to survival apart from such a profession. In the first instance, submitting to Islām is unacceptable. In the second instance, a cruel suggestion is made for them to murder their own wives and children so as to enter battle with abandon. And in the third instance, the B. Qurayẓa have already refused to break the Sabbath at the battle of the Trench. They are exhausted by the siege, and see no escape or ability to engage in battle. After further variables, Sa'd b. Mu'ādh, who earlier refused to allow Muḥammad to compromise at the Battle of the Trench, is selected by Muḥammad to render judgment on the Banū Qurayẓa. He decrees death for the men and captivity for the women and children, fulfilling Jibrīl's call to fight the Qurayẓa.
"Then they surrendered, and the apostle confined them in Medina in the quarter of d. al-Ḥārith, a woman of B. al-Najjār. Then the apostle went out to the market of Medina (which is still its market today) and dug trenches in it. Then he sent for them and struck off their heads in those trenches as they were brought out to him in batches. Among them was the enemy of Allah Ḥuyayy b. Akhṭab and Ka'b b. Asad their chief. There were 600 or 700 in all, though some put the figure as high as 800 or 900. As they were being taken out in batches to the apostle[,] they asked Ka 'b what he thought would be done with them. He replied, 'Will you never understand? Don't you see that the summoner never stops and those who are taken away do not return? By Allah it is death!' This went on until the apostle made an end of them."
Ḥussay says to Muḥammad: " 'By God, I do not blame myself for opposing you, but he who forsakes God will be forsaken.' Then he went to the men and said, 'God's command is right. A book and a decree, and massacre have been written against the Sons of Israel.' Then he sat down and his head was struck off." In other words, Ḥussay knows he has broken his military treaty with Muḥammad, and even though Muḥammad breaks treaties elsewhere, Ḥussay will not. He has a moral conscience in the sight of Yahweh Elohīm and goes to his death in this sense of accountability. One woman is also beheaded, one who laughs "immoderately as the apostle was killing her men."
As Muḥammad earlier supervises the beheading of 400 men, and approving of the pleasure of the Khazraj in so doing, he is here in his fifties, and bearing some extra weight in his age as he engages in the extraordinary vengeance of physical strength in wielding heavy sword, again and again, to the necks of 600-900 men. Surrender means death in much ancient warfare, and Islām does not break this pattern. Massacre is permitted in the name of Allāh. No prisoners are taken except as the de facto reality dhimmīs, those who submit to Islām, not in confession, but in paying the jizya or poll-tax.
"Then the apostle divided the property, wives, and children of B. Qurayẓa among the Muslims, and he made known on that day the shares of horse and mule, and took out the fifth … It was the first booty on which lots were cast and the fifth was taken. According to its precedent and what the apostle did, the divisions were made, and it remained the custom for raids."
As part of the booty, Muḥammad chooses a Qurayẓa woman for himself, Rayḥāna d. 'Amr b. Khunāfa, proposing to "marry her and put the veil on her." But "[s]he had shown some repugnance towards Islam when she was captured and clung to Judaism. So the apostle put her aside and felt some displeasure."

redpajama_set_name: RedPajamaCommonCrawl


Sam Blakelock is an amazing guitarist based out of New York and brilliant when it comes to social media marketing. Using marketing skills that he learned on his own, Sam quickly grew his Instagram page @pickupjazz from 0 to over 240,000 followers in very little time. If that wasn't impressive enough, he also consults various companies on social media marketing. He is absolutely BRILLIANT and wanted to learn more about his strategies, thought process and work ethic.


FreshWorks is a mobile app design and development studio based in Victoria & Vancouver, BC., delivering elegant mobile apps (iOS & Android), product strategies, responsive web applications, and delightful user experiences driven by narrative story-telling for bold entrepreneurs.
Why Nudge.ai? Reach out to the decision makers at FreshWorks Studio with a personalized email - let them know that you found their recent news mention interesting.
Nudge.ai's relationship intelligence platform can help you find the right executive to reach out to at FreshWorks Studio, such as Samarth Mod, Ceo & Co-Founder. Then get in touch and start a conversation.
Executives can be difficult to reach, but every company has connectors who bring together people and ideas. At FreshWorks Studio, people like Samarth Mod, and Harpreet Parhar are likely connectors.
When FreshWorks Studio is mentioned in the news, Nudge.ai often sees topics like Community. If your conversation focuses on these topics, you'll probably get FreshWorks Studio's attention.
When FreshWorks Studio is mentioned in the news, companies such as Concordia University, Asian Development Bank, and Government Of Manitoba are often also mentioned. These companies can serve as customer examples.
Employees who are often in the news are usually open to starting conversations or exploring ideas. Nudge.ai's sales AI saw Samarth Mod from FreshWorks Studio recently mentioned in the news.
Who in your network can introduce you to FreshWorks Studio? Nudge.ai finds the best pathway to reach contacts at this company.


An abbreviation for Peripheral Component Interconnect Express. For questions related to the connectivity between a motherboard and its bus devices (e.g., a graphics card).
Is there such a thing as an M.2 Thunderbolt / USB-C controller?
LGA1366 Motherboard compatiable with Xeon E5640 Quad Core CPU?
How Do I Go From PCI-E 16 To PCI Graphics?
What is the best drive supported by the MSI GE62 Apache Pro-004?


text:
#include "BookCompany.h"
#include <fstream>
#include<algorithm>

int main()
{
	BookCompany company;
	company.addBook("Algorithms in C", 1998,"Matthews",3);
	company.addBook("C", 1988,"Smith",2);
	company.addBook("C", 1978,"Jacob",4);
	company.addBook("Forth", 1972,"Cohen",1);
	company.addBook("Discrete Mathematics", 1999,"Burger",2);
	company.addBook("Algorithms", 1990,"Moran",3);
	company.addBook("Art of Computer Programming", 1978,"Andrews", 4);
	company.addBook("Fortran", 1965,"Banker", 2);
	company.addBook("The Practice of Programming", 1999,"Costeau", 6);
	//Test of removeBook
	company.removeBook("Algorithms", 1990,"Matthews",3);
	company.removeBook("none", 1234);

	cout << "Unsorted inventory" << company;

	// Anonymous function object from function class, JustTitle
	// is created and passed to sortInventory()
	// See Book.h for definition of JustTitle
	company.sortInventory(JustAuthor() );
	cout << "Sorted by Author" << company;

	// LangThenAuthor  is a function class, c2 the function object.
	// See Book.h for definition of YearThenTitle
	LangThenAuthor c2;
	company.sortInventory(c2);
	ofstream myout("outputFile");
	myout << "Sorted by Lang then Author" << company;

	// See Book.cpp for overloaded "<"
	// See BookCompany.h for definition of sortBook
	company.sortBooks();
	cout << "Sorted by Author then Lang" << company;
	return 0;
}

redpajama_set_name: RedPajamaGithub


text:
require "fit_commit/validators/base"

module FitCommit
  module Validators
    class LineLength < Base
      MERGE_COMMIT = /\AMerge branch '[^']+' into ./
      URL = %r{[a-z]+://}

      def validate_line(lineno, text)
        if lineno == 1
          if text.empty?
            add_error(lineno, "Subject line cannot be blank.")
          elsif text !~ MERGE_COMMIT
            if text.length > max_line_length
              add_error(lineno, format("Lines should be <= %i chars. (%i)",
                max_line_length, text.length))
            elsif text.length > subject_warn_length
              add_warning(lineno, format("Subject line should be <= %i chars. (%i)",
                subject_warn_length, text.length))
            end
          end
        elsif lineno == 2
          unless text.empty?
            add_error(lineno, "Second line must be blank.")
          end
        elsif text.length > max_line_length && !(allow_long_urls? && text =~ URL)
          add_error(lineno, format("Lines should be <= %i chars. (%i)",
            max_line_length, text.length))
        end
      end

      def max_line_length
        config.fetch("MaxLineLength")
      end

      def subject_warn_length
        config.fetch("SubjectWarnLength")
      end

      def allow_long_urls?
        config.fetch("AllowLongUrls")
      end
    end
  end
end


text:
Lack of Diversity on State Courts Threaten LGBT Rights, Study
- in National
Lambda Legal Senior Counsel and National Director of Constitutional Litigation Susan Sommer and Lambda Legal then-Executive Director Kevin Cathcart, with Nicole and Pam Yorksmith and their children standing in front of the U.S. Supreme Court on decision day, June 26, 2015.
Explosion in Judicial Campaign Spending Undermines Impartiality of Courts, Gives Way to Decisions Increasingly Driven by Ideology
A new report published Wednesday, Justice Out of Balance, finds that the election of judges and stunning lack of diversity on state courts threaten the rights of the lesbian, gay, bisexual and transgender (LGBT) community. The report, which was published by Lambda Legal, examines all cases involving LGBT issues in state high courts since 2003.
The study finds that courts whose judges stand for election are less supportive of LGBT rights. High courts with partisan elections supported a pro-LGBT claim in only 53 percent of cases; where judges were appointed and never stand for election, courts ruled in favor of LGBT rights in 82 percent of cases. Moreover, the results suggest that the ideology of justices was a primary factor in courts' rulings on LGBT rights, with ideology playing a larger role when judges are subject to competitive elections.
"Judicial elections are putting LGBT equality – and our country's very democratic principles – in increasing danger," said Rachel B. Tiven, CEO of Lambda Legal, the nation's oldest and largest legal LGBT organization. "Queer people across the country are seeing their constitutional rights and liberties thrown out the window by judges beholden to special interest groups."
The negative impacts on LGBT rights from judicial elections are exacerbated by the serious lack of judicial diversity in U.S. courts. The report finds that:
Of 340 currently-serving state high court justices, only 10 identify as openly gay or lesbian – and all of them were initially appointed to the bench
There are only 2 transgender judges in the country, and there are no openly bisexual judges
In state appellate courts, white males are overrepresented by nearly double their proportion of the national population
While a majority of law students are female, nowhere do women account for more than 34% of state judges – and in some states as few as 16%.
"The United States is more diverse than ever, yet our nation's courts are painfully homogenous," said Eric Lesh, Lambda Legal Fair Courts Project Director and author of the report. "This lack of diversity results in a biased system – both in perception and reality. For our courts to render fair decisions and be seen as legitimate, they must reflect the rich diversity of the communities they serve."
The report comes as judicial campaign spending has exploded in recent years, further affecting the impartiality of courts. Recent studies reveal that this flood of money in judicial races causes judges to issue more pro-business rulings, send more people to jail and sentence more people to death, among other effects. More than three out of four Americans believe that campaign cash affects court decisions, and roughly half of judges agree.
"The problem with judicial elections is that judges should not be politicians," added Tiven. "By design, judges are supposed to decide individual cases based on the law and the facts in front of them. Each day, however, thousands of elected judges across America know that they could lose their jobs if they make unpopular decisions that anger a majority of voters or inspire special interest attacks. It's a fundamentally flawed system that is ceding justice to politics."
The report concludes with recommendations to restore a fair and impartial court system, including:
Putting an end to judicial elections in favor of merit-based appointment by committee
Promoting greater diversity on court benches
Strengthening judicial codes of conduct, including more stringent judicial campaign regulations
Increasing anti-bias and cultural competency legal trainings, in addition to public awareness
"The scales of justice are clearly out of balance, but they don't have to be. The United States should join virtually every other country in the world that says 'no' to selecting judges by popular election," said Lesh. "LGBT people and other minorities need to know that there isn't a thumb on the scales, that they haven't been shut out of the process. Reforming the selection and retention of judges is a critical step toward ensuring fair, impartial courts and equal access to justice for all people."
Read the full findings from the Justice Out of Balance report here.
TagsCourt SystemGay RightsJudgeJustice Out of BalanceLambda LegalLesbianLGBTMarriageMarriage EqualityState CourtsSupreme Court
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text:
..::Iranian Truss Structures Simulation Association::..::انجمن شبیه سازی سازه های خرپایی ایران ::.. شبیه سازی سازه های خرپایی شبیه سازی سازه های خرپایی با استفاده از ماکارونی و پرسش و پاسخ پیرامون آن Turin, Italy - Juventus and Italy international midfielder Andrea Pirlo signed a new
Turin, Italy - Juventus and Italy international midfielder Andrea Pirlo signed a new
Turin, Italy - Juventus and Italy international midfielder Andrea Pirlo signed a new two-year contract with the Serie A champions on Wednesday, which will keep him at Juventus Stadium until 2016. Raphael Varane Jersey . Pirlo, 35, joined the Old Lady in 2011 after leaving AC Milan and helped the club win three consecutive Serie A titles. Last season, Pirlo played 43 games and scored six goals as Juventus won the league with a record 102 points. Overall, he has made 99 appearances for the Bianconeri, scoring 12 goals. Italy manager Cesare Prandelli will be counting on his midfield maestro to guide the club through a difficult Group D against England, Costa Rica and Uruguay. Pirlo has been capped 109 times for Italy, scoring 13 goals. The Italians open their Group D campaign on June 14 against England in Manaus. Steve Mandanda France Jersey .ca! There is plenty of blame to be shared as a result of the most recent NHL player (Pittsburghs Brooks Orpik) to be evacuated from the ice on a stretcher following an ugly incident Saturday night in Boston. Antoine Griezmann Jersey .The other side of the Bruins special teams unit delivered in the end.Gregory Campbell netted the game-winner with less than 10 minutes remaining and the Bruins fended off a late four-minute San Jose power play to earn a wild 5-3 win over the Sharks. http://www.soccerfrancenationalshop.us/P...ce-Jersey/ . Murray beat Sam Querrey 7-6 (5), 6-7 (3), 6-1, 6-3 to clinch Britains opening-round victory against the United States on Sunday at Petco Park. "Im proud of the way Im playing just now, because I had to do a lot of work to get back to where I want to be," Murray said after celebrating with his teammates on the red clay court in a temporary stadium in left field of the downtown home of baseballs San Diego Padres.The New York Rangers re-signed forward Dominic Moore to a two-year, $3 million contract and signed unrestricted free-agent forward Tanner Glass to a three-year contract worth $1.45 million per season. Moore, 33, sscored six goals and added 12 assists for 18 points in 73 games with the Rangers in 2013-2014. Jeremy Mathieu Jersey. Glass, 30, scored four goals and added nine assists for 13 points in 67 games with the Pittsburgh Penguins in 2013-2014. Black China NFL Jerseys China NFL Jerseys Jerseys Wholesale Wholesale NFL T-shirts Cheap NFL T-shirts Wholesale Jerseys 2018 Cheap NFL Jerseys Camo ' ' '


Admittedly, Adobe Illustrator is often most certainly not the first choice that comes to mind when it comes to Web design. Fireworks and Photoshop are used much more often, and there are some good reasons for that. Still, although Illustrator has traditionally been used for drawing illustrations and logos, you can use it to design layouts and user interfaces, too. In fact, in my opinion, you can utilize Illustrator to solve some regular design tasks better and more easily than you would do with other tools. With the techniques and tips I'd like to present in this article, I am certain that you will be able to build modular, flexible websites in less time and with less work.
Five Minute Upgrade – Making Your Design Pop | Build Internet!


If you pick the right plants to grow, you can create a great atmosphere in your home, literally speaking. They improve the air and add a touch of comfort to your house.
Prayer? Really? Well, this plant folds its leaves life those praying hands and unfolds them to praise the Lord. At this, the plant loves lots of moisture with little sunlight, which is just perfect for indoor growth. Plus it looks just great and makes your room look cozier.
Such a lousy name for beautiful and useful indoor plants! The proper one is Chlorophytum comosum. They reproduce fast and improve the quality of air indoors. What is more, they can survive pretty low sunlight levels!
The other name is even funkier than the first one! It's also called Mother-in-Law Tongue! The plants stand low levels of light and moisture and make perfect floor decor.
What is a low level of sunlight?
The level of light should not be considered as a stand-alone criterion for your plants. You should also take into the account such things as humidity, indoor temperature and the demand for moisture.
Low level means your room faces north. In this kind of light, a person can read or sew, but you would not feel comfortable doing without the additional illumination. It's the easiest test you can run to determine the level of light you have and pick the right indoor plants to grow in it.


text:
\section{Watts' formula}

When Langlands, Pichet, Pouliot and Saint-Aubin (\citeyear{LPPS}) were
doing computer simulations to test the conformal invariance of
percolation, there were several different events whose probability
they measured. The first event that they studied was the probability
that there is a percolation crossing connecting two disjoint boundary
segments. Using conformal field theory, \citet{Cardy}
derived his now-famous formula for this crossing probability, and the
formula was later proved rigorously by \citet{Smirnov} for site
percolation on the hexagonal lattice. The next event
that Langlands et al. tested was the probability that there
is both a percolation crossing connecting the two boundary segments
and a percolation crossing connecting the complementary boundary
segments (see Figure~\ref{fig:cardy-watts}). This probability also appeared to be conformally invariant,
but finding a~formula for it was harder, and it was not until several
years after Cardy's formula that  \citet{Watts} proposed his formula
for the probability of this double crossing. Watts
considered the derivation of the formula unsatisfactory, even by the
standards of
physics, but it matched the data of Langlands et
al. very well, which lent credibility to the formula. Watts'
formula was proved rigorously by \citet{DubedatWatts}.

\begin{figure}

\includegraphics{652f01.eps}

\caption{In the left panel, there is no left--right
crossing in blue hexagons. In the second panel there is a blue
left--right crossing, but no blue up--down crossing. In the third panel,
there are both blue left--right and blue up--down crossings. Cardy's
formula gives the probability of a left--right crossing in a domain,
while Watts' formula gives the probability that there is both a
left--right crossing and an up--down crossing.} \label{fig:cardy-watts}
\end{figure}

To express Cardy's formula and Watts' formula for the two types of
crossing events, since the scaling limit of percolation is conformally
invariant, it is enough to give these probabilities for one canonical
domain, and this is usually taken to be the upper half-plane.
There are four points on the boundary of the domain (the real line).
Label them in increasing order $x_1$, $x_2$, $x_3$ and $x_4$. Cardy's
formula is then the
probability that there is a percolation crossing from the interval
$[x_1,x_2]$ to the interval $[x_3,x_4]$. Again by conformal invariance,
we may map the upper half-plane to itself so that $x_1\to0$, $x_3 \to1$
and $x_4\to\infty$. The remaining point $x_2$ gets mapped to
\begin{equation} s =\cratio(x_1,x_2,x_3,x_4) := \frac
{(x_2-x_1)(x_4-x_3)}{(x_3-x_1)(x_4-x_2)},
\end{equation}
which is a point in $(0,1)$ known as the cross-ratio. Both Cardy's
formula and
Watts' formula are expressed in terms of the cross-ratio.
Cardy's formula for the probability of a percolation crossing is
\begin{equation}\label{e.cardy} \cardy(s) := \frac{\Gamma(
2/3)}{\Gamma(4/3)\Gamma(1/3)}  s^{1/3}\,\hspace*{0.4pt}{}_2\hspace*{-0.4pt}F_1\biggl(\frac
13,\frac23;\frac43;s\biggr),
\end{equation}
where $\Gamma$ is the gamma function, and ${}_2\hspace*{-0.4pt}F_1$ is the hypergeometric
function defined by
\[
{}_2\hspace*{-0.4pt}F_1(a,b;c;z) = \sum_{n=0}^\infty\frac{(a)_n (b)_n}{(c)_n n!} z^n,
\]
where $a,b,c\in\C$ are parameters, $c\notin-\N$
(where $\N=\{0,1,2,\ldots\}$), and $(\ell)_n$ denotes $\ell
(\ell+1)\cdots(\ell+n-1)$. This series converges for $z\in\C$ when $|z|<1$,
and the hypergeometric function is defined by analytic continuation elsewhere
(though it is then not always single-valued).

By comparison, Watts' formula for the probability of the two crossings is
the same as Cardy's formula minus another term
\begin{eqnarray}
\label{e.watts}
\watts(s)&:=& \frac{\Gamma(2/3)}{\Gamma(4/3)\Gamma(
1/3)}   s^{1/3}\,\hspace*{0.4pt}{{}_2\hspace*{-0.4pt}F_1\biggl(\frac13,\frac23;\frac43;s\biggr)}
\nonumber\\[-8pt]\\[-8pt]
&&{}- \frac{1}{\Gamma(1/3)\Gamma(2/3)}
s\,{\hspace*{0.4pt}{}_3\hspace*{-0.4pt}F_2\biggl(1,1,\frac43;2,\frac53;s\biggr)},\nonumber
\end{eqnarray}
where ${}_3\hspace*{-0.4pt}F_2$ is the generalized hypergeometric function.
The functions $\operatorname{cardy}(s)$ and $\operatorname{watts}(s)$
are shown in Figure \ref{fig:plot}.
[The reader should not be intimidated by these formulae; the parts of
the proof involving hypergeometric functions can be handled mechanically
with the aid of Mathematica.
See also \citet{Watts} and \citet{Maier}
for equivalent double-integral formulations of Watts' formula.]

\begin{figure}

\includegraphics{652f02.eps}

\caption{Cardy's formula (upper curve), Watts'
formula (lower curve), and a tripod probability (defined in
Section \protect\ref{s.tripodreduction}) as a function of the
cross-ratio $s$.}
\label{fig:plot}
\end{figure}

Schramm thought that Dub\'{e}dat's paper on Watts' formula was an
exciting development and started reading it as soon as it appeared in
the arXiv. Schramm sometimes presented papers to interested people at
Microsoft Research: for example, he presented Smirnov's proof of
Cardy's formula when it came out [\citet{Smirnov}],
as well as Dub\'{e}dat's paper on Watts' formula [\citet{DubedatWatts}],
and later Zhan's paper on the
reversibility of SLE$_\kappa$ for $\kappa\leq4$ [\citet{Zhan}]. In the
course of
reaching his own understanding of Watts' formula, Schramm simplified
Dub\'{e}dat's proof, with the help of a Mathematica notebook, and it was
this version that he presented at Microsoft on May 17, 2004. This
proof did not come up again until an August 2008
Centre de Recherches Math\'{e}matiques (CRM) meeting on SLE in
Montr\'{e}al, after a talk by Jacob Simmons on his work with Kleban and
Ziff on ``Watts' formula and logarithmic conformal field theory''
[\citet{SKZ}]. Schramm mentioned that he had an easier proof of Watts'
formula, which he recalled after just a few minutes. The people who
saw his version of the proof thought it was very elegant and strongly
encouraged him to write it up. The next day Oded wrote down an
outline of the proof, but he tragically died a few weeks later. There
is interest in seeing a written version of Schramm's version of the
proof, so we present it here.

\section{Outline of proof}
This is a slightly edited version of the proof outline that Oded wrote
down at the CRM. Steps 1 and 2 are the same as in Dub\'{e}dat's proof,
but with step 3 the proofs diverge. We will expand on these steps of
the outline (with slightly modified notation) in subsequent sections.

\begin{itemize}
\item Reduce to the problem of calculating the probability that there
is a crossing up--down
which also connects to the right.
\item Further reduce to the following problem. In the upper half-plane,
say, mark points
$-\infty<y_1<x_0<y_2<y_3=\infty$. Let $\gamma$ be the SLE$_6$
interface started from~$x_0$.
Let $\tau:=\inf\{t\ge0\dvtx\gamma_t\in\R\setminus[y_1,y_2]$ and
$\sigma:=\sup\{t<\tau\dvtx\gamma_t\in\R\}$.
Calculate $\mathbf{P}[\gamma_\tau\in[y_2,y_3], \gamma_\sigma\in
[x_0,y_2]]$.
\item Let $\sigma_1:=\sup\{t<\tau\dvtx\gamma_t\in[y_1,x_0]\}$
and $\sigma_2:=\sup\{t<\tau\dvtx\gamma_t\in[x_0,y_2]\}$.
Now calculate the probability density of the event
$\gamma_{\sigma_1}=z_1,\gamma_{\sigma_2}=z_2,\gamma_\tau=z_3$
as $h(z_1,x_0,z_2,z_3):=\p_{z_1}\p_{z_2}\p_{z_3}\operatorname
{Cardy}(z_1,x_0,z_2,z_3)$.
\item Now, $h$ [times certain derivatives] is a martingale for the
corresponding diffusion. Consider the
Doob-transform ($h$-transform) of the diffusion with this $h$. This
corresponds to
conditioning on this probability zero event. For the Doob-transform, calculate
the probability that $\sigma_2>\sigma_1$. This comes out to be a
hypergeometric
function $g$. Finally,
\[
\operatorname{Watts}(y_1,x_0,y_2,y_3)=
\int_{[y_1,x_0]}dz_1\int_{[x_0,y_2]} dz_2 \int_{[y_2,y_3]} g h\,
dz_3,
\]
(or more precisely, the three-arm probability),
and use integration by parts.
\end{itemize}

\section{Reduction to tripod probabilities} \label{s.tripodreduction}

$\!\!\!$The initial reduction, which is step~1 of the proof, has been derived
by multiple people independently. The first place that it appeared in
print appears to be in Dub\'{e}dat's (\citeyear{dubedatBM}) paper, where it
is credited to
Werner, who, in turn, is sure that it must have been known earlier.
In the interest of keeping the exposition self-contained, we explain
this reduction.

It is an elementary fact that exactly one of the following two events occurs:
\begin{longlist}[(2)]
\item[(1)]
there is a horizontal blue crossing in the rectangle (i.e., a path of
blue hexagons connecting the left
and right edges of the rectangle), which we denote by $H_b$;
\item[(2)] there is a vertical yellow crossing, which we denote by $V_y$.
\end{longlist}

\begin{center}

\includegraphics{652i01.eps}
\vspace*{3pt}
\end{center}

If there is a horizontal crossing, then by considering the region
beneath it, using the above fact, either it connects to the bottom
edge (forming a T shape) or else there is another crossing beneath it
of the opposite color. Since there are finitely many hexagons, there
must be a bottom-most crossing, which then necessarily forms a T shape.
Thus exactly one of the following three events occurs:
\begin{longlist}[(2)]
\item[(1)] there is no horizontal crossing of either color (denoted by $N$);
\item[(2)] there is a blue T (denoted $T_b$);
\item[(3)] there is a yellow T (denoted $T_y$).
\end{longlist}

\begin{center}

\includegraphics{652i02.eps}
\vspace*{3pt}
\end{center}

Of course the latter two events have equal probability, so we have
\[
\Pr[N] + 2 \Pr[T_b] = 1.
\]

Recall again that there is either a blue horizontal crossing or a
yellow vertical crossing but not both. We can decompose the yellow
vertical crossing event into two subevents according to whether or not
there is also a~yellow horizontal crossing. The first subevent is,
of course, the event we are interested in (with blue and yellow
reversed), and the second subevent is identical to the event $N$.\vspace*{3pt}

\begin{center}

\includegraphics{652i03.eps}
\end{center}

Thus we have
\[
\Pr[H_b] + \Pr[H_y \wedge V_y] + \Pr[N] = 1.
\]
Combining these equations, we see that
\[
\Pr[H_b \wedge V_b] = 2 \Pr[T_b] - \Pr[H_b].
\]
In the limit of large grids with cross ratio $s$, the third term is
given by Cardy's formula, $\cardy(s)$,
and we seek to show that the left-hand side is given by Watts'
formula, $\watts(s)$. Let us give another name for
what we expect to be the limit of the second term. Define $\tripod(s)$
to satisfy
\[
\watts(s)=2\tripod(s) - \cardy(s),
\]
that is [substituting (\ref{e.cardy}) and (\ref{e.watts})],
\begin{eqnarray*}\tripod(s) &=& \frac{\watts(s) + \cardy(s)}{2} \\
&=& \frac{\Gamma(2/3)}{\Gamma(4/3)\Gamma(1/3)}
s^{1/3}\,\hspace*{0.4pt}{{}_2\hspace*{-0.4pt}F_1\biggl(\frac13,\frac23;\frac43;s\biggr)}
\\
&&{} - \frac{1}{2\Gamma(1/3)\Gamma(2/3)}
s\,\hspace*{0.4pt}{{}_3\hspace*{-0.4pt}F_2\biggl(1,1,\frac43;2,\frac53;s\biggr)}.
\end{eqnarray*}
Then in light of Cardy's formula, proving Watts' formula is equivalent
to showing that $\Pr[T_b]$ is given by $\tripod(s)$ in the fine mesh limit.\vadjust{\goodbreak}

\section{Comparison with SLE$_6$}
\subsection{Discrete derivatives of the tripod probability}

Consider percolation on the upper half-plane triangular lattice, and
let $P_T[x_1,x_2,x_3,x_4]$ be the probability of a blue tripod
connecting the intervals $(x_1,x_2)$ and $(x_2,x_3)$ and $(x_3,x_4)$
when the four (here discrete) locations are $x_1 < x_2<x_3<x_4$ (each
of which is a point between two boundary hexagons; see the upper image
in Figure \ref{fig:dddtripod}).

\begin{figure}

\includegraphics{652f03.eps}

\caption{The discrete triple partial derivative of the
tripod probability is the probability of a~multi-arm event. The top
panel illustrates the event whose probability is $P_T[x_1,x_2,x_3,x_4]$,
the next panel illustrates $\Delta_{x_4} P_T[x_1,x_2,x_3,x_4]$, the
third panel illustrates $\Delta_{x_3} \Delta_{x_4}
P_T[x_1,x_2,\allowbreak x_3,x_4]$ and the bottom panel illustrates
$-\Delta_{x_1} \Delta_{x_3} \Delta_{x_4} P_T[x_1,x_2,x_3,x_4]$.}
\label{fig:dddtripod}
\end{figure}

Then $\Delta_{x_4} P_T[x_1,x_2,x_3,x_4] := P_T[x_1,x_2,x_3,x_4] -
P_T[x_1,x_2,x_3,x_4-1]$ gives the probability that there is a crossing
tripod for $(x_1,x_2,x_3,x_4)$ but \textit{not} for
$(x_1,x_2,x_3,x_4-1)$. (Here we assume that the lattice spacing is $1$.)
Since the crossing tripod for
$(x_1,x_2,x_3,x_4)$ does not extend to a crossing tripod for
$(x_1,x_2,x_3,x_4-1)$, there must be a path of the opposite color from
the hexagon just to the left of $x_4-1$ to the interval between
$x_2+1$ and $x_3+1$; this event is represented by the second image in
Figure \ref{fig:dddtripod}. Similarly,
\[
-\Delta_{x_1} \Delta_{x_3} \Delta_{x_4}P_T[x_1,x_2,x_3,x_4]
\]
gives
the probability of a multi-arm event such as the one in the bottom
image in Figure \ref{fig:dddtripod}.

By summing these discrete differences, it is straightforward to write
\[
P_T[x_1,x_2,x_3,x_4] = \sum_{c\in(x_3,x_4]} \sum_{b\in(x_2,x_3]}
\sum_{a\in(x_1,x_2]}
-\Delta_{x_1} \Delta_{x_3} \Delta_{x_4}P_T[a,x_2,b,c].
\]
If there is a blue tripod connecting the intervals $(x_1,x_2)$,
$(x_2,x_3)$ and $(x_3,x_4)$, then there is only one cluster containing
such a tripod.
This formula can be interpreted as partitioning the tripod event into
multi-arm events of the type shown in the bottom panel
of Figure \ref{fig:dddtripod}.
The triple $(a,b,c) \in(x_1,x_2] \times(x_2,x_3] \times(x_3,x_4]$ is
uniquely determined by the tripod:
$a$ is (half a lattice spacing to the right of) the rightmost boundary
point of the tripod cluster in the interval $(x_1,x_2)$, $b$ is (just
right of) the rightmost point of the tripod cluster in $(x_2,x_3)$ and
$c$ is (just right of) the leftmost point of the tripod cluster in $(x_3,x_4)$.

\subsection{Discrete derivatives of the crossing probability}

Consider percolation on a half-plane triangular lattice, as in the
previous subsection, and let $P_C[x_1,x_2,x_3,x_4]$ be the probability
of at least one blue cluster spanning the intervals $(x_1,x_2)$ and
$(x_3,x_4)$; see the upper image in Figure \ref{fig:dddcardy}. Then
\begin{figure}

\includegraphics{652f04.eps}

\caption{The discrete triple partial derivative of the crossing
probability is sum of the probabilities of two multi-arm events.
The panels illustrate the events whose probability is
$P_C[x_1,x_2,x_3,x_4]$ (top), $\Delta_{x_4} P_C[x_1,x_2,x_3,x_4]$
(second), $\Delta_{x_3} \Delta_{x_4} P_C[x_1,x_2,x_3,x_4]$ (third row),
and $-\Delta_{x_1} \Delta_{x_3} \Delta_{x_4} P_C[x_1,x_2,x_3,x_4]$
(bottom row).}
\label{fig:dddcardy}
\end{figure}
$\Delta_{x_4} P_C[x_1,x_2,x_3,x_4] = P_C[x_1,x_2,x_3,x_4] -
P_C[x_1,x_2,x_3,x_4-1]$ gives the probability that there is a crossing
for $(x_1,x_2,x_3,x_4)$ but \textit{not} $(x_1,x_2,x_3,\allowbreak x_4-1)$.
This event is represented by the second image in Figure \ref{fig:dddcardy}.
Similarly,
\[
-\Delta_{x_1} \Delta_{x_3} \Delta_{x_4}P_C[x_1,x_2,x_3,x_4]
\]
gives the probability that
\textit{one of the two} multi-arm events in the bottom image in
Figure \ref{fig:dddtripod} occurs. The event of a crossing cluster is
equivalent
to the event that one of these multi-arm events occurs for
\textit{some} (necessarily unique) set of three points $(a,b,c) \in
(x_1,x_2] \times(x_2,x_3] \times(x_3,x_4]$:
$a$ is (just right of)
the rightmost boundary point of the crossing cluster(s) in the
interval $(x_1,x_2)$; $b$ is (just right of) the rightmost point of
the crossing cluster in $(x_2,x_3)$ [if it exists; otherwise $b$ is
the rightmost boundary point in $(x_2,x_3)$ of a crossing yellow
cluster, as shown]; and $c$~is (just right of) the leftmost point of
the cluster(s) in $(x_3,x_4)$.

Thus $-\Delta_{x_1} \Delta_{x_3}
\Delta_{x_4}P_C[x_1,x_2,x_3,x_4]$ decomposes into the probabilities of two
multiarm events, the first of which is $-\Delta_{x_1} \Delta_{x_3}
\Delta_{x_4}P_T[x_1,x_2,x_3,x_4]$.

\subsection{Multi-arm events and the interface} \label{ss:interface}

Consider the setting of Figures~\ref{fig:dddtripod}
and~\ref{fig:dddcardy}, and suppose we add an additional boundary
layer\vadjust{\goodbreak} of blue hexagons to the left of $x_2$ and yellow hexagons to the
right of $x_2$. Then let~$\gamma_{\mathrm{discrete}}$ be the discrete
interface starting at $x_2$. (See Figure \ref{fig:interface}.)

\begin{figure}

\includegraphics{652f05.eps}

\caption{The interface interpretation of the multi-arm events.}
\label{fig:interface}
\end{figure}

Then the union of the two multi-arm events at the bottom of
Figure \ref{fig:dddcardy}~de\-scribes the event that that $c$ is the first
boundary point that $\gamma_{\mathrm{discrete}}$ hits outside the
interval $(x_1,x_3)$, and that $a$ and $b$ are the leftmost and
rightmost boundary points hit by $\gamma_{\mathrm{discrete}}$
\textit{before} $c$. The left figure corresponds to~the~ca\-se that
$a$ is hit before $b$, and the right figure to the case that $b$ is
hit before~$a$.

\subsection{Continuum Watts' formula: A statement about SLE} \label
{ss.SLEstatement}

Like Cardy's formula, Watts' formula has a continuum analog, which is
a statement strictly about SLE$_6$. Fix real numbers $x_1 < x_2 < x_3
< x_4$, and let $s$ be their cross ratio. Consider the usual SLE$_6$
in the upper half-plane, where the starting point of the path is
$x_2$. Before Smirnov proved Cardy's formula for the scaling limit of
triangular lattice percolation, it was already known by Schramm that
$\cardy(s)$ represents the probability that $\gamma$ hits $(x_3,x_4)$
\textit{before} hitting $\R\setminus[x_1, x_4]$. [In the discrete
setting of Section \ref{ss:interface}, having $\gamma_{\mathrm{discrete}}$
hit $(x_3,x_4)$ before the complement of $(x_1,x_4)$ is equivalent to
the existence of a crossing.] In light of Section \ref{ss:interface}, the
following is the natural continuum analog of the tripod formula.
\begin{theorem} \label{t:continuumtripod}
Let $\SLEtripod(s)$ be the probability that both:
\begin{longlist}[(2)]
\item[(1)] $\gamma$ first hits $(x_3,x_4)$ (at some time $t$) \textit
{before} it first hits $\R\setminus(x_1,x_4)$, and
\item[(2)] $\gamma$ hits the leftmost point of $\R\cap\gamma[0,t)$
before it hits the rightmost point.
\end{longlist}
Then $\SLEtripod(s) = \tripod(s)$.\vadjust{\eject}
\end{theorem}

Theorem \ref{t:continuumtripod} is the actual statement that was
proved by
Dub\'{e}dat, and the statement whose proof was sketched by Oded.
Dub\'{e}dat claimed further that Theorem~\ref{t:continuumtripod} would imply
the tripod formula (and hence Watts' formula) for the scaling limit of
critical triangular lattice percolation if one used the (at the time
\mbox{unpublished}) proof that SLE$_6$ is the scaling limit of the interface
[\citet{DubedatWatts}]. To be fully precise, one needs slightly more
than the fact that the interface scaling limit is SLE$_6$: it is
important to know that the discrete interface is unlikely to get close
to the boundary without hitting it. [Similar issues arise when using
Cardy's formula to prove SLE$_6$ convergence; see, e.g.,
\citet{CM}.]
Rather than address this (relatively minor technical) point here, we
will proceed to prove Theorem \ref{t:continuumtripod} in the manner outlined
by Oded and defer this issue until Section \ref{s.percolationstatement}.

It is convenient to have a name for the SLE versions of the multi-arm
events in Figure \ref{fig:dddcardy}. Say that a triple of distinct
real numbers $(a,b,c)$ with $a<0<b$
constitutes a \textit{tripod set} for $\gamma$ if for some $t > 0$ we
have:
\begin{longlist}[(2)]
\item[(1)] $\gamma(t) = c$;
\item[(2)] $\inf(\gamma[0,t)\cap\R) = a$;
\item[(3)] $\sup(\gamma[0,t)\cap\R) = b$.
\end{longlist}
There are a.s. a countably infinite number of tripod sets, but if
$x_1<0$ and $x_3 > 0$ is fixed, there is a.s. exactly one for which
$x_1<a<0<b<x_3$ and $c\notin(x_1,x_3)$. Let $\mathcal P_C\dvtx \R^3\to\R$
be the probability density function for this $(a,b,c)$.
(We see in Lemma \ref{density} that this density function exists.)
There are
also two types of tripod sets $(a,b,c)$: those for which $\gamma$ hits
$a$ first and those for which $\gamma$ hits $b$ first. Write
$\mathcal P_C = \mathcal P_A + \mathcal P_B$, where $\mathcal P_A$ and
$\mathcal P_B$ are the corresponding probability densities for
$a$-first and $b$-first tripod sets.

Now, we claim the following:
\begin{lemma}\label{density}
Using the notation above, the density functions $\mathcal P_C$ and
$\mathcal P_A$ exist, and
\[
\cardy(s) = \int_{x_1}^{0}\int_{0}^{x_3} \int_{x_3}^{x_4} \mathcal
P_C(a,b,c) \,dc \,db \,da
\]
and
\[
\SLEtripod(s) = \int_{x_1}^{0}\int_{0}^{x_3} \int_{x_3}^{x_4}
\mathcal P_A(a,b,c) \,dc \,db \,da.
\]
\end{lemma}
\begin{pf}
In the event that the density functions do not exist, we abuse
notation and let ``$\mathcal P_C(a,b,c) \,dc \,db \,da$'' and
``$\mathcal P_A(a,b,c) \,dc \,db \,da$'' denote the relevant measures,
which must exist. It is easy to see that the event that $(x_3,x_4)$
is hit before $\R\setminus[x_1,x_4]$ is equivalent to the event
that $(a,b,c) \in(x_1,0) \times(0,x_3) \times(x_3,x_4)$. By
definition, $\SLEtripod(s)$ is the probability of the same event
intersected with the event that $\gamma$ hits $a$ first. Finally,
observe that Cardy's formula of the cross-ratio of 4-points is
three-times differentiable, so the density function $\mathcal
P_C(a,b,c)$ exists and, consequently, the density function $\mathcal
P_A(a,b,c)$ also exists.
\end{pf}

Of course, from this, one has the immediate corollary:
\begin{corollary} \label{c.triplederivatives}
Using the above notation,
\[
\partial_{x_1} \partial_{x_3} \partial_{x_4} \cardy( \cratio
(x_1,0,x_3,x_4)) = \mathcal P_C(x_1,x_3,x_4)
\]
and
\[
\partial_{x_1} \partial_{x_3} \partial_{x_4} \SLEtripod(\cratio
(x_1,0,x_3,x_4)) = \mathcal P_A(x_1,x_3,x_4).
\]
\end{corollary}

If we could show further that
\begin{equation}\label{PA} \mathcal P_A(x_1,x_3,x_4) = \partial_{x_1}
\partial_{x_3} \partial_{x_4} \tripod(\cratio(x_1,0,x_3,x_4)),
\end{equation}
then this corollary and standard integration would imply
Theorem \ref{t:continuumtripod}, since we know that $\tripod(\cratio
(\cdot)) =
\SLEtripod(\cratio(\cdot))$ on the bounding planes \mbox{$x_1=0$}, \mbox{$x_2=0$}
and $x_3 = x_4$. Since we already have an explicit formula for
$\tripod$, the only remaining step is to explicitly compute $\mathcal P_A$.
Oded's approach is to compute the ratio $\mathcal P_A/\mathcal P_C$ as
the conditional probability [given that $(a,b,c)$ form a tripod set]
that $\gamma$ hits $a$ before $b$. Since $\mathcal P_C$ is known,
this determines~$\mathcal P_A$.

\section{Conditional probability that $a$ is hit first} \label{sec:cond-prob}

Schramm was very adept with using Mathematica to calculate all manner
of things. He probably would have considered this last step to be
routine, since it was for him straightforward to set up the right
equations and then let Mathematica solve them. At this point we refer
to his original Mathematica notebook from 2004, and explain the
various steps in the calculation. To be consistent with Oded's
notation, we now make the following substitutions:
\[
v_3 = a,  \qquad   W = x_2,  \qquad   v_1 = b,  \qquad    v_2 = c.
\]
(We assume $v_3 < W < v_1 < v_2$. Oded apparently chose this notation
because under cyclic reordering it was the same as $W=v_0,v_1,v_2,v_3$.)

First we formally define the function $\cardy(s)$ as in
(\ref{e.cardy}). The Mathematica function \Mcardy defined here
involves an additional parameter $\kappa$, but it specializes to
$\cardy(s)$ when $\kappa=6$. This more general formula is analogous
to Cardy's formula but gives the (conjectural) crossing probability
for the critical Fortuin--Kasteleyn random cluster model [with
$q=4\cos^2(4\pi/\kappa)$] with alternating wired-free-wired-free
boundary conditions. (See Rohde and Schramm [(\citeyear{RS}), conjecture 9.7],
for some background.)
This formula was known at Microsoft in 2003,
and most likely Oded copied it from another Mathematica notebook.
This formula was later independently discovered [\citet{BBK}]
(nonrigorously) and [\citet{DEuler}] (rigorously).\vspace*{6pt}

\begin{center}

\includegraphics{652i04.eps}
\vspace*{2pt}
\end{center}

Consider the evolution of chordal SLE$_6$ started from $W$ and run to
$\infty$, when at time zero there are 3 marked points at positions
$v_1$, $v_2$ and $v_3$. We then let $W(t)$ represent the SLE$_6$
driving function [i.e., $W(t) = W(0) + \sqrt6 B_t$ where $B_t$ is a
standard Brownian motion] of the Loewner evolution
\[
\partial_t g_t(z) = \frac{2}{g_t(z)-W(t)},
\]
and interpret the $v_i$ as functions of $t$, evolving under the
Loewner flow, that is, $v_i(t):= g_t(v_i(0))$.

If $f$ is any function of $v_1, v_2, v_3, W$, we define
\begin{equation} \label{e.generator}
L(f):=  \frac{\partial}{\partial t} \mathbb E [ f(W(t),
v_1(t), v_2(t), v_3(t))] \bigg|_{t=0}.
\end{equation}
This is a new function of the same four variables which can be
calculated explicitly using It\^{o}'s formula as
\[
L(f) = \frac{\kappa}{2} \,\frac{\partial^2}{\partial W^2}f + \sum
_{i=1}^3 \frac{2 ({\partial}/{\partial v_i})f}{v_i - W}.
\]
This operator is defined as \ML in the Mathematica code below.\vspace*{6pt}
\begin{center}

\includegraphics{652i05.eps}
\vspace*{2pt}
\end{center}
Similarly in the Mathematica code, \Mcr is the cross-ratio [defined in
(\ref{e.cratio})], that is,
\begin{equation}\label{e.cratio} \cratio:=\cratio(v_3,W, v_1, v_2)=
\frac{(W-v_3)(v_1-v_2)}{(W - v_2)(v_1-v_3)}.
\end{equation}

In the next line, Oded performed a consistency check.
Cardy's formula should be a martingale for the SLE$_\kappa$ diffusion,
hence $L(\cardy(\cratio(\cdot))) = 0$.\vspace*{3pt}
\begin{center}

\includegraphics{652i06.eps}
\vspace*{3pt}
\end{center}
Next, Oded computes the triple derivative \Mh of Cardy's formula
(which is the same as the $\mathcal P_C$ defined in
Section \ref{ss.SLEstatement}).\vspace*{3pt}

\begin{center}

\includegraphics{652i07.eps}
\vspace*{3pt}
\end{center}

That is, he computes
\[
h(v_3,W,v_1,v_2) := \partial_{v_1} \partial_{v_2} \partial_{v_3}
\cardy\biggl(\frac{(W-v_3)(v_1-v_2)}{(W-v_2)(v_1-v_3)}\biggr).
\]
The result is somewhat complicated, but we may ignore it, since it is
just an
intermediate result.

The next step involves conditioning on an event of zero probability,
the event that $(v_3,v_1,v_2)$ is a tripod set. We can make sense of
this by introducing a triple difference of Cardy's formula and
recalling the results of Section \ref{ss.SLEstatement}. First we introduce
notation to describe some small evolving intervals. For given values
$v_1(0), v_2(0), v_3(0), W(0)$, pick $\eps$ small enough so that the
intervals $(v_i(0), v_i(0)+\eps)$ are disjoint and do not contain
$W(0)$. Write $\tilde v_i(0) = v_i(0) + \eps$. Define $\tilde
v_i(t)$ using the Loewner evolution, and write $\eps_i(t) := \tilde
v_i(t) - v_i(t)$. Let us write
\begin{eqnarray}\label{heps}
&&h_{\eps_1, \eps_2, \eps_3}(v_3,W,v_1,v_2)\nonumber\\[-8pt]\\[-8pt]
&&\qquad := \Delta_{v_1}^{(\eps
_1)} \Delta_{v_2}^{(\eps_2)} \Delta_{v_3}^{(\eps_3)} \cardy
(\cratio(v_3,W,v_1,v_2)),\nonumber
\end{eqnarray}
where $\Delta_v^{(\eps)}$ is the difference operator defined by
\[
\Delta_v^{(\eps)} f(v) = f(v+\eps)-f(v).
\]
Note that the $\Delta^{(\eps_i)}_{v_i}$ depend on $t$. By Corollary
\ref{c.triplederivatives}, equation (\ref{heps}) at time~$t$
represents the conditional probability (given the Loewner evolution up
to time $t$) that there is a tripod set in $[v_3(0), \tilde v_3(0)]
\times[v_1(0), \tilde v_1(0)] \times[v_2(0), \tilde v_2(0)]$.
By Girsanov's theorem,
conditioning on this event induces a~drift on the Brownian motion $W_t$
driving the SLE, where the drift is
\[
\kappa  \,\partial_W \log h_{\eps_1, \eps_2, \eps_3}(v_3,W,v_1,v_2).
\]
Observe that
\begin{eqnarray*}
\frac{\partial}{\partial W} \log h_{\eps_1, \eps_2, \eps_3} &=&
\frac{({\partial}/{\partial W}) h_{\eps_1,\eps_2,\eps
_3}}{h_{\eps_1,\eps_2,\eps_3}} =
\frac{\iiint({\partial}/{\partial W}) h}{\iiint h} \\
&\approx&\frac
{\eps_1 \eps_2 \eps_3 ({\partial}/{\partial W}) h}{\eps_1\eps
_2\eps_3 h} = \partial_W \log h,
\end{eqnarray*}
where there triple integral is over $\prod[v_i, \tilde v_i]$.
Thus upon taking the limit $\eps\to0$, the drift becomes
\begin{equation} \label{e.drift}\drift(t):= \kappa  \partial_W
\log h(v_3,W,v_1,v_2).
\end{equation}
The next Mathematica code explicitly computes (\ref{e.drift}).\vspace*{3pt}

\begin{center}

\includegraphics{652i08.eps}

\end{center}

The expression above is complicated, but again it is an intermediate
result that we do not need to calculate or read ourselves. The first
line of the Mathematica code below defines the generator \MLd(which
we will write as~$L_1$) for the conditioned SLE$_6$, where the driving
function $W_t$ has the drift given above. Here $L_1$ is defined as in
(\ref{e.generator}) except that the expectation is with respect to the
law of $W_t$ with the drift term (\ref{e.drift}). Thus
\[
L_1(f) :=
L(f) + \drift(t)  \,\partial_W f.
\]
As before, if $f$ is a real function
of $W,v_1,v_2,v_3$, then $L_1(f)$ will be a function of the same four
variables.

We now compute the probability in the modified diffusion that
$v_3$ is absorbed before $v_1$, that is, that $W(t)$ collides with $v_3(t)$
before colliding with~$v_1(t)$. This probability will be a martingale
that only depends upon the cross-ratio $s$. Thus, in the next
paragraph, we specialize and consider functions of~$W,v_1,v_2,v_3$
that have the form $f(\cratio(v_3,W,v_1,v_2))$ where $f\dvtx\R\to\R$ is a
function of \textit{one} variable. We would like to find a one-parameter
function~$f$ for which $f(\cratio(v_3,W,v_1,v_2))$ is a
martingale with respect to this modified diffusion, so we will require
that $L_1(f(\cratio(v_3,W,v_1,v_2))) = 0$. What one-parameter
functions $f$
have this property?

Oded answers this question with some clever Mathematica work. First,
he re-expresses the differential equation $L_1(f(\cratio
(v_3,W,v_1,v_2))) = 0$---which involves the four parameters
$W,v_1,v_2,v_3$---in terms of the
parameters $s,v_1,v_2,\allowbreak v_3$. He does this by setting $s$ equal to the
expression for $\cratio$ given in (\ref{e.cratio}), solving to get $W$
in terms of the other variables and plugging this new expression for
$W$ into the expression $L_1(f(\cratio(v_3,W,v_1,v_2)))$.\vspace*{6pt}

\begin{center}

\includegraphics{652i09.eps}
\vspace*{6pt}
\end{center}

This expression for $L_1(f(\cratio(v_3,W,v_1,v_2)))$ depends on
$f'$ and $f''$, and equating it to zero yields a differential
equation for $f$, the unknown one-parameter function of the
cross-ratio that we seek,
\[
2(1-6s^2+4s^3)f'(s) + 3 s (-1+2s-2s^2+s^3) f''(s) = 0.
\]
Oded solves this differential equation, which yields the function
$f$ up to two free parameters $C_1$ and $C_2$.

\begin{center}

\includegraphics{652i10.eps}
\vspace*{3pt}
\end{center}

Here Mathematica gives
\[
f(s) = C_2 + C_1 s^{2/3} \frac{1-3 s+2 s^2-(1-s)^{1/3} (1-s+s^2)
\,\hspace*{0.4pt}{}_2\hspace*{-0.4pt}F_1(2/3,1/3;5/3;s)}{3 (-1+s)^{1/3} (1-s+s^2)}.
\]

The conditional probability that we seek tends to $1$ when $s\to0$ and
tends to $0$
when $s\to1$, and this determines $C_1$ and $C_2$: $C_2$ must be $1$,
and~$C_1$ follows from Gauss's hypergeometric formula,
\[
{}_2\hspace*{-0.4pt}F_1(a,b;c;1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma
(c-b)}.
\]
Solving for $C_1$ and substituting, we find that the conditional
probability that $\gamma$ hits $v_3$ before $v_1$, given that
$(v_3,v_1,v_2)$ is
a tripod set, is given by
\[
f(s) = 1 - \frac{\Gamma(4/3)}{\Gamma(2/3)\Gamma(
5/3)}   s^{2/3}  \biggl[\frac{-1+3 s-2 s^2}{(1-s)^{1/3} (1-s+s^2)} +
{}_2\hspace*{-0.4pt}F_1\biggl(\frac23,\frac13;\frac53;s\biggr)\biggr],
\]
where $s$ is the cross-ratio of $v_3,0,v_1,v_2$.

\section{Comparison of triple derivatives}

Taking $\mathcal P_A$ and $\mathcal P_C$ as defined in
Section \ref{ss.SLEstatement}, and $f$ and $h$ as defined in the previous
section, we now have
\[
h(v_3,0,v_1,v_2) = \mathcal P_C(v_3,v_1,v_2)
\]
and
\[
f(\cratio(v_3,0,v_1,v_2))=\mathcal P_A(v_3,v_1,v_2)/\mathcal P_C(v_3,v_1,v_2).
\]
In principle the next step toward proving (\ref{PA}) (and hence
Theorem \ref{t:continuumtripod}) would be to integrate $\mathcal P_A =
f(\cratio(\cdot))h$ over the three variables $v_1,v_2,v_3$ and show
that one obtains $\tripod(\cratio(\cdot))$. In Oded's original notes,
he stated that this could be done using integration by parts.
Fortunately (for those who lack Oded's skill at integrating) we
already know (thanks to Watts) what we expect $\tripod$ to be, so we
can instead differentiate $\tripod(\cratio(\cdot))$ three times
(w.r.t. $v_1,v_2,v_3$), and check that it equals
$f(\cratio(\cdot))h$. The Mathematica code in this final section was
generated by the authors of this paper, not by Schramm.

First we redefine \Mcardy to have an explicit constant and define the
purported tripod probability.

\begin{center}

\includegraphics{652i11.eps}
\vspace*{3pt}
\end{center}

Next we differentiate Cardy's formula three times.\vspace*{6pt}

\begin{center}

\includegraphics{652i12.eps}
\vspace*{3pt}

\end{center}

This is the same as \Mh defined earlier, but with the trick of
eliminating the variable $W$ and expressing the formula in terms of $s$.
Next we triply differentiate the purported tripod probability.\vspace*{6pt}

\begin{center}

\includegraphics{652i13.eps}
\vspace*{3pt}

\end{center}

Notice that the triple derivative of the tripod probability is
expressed in terms of two different hypergeometric functions. In
order to compare this expression with the conditional probability
computed in Section \ref{sec:cond-prob}, we need to use some
hypergeometric identities.
We use one of Gauss's relations between ``contiguous''
hypergeometric functions [Erd{\'e}lyi et al.
(\citeyear{EMOT}), Section~2.8, equation 33], to write
\[
-\tfrac13 \,\hspace*{0.4pt}{}_2\hspace*{-0.4pt}F_1\bigl(\tfrac23,\tfrac43;\tfrac53;s\bigr) + \tfrac23 (1-s)
\,\hspace*{0.4pt}{}_2\hspace*{-0.4pt}F_1\bigl(\tfrac53,\tfrac43;\tfrac53;s\bigr) - \tfrac13
\,\hspace*{0.4pt}{}_2\hspace*{-0.4pt}F_1\bigl(\tfrac23,\tfrac
13;\tfrac53;s\bigr)=0.
\]
But ${}_2\hspace*{-0.4pt}F_1(c,b;c;s) = (1-s)^{-b}$ \citet{EMOT}, Section 2.8,
equation 4, so
\begin{equation}
{}_2\hspace*{-0.4pt}F_1\bigl(\tfrac23,\tfrac43;\tfrac53;s\bigr) = 2 (1-s)^{-1/3}
- {}_2\hspace*{-0.4pt}F_1\bigl(\tfrac
23,\tfrac13;\tfrac53;s\bigr).
\end{equation}

\begin{center}

\includegraphics{652i14.eps}
\vspace*{3pt}

\end{center}

Next we compare the two expressions for the conditional probability and
verify that they are the same.\vspace*{6pt}

\begin{center}

\includegraphics{652i15.eps}
\vspace*{3pt}

\end{center}

Therefore the triple derivatives agree, and we have established (\ref{PA}).

\section{Percolation statement} \label{s.percolationstatement}

\mbox{\hspace*{8pt}}We have the established equivalence of\break $\SLEtripod(s)$ and $\tripod
(s)$, but we still need
to make the connection to percolation.
\begin{theorem} \label{t.watts}
Let $D \subset\C$ be a fixed bounded
Jordan domain with marked points $x_1,x_2,x_3,x_4$ on its boundary.
For any $\eps$, we may consider the hexagonal lattice rescaled to
have side length $\eps$ and color the faces blue and yellow
according to site percolation. Let $B$ be the closure of the set of
blue faces, and let $P^\eps$ be the probability that $B \cap\overline D$
contains a connected component that intersects all four boundary
segments $(x_1,x_2)$, $(x_2,x_3)$, $(x_3,x_4)$ and $(x_4,x_1)$.
Then
\[
\lim_{\eps\to0} P^\eps= \watts(s).
\]
\end{theorem}

Proving Theorem \ref{t.watts} solves the problem addressed by Watts.
However, we remark that more general statements are probably possible.
Any domain~$D$ with four marked boundary points has a ``center''
$c(D)$ with the property that a conformal map taking the domain to a
rectangle (and the points to the corners) sends $c$ to the center of
the rectangle. Oded would probably have preferred to show that for any
sequence $D_n$ of simply connected \textit{marked hexagonal domains}
(domains comprised of unions of hexagons within a~fixed hexagonal lattice $H$ with four
marked boundary points of cross ratios $s_n$ converging to $s$), the
probability of the Watts event tends to $\watts(s)$ provided that the
distance from $c(D_n)$ to $\partial D_n$ tends to~$\infty$. (Oded's
SLE convergence results are similarly general
[\citet{LSW}, Schramm and Sheffield
(\citeyear{SchrammSheffieldHE,SchrammSheffieldGFF})].) However, Oded's
derivation of Watts' formula (like Dub\'{e}dat's derivation) depends on
SLE$_6$ convergence, and existing SLE$_6$ convergence statements [e.g.,
\citet{CM}] are not quite general enough to imply this.
\begin{pf*}{Proof of Theorem \ref{t.watts}}
As shown in Section \ref{s.tripodreduction}, it suffices to prove the
analogous statement about tripod events $(x_1,x_2)$, $(x_2,x_3)$,
and $(x_3,x_4)$ and the function $\tripod(s)$.

Let $\eps_n$ be a sequence of positive reals tending to zero, and
define $\gamma_n$ to be the random interface in $D$ obtained from
percolation on $\eps_n$ times the hexagonal lattice, between the
lattice points closest to $x_2$ and $x_4$. Let~$a_n,\allowbreak b_n,c_n$ be the
tripod set for this interface and the points $x_1$ and $x_3$, that
is,~$c_n$ is the first point on $\gamma_n \cap\partial D_n$
outside the boundary segment $(x_1,x_3)$, and the interface
$\gamma_n$ up to point $c_n$ last hits the boundary intervals
$(x_1,x_2)$ and $(x_2,x_3)$ at $a_n$ and $b_n$, respectively. From the work
of \citet{CM}, we can couple the $\gamma_n$ and
$\gamma$ in such a way that $\gamma_n \to\gamma$ almost surely in
the uniform topology (in which two curves are close if they can be
parameterized in such a way that they are close at all times). By
the compactness of $\partial D$ (and the corresponding compactness---in
the topology of convergence in law---of the space of
measures on $\partial D$) it is not hard to see that there must be a
subsequence of the $n$ values and a coupling of the $\gamma_n$
with~$\gamma$ in which the entire quadruple $(\gamma_n, a_n, b_n, c_n)$
converges almost surely to some limit. If we could show further
that this limit must be $(\gamma, a,b,c)$ almost surely, this would
imply the theorem, since uniform topology convergence would imply
that if $\gamma$ hits $a$ before $b$ then $\gamma_n$ hits $a_n$
before $b_n$ for large enough $n$ almost surely. However, it is not
clear a priori that this limit is $(\gamma, a,b,c)$
almost surely (even though the $\gamma_n$ converge to $\gamma$),
since while $\gamma$ touches the boundary at $a$, $b$ and $c$, it
could be that $\gamma_n$ comes close to the boundary at these points
without touching it.

To obtain a contradiction, let us suppose that there is a uniformly positive
probability (i.e., bounded away from $0$ as $n\to\infty$) that,
say, the limit of the~$a_n$ is not $a$. (The
argument for the $b_n$ and the $c_n$ is essentially the same.) Then
there must be an open interval $(\alpha_1, \alpha_2)$ of the
boundary and an open subinterval
$(\beta_1,\beta_2)\subset(\alpha_1,\alpha_2)$ of the boundary (with
$\beta_1\neq\alpha_1$ and $\beta_2\neq\alpha_2$) such that there is
a uniformly positive probability that $a$ lies $(\beta_1,\beta_2)$
but the limit of the $a_n$ does not lie in that\vadjust{\eject}
$(\alpha_1,\alpha_2)$. Now we can expand the Jordan domain $D$ to a
larger Jordan domain $\tilde D$ that includes a neighborhood of
$(\beta_1,\beta_2)$, but where the boundary of $\tilde D$ agrees
with boundary of $D$ outside of $(\alpha_1,\alpha_2)$. Let $\tilde
\gamma_n$ denote the discrete interfaces in this expanded domain.
We can couple the $\tilde\gamma_n$ with the $\gamma_n$ in such a
way that the two agree whenever~$\tilde\gamma_n$ does not leave $D$ (by
using the same percolation to define both). But now we have a
coupling of the $\tilde\gamma_n$ sequence with the property that
there is a~positive probability that the limit of the $\tilde
\gamma_n$ is a path that\vspace*{2pt} hits the boundary of $\tilde D \setminus D$
without entering $\tilde D \setminus D$. This implies that if the
$\tilde\gamma_n$ converge in law, they must converge to a random
path that\vspace*{2pt} with positive probability hits the boundary of $\tilde D
\setminus D$ without entering $\tilde D \setminus D$. By the
Camia--Newman theorem, applied to the domain $\tilde D$, the
$\tilde\gamma_n$ converge in law to chordal SLE$_6$ in $\tilde D$,
and on the event that SLE$_6$ hits $\partial(\tilde D \setminus D)$,
it will a.s. enter $\tilde D \setminus D$, a~contradiction.
\end{pf*}

\section*{Acknowledgments}
We thank Nike Sun, Kalle Kyt\"{o}l\"{a} and Stas Smirnov for their
comments on an earlier version of this article.

redpajama_set_name: RedPajamaArXiv


text:
Oblężenie Tulonu (18 września – 18 grudnia 1793) – jedno z pierwszych zwycięskich dla strony republikańskiej starć z rojalistycznymi buntownikami. Miało miejsce w portowym mieście Tulon, położonym na południu Francji, w okresie rewolucji francuskiej.

Okoliczności
Po aresztowaniu żyrondystowskich posłów w dniu 31 maja 1793 w różnych miejscach na terenie Francji ożywiły się ruchy reakcyjne – wywołując bunty, obejmujące swym zasięgiem takie miasta jak: Lyon, Awinion, Nîmes i Marsylia. W Tulonie federaliści uzyskali przewagę nad frakcją jakobinów, jednak wkrótce sami zostali zdominowani przez przeważających liczebnie rojalistów. Po obwieszczeniu odzyskania Marsylii przez republikanów oraz w reakcji na akcje odwetowe przeprowadzone tam przez rewolucjonistów, baron d'Imbert wezwał na pomoc floty angielską i hiszpańską. W dniu 28 sierpnia admirał Samuel Hood z angielskiej floty królewskiej (Royal Navy) wraz z admirałem  Juanem de Lángarą wyruszyli do Tulonu ze wsparciem dla rojalistów, na czele 13-tysięcznych oddziałów, złożonych z żołnierzy brytyjskich, hiszpańskich, neapolitańskich oraz piemonckich. Przybycie zagranicznych popleczników rojalistów stanowiło dotkliwy cios dla wojsk republikańskich. Tulon stanowił bowiem kluczową bazę floty francuskiej. Jego utrata oznaczałaby kres morskich ambicji republikańskiej Francji, która nie mogłaby rywalizować z państwami koalicji antyfrancuskiej, a w szczególności z Anglią o panowanie na morzach. Ponadto porażka sił rewolucyjnych w Tulonie mogłaby stać się groźnym precedensem, który ośmieliłby do buntu ruchy reakcyjne w innych miejscach kraju. Tak więc dalsze istnienie republiki stało pod znakiem zapytania i w dużej mierze zależało od losów oblężenia. W dniu 1 października baron d'Imbert ogłosił młodego Ludwika XVII Burbona królem Francji. Wywiesił także królewską flagę Fleur-de-lis, jednocześnie powierzając ochronę Tulonu flocie brytyjskiej.

Oblężenie

Oddziały republikańskie pod komendą Jeana François Carteaux pojawiły się pod Tulonem w dniu 8 września, odbiwszy uprzednio z rąk buntowników Awinion, Marsylię oraz Ollioules. Połączyły się z 6-tysięcznymi oddziałami Marynarki Alpejskiej, dowodzonej przez generała Jeana François Cornu de La Poype, który był świeżo po zdobyciu La Valette-du-Var i szykował się do uderzenia na fortyfikacje w Mount Faron, górujące ponad wschodnią stroną miasta. Do armii tej dołączyły posiłki w sile trzech tysięcy marynarzy pod dowództwem admirała de Saint Juliena, który w przeciwieństwie do swojego zwierzchnika, Trogoffa, odmówił współpracy z Brytyjczykami. Kolejnych 5 tysięcy żołnierzy do realizacji planu zdobycia Tulonu dostarczyła Armia Włoch. Na czele tego oddziału stał generał La Poype.

Dowódca artylerii Elzear Auguste Donmartin został ranny pod Ollioules, co wykluczyło go z udziału w działaniach oblężniczych. W jego miejsce specjalni wysłannicy Konwentu – Augustin Robespierre oraz Antoine Christophe Saliceti – wyznaczyli młodego kapitana artylerii, Napoleona Bonaparte, który był ich znajomym. Bonaparte znajdował się wówczas w okolicach, jako że zajmował się eskortą konwoju wiozącego proch na trasie z Nicei. Zatrzymał się w pobliżu Tulonu, by spotkać się ze swoim rodakiem z Korsyki, Salicetim. Bonaparte służył w armii już podczas walk w Awinionie. Zastąpił na stanowisku dowódcy artylerii Donmartina, chociaż obaj panowie nie darzyli się wzajemnie sympatią.

Pomimo obopólnej niechęci pomiędzy Bonapartem i jego zwierzchnikiem, młody oficer artylerii zdołał zgromadzić odpowiednie siły godne oblężenia Tulonu i zdolne przeciwstawić się pospiesznie wzniesionym przez Anglików fortyfikacjom. Na potrzeby oblężenia ściągnięto działa z Marsylii, Awinionu oraz z Armii Włoch.
Miejscowa ludność starała się udowodnić swoją lojalność wobec republikanów, przeciw którym niedługo wcześniej wnosiła bunty. Okolicznych wytwórców szantażem przymuszono do dostaw zaopatrzenia oraz zwierząt dla oddziałów republikańskich. Działania Napoleona doprowadziły do pozyskania 100 dział na potrzeby oblężenia. Dzięki wsparciu swoich przyjaciół, komisarzy Salicetiego oraz Augustina Robespierre'a, obdarzonych przez Konwent Narodowy olbrzymią władzą w terenie, Bonaparte zdołał przekonać emerytowanych oficerów artylerii do powrotu do wojska. Rozwiązanie to nie wystarczało jednak, by zapewnić dostateczną obsadę dział. Dlatego też Bonaparte zorganizował intensywne szkolenie piechurów w obsłudze armat, które dostarczyło mu żołnierzy zdolnych do wykorzystania nabytego potencjału artyleryjskiego.

Jednak pomimo wysiłku włożonego w przygotowanie operacji, Bonaparte nie był tak pewien jej powodzenia, jak przy kolejnych operacjach militarnych, które przeprowadzał w późniejszych latach. Oficerowie współpracujący z nim wykazywali się niekompetencją. Bonaparte był zaniepokojony niepotrzebnymi opóźnieniami, wynikającymi z błędów przez nich popełnianych. Jego niepokój znalazł wyraz w listownym apelu do Komitetu Ocalenia Publicznego. Proponował, by na jego miejsce postawiono na czele artylerii dowódcę w randze generała, gdyż uważał, że on sam nie zdoła poradzić sobie ze swoimi przełożonymi, przejawiającymi niedostatki umiejętności: "... (ktoś, kto potrafiłby) wzbudzić respekt i poradzić sobie ze zgrają głupców ze sztabu, z którymi trzeba się nieustannie kłócić i zmuszać ich, by porzucili swoje przesądy i podjęli kroki, które zarówno w teorii, jak i w praktyce są aksjomatyczne dla każdego wyszkolonego oficera w tym korpusie".

Rozeznawszy się w sytuacji, Bonaparte stworzył plan, który przewidywał zdobycie fortów l'Eguillette oraz Balaguier położonych na wzgórzu Cairo, przez co zablokowany zostałby przejazd pomiędzy mniejszymi a większymi przystaniami w porcie. W ten sposób przeciwnik nie byłby w stanie dostarczać zaopatrzenia oblężonemu miastu. Carteaux, pozbawiony entuzjazmu, wysłał tylko niewielki oddział pod komendą generała-majora Delaborde'a, którego atak z 22 września zakończył się niepowodzeniem. Stał się jednocześnie sygnałem dla dowódców sił koalicji antyfrancuskiej o nadciągającym zagrożeniu. Aby ustrzec się przed nim, zbudowali na szczycie wzgórza Fort Mulgrave, nazwany tak na cześć brytyjskiego dowódcy Henry'ego Phippsa, 1. hrabiego Mulgrave. Wsparcie dla niego stanowiły trzy mniejsze forty o nazwach: Saint-Philippe,  Saint-Côme oraz Saint-Charles. Ten, sprawiający wrażenie niemożliwego do zdobycia, zespół fortyfikacji, Francuzi okrzyknęli "Małym Gibraltarem".

Bonapartemu nie wystarczała pojedyncza bateria artyleryjska, zwana "Górą" usytuowana na wzgórzu Saint-Laurent, którą rozstawiono tam 19 września. Postanowił umiejscowić kolejną na wybrzeżu Brégallion. W historii  zapisała się ona pod nazwą "Sankiuloci". Brytyjski admirał Hood bez powodzenia usiłował ją zlikwidować. Ze względu na obecność mielizn pomiędzy Mourillon a Tour Royale flota brytyjska musiała się rozciągnąć wzdłuż wybrzeża. W dniu 1 października, po nieudanym dla generała La Poype'a uderzeniu na Wschodni Fort Faron, Bonaparte otrzymał zadanie zbombardowania wielkiego fortu w Malbousquet, którego neutralizacja miała kluczowe znaczenie dla zdobycia Tulonu. Dlatego też Napoleon ściągnął artylerię z całej okolicy, gromadząc pod swoją komendą pięćdziesiąt baterii – po sześć armat w każdej z nich. W dniu 19 października awansowano go na stopień dowódcy batalionu (chef de bataillon). Sformował wielką baterię, którą nazwano "Konwentową". Została ona rozmieszczona na wzgórzu Arènes bezpośrednio naprzeciw fortu. Za wsparcie służyły jej baterie: z "Obozu Republikanów" na wzgórzu Dumonceau, "Farinière" na wzgórzu Gaux, a także "Poudrière" na wzgórzu Lagoubran.

W dniu 11 listopada Carteaux otrzymał dymisję, a zwolnione przez niego stanowisko objął Doppet, z wykształcenia lekarz, którego brak zdecydowania doprowadził do nieudanego ataku z zaskoczenia na Fort Mulgrave w dniu 16 listopada. Świadomy własnej niekompetencji, podał się do dymisji. Na stanowisku zastąpił go zawodowy żołnierz, Dugommier, który szybko docenił zalety planu Bonapartego i przygotował atak na fortyfikacje "Małego Gibraltaru". Tuż po jego przyjeździe, w dniu 20 listopada na górskim grzbiecie l'Evescat sformowano baterię o kryptonimie "Jakobini". Później, w dniu 28 listopada na lewym skrzydle powstała bateria "Nieustraszonych". Wreszcie 14 grudnia lukę pomiędzy nimi wypełniono, formując baterię "Chasse Coquins". Kolejne dwie baterie utworzono na potrzeby odparcia ewentualnego ataku ze strony koalicyjnej marynarki. Nadano im kryptonimy: "Wielki Port" oraz "Cztery wiatraki".

Przyciśnięte ostrzałem artyleryjskim oddziały angielsko-neapolitańskie przeprowadziły udany wypad przeciwko baterii "Konwentowej", uwieńczony przechwyceniem dział. Francuskie kontrnatarcie, kierowane przez Dugommiera oraz Bonapartego doprowadziło do odepchnięcia Brytyjczyków oraz wzięcia do niewoli brytyjskiego generała O'Hara. Zapoczątkował on negocjacje dotyczące kapitulacji sił koalicyjnych z młodszym Robespierre'em oraz Antoine'em Louisem Albitte'em, reprezentującymi republikańską Francję. W trakcie rozmów uzgodniono rozbrojenie batalionów rojalistów, federalistów oraz ich koalicjantów.

Wziąwszy za jeńca generała O'Hara, nocą z 16-17 grudnia republikańscy dowódcy: Dugommier, Lapoype oraz Bonaparte (już w randze pułkownika) przypuścili główne natarcie. Około północy rozpoczął się atak na "Mały Gibraltar". Walki ciągnęły się przez całą noc. Bonaparte został ugodzony bagnetem w udo przez brytyjskiego sierżanta. Jednak już nad rankiem, gdy republikanie zdobyli pozycje na "Małym Gibraltarze", Marmont uzyskał możliwość rozmieszczenia tam artylerii, skierowanej przeciwko fortom l'Eguillette oraz Balaguier. Jeszcze tego samego dnia Brytyjczycy oddali te pozycje bez walki. Tymczasem La Poype mógł wreszcie zająć fortyfikacje Faron oraz Malbousquet. Przyciśnięci alianci postanowili ewakuować się drogą morską. Brytyjski komandor Sydney Smith otrzymał od Hooda rozkaz spalenia statków transportowych oraz arsenału; incydent ten jest uznawany za "najbardziej dotkliwy cios dla francuskiej marynarki w drugiej połowie XVIII wieku".

Pacyfikacja
Żołnierze republikańscy wkroczyli do miasta 19 grudnia. Pacyfikacja dowodzona przez Paula Barrasa oraz Stanislasa Frérona była wyjątkowo krwawa. Szacuje się, że około 700 – 800 jeńców rozstrzelano lub zadźgano bagnetami na tulońskim Polu Marsowym. Bonaparte, który w tym czasie leczył rany u doktora Jeana François Hernandeza, nie uczestniczył w masakrze. W dniu 22 grudnia awansowano go do stopnia generała brygady. Udał się wówczas w drogę do Nicei, by objąć tam stanowisko dowódcy artylerii w republikańskiej Armii Włoch. Na bramie, stanowiącej element dawnych murów wokół Tulonu zamontowano tablicę upamiętniającą wyjazd Napoleona do Armii Włoch. Bramę tę okrzyknięto mianem "Porte d'Italie" (Bramy Włoskiej).

W literaturze pięknej
Opis brutalnej pacyfikacji Tulonu przez zwycięskie wojska republikańskie został przedstawiony w powieści Josepha Conrada pt. Korsarz.

Przypisy

Bibliografia

Bitwy I koalicji antyfrancuskiej
Tulon

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