Thomas Bolli & Stefanie Hof, 2014. "The Impact of Apprenticeship Training on Personality Traits: An Instrumental Variable Approach," KOF Working papers 14-350, KOF Swiss Economic Institute, ETH Zurich.
Thomas Bolli & Martin Woerter, 2013. "Technological Diversification and Innovation Performance," KOF Working papers 13-336, KOF Swiss Economic Institute, ETH Zurich.
Thomas Bolli & Joerg Schlaepfer, 2013. "Job Mobility, Peer Effects, and Research Productivity in Economics," KOF Working papers 13-342, KOF Swiss Economic Institute, ETH Zurich.
Thomas Bolli & Jörg Schläpfer, 2015. "Job mobility, peer effects, and research productivity in economics," Scientometrics, Springer;Akadémiai Kiadó, vol. 104(3), pages 629-650, September.
Thomas Bolli & Martin Woerter, 2011. "Competition and R&D cooperation with universities and competitors," KOF Working papers 11-275, KOF Swiss Economic Institute, ETH Zurich.
Thomas Bolli & Martin Woerter, 2013. "Competition and R&D cooperation with universities and competitors," The Journal of Technology Transfer, Springer, vol. 38(6), pages 768-787, December.
Thomas Bolli, 2011. "The Global Production Frontier of Universities," KOF Working papers 11-272, KOF Swiss Economic Institute, ETH Zurich.
Thomas Bolli & Mehdi Farsi, 2011. "The dynamics of labor productivity in Swiss universities," KOF Working papers 11-278, KOF Swiss Economic Institute, ETH Zurich.
Spyros Arvanitis & Thomas Bolli, 2009. "A comparison of firm-level innovation cooperation in five European countries," KOF Working papers 09-232, KOF Swiss Economic Institute, ETH Zurich.
Thomas Bolli & Frank Somogyi, 2009. "Do Competitively Acquired Funds Induce Public Research Institutions to Behave Efficiently?," KOF Working papers 09-242, KOF Swiss Economic Institute, ETH Zurich.
Thomas Bolli & Mathias Zurlinden, 2008. "Measuring growth of labour quality and the quality-adjusted unemployment rate in Switzerland," Working Papers 2008-13, Swiss National Bank.
Thomas Bolli & Mathias Zurlinden, 2009. "Measuring Growth of Labor Quality and the Quality-Adjusted Unemployment Rate in Switzerland," Applied Economics Quarterly (formerly: Konjunkturpolitik), Duncker & Humblot, Berlin, vol. 55(2), pages 121-145.
Mathias Zurlinden & Thomas Bolli, 2008. "Measurement of Labor Quality Growth: Caused by Unobservable Characteristics," KOF Working papers 08-203, KOF Swiss Economic Institute, ETH Zurich.
Thomas Bolli & Mathias Zurlinden, 2012. "Measurement of labour quality growth caused by unobservable characteristics," Applied Economics, Taylor & Francis Journals, vol. 44(18), pages 2297-2308, June.
Thomas Bolli & Mathias Zurlinden, 2009. "Measurement of labor quality growth caused by unobservable characteristics," Working Papers 2009-01, Swiss National Bank.
Spyros Arvanitis & Thomas Bolli, 2013. "A Comparison of National and International Innovation Cooperation in Five European Countries," Review of Industrial Organization, Springer;The Industrial Organization Society, vol. 43(3), pages 163-191, November.
Spyros Arvanitis & Marius Ley & Tobias Stucki & Martin Woerter & Thomas Bolli & Christian Soltmann, 2012. "Potenziale für Cleantech in der Schweizer Industrie," KOF Analysen, KOF Swiss Economic Institute, ETH Zurich, vol. 6(1), pages 73-92, March.
Bolli, Thomas & Somogyi, Frank, 2011. "Do competitively acquired funds induce universities to increase productivity?," Research Policy, Elsevier, vol. 40(1), pages 136-147, February.
Spyros Arvanitis & Heinz Hollenstein & Marius Ley & Thomas Bolli & Martin Woerter, 2010. "Starke Position der Schweizer Wirtschaft im internationalen Innovationswettbewerb," KOF Analysen, KOF Swiss Economic Institute, ETH Zurich, vol. 4(1), pages 87-97, March.
Thomas Bolli, 2010. "Welche Rolle spielt Asien als Wissensstandort für die Schweizer Wirtschaft?," KOF Analysen, KOF Swiss Economic Institute, ETH Zurich, vol. 4(2), pages 57-71, June.
Gao, Lei & Brink, Alisa G., 2017. "Whistleblowing studies in accounting research: A review of experimental studies on the determinants of whistleblowing," Journal of Accounting Literature, Elsevier, vol. 38(C), pages 1-13.
Uschi Backes-Gellner & Agnes Bäker & Kerstin Pull, 2018. "The Opportunity Costs of Becoming a Dean: Does Leadership in Academia Crowd Out Research?," Schmalenbach Business Review, Springer;Schmalenbach-Gesellschaft, vol. 70(2), pages 189-208, May.
Agnes Bäker & Susanne Breuninger & Julia Muschallik & Kerstin Pull & Uschi Backes-Gellner, 2016. "Time to Go? (Inter)National Mobility and Appointment Success of Young Academics," Schmalenbach Business Review, Springer;Schmalenbach-Gesellschaft, vol. 17(3), pages 401-421, December.
Irene Ramos-Vielba & Mabel Sánchez-Barrioluengo & Richard Woolley, 2016. "Scientific research groups' cooperation with firms and government agencies: motivations and barriers," The Journal of Technology Transfer, Springer, vol. 41(3), pages 558-585, June.
Zara Daghbashyan & Enrico Deiaco & Maureen McKelvey, 2014. "How and why does cost efficiency of universities differ across European countries? An explorative attempt using new microdata," Chapters,in: Knowledge, Diversity and Performance in European Higher Education, chapter 9, pages iii-iii Edward Elgar Publishing.
Calogero Guccio & Marco Ferdinando Martorana & Luisa Monaco, 2016. "Evaluating the impact of the Bologna Process on the efficiency convergence of Italian universities: a non-parametric frontier approach," Journal of Productivity Analysis, Springer, vol. 45(3), pages 275-298, June.
Marcel Clermont, 2016. "Effectiveness and efficiency of research in Germany over time: an analysis of German business schools between 2001 and 2009," Scientometrics, Springer;Akadémiai Kiadó, vol. 108(3), pages 1347-1381, September.
Maria Olivares & Andrea Schenker-Wicki, 2012. "The Dynamics of Productivity in the Swiss and German University Sector: A Non-Parametric Analysis that Accounts for Heterogeneous Production," Working Papers 309, University of Zurich, Department of Business Administration (IBW).
Martin Srholec, 2014. "Cooperation and Innovative Performance of Firms: Panel Data Evidence from the Czech Republic, Norway and the UK," Journal of the Knowledge Economy, Springer;Portland International Center for Management of Engineering and Technology (PICMET), vol. 5(1), pages 133-155, March.
Michael Graff & Massimo Mannino & Michael Siegenthaler, 2014. "The Swiss "Job Miracle"," KOF Working papers 14-368, KOF Swiss Economic Institute, ETH Zurich.
Barbara Rudolf & Mathias Zurlinden, 2010. "Productivity and Economic Growth in Switzerland 1991-2006," Swiss Journal of Economics and Statistics (SJES), Swiss Society of Economics and Statistics (SSES), vol. 146(III), pages 577-600, September.
Laura-Maria DINDIRE, 2012. "Assessment Model Of The Nations? Human Capital - The Case Of The Eu Countries," Management Strategies Journal, Constantin Brancoveanu University, vol. 18(4), pages 21-27.
Øivind A. Nilsen & Arvid Raknerud & Marina Rybalka & Terje Skjerpen, 2011. "The Importance Of Skill Measurement For Growth Accounting," Review of Income and Wealth, International Association for Research in Income and Wealth, vol. 57(2), pages 293-305, June.
Nilsen, Øivind Anti & Raknerud, Arvid & Rybalka, Marina & Skjerpen, Terje, 2010. "The Importance of Skill Measurement for Growth Accounting," IZA Discussion Papers 4997, Institute for the Study of Labor (IZA).
Spyros Arvanitis, 2012. "How do different motives for R&D cooperation affect firm performance? – An analysis based on Swiss micro data," Journal of Evolutionary Economics, Springer, vol. 22(5), pages 981-1007, November.
Erika & Rosina, 2016. ""Technological cooperation in Spanish firms"," IREA Working Papers 201601, University of Barcelona, Research Institute of Applied Economics, revised Jan 2016.
Martin Srholec, 2015. "Understanding the diversity of cooperation on innovation across countries: multilevel evidence from Europe," Economics of Innovation and New Technology, Taylor & Francis Journals, vol. 24(1-2), pages 159-182, March.
Erika Raquel Badillo & Rosina Moreno, 2014. ""Does absorptive capacity determine collaborative research returns to innovation? A geographical dimension"," AQR Working Papers 201416, University of Barcelona, Regional Quantitative Analysis Group, revised Oct 2014.
Erika Raquel Badillo & Rosina Moreno, 2014. ""Does absorptive capacity determine collaborative research returns to innovation? A geographical dimension"," IREA Working Papers 201428, University of Barcelona, Research Institute of Applied Economics, revised Nov 2014.
Amelie Cloutier & Nabil Amara, 2018. "DETERMINANTS OF PROPENSITY VS. INTENSITY OF INNOVATION CO-OPERATION FOR SMEs," International Journal of Innovation Management (ijim), World Scientific Publishing Co. Pte. Ltd., vol. 22(05), pages 1-31, June.
Spyros Arvanitis & Areti Gkypali & Kostas Tsekouras, 2014. "Knowledge Base, Exporting Activities, Innovation Openness and Innovation Performance," KOF Working papers 14-361, KOF Swiss Economic Institute, ETH Zurich.
Gkypali, Areti & Arvanitis, Spyros & Tsekouras, Kostas, 2018. "Absorptive capacity, exporting activities, innovation openness and innovation performance: A SEM approach towards a unifying framework," Technological Forecasting and Social Change, Elsevier, vol. 132(C), pages 143-155.
Erika Raquel Badillo & Rosina Moreno, 2016. "Are Collaborative Agreements in Innovation Activities Persistent at the Firm Level? Empirical Evidence for the Spanish Case," Review of Industrial Organization, Springer;The Industrial Organization Society, vol. 49(1), pages 71-101, August.
Irina Ervits & Malgorzata Zmuda, 2018. "A cross-country comparison of the effects of institutions on internationally oriented innovation," Journal of International Entrepreneurship, Springer, vol. 16(4), pages 486-503, December.
Luísa Carvalho & Maria José Madeira & João Carvalho & Dulcineia Catarina Moura & Filipe P. Duarte, 2018. "Cooperation for Innovation in the European Union: Outlook and Evidences Using CIS for 15 European Countries," Journal of the Knowledge Economy, Springer;Portland International Center for Management of Engineering and Technology (PICMET), vol. 9(2), pages 506-525, June.
Erika Raquel Badillo & Rosina Moreno, 2018. "Does absorptive capacity determine collaboration returns to innovation? A geographical dimension," The Annals of Regional Science, Springer;Western Regional Science Association, vol. 60(3), pages 473-499, May.
Avery Sen, 2017. "Island + Bridge: how transformative innovation is organized in the federal government," Science and Public Policy, Oxford University Press, vol. 44(5), pages 707-721.
Alona Zharova & Janine Tellinger-Rice & Wolfgang Karl Härdle, 2018. "How to Measure a Performance of a Collaborative Research Centre," SFB 649 Discussion Papers SFB649DP2018-001, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
Claudia Fuentes & Gabriela Dutrénit, 2016. "Geographic proximity and university–industry interaction: the case of Mexico," The Journal of Technology Transfer, Springer, vol. 41(2), pages 329-348, April.
Kai Rao & Andrea Piccaluga & Xian-fei Meng, 2013. "The impact of human resource factors on university patent technology transfer activities in China – based on the analysis of provincial panel data," Chapters,in: Knowledge Commercialization and Valorization in Regional Economic Development, chapter 12, pages 258-283 Edward Elgar Publishing.
Grimpe, Christoph, 2012. "Extramural research grants and scientists' funding strategies: Beggars cannot be choosers?," Research Policy, Elsevier, vol. 41(8), pages 1448-1460.
Alona Zharova & Janine Tellinger-Rice & Wolfgang Karl Härdle, 2018. "How to measure the performance of a Collaborative Research Center," Scientometrics, Springer;Akadémiai Kiadó, vol. 117(2), pages 1023-1040, November.
Tüzin Baycan (ed.), 2013. "Knowledge Commercialization and Valorization in Regional Economic Development," Books, Edward Elgar Publishing, number 14868, April.
Mu-Hsuan Huang & Mei-Jhen Huang, 2018. "An analysis of global research funding from subject field and funding agencies perspectives in the G9 countries," Scientometrics, Springer;Akadémiai Kiadó, vol. 115(2), pages 833-847, May.
Чернова Е. Г. & Ахобадзе Т. Д. & Малова А. С. & Салтан А. А., 2017. "Модели Финансирования Высшего Образования И Эффективность Деятельности Университетов Эмпирическое Исследование Европейского Опыта И Отечественная Практика," Вопросы образования // Educational Studies, НИУ ВШЭ, issue 3, pages 37-82.
De Fuentes, Claudia & Dutrénit, Gabriela, 2012. "Best channels of academia–industry interaction for long-term benefit," Research Policy, Elsevier, vol. 41(9), pages 1666-1682.
Erik E. Lehmann & Matthias Menter, 2016. "University–industry collaboration and regional wealth," The Journal of Technology Transfer, Springer, vol. 41(6), pages 1284-1307, December.
Alona Zharova & Wolfgang K. Härdle & Stefan Lessmann, 2017. "Is Scientific Performance a Function of Funds?," SFB 649 Discussion Papers SFB649DP2017-028, Sonderforschungsbereich 649, Humboldt University, Berlin, Germany.
Wolszczak-Derlacz, Joanna, 2017. "An evaluation and explanation of (in)efficiency in higher education institutions in Europe and the U.S. with the application of two-stage semi-parametric DEA," Research Policy, Elsevier, vol. 46(9), pages 1595-1605.
Bolli, Thomas & Olivares, Maria & Bonaccorsi, Andrea & Daraio, Cinzia & Aracil, Adela Garcia & Lepori, Benedetto, 2016. "The differential effects of competitive funding on the production frontier and the efficiency of universities," Economics of Education Review, Elsevier, vol. 52(C), pages 91-104.
To update listings or check citations waiting for approval, Thomas Bolli should log into the RePEc Author Service.


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Here at Modern Tortilla we are continuing our customary tradition of making incredible Mexican food with only the freshest of ingredients. Each dish we create starts from what we enjoy eating, and we make sure to pass on our favorite flavors to you. If you have yet to try something from our delicious menu, now is the time! Introducing the Modern Mayan, a masterpiece of two of our most popular tacos offered side by side from our taco truck in Phoenix.
When coming up with the Modern Mayan, we just had to decide what our customers would love. We knew it was easy when starting with freshly made corn tortillas fresh out of the oven. Each tortilla is made by hand every single day for every dish we create. Next up, we top the tortillas with pork that is slowly braised in achiote for a full day; we believe this really brings out the flavors you will crave in these delicious tacos. Finally, we bring the decadence together with a fresh mixture of roasted tomatoes, garlic, pineapple, fresh avocado, pickled red onion, and of course, cotija cheese.
At the Modern Tortilla food truck in Phoenix, we understand that every taco needs to get paired with an equally delicious side to create the perfect meal. You can pick from Spanish rice or chips & fire roasted salsa. We handmake each side every day with the same fresh ingredients you have come to expect from main dishes. Don't forget to cap your meal off with some crème-filled churros for dessert!
There has never been a better time to check out the delicious menu at Modern Tortilla. In fact, the verdict is out that Modern Tortilla is the best Arizona food truck! Since 2016, we have has been specializing in Phoenix street food. When looking to try our delicious food, check out our calendar to see where we will be next. You can even request us at your next event by completing a simple form online!
Now is the time to try the Modern Mayan and other delicious entrees we continue to serve up at Modern Tortilla. Come check us out during one of our next scheduled events in the Phoenix valley. Your taste buds will not know what hit them after one of these delicious tacos!


text:
PIH6 Economic Evaluation of Carbetocine for the Prevention of Uterine Atony in Patients with Risk Factors in Mexico
German Del Angel Garcia, Fernando Garcia-Contreras, Patricia Constantino-Casas, Armando Nevarez-Sida, Noe Lopez-Gonzalez, Matias Garcia-Constantino, Manuel Zuniga
OBJECTIVE To evaluate the cost-effectiveness of IMSS formulary drugs for preventing uterine atony in patients with risk factors. METHODS A final report of a randomized pragmatic clinical trial at the Mexican Institute of Social Security (IMSS) is presented. Carbetocine and oxytocin are the only drugs within IMSS formulary for preventing uterine atony. Risk factor included fetal macrosomia, polyhydramnios, low insertion of the placenta, multiple gestation, prolonged labor, uterine myomas and chorioamnionitis. The effectiveness was defined as the reduction of the number of patients with uterine atony. The use of resources was obtained from the clinical trial and the costs were gotten from financial information from IMSS, and are expressed in US 2006 dollars. Squared ji and U de Mann Whitney test were used. Univariate and probabilistic sensitivity analyses, a Monte Carlo microsimulation with 10,000 iterations was performed using probability distribution data from the clinical trial. A 95% confidence interval of ICER was calculated by ellipse method. RESULTS Seventy-seven patients received carbetocine and 75 oxytocin. Both groups had similar obstetrics and sociodemographic characteristics. Uterine atony was reported in 19% in the oxytocin group compared to 8% in the carbetocine one (p <0.0001). Multiple gestation was the most frequent diagnosis, 35% (p <0.0001). Bleeding was less than 500 mL in the carbetocine group and 500–1000 mL in the oxytocin one (p <0.0001).Mean cost per patient treated with carbetocine was \$3525 vs.\$4054 for oxytocine (p <0.0001). Mean cost-effectiveness ratio for oxytocin was \$4944, while for carbetocine $3874; ICER showed that carbetocine was dominant. Univariate analysis sup-ported those results. The acceptability curve and health net benefits showed that carbetocine group was superior independently of WTP. CI 99% by ellipse method showed that carbetocine was dominant in 100% of cases. CONCLUSIONS Carbetocine was dominant in preventing uterine atony in patients with risk factors.
In Value In Health Journal

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: jQuery - divs not fading out completely before next starts fading in The problem is a simple one, I know, but I'm having a heck of a time trying to figure it out. I have 8 divs on a page that hold image galleries which start off hidden. When a user clicks on an icon to view a gallery, the first gallery they chose fades in all shiny. But when they choose another, the one they were viewing begins to fade out while the other fades in below it, and then moves up into its correct position as the first div is hidden.
See jsfiddle.
So, my question is apparent:
How do I get the gallery they were viewing to fade out completely before the next one fades into the correct space.

A: Your problem is explained by this sentence in the documentation (fadeOut):

If multiple elements are animated, it is important to note that the callback is executed once per matched element, not once for the animation as a whole.

In other words, because you are animating two things, the callback will be called twice.  Because one of your divs is already hidden, the fadeOut completes instantly*, and immediately calls back.
*Instantly completing if already shown/hidden is unique to the show/hide type helper functions.  If you constructed your own fade animation it would always take the specified amount of time, regardless of the current state.
You can fix this a couple of ways.  The easiest is to chain your fadeOut() calls inside of each other, instead of calling one on both elements:
$('.icon_one').click(function() {
    $('#image_two').fadeOut(function() {
        $('#image_three').fadeOut(function() {
            $('#image_one').fadeIn();
        });
    });
});

Because one will usually be hidden already, it will fall through instantly, but the other will take time.  Either way, only after both are complete will the fade in happen.
Demo: http://jsfiddle.net/jtbowden/XQnhs/
Of course, with a little change to your HTML, you can do all three with one click handler:

Red

Blue

Yellow

Script:
$('.icon_show').click(function() {
    var showID = '#image_' + $(this).data('num');
    $('.imageBox:visible').fadeOut(function() {
        $(showID).fadeIn(400);
    });
});

Demo: http://jsfiddle.net/jtbowden/NAcPW/
The key being that you automatically figure out which one to show based on the link you click.  In this case, I do it by storing a number in a data attribute.  You could also just use part of the ID, or do it by figuring out the offset using index and then feeding that into eq.
Demo: http://jsfiddle.net/jtbowden/NnN58/
There are lots of ways to simplify this code.

A: This was interesting... I like the way you set up your jsfiddle- it made it easier to play around with. see if this helps.. fadeOuts are bound to more than element. Only fade out the visible one. I think your issue is fadeout is getting called on elements that already faded out.
here is a slight variation on your jsfiddle..
$(function(){
        $('#image_two,#image_three').hide();

//1//

          $('.icon_one').click(function(){
              $('#image_two:visible,#image_three:visible').fadeOut(
                  function(){

                     $('#image_one').fadeIn();
                  }
              );
           });

//2//
alert("binding");
   $('.icon_two').click(function(){
       $('#image_one:visible,#image_three:visible').fadeOut(
            function(){
                $('#image_two').fadeIn();
             }
                                            );
  });

    //3//

   $('.icon_three').click(function(){
       $('#image_one:visible,#image_two:visible').fadeOut(function(){
        $('#image_three').fadeIn();
        });
  });

});

A: You can place divs on top of each other with position:absolute;
Working example: http://jsfiddle.net/Ghokun/wnTte/13/

A: how about replacing
$('#image_one').fadeIn();

with
$('#image_one').delay(500).fadeIn();

and like wise....

redpajama_set_name: RedPajamaStackExchange


Costa Teguise Golf enLatest newsEl CampoFrom 1978 to 2018….and many more years!
From 1978 to 2018….and many more years!
Last weekend took place the celebration of the 40th Anniversary of Costa Teguise Golf. From the afternoon of Thursday and throughout the Friday, all the guests and regular players of the golf course were invited to a Fitting organised by Golf Shop Canaries.
The 40th Anniversary golf tournament was an exclusive tournament that began on Friday the 26th at 3:45 pm with the longest Drive on the 10th Hole, Approach on the 19th Hole, Bunker on the Pitching Green and Putt on the Putting Green. After that, all the participants were invited to a Barbecue at the Tres Lunas Golf Restaurant in the Club House.
On Saturday, October 27, the tournament was played in the TEXAS SCRAMBLE modality with a participation of 92 players who enjoyed the day and had very good results.
At 14:00 pm was celebrated the Great Lunch offered to the guests of this 40th Birthday. A lot of relevant personalities were in the event, such as Ángel Vázquez, Tourism Promotion Advisor, Oswaldo Betancort, Mayor of Teguise, Héctor Fernández, Manager Lanzarote Tourist Board and SPEL, Lucas Montaner, Manager of the Canary Islands Golf Federation and Mr. Pablo Rivero, Delegate for Lanzarote of the Canary Islands Golf Federation.
During the awards ceremony, Mr Jaime G Canomanuel, Costa Teguise Golf Manager, made an emotional review of the club's 40-year history and 3 very important people for this club, such as Miguel Mestre, Rodney Barr and Nicolás García were awarded by this great contribution to the History of Costa Teguise Golf.


The once ubiquitous DF4s are now almost as rare as the steam locomotives they replaced in the 1980s and 1990s. A number remain in CNR service in the Haila'er area but are unlikely to survive once electric working starts on the main line from Qiqiha'er to Manzhouli, displacing large numbers of modern diesels.
DF4 1584 approached Meitian with train 4181/4184, the 06:12 passenger from Haila'er to Mangui, on 20 November 2016. Note the makeshift snow brushes and LED headlight.


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1) From Cobbler's Pose lean backward, bringing your elbows to the floor.
2) Lower the back all the way to the floor.
3) Stay here several minutes. To come out, roll over to your side and sit up, using your hands to support you.
If you are not comfortable reclining on the floor, you may use a bolster or several folded blankets to support the spine. You may also want to place a block under each knee for support.


A forensic investigator in the Committee on Ethics and Anti-Corruption has been accused of hacking into data ad agency at the Center for Integrity in Nairobi.
Mr. Weldon Kiptoo Rotich Monday denied a charge of unauthorized access to computer data in violation of the Law of Information and Communications of Kenya, 2009.
The offense was allegedly committed on 11 June.
The indictment alleges that caused the computer system to perform an illegal commission knowing that access was secured and authorized him function.
The indictment says the researcher on the material, while the offices on the ground floor of the commission, was discovered around 16:00 for their fellow staff members in the workstation to a colleague while out of the country on official business.
It is alleged that staff of the ICT unit were called to establish how the suspect had agreed user profile to a John Kiilu.
It was established that was entered using an administrator password that was not supposed to and that is not an administrator and not authorized by the ICT team, states the dossier submitted to Judge Daniel Ogembo Nairobi.
Prosecutors say Mr. Rotich was found to have logged on user window John Kiilu and a flash drive is inserted into the central processing unit of the computer (CPU) to retrieve information.
A screen on the monitor and saved a copy in printed and delivered to the police. The flash disk has also been taken to a police forensic laboratory in the Department of Criminal Investigation headquarters to establish its content.
On Monday, a lawyer requested copies of witness statements and material evidence the prosecution will use at trial.
Meanwhile, the suspect was ordered to pay a deposit of Sh50,000 pending a hearing.
The case will be heard on July 14.
Mr. Rotich has been ordered to appear in court on June 29, when his case was mentioned.


text:
\section{Background}
In this section we formally present the parking process on $ \mathbb{B}$ and gather a few ``rough'' probabilistic results (mostly adapted from \cite{GP19,BBJ19}).
\subsection{Parking on infinite trees}
\label{sec:parkingpropre}
Let $\tau$ be a rooted locally finite (plane) tree decorated with car arrivals $(a_u : u \in \tau)$. As described in the introduction, cars try to park on their arrival node, and if the spot is taken they travel downwards in search of the first empty spot and, in case there is no such spot, exit at the root. In the case $\tau$ is finite, an easy Abelian property shows that the number of cars visiting each vertex of the tree does not depend on the order in which we park the cars, see Section 2.1 of \cite{LaP16}.

On infinite trees, to prevent cumbersome issues,
 we shall stick to a given parking procedure: \emph{park the lowest cars first}. More precisely, for each $ n \geq 0$ let us consider the finite tree $[\tau]_{n}$ made of the first $n$ generations above the root $\varnothing$ (recall that $\tau$ is supposed locally finite) together with the restriction of the car arrivals on these vertices. We can then perform the parking on $ [\tau]_{n}$ and construct variables
 $$ x_{n}(u) \quad u \in [\tau]_{n},$$
 representing the number of cars that visited the vertices of $[\tau]_{n}$ in the parking process (recall that those variables do not depend on the order in which we parked the cars on $[\tau]_{n}$). Notice that for a given vertex $u \in \tau$, the function $ n \mapsto x_{n}( u)$ is non-decreasing (it is defined for $n$ larger than the height of $u$) so that we can let $n \to \infty$ and define
 $$ x(u) = \lim_{n \to \infty} x_{n}(u),$$
 as the limiting number of cars visiting $u$ in the parking process on $\tau$. This morally corresponds to parking the lowest cars first\footnote{In fact, we could equivalently fix an exhaustion of $\tau$ by finite trees $\tau_1 \subset \tau_2 \subset \cdots$ and define the parking on $\tau$ as the limit of the parking procedure over the $\tau_n$'s.}. In particular we say that $u$ is void if $ x(u)= 0$, that $u$ is occupied if $ x(u) \geq 1$ and the flux of outgoing cars at $u$ is $f(u) = (x(u)-1)_{+}$.

  \subsection{Rough phase transition}
 We now focus on the case of the binary tree $ \mathbb{B}$ with i.i.d.\ car arrivals $(A_u : u \in \mathbb{B})$ with law $\mu$ satisfying $ \mu(\{0,1\})<1$. We denote by $X(u)$ the number of cars that visited vertex $u \in \mathbb{B}$ as defined in the preceding section  and will use the shorthand notation $X= X(\varnothing)$. We first establish a dichotomy on $X$ in the next lemma, which we then interpret in more geometric terms by proving that there cannot be infinite black clusters with a finite flux.

\begin{lemma}[Dichotomy subcritical/supercritical] \label{lem:sub/super}\label{phase-transition} We have the following dichotomy:\\
 \textbf{Subcritical case.}  Either the sequence $(2^n \P(X>n) : n \geq 0)$ is bounded.\\
 \textbf{Supercritical case.} Or $X=\infty$ a.s, in which case all vertices are parked a.s.
\end{lemma}

\begin{proof}[Proof of the lemma.] Assume that $(2^n \P(X>n) : n \geq 0)$ is not bounded, and observe that the same is then true of the sequence $(2^n \P(X>n+k) : n \geq 0)$ for any integer $k$.
Then consider the collection of the $2^n$ i.i.d.~variables  $X({u})$  attached to the vertices $u$ of $\mathbb{B}$ at height $n$. We have the upper bound
$$ \P\Big(\bigcap_{u : |u| =n } \{X(u)\le n+k\}\Big) \le   \mathrm{e}^{- 2^n \P(X>n+k))},$$
with the right-hand side going to $0$ along a subsequence. But on the complement of the event on the left-hand side, one of the variables $X(u)$ is strictly larger  $n+k$ and this contribution only suffices to imply that $X= X( \varnothing) \geq k$.  Combined with our assumption, this implies that, almost surely, $X \geq k$, hence, $k$ being arbitrary, $X=\infty$. This means that the root of $ \mathbb{B}$ almost surely contains a car, and it is the same for any other vertex. \end{proof}

The next lemma says that the above dichotomy is equivalent to the existence of infinite black clusters. In particular, it rules out the possibility of having an infinite black cluster and a finite flux.
\begin{proposition} \label{prop:noblack} In the subcritical regime, there is no infinite black cluster.
\end{proposition}
\begin{proof} Suppose that $\mu$ is subcritical, so that all variables $X(u)$ are finite after the parking process. It suffices to prove that the probability that the cluster of the origin $  \mathcal{C}(\varnothing)$ is infinite is $0$.
Fix $p \geq0$ and let us consider the event $ \mathcal{E} = \{ X( \varnothing) =p \mbox{ and }  \mathcal{C}( \varnothing) \mbox{ is infinite}\}$. We shall explore the process by parking on the first $n$ levels of $ \mathbb{B}$ as in the preceding section. More precisely, let $ \mathcal{F}_{n}$ be the sigma field generated by the variables $X_{n}(u)$ for $X_{n}(u)$ the number of cars visiting the vertex $u$ when restricting the parking on $[ \mathbb{B}]_{n}$. We then construct a sequence of stopping times $\theta_{1} < \theta_{2}  <  \cdots$ obtained as follows: $\theta_{1} = \inf\{n \geq 0 : X_{n}( \varnothing) =p \}$ and then by induction $\theta_{i+1} = \inf\{ n > \theta_{i} : \varnothing \leftrightarrow \partial [\mathbb{B}]_{n}\}$ where $\varnothing \leftrightarrow \partial [\mathbb{B}]_{n}$ means that $ \varnothing$ is connected to the level $n$ by a path whose vertices satisfy $ X_{n}(u) \geq 1$. A moment's though shows that  on the event $ \mathcal{E}$ all these stopping times are finite for otherwise the black cluster of the origin would not be infinite. For $n \geq 2$, on the event $\{\theta_n <\infty\}$, let $v_n$ be the (first, for definiteness) vertex of $\partial [\mathbb{B}]_{\theta_n}$ to be connected to the root when parking on $[\mathbb{B}]_{\theta_n}$.  Set $\mathcal{D}_n = \{  A_{v_n 0},A_{v_n 1} \in \{0,1\}\}$ the event that the two children of $v_n$ have car arrivals $\leq 1$. Plainly, $\mathcal{E} \subset  \mathcal{D}_n \cap \{\theta_n <\infty\}$ since
otherwise, the flux coming from these two vertices would go down all the way to $ \varnothing$ and we would have $ X( \varnothing) > p$. In particular we have
$$ \mathbb{P}( \mathcal{D}_n  \mathbf{1}_{\theta_n < \infty}\mid \mathcal{F}_{\theta_n}) = (\mu_0+ \mu_1)^{2}<1.$$  Notice then that $ \mathcal{D}_{1}, ... , \mathcal{D}_{n-1}$ are $ \mathcal{F}_{\theta_n}$-measurable so that by induction we have
$$\P(\mathcal{E}) \leq \P(\theta_n<\infty)  \prod_{k=2}^n \P(\mathcal{D}_k |\theta_n <\infty ) \leq (\mu_0+ \mu_1)^{2(n-1)}, $$  which implies $\P(\mathcal{E})=0$ since $n$ is arbitrary and we assumed the distribution $\mu$  satisfies $\mu_0+ \mu_1<1$. \end{proof}

As a consequence of the (proof of)  Lemma \ref{lem:sub/super}, there is no lower bound for $\mathbb{E}[A]$ for supercritical parking, since one may cook up distributions $\mu$ with arbitrarily small expectation but $\mathbb{E}[2^A]=\infty$.
However, if the car arrival distribution is bounded, one can obtain  a lower bound for the expectation $\mathbb{E}[A]$ of the car arrival distribution for supercritical parking using a first moment method, see  \cite[Proposition 3.5]{GP19} and \cite{BBJ19} for details

To speak of a phase transition,  one may imagine a family $(\mu^{\alpha})_{\alpha \geq 0}$ of car arrival distributions that is stochastically increasing in the mean $\alpha$. In this case, the subcritical phase is  identified with a \emph{closed set} $ \alpha \in [0,\alpha_c]$, and the supercritical phase with the set $]\alpha_c,1]$. The fact that $\alpha_{c}$ is actually subcritical (i.e.~satisfies the first alternative of the dichotomy) can be seen by monotone convergence since the expectation of the flux is bounded above by $2$ in the whole subcritical phase as recalled in the introduction (see \cite{BBJ19} for details).

\section{Decomposition into fully-parked components}
In this section we present our combinatorial decomposition which underlies our main results. The idea is very simple: we decompose the final configuration on $ \mathbb{B}$ into the black clusters of parked vertices and the white empty vertices. This shows that we can decompose the final configuration as a two-type Bienaym\'e--Galton--Watson tree whose offspring distribution is related to the generating function $F$ of fully parked trees studied in detail in the next section.
\subsection{Fully parked trees}
\label{def-fpt}
Suppose that we performed the parking process on $ \mathbb{B}$, and recall that the black vertices are those $u \in \mathbb{B}$ satisfying $ X(u) \geq 1$, the other ones being the empty or white vertices. The \emph{finite} black connected components are \emph{fully parked trees} $\mathbf{t}$, i.e.\ connected subsets of the binary tree decorated by car arrivals $(a_u)_{u \in \mathbf{t}}$ such that after parking all vertices are occupied. If such a tree appears as the black component of the root $ \varnothing$, then the fully parked tree may have an outgoing flux at the root (i.e.~containing more cars than vertices), otherwise it contains as many cars as vertices. See Figure \ref{fig:FPT}.

\begin{figure}[!h]
 \begin{center}
 \includegraphics[width=12cm]{fpt}
 \caption{Three examples of fully parked trees. Notice that the first two have no outgoing flux and represent the same plane tree but their embeddings in $ \mathbb{B}$ is different. The last fully parked tree on the right has an outgoing flux of $2$ cars. \label{fig:FPT}}
 \end{center}
 \end{figure}

 For the enumeration of the fully parked trees we shall always consider that their bottom vertex is $\varnothing$. Each plane rooted structure of a fully parked tree with $m$ vertices with $1$ child actually corresponds to $2^m$ different embeddings as a subset of $  \mathbb{B}$ (with $ \varnothing$ as the root): for this reason, later in the decomposition we shall put a weight of $2$ for vertices with outdegree $1$. Let us denote by $ \mathbb{T}_n^{(p)}$ the set of all fully parked trees with root $\varnothing$, with $n$ vertices and having outgoing flux $p \geq 0$ (i.e.~$p+1$ cars have visited the root vertex). The weight $w({\mathbf{t}})$ of a fully parked tree $ \mathbf{t} \equiv (\mathbf{t} : (a_u : u \in \mathbf{t}))$ is the weight of its car decoration, that is
 $$w({\mathbf{t}})= \prod_{u \in \mathbf{t}} \mu_{a_u}.$$
 We can then form the bivariate generating series of fully parked trees (with flux) as
 \begin{eqnarray*} F(x,y)  \equiv F_{\mu}(x,y) &:=& \sum_{n \geq 1}\sum_{p \geq 0} \sum_{\mathbf{t} \in \mathbb{T}_n^{(p)}} x^n y^p w({\mathbf{t}})\\ &=&  x  \big(\mu_1+ \mu_2 y + \mu_3 y^2 + \cdots\big) + x^2 \big(2(\mu_2+ \mu_1^2) +  2y(\mu_3+ 2\mu_1\mu_2) + \cdots \big)  + \cdots   \end{eqnarray*}
 Section \ref{sec:enumeration} is devoted to the study of $F$ via a functional equation obtained by splitting a fully parked tree at the root, see \eqref{eq:tuttesimple}. But before doing so, let us present the combinatorial decomposition and the characterization of subcriticality in terms of $F$. It turns out that most equations simplify if one introduces
 $$ \mathbf{F}(x,y) := 1 + F(x,y) \quad \mbox{ and }\quad  \mathbf{F}_{0}(x) := \mathbf{F}(x,0) := 1+ F(x,0).$$

 \subsection{Decomposition}
Recall that the law of $X$ is $(p_k : k \geq 0)$ and that we gave a short-hand notation $p_\circ =p_0$ for the probability that the root vertex is empty. We write $ \mathcal{C}( \varnothing)$ for the monochromatic cluster of the origin in $ \mathbb{B}$ after parking. Notice that the number of vertices adjacent from above to a fully parked tree $\mathbf{t} \subset \mathbb{B}$ with $n$ vertices is $n+1$, regardless of the shape of $\mathbf{t}$. Recalling Proposition \ref{prop:noblack} we have for $k \geq 0$
 \begin{eqnarray}  \mathbb{P}( X( \varnothing) = k+1) &\underset{ \mathrm{Prop.} \ref{prop:noblack}}{=} &\mathbb{P}( X( \varnothing) = k+1 \mbox{ and } \# \mathcal{C}( \varnothing) < \infty) \nonumber \\& =& \sum_{\mathbf{t} \in \mathbb{T}_n^{(k)}} \mathbb{P}(\mathcal{C}( \varnothing) = \mathbf{t})\nonumber \\&=&  \sum_{n \geq 1} \sum_{\mathbf{t} \in \mathbb{T}_n^{(k)}} w( \mathbf{t}) p_\circ^{n+1}   = p_\circ [y^k]F(p_\circ,y). \label{eq:funda1}\end{eqnarray}
The other fundamental equation is obtained by noticing that the event $\{X(\varnothing)=0\}$ occurs if and only if $A_\varnothing=0$ and $\{X(u) \in \{0,1\} : \mbox{ for } u \in \{0,1\}\}$ which turns into
 \begin{eqnarray} p_\circ &=& \mu_0 ( p_\circ+ p_\bullet)^2.  \label{eq:funda2}\end{eqnarray}
Specializing  \eqref{eq:funda1} to $k=0$, we recover together with the previous display the fixed point equation $p_\bullet = p_\circ {F} (p_\circ,0)$, mentioned in the introduction.  In particular, re-injecting in  \eqref{eq:funda2} we obtain
  \begin{eqnarray*} \quad p_{\circ} = \mu_{0} p_{\circ}^{2}\big(  \mathbf{F}_{0}( p_{\circ})\big)^{2}, \quad \mbox{ where we recall that } \mathbf{F}_{0}(x) = 1+F(x,0).   \label{eq:funda3}
  \end{eqnarray*}
Notice that the function $ x \mapsto \mu_{0} x^{2} (\mathbf{F}_{0}(x))^{2}$ is strictly convex and that $p_{\circ}=0$ is a trivial solution to the above equation, so there is \textbf{at most one positive solution} $p_{\circ}$. Under the same hypothesis, splitting according to the values of $X(\varnothing)$ we also obtain thanks to \eqref{eq:funda1}
 \begin{eqnarray} 1 &=& p_{\circ} + p_{1}+ p_{2}+ \cdots \underset{\eqref{eq:funda1}}{=} p_{\circ} \mathbf{F}(p_{\circ},1). \label{eq:funda4}  \end{eqnarray}

 \begin{proposition}[$F$-characterization of subcriticality] \label{prop:criti} \label{prop:decomp}
 The law $\mu$ is subcritical if and only if there is a positive solution to the
 equation  \begin{eqnarray} \quad 1 = \mu_{0} x(  \mathbf{F}_{0}(x))^{2}. \label{eq:fundabis}
 \end{eqnarray}\end{proposition}

\begin{proof} Let $\mu$ be a subcritical law for the parking on $ \mathbb{B}$. Since $p_{\circ} \ne 0$, the above calculations show that $p_{\circ}$ is indeed a solution to the equation \eqref{eq:fundabis}.

Conversely, suppose that there is a positive solution $x_{\circ}$ to  \eqref{eq:fundabis}. As a special case of equation \eqref{eq:tuttesimple} below for $ \mathbf{F}(x,y)$, we know that the series $  \mathfrak{f}(y) = \mathbf{F}(x_{\circ},y)$ is a solution to $ y +  \mathfrak{f}^2 x_{\circ} G(y) -  y \mathfrak{f}- 1 =0$.  Solving the quadratic equation and taking the combinatorial solution we have
$$ \mathfrak{f}(y) = \frac{y + \sqrt{y^{2}+ 4x_{\circ}(1-y)G(y)}}{2x_{\circ} G(y)}.$$
At first, the above equality holds only as a formal power series in $y$. But notice that the function $y^{2}+ 4x_{\circ}(1-y)G(y)$ inside the square-root does not vanish over $y \in [0,1]$ so that the solution above is analytic over $[0,1]$. By Pringsheim's theorem  \cite[Theorem IV.6 p.240]{Flajolet:analytic}, the function $ \mathfrak{f}$ has radius of convergence at least $1$ and we have $ \mathfrak{f}(1) = \frac{1}{x_{\circ}}$ which is   $x_\circ \mathbf{F}(x_\circ,1)=1.$
This in turn ensures that there exists a random variable $Z$ (the outgoing flux of cars) whose generating function is
$$  \mathbf{Z}(y) =  x_{\circ}  \mathbf{F}(x_{\circ},y).$$
We then compute, using Tutte's equation \eqref{eq:tutteintro} (see \eqref{eq:tuttesimple} below) as well as  \eqref{eq:fundabis}:
\begin{align*}
& \frac{1}{y} \big( \mathbf{Z}(y)^{2} G(y) -  \mathbf{Z}(0)^{2} G(0) \big) + \mathbf{Z}(0)^{2} G(0)  \\
=&   x_{\circ} \left(\frac{x_{\circ}}{y} \big(   \mathbf{F}(x_{\circ},y) ^2 G(y) -  \mathbf{F}(x_{\circ},0)^2 G(0) \big) \right)+  x_{\circ}^2  \mathbf{F}(x_{\circ},0)^2 G(0)\\
=& x_\circ F(x_\circ,y) +  x_{\circ}
 = \mathbf{Z}(y),
 \end{align*}
but this identity is equivalent to the  following recursive distributional equation for $Z$:
$$Z \overset{(d)}{=} (Z_{1}+Z_{2} +A-1)_{+},$$ where on the right-hand side the variables are independent and $Z_{1},Z_{2}$ are two copies of law $Z$.

This recursive distributional equation enables us to decorate the vertices of $ \mathbb{B}$ by i.i.d.\ variables $A_{u}$ in such a way that for every $n$, the parking on $[ \mathbb{B}]_{n}$ together with i.i.d.\ fluxes on $\partial [ \mathbb{B}]_{n}$ yields a flux of law $Z$ at the root (in a coherent manner). Replacing  the i.i.d.\ fluxes on $\partial [ \mathbb{B}]_{n}$ by null fluxes on $\partial [ \mathbb{B}]_{n}$, and writing $X_n = X([ \mathbb{B}]_{n})$
for the number of cars visiting the root for the parking on $[ \mathbb{B}]_{n}$,
we deduce by comparison that the flux at the root $\mathrm{F_n} = (X_n-1)_{+}$ is dominated by the a.s. finite random variable $Z$, which implies in particular that the parking on $ \mathbb{B}$ with law $\mu$ is subcritical. Et voil\`a.
\end{proof}

\section{Enumeration of fully parked trees} \label{sec:enumeration}
This section is the analytic core of the paper. We write the recursive equations (Tutte's equation) for fully parked trees and solve them using the kernel method of Bousquet-M\'elou \& Jehanne \cite{BMJ06}. Combined with Proposition \ref{prop:decomp} this enables us to prove our main results easily. The results are similar to the work of Chen \cite{chen2021enumeration} which considered plane fully-parked trees (as opposed to our binary case). Notice also that the technical part of \cite{chen2021enumeration} consists in obtaining asymptotics for the coefficients, a goal that we did not pursue in these pages.
\subsection{Solving Tutte's equation}

Recall that $F(x,y)$ is the bivariate generating function of the fully parked trees where $x$ encodes the number of vertices of the tree and $y$ the \textbf{flux} of cars and $G$ is the generating function of the car arrivals. To enumerate  fully parked trees, we decompose them at their root vertices. Take a fully parked tree with $n \geq 1$ vertices and $n+p$ cars in total (the flux of cars is $p$). Then
\begin{itemize}
\item either $n=1$ which means that the root vertex has no vertex above it. In this case, at least one car arrives on this vertex (since the root vertex should contain a parked car) and the number of cars arriving on this vertex is $1+p$. Summing over $p$  gives the term $ x (G(y)- G(0))/y$.
\item Another possibility is that the root vertex has a unique child in the fully parked tree, which can be the left or right neighbor in $ \mathbb{B}$. In that case, the subtree above this child is a fully parked tree with $n-1$ vertices and a flux of cars $p_1$ where $p_1+ \ell -1 =p$ if there are $ \ell$ cars arriving on the initial root vertex. Notice that the case $p_1 = \ell =0$ is excluded since otherwise the root vertex is not parked. Summing over $p$ yields the term $ 2x(F(x,y)G(y)- F(x,0)G(0))/y$.
\item The last case is when the root vertex has two parked children, each carrying a fully parked tree above it with respective sizes $k \geq 1$ and $n-k-1 \geq 1$ and flux of cars $p_1$ and $p_2$. To obtain a flux of cars $p$ at the root, one must have $p_1+p_2+\ell - 1 = p$ where $ \ell$ is the number of cars arriving at the root vertex. Again the case $p_1=p_2= \ell=0$ is excluded.  We thus obtain a term $x (F(x,y)^2 G(y) - F(x,0)^2G(0))/y.$
\end{itemize}

\begin{figure}[!h]
 \begin{center}
 \includegraphics[width=15cm]{tutte-parking}
 \caption{\label{fig:tutte} Illustration of Tutte's recursive decomposition at the root vertex. On the left a fully parked tree with flux $p$. If $n=1$, then the tree is just a vertex with $p+1$ cars arriving on it. Otherwise, it has one or two children which are the root of smaller fully parked tree. }
 \end{center}
 \end{figure}
Summing these three terms, we obtain the following recursive equation for $F$:
\begin{equation} \label{eq:Tutte} yF(x,y) = x(G(y)-G(0)) + 2x(F(x,y)G(y)- F(x,0)G(0))+ x (F(x,y)^2 G(y) - F(x,0)^2G(0))\end{equation}
 With our notation $ \mathbf{F} = F+1$ and $ \mathbf{F}_0 (x) = \mathbf{F}(x, 0) = F(x,0)+1$, this equation simplifies to
  \begin{eqnarray} P( \mathbf{F}(x,y), \mathbf{F}_0(x),x,y)=0 \quad \mbox{ where } \quad  P(f,f_0,x,y) = y + f^2 x G(y) - f y - f_0^2 x G(0).  \label{eq:tuttesimple}  \end{eqnarray}

To solve this equation, we apply the kernel method of Bousquet-M\'elou and Jehanne \cite{BMJ06} and look for a (formal) power series $ Y=Y(x)$ such that $\partial_{f} P( \mathbf{F}(x, Y(x)), \mathbf{F}_0(x),x,Y(x))=0$ so that combined with \eqref{eq:tuttesimple} we also find automatically $\partial_{y} P( \mathbf{F}(x, Y(x)), \mathbf{F}_0(x),x,Y(x))=0$.
This introduction may seem ad-hoc, but it enables us to find a system of three equations on the three unknowns $ \mathbf{F}, \mathbf{F}_{0}$ and $Y$, so that with a little luck we will find an ``expression'' for those. Actually, as we will see below $x \mapsto Y(x)$ is a convenient change of variable which simplifies our expressions. To summarize, we are looking for a solution $Y \equiv Y(x)$ to the following system
\begin{equation}  \label{eq:system}
\left\lbrace
\begin{array}{l}
Y - 2 x \mathbf{F} G(Y)=0, \\
1 +  x  G'(Y) \mathbf{F}^2= \mathbf{F}, \\
Y + xG(Y) \mathbf{F}^2= Y \mathbf{F}+xG(0) \mathbf{F}_0^2 .\end{array}\right.
\end{equation}
Thanks to the first equation, we know that $ \mathbf{F}= Y/(2xG(Y)).$ Replacing $ \mathbf{F}$ by this quantity in the second equation, we obtain
 \begin{eqnarray} 1 + \frac{Y^2 G'(Y)}{4xG(Y)^2} = \frac{Y}{2xG(Y)} \quad \mbox{ that is } \quad  Y = x \left( \frac{4G(Y)^2}{2G(Y) - YG'(Y)}\right),   \label{eq:lagrange}\end{eqnarray} which makes it clear that $Y \equiv Y(x)$ exists as a power series (and even with a positive radius of convergence in a neighborhood of $0$). Once the existence of $Y$ is granted, we use again the system of equations \eqref{eq:system} to obtain an equation that only involves $\mathbf{F}_0(x)$ and $Y (x). $
If we replace in the third equation $ \mathbf{F}$ by $Y/(2xG(Y)) $ (which is a consequence of the first equation) and $x$  by $\frac{Y (2G(Y) - YG'(Y))}{4G(Y)^2}$, we obtain
$$ \frac{4YG(Y)}{2G(Y) -Y G'(Y)}+ \frac{ \mathbf{F}_0^2 Y G(0) (2G(Y) -Y G'(Y))}{G(Y)^2}= 4 Y.$$
This equation is quadratic in $\mathbf{F}_0(x)$ and using the fact that it has non negative coefficients we obtain

\begin{equation} \label{eq:F1}\mathbf{F}_0(x) =   \frac{2 G( Y) \sqrt{G( Y) - Y G'( Y)}}{  (2 G( Y) - YG'( Y))\sqrt{G(0) }} \quad \mbox{with} \quad Y\equiv Y(x) \mbox{ as in \eqref{eq:lagrange}}.
\end{equation}
We have found here an ``explicit" solution for $\mathbf{F}_0(x)$ around $x =0$.
Coming back to Tutte's equation, once $ \mathbf{F}_{0}$ is known this equation is quadratic in $ \mathbf{F}$ and we can solve it into
\begin{eqnarray}
\label{Fxy}
\mathbf{F}(x,y) = \frac{y \pm \sqrt{y^2+ 4 G(y) x (G(0)  \mathbf{F}_0(x)^2 x - y) }}{2 G(y) x}.
\end{eqnarray}
The sign in front of the square root can actually change since the function inside the square root vanishes when  $y = Y(x)$ and we need to change branch to keep an analytic function. But we shall not use the exact expression in what follows.

\subsection{Radius of convergence} \label{sec:Y(x)}
In this section we use the explicit resolution of the functional equation \eqref{eq:tuttesimple} to  determine the radius of convergence $x_{c}$ of $ \mathbf{F}_{0}$ and the value of $ \mathbf{F}_{0}(x_{c})$. The important fact for our application to parking being that under the condition $(\star)$ on the existence of $t_c$ in Theorem \ref{thm:phase} we have
$$  \mathbf{F}_0(x_c) =  \frac{2 G( t_c) \sqrt{G( t_c) - t_c G'( t_c)}}{  (2 G( t_c) - t_cG'(t_c))\sqrt{G(0) }} .$$
Recall from \eqref{eq:F1} that $ \mathbf{F}_0(x)$ is an explicit function of $Y(x)$ itself given by the implicit equation \eqref{eq:lagrange}.

\paragraph{Analyticity of $Y$.} We first determine the analytic properties of the change of variable $x \mapsto Y(x)$. Recall from \eqref{eq:lagrange} that $x$ and $Y=Y(x)$ are linked by the equation

\begin{equation} \label{eq:xfcty} x= \frac{Y (2G(Y) - YG'(Y))}{4G(Y)^2}, \quad \mbox{ equivalently}\quad  x = \psi(Y(x)) \quad \mbox{ with } \psi(y) =\frac{y (2G(y) - yG'(y))}{4G(y)^2}.  \end{equation
Note that
$$ \partial_{y} \psi(y) = \frac{2(G(y)-yG'(y))^{2}-y^{2}G(y)G''(y)}{4G(y)^{3}}$$ and in particular $\psi(0)=0$ and $\psi'(0) >0$ so that by the implicit function theorem, we can define $ Y$ in a neighborhood of $0$ such that $ x = \psi(Y(x))$. Recall the condition $(\star)$ from Theorem \ref{thm:phase} which says that the function $ y \mapsto y^2G''(y)G(y)-2(G(y)-yG'(y))^2$ at the numerator of $\partial_{y} \psi(y)$ reaches $0$ at time $t_c \in (0,\infty)$, see Figure	 \ref{fig:scenarii}.
\begin{remark}  \label{rek:RCVinfty} In particular if $G$ has an infinite radius of convergence then $(\star)$ holds. Indeed, the quantity $ G(y) - y G'(y) = \sum_{k \geq 0} \mu_k (1-k) y^k$ equals $G(0) = \mu_0 >0$ at $y=0$ and is bounded from above by $ \mu_0 - (1- \mu_0 - \mu_1) y^2$ for $y >1$ which goes to $- \infty$ as $ y \to + \infty$. Thus, there exists $z_c$ such that $ G(z_c) - y_c G'(z_c) = 0$ and the function $ y \mapsto y^2G''(y)G(y)-2(G(y)-yG'(y))^2$ is positive at $y = z_c$. Since it is negative at $y=0$, then ($\star$) holds and $t_c \in (0,z_c).$ \\
The assumption $(\star)$ is also satisfied when $G$ has a finite radius of convergence $y_c$
and (at least) one of the three quantities $G(y_c)$, $G'(y_c)$, $G''(y_c)$ is infinite.
In case $G(y_c)=\infty$, starting from $y G'(y)-G(y) = y  \sum_{k \geq 0} \mu_k (k-1) y^{k-1}$, and noting that $\mu_k (k-1) y^{k-1}\sim \mu_k k y^{k-1}$, we deduce that $y G'(y)-G(y) \sim y G'(y)$ as $y \to y_c$, hence $G(y) - y G'(y) $ again has limit $-\infty$ as $y \to \infty$, and the same argument as above applies.
The last cases are obvious : in case $G'(y_c)=\infty$ but $G(y_c)<\infty$, $G(y) - y G'(y) $ plainly has limit $-\infty$; last, in case $G''(y_c)=\infty$ but $G'(y_c)<\infty$, we directly have $\lim_{y \to y_c} y^2G''(y)G(y)-2(G(y)-yG'(y))^2 = + \infty.$
\end{remark}
To clarify the reader's spirit and for latter discussion, let us classify the possible scenarios according to the three cases identified in  Chen \cite[Figure 1]{chen2021enumeration}, see Figure \ref{fig:scenarii}:
\begin{itemize}
\item the most common case is when $t_{c}$ exists and is strictly less than the radius of convergence $y_{c}$ of $G$. At this point we have
$$ \left. \frac{ \partial ^{2}}{\partial y \partial y}\psi(y)\right|_{y=t_{c}} =  \frac{t_{c}}{4G(t_{c})^{3}}  \left( 2(t_{c}G'(t_{c})-2G(t_{c}))G''(t_{c})-t_{c}G(t_{c})G'''(t_{c})\right),$$ and since
$G(t_{c}) \geq t_{c}G'(t_{c})$ this second derivative is strictly negative so that the function $ y \mapsto \psi(y)$ reaches a local maxima at this point. We call this situation the \textbf{generic} situation.
\item it could also happen that $t_{c}$ exists and is equal to the radius of convergence of $G$. In this case, although $y \mapsto \psi(y)$ reaches a maxima at $t_{c}$, the local behavior around the maximum may not be quadratic. We call this situation the \textbf{non-generic situation}.
\item Otherwise $t_{c}$ does not exists and in particular the radius $y_{c}$ of convergence of $G$ is finite and $y \mapsto \psi(y)$ has a finite positive derivative at $y_{c}$. This is the \textbf{dense} situation which leave aside for the moment.
\end{itemize}

\begin{figure}[!h]
 \begin{center}
 \includegraphics[width=15cm]{non-generic}
 \caption{Illustration of the three scenarios  for the change of variable $ x \mapsto Y(x)$. Our standing assumption $(\star)$ holds in the first two cases. \label{fig:scenarii}}
 \end{center}
 \end{figure}

Under the assumption $(\star)$ --i.e.\ except in the dense situation-- we introduce
 \begin{eqnarray} \label{eq:defxc} x_{c} = \psi(t_{c})=\frac{t_{c} (2G(t_{c}) - t_{c}G'(t_{c}))}{4G(t_{c})^2}. \end{eqnarray}
\begin{lemma} Under assumption $( \star)$ the function $x \mapsto Y(x)$ is increasing and  analytic over $[0,x_c)$ and furthermore
$$ \lim_{x \to x_c} Y'(x) = \infty.$$
\end{lemma}
\begin{proof} Under the assumption $(\star)$ the function $y \mapsto \psi(y)$ is increasing and analytic over $[0,t_{c})$ so that by the analytic version of the implicit function theorem one can define its increasing  inverse function $ x \mapsto Y(x)$ over $[0,x_{c})$. Note that since $\psi'(y) \to 0$ as $y \uparrow y_{c}$ we have $Y'(x) \to \infty$ as $x \uparrow x_{c}$
 \end{proof}

\paragraph{Radius of convergence of $ \mathbf{F}_0$.} We still suppose $ (\star)$. Coming back to $ \mathbf{F}_0$, notice  that $ (G(Y(x))-Y(x)G'(Y(x)))$ is always positive for $x \in [0,x_c]$,  thus by Equation \eqref{eq:F1} the function $ \mathbf{F}_0$ is also analytic over $[0,x_c)$. Since $ \mathbf{F}_0$ only has positive coefficients, by  Pringsheim's Theorem (see for example \cite[Theorem IV.6 p.240]{Flajolet:analytic}), its radius of convergence is \emph{at least} $x_c$. The following lemma shows that it actually coincides with it
\begin{lemma} Suppose $( \star)$ then we have
$$ \lim_{x \to x_c} \mathbf{F}''_0(x)= \infty,$$
in particular the radius of convergence of $ \mathbf{F}_0$ is equal to $x_c$.
\end{lemma}
\begin{proof
We use our explicit computations of $ \mathbf{F}_0$ and $ Y$ to derive formulas for the first two derivatives of $x$. Even if we don't need it for this lemma, we start with the first derivative. We take the expression of $ \mathbf{F}_0$ given by Equation \eqref{eq:F1}, differentiate it with respect to $x$ and replace the occurence of $ Y'(x)$ by $1/ (\partial_{y} \psi (Y(x)))$ thanks to Equation \eqref{eq:xfcty}. We obtain
$$ \mathbf{F}'_{0} (x) = \frac{4 G(Y(x))^3 G'(Y(x))}{\sqrt{G(Y(x))-Y(x) G'(Y(x))} (2 G(Y(x)) -Y(x) G'(Y(x)))^2}.$$
This quantity has a finite limit when $ Y(x)$ converges to $t_c$. We thus need to compute the second derivative of $ \mathbf{F}_0$. To do so, we differentiate the above expression of $ \mathbf{F}'_0$ and again replace the occurence of $ Y'(x)$ by $1/ (\partial_{y} \psi (Y(x)))$. We then obtain a fraction involving the derivatives of $G$ at $Y(x)$.
Using the definition of $t_c$ under assumption $(\star)$, we can show that
\begin{eqnarray*} \lim_{x \to x_c} \mathbf{F}_0'' (x) = \lim_{t \to t_c}\frac{16 G(t)^6 (2 G(t) - t G'(t))^3 (G(t) -  t G'(t)) (2 G(t) - t G'(t))^2}{t^2 (G(t) - t G'(t))^{3/2} \cdot (2 (G(t) - t G'(t))^2 -  t^2 G(t) G''(t))} = + \infty,
  \end{eqnarray*}
since all but the factor $(2 (G(t) - t G'(t))^2 -  t^2 G(t) G''(t)$ are positive for $t< t_c$ and have a positive limit as $t \to t_c$.
\end{proof}

\section{Probabilistic consequences}

\label{sec:proba}
Armed with our enumeration results and the  criterion  of Proposition \ref{prop:criti}, we can now proceed to the proof of our main results.
\subsection{Theorem \ref{thm:phase}: Location of the threshold}

Recall that by Proposition \ref{prop:criti}, the parking process is subcritical if and only if there exists a positive solution to \eqref{eq:fundabis}. When $(\star)$ holds, since the function $ x \mapsto G(0) x \mathbf{F}_{0}(x)$ is strictly increasing, Equation \eqref{eq:fundabis} has a solution if and only if $G(0) x_c \mathbf{F}_{0}(x_c)^2 \geq1 $ where $x_c$ is the radius of convergence of $ \mathbf{F}_0$ found in the previous paragraph.  Now, since $t_c = Y (x_c)$ and plugging the value of $\mathbf{F}_0(x_c)$ given by Equation \eqref{eq:F1} in $G(0) x_c \mathbf{F}_{0}(x_c)^2 \geq1$, we obtain $t_c+ \frac{t_c G(t_c)}{t_c G'(t_c) - 2 G(t_c)} \geq 1$.

By definition of $t_c$ under Assumption ($\star$), the quantity $t_cG'(t_c) - 2G(t_c)<t_cG'(t_c) - G(t_c)$ is always negative. Hence there is a solution $x$ to \eqref{eq:fundabis} if and only if  $t_c$ satisfies $$(t_c-2)G(t_c) \geq t_c(t_c-1)G'(t_c), $$
which together with Proposition \ref{prop:criti} concludes the proof of Theorem \ref{thm:phase}.

\begin{remark}\label{tcgeq2} Notice that if $t_c$ satisfies the condition of Theorem \ref{thm:phase}, then it is greater than $2$. This implies that if the parking process is subcritical, then the radius of convergence of the generating function $G$ of the car arrivals is at least $2$. To see it, first note that the inequality $(t-2)G(t) \geq t(t-1)G'(t)$ can not be satisfied for $t \in [1,2]$ since the left-hand side is non-positive and equals $0$ only for $t=2$, whereas the right-hand side is non-negative and equals $0$ for $t=0$ and $t=1$ only.
Neither can this inequality be satisfied for $ t \in (0,1)$, because for such $t$, we can bound from above the quantity $  t(t-1)G'(t)/(t-2)$ by $ (3 -2\sqrt{2})G'(1) \leq  (3 -2\sqrt{2})G'(1) \leq 1/5$ since $ G'(1) = \mathbb{E}[A] \leq 1$ in the subcritical case. On the other side, the quantity  $G(t)$ is bounded from below by $G(0) \geq 1/2.$
\end{remark}

\subsection{Theorem \ref{thm:critical}: critical computations}
Before moving on to the critical computation (Theorem \ref{thm:critical}) let us prove that the critical case caracterized by the equality in the second display of Theorem \ref{thm:phase} actually corresponds to the natural fact that one "cannot increase the number of cars" and stay subcritical:

\begin{lemma}[Criticality] Suppose $(\star)$. Then we have equality in \eqref{eq:threshold} iff for any $ \varepsilon>0$ the law with generating function $G_{ \varepsilon} (t)= G (t) + \varepsilon t - \varepsilon$ is supercritical.
\end{lemma}
\proof
Suppose that $ \mu$ is subcritical in the sense of Theorem \ref{thm:phase} and look at the probability measure $ \mu_{ \varepsilon}$ such that its generating function is given by $ G_{ \varepsilon} (t)= G (t) + \varepsilon t - \varepsilon$ for some $ \varepsilon >0$. First notice that $ \mu_{ \varepsilon}$ satisfies Assumption ($\star$) for $ \varepsilon$ small enough. Indeed the radius of convergence of $ G_{\varepsilon}$ is that of $G$ and the quantity
\begin{eqnarray*}2 (G_{ \varepsilon}(t) - t G_{ \varepsilon}'(t))^2 -  t^2 G_{ \varepsilon}(t) G_{ \varepsilon}''(t) &=& (2 (G(t) - t G'(t) - \varepsilon)^2 -  t^2 (G(t) + \varepsilon(t-1))G''(t) \\
&=&2 (G(t) - t G'(t))^2  -  t^2 G(t)G''(t) \\
&\quad& \qquad+ 2 \varepsilon^2  - 2\varepsilon (G(t) - t G'(t)) -  \varepsilon t^2 (t-1)G''(t)
\end{eqnarray*}
is negative at $ t_c$ when $ \varepsilon$ is small enough, so that $ t_c^{ \varepsilon} := \min \{ t \geq 0,\ 2 (G_{ \varepsilon}(t) - t G_{ \varepsilon}'(t))^2 =  t^2 G_{ \varepsilon}(t) G_{ \varepsilon}''(t) \} < t_c$ and the function $ \varepsilon \mapsto t_{c}^{ \varepsilon}$ is continuous in a positive neighborhood of $0$. To determine whether $ \mu_{ \varepsilon}$ is subcritical or not, we then need to determine the sign of
$$ (t_c^{ \varepsilon}-2)G_{ \varepsilon}(t_c^{ \varepsilon}) - t_c^{ \varepsilon}(t_c^{ \varepsilon}-1)G_{ \varepsilon}'(t_c^{ \varepsilon}), $$
which is then continuous is $ \varepsilon$. Thus if Equation \eqref{eq:threshold} is not an equality, we can increase $ \mu$ and remain subcritical.

Suppose now that  \eqref{eq:threshold} is an equality. We can show that
$$ \partial_{ y} \varphi_{ \varepsilon} (t_c - \delta)  = \frac{6 G(t_c)+ t_c^2 G^{(3)}(t_c) (t_c - 1)^3 }{4 G(t_c)^2(t_c - 1)^3 } \delta - \frac{3}{2 (t_c-1) G(t_c)^2} \varepsilon + o(\varepsilon) + o(\delta). $$
When $t_c - \delta = t_c^{\varepsilon}$, then the left hand side is zero so that the main asymptotic on the right hand side should be $0$. Hence $ \delta = t_c - t_c^{ \varepsilon}$ is of order $\varepsilon.$
Moreover,
$$(t_c^{ \varepsilon}-2)G_{ \varepsilon}(t_c^{ \varepsilon}) - t_c^{ \varepsilon}(t_c^{ \varepsilon}-1)G_{ \varepsilon}'(t_c^{ \varepsilon}) = -2 \varepsilon ( t_c^{ \varepsilon}-1 ) + O( (t_c^{ \varepsilon} - t_c)^2). $$
is negative when $ \varepsilon$ small enough and $ \mu_{ \varepsilon}$ is supercritical by Theorem \ref{thm:phase}. \endproof

\begin{proof}[Proof of Theorem \ref{thm:critical}]. Let us focus now on the case when \eqref{eq:threshold} is an equality. In that case, since \eqref{eq:fundabis} is an equality for $x=x_c$, then $x_c = p_{ \circ}.$  If $x$ satisfies Equation \eqref{eq:xfcty} and the parking is critical, so that $y =t_c$ is solution of $(t-2)G(t) = t(t-1)G'(t)$, then \begin{equation}\label{eq:xf0crit}x = x_c = \frac{t_c^2}{4(t_c-1) G(t_c)} \qquad \mbox{and} \qquad \mathbf{F}_0(x_c)^2 = \frac{1}{p_{ \circ} G(0)} =  \frac{1}{p_{ \circ} \mu_0}.\end{equation}
Moreover, using Equation \eqref{eq:funda2}, we obtain
$$p_{ \bullet} = \sqrt{ \frac{p_{ \circ}}{ \mu_0}}- p_{ \circ},$$
and this concludes the proof of Theorem \ref{thm:critical}.
\end{proof}

\begin{remark} \label{ref:p0} The fact that $ p_{ \circ} >1/2$ is a consequence of the equation for $t$ given by Theorem \ref{thm:phase}. Indeed, in the critical case, then $ p_{ \circ} = x_c = t_c^2/(4(t_c-1) G(t_c))$ where $t_c$ is given by Theorem \ref{thm:phase}. But since $ t_c > 1$, we have $ G(t_c) - G(1) \leq (t_c- 1)G'(t_c) = (t_c-2)G(t_c)/t_c$. Thus $G(t_c) \leq t_c/2$ and
 $$ p_{ \circ} \geq \frac{2t_c}{t_c-1} > 1/2.$$
\end{remark}

\subsection{Examples}
Let us proceed to the computation of the critical threshold for parking for various families of stochastically  increasing laws. In the first four cases below, it is easy to check that condition $(\star)$ holds so that we can just apply the general formulas. In the last example we explain how our techniques can be applied even if $(\star)$ does not hold.
\paragraph{Binary$_{0/2}$ car arrivals.} As a first example, we can imagine that either $0$ or $2$ cars arrive at each spot, i.e.\ the law of the car arrivals is $ \mu = (1- \frac{ \alpha}{2}) \delta_0 + \frac{ \alpha}{2} \delta_2$, so that $ G(t) = (1- \frac{ \alpha}{2}) +  \frac{ \alpha}{2}t^2$. This is the example considered in  \cite[Proposition 3.5]{GP19} and \cite[Proposition 4]{BBJ19}. Explicit computations show in this case that

$$ t_{c}=Y(x_c) = \sqrt{ \frac{2- \alpha}{3 \alpha}}, \qquad x_c = \frac{3 \sqrt{3}}{16 \sqrt{ \alpha(2- \alpha)}} \qquad \mbox{and} \qquad \mathbf{F}_0(x_c)= \frac{4\sqrt{6}}{9}\cdot $$
Note that $ \mathbf{F}_{0} (x_c)$ does not depend on $ \alpha$, see the remark below.
We then see that for $t = t_c$, the Inequality \eqref{eq:threshold} is quadratic in $ \alpha$ and is satisfied as soon as  $ \alpha < \alpha_c = 1/14$.
\paragraph{Binary$_{0/k}$ car arrivals.} In the case when $ \mu = (1- \frac{ \alpha}{k}) \delta_0 + \frac{ \alpha}{k} \delta_k$, so that $ G(t) = (1- \frac{ \alpha}{k}) +  \frac{ \alpha}{k}t^k$ with $k \geq 3$, we have
$$ Y(x_c) = \left(\frac{ (k - a) (-4 + k (3 + k - \sqrt{(k-1) (k+7)})}{2a ( k-2) ( k-1)}\right)^{1/k} , $$
 $$ \mathbf{F}_0 (x_c) = \frac{\sqrt{2} (3 k - \sqrt{(k-1) (k+7)}-3 )}{( k-2) (k-1) \sqrt{\frac{5 - k + \sqrt{(k-1) ( k+7)}}{(k-1) k}}}. $$
so that the model is critical at $$ \alpha_c ( \mathrm{Binary}_{0/k}) = \frac{k}{1 + 2^{-k-2} \left( 3 + \sqrt{ \frac{k+7}{k-1}}\right)^k \left( (k-1)(k+4)+ k \sqrt{(k+7)(k-1)}\right)} . $$
\paragraph{Poisson.} Suppose the law of the car arrivals is Poisson with mean $ \alpha> 0$, so that in this case $G(t) = e^{ \alpha(t - 1)}. $ Again, explicit computation show that

$$ Y(x_c) = \frac{2- \sqrt{2}}{ \alpha}, \qquad x_c = \frac{(\sqrt{2}-1) \mathrm{e}^{ \alpha-2 + \sqrt{2} }}{2 \alpha} \qquad \mbox{and} \qquad \mathbf{F}_0(x_c) =  \sqrt{2 ( \sqrt{2}-1)} \mathrm{e}^{1 - 1/\sqrt{2}}, $$
so that the model is subcritical for parking as long as $\alpha \leq \alpha_c $ with $$ \alpha_c = 3 - 2 \sqrt{2}.$$

\paragraph{Geometric.} Consider here the case when $ G(t) = 1/(1+ \alpha- \alpha t)$. Then
$$ Y(x_c) = \frac{1+ \alpha}{3 \alpha}, \qquad x_c = \frac{(1+ \alpha)^2}{12 \alpha} \qquad \mbox{and} \qquad \mathbf{F}_0(x_c) = \frac{2}{ \sqrt{3}} \cdot$$
so that the model is subcritical for parking as long as $ \alpha \leq \alpha_c $ with $ \alpha_c = 1/8.$

\begin{remark}[Combinatorial counting]
The reader may be puzzled by the fact that in the last four cases the value $ \mathbf{F}_{0}( x_{c})$ does not depend on the parameter $\alpha$.
This is because in each case, the dependence on $\alpha$ of the  $\mu_{\alpha}$-weight of a fully parked tree of size $n$ is of the form $(c_{\alpha})^{n}$, for a constant $c_{\alpha}$ depending on $\alpha$ that cancels out at criticality. To wit, consider fully-parked trees associated with $0/k$ arrivals : in this case, for $n$ a multiple of $k$, fully parked trees of size $n$ have weight $\prod_{v} \mu_{a(v)} = \mu_{0}^{n/k} \mu_{k}^{(1-1/k)n}  =(\mu_{0}^{1/k} \mu_{k}^{(1-1/k)})^n$.

\end{remark}

\paragraph{Without $ (\star)$.} \label{sec:non-generic} When hypothesis $(\star)$ is not satisfied we can still apply our method: If the generating series $G$ has a radius of convergence $y_{c}$ then the value $t_{c}$ is then replaced by
$$\tilde{t} _{c} = \min\left\{ y_{c} , \min \{ t \geq 0:\ 2(G(t)- t G'(t))^2 = t^2 G(t) G''(t) \}\right\},$$
and we need to check that $\tilde{x}_{c}$ defined analogously by \eqref{eq:defxc} is again the radius of convergence of the series $ \mathbf{F}_{0}$ (we did not try to prove such a general statement here). Then using Proposition \ref{prop:criti}, the subcriticality of the parking is equivalent to the fact that $ \mu_{0} \tilde{x}_{c}(  \mathbf{F}_{0}(\tilde{x}_{c}))^{2} \geq 1$, the only different point is that we cannot use the equality $2(G(\tilde{t}_{c})- \tilde{t}_{c} G'(\tilde{t}_{c}))^2 = \tilde{t}_{c}^2 G(\tilde{t}_{c}) G''(\tilde{t}_{c})$ to further simplify the expression. As an example of such law, consider the generating function
  \begin{eqnarray} \label{eq:ng} G(t) = 1+ \frac{1+ t^2}{26} - \frac{1}{13} \left( \frac{3-t}{2}\right)^{7/3}.  \end{eqnarray}
The radius of convergence of $G$ is equal to $3$ but $G'(3)$ and $G''(3)$ exist. An explicit computation shows that $(\star)$ holds with $t_{c}=y_{c}= \tilde{t}_{c}=3$ for $G$, and furthermore $G$ is critical for the parking process.  However if one considers $\tilde{G} = 0.9 + 0.1G$ then $\tilde{t}_{c} = 3$ and $(\star)$ does not hold but still $ \tilde{G}$ is subcritical for the parking process.

\section{Extensions and comments}

\label{sec:comments}
In this work, we voluntarily stick to the simplest case of the binary tree with i.i.d.\ arrivals without specific conditions to keep the paper accessible to a wide audience. Let us mention a few perspectives that our approach opens:
\subsection{Non-generic and dense  case} In this work, we focused on the localization of the threshold and on the computation of some critical quantities. One could also try to get scaling limits of critical components and compute several critical or near-critical exponents. As mentioned in the introduction, we expect that  a large family of critical car arrivals (say, with bounded support) belong to a common universality class where we expect that the cluster size of the root has a tail that decays as $ n^{-5/2}$ as $ n \to \infty$ and where the scaling limits of the critical components are given by $3/2$-stable Growth-Fragmentation trees. But when the car arrivals have a heavy tail (and when the parameters are fine-tuned so that the law is critical), we hope to see different universality classes. Actually, as seen in Section \ref{sec:Y(x)}, the singular behavior of $ \mathbf{F}_{0}$ near its radius of convergence is linked to the behavior of the $Y(x)$ near near $x_{c}$, see Figure \ref{fig:scenarii}.  For instance, in the example \eqref{eq:ng} an explicit computation shows that the singular behavior of $ \mathbf{F}_{0}(3-x) - \mathbf{F}_0 (3)$ is of the form $C x^{4/3}$  which indicates a polynomial decay of the critical cluster with exponent $n^{-7/3}$. This is very similar to the scenarios that happened in the enumeration of plane fully parked trees in Chen \cite{chen2021enumeration}, with the notable difference that in our model the dense case can be critical for the parking process.
See also \cite[Theorem 1.2]{chen2019derrida} for the Derrida-Retaux model with heavy-tailed distributions where the free-energy has a peculiar behavior. We plan on studying those different behaviors in forthcoming works.

\subsection{General case of $d$-ary tree and GW trees}
Our work may be extended to parking on more general trees such as $d$-ary trees and perhaps supercritical Bienaym\'e--Galton--Watson trees. The crux is of course the enumeration of fully parked trees. In the case of supercritical Bienaym\'e--Galton--Watson trees, one would probably need the addition of another variable $z$ counting the number of adjacent vertices of the fully parked tree inside the global tree (in our case, we had a fixed number $n+1$ of  vertices adjacent from above to a fully parked tree of $ \mathbb{B}$ of size $n$). We wonder whether the randomness of the underlying tree may yield to different universality classes compared to the case of $d$-ary trees.

\subsection{Links with Derrida-Retaux model} As mentioned several times in the paper, the Derrida-Retaux model is closely related to the parking process on $ \mathbb{B}$. We wonder whether a firm connection can be made between the two models.

\bibliographystyle{siam}

redpajama_set_name: RedPajamaArXiv


text:
> *@param* **list** _{Object}_  - valid Keystone list
> *@param* **path** _{String}_  - path
> *@param* **options** _{Object}_  - options
> *@api* **public**

defaults

this._fixedSize = 'medium';

// TODO: implement filtering, usage disabled for now
options.nofilter = true;

  code

---
 .Field.Types.GeoPoint.addToSchema(  )
> *@api* **public**

Registers the field on the List's Mongoose Schema.
Adds a 2dsphere indexed lat/lng pair.

---

 .Field.Types.GeoPoint.getData( item )
> *@param* **item** {Object} - valid item object
> *@api* **public**

Gets the field's data from an Item, as used by the React components

---
 .Field.Types.GeoPoint.format( item )
> *@param* **item** {Object} - valid field object
> *@api* **public**

Formats the field value.
```javascript
return item.get(this.paths.html);
```

---
 .Field.Types.GeoPoint.validateInput( data [, required, item ] )
> *@param* **data** {Object} - valid data object
> *@param* **required** {Boolean} - true if this is a required field
> *@param* **item** {Object} - valid item object
> *@api* **public**

Validates that a value for this field has been provided in a data object.

---
 .Field.Types.GeoPoint.updateItem( item, data )
> *@param* **item** {Object} - valid field object
> *@param* **data** {Object} - data to update
> *@api* **public**

Updates the value for this field in the item from a data object.

redpajama_set_name: RedPajamaGithub


The Dover Merchant Navy Memorial Fund was set up to finance a permanent memorial in Dover to all merchant seamen who were lost during World War 2. Planning permission was obtained for a life size bronze figure to be installed on a granite plinth overlooking the harbour entrance. The figure is depicted on watch with a pair of binoculars, and could be either an ordinary seaman or an officer.
The memorial was unveiled by Sir Winston Churchill in September 2008.


Bramell, Party of Five: Tick Tock, Tick Tock....watch out, she might EXPLODE!
I always look forward to your updates.... It won't be long now. Two weeks and three days is nothing compared to the time that has already passed. Looking forward to meeting those 2 new precious babies! Take care. Oh yeah, I'm also looking forward to seeing you drive that dreaded mini-van!
I am glad that you are going to go ahead and take off work. I think it is a good idea. You need the rest before the babies get here anyway. I am so excited to meet them! I think about who they will look like and what color hair they will have all the time! Not long now! And I agree, the mini-van will be the best. Hopefully you will call me soon to get up there quick! I love you!
Wow Rhonda! Talk about lots of major changes happening. It sounds like you have done a great job planning and getting prepared for the big arrival, which is a task in itself when your carrying 2 and already have one. You're going to be so amazing to these new babies!! What a great family!
Trust me about lots of changes quickly, but I just remind myself to take it one day at a time sometime and let the little stuff slide. :) I'm going to be "blog stalking" from now on waiting for the big post. :) Take care!!


Mr & Mrs JH are off celebrating their 28th wedding anniversary this weekend.
However, before you dash off, I'd love you to take a peek at a brand new innovative writer/reader website which is launching on 29th June.
More to come on this exciting venture soon.
Until next week, have a great bank holiday weekend everyone!
Aw, lovely photos and you get lovelier each year, Janice! Have a wonderful anniversary. The new website looks great.
Congratulations on 28 years of marriage! Great picture of the two of you. Have a wonderful weekend break.
Aw! Happy anniversary. I'm glad you had a good weekend!
Thank you Talli. It was a quiet weekend which was great - the boys were all away doing their own thing and we got the house to all to ourselves. Yay!
Glad you had a good Anniversary. DH and I celebrate 32 years 2nd June.