text:
This page lists student organizations of Johns Hopkins University.

Registered Student Organizations

Johns Hopkins has over 500 student-run organizations, each one providing a unique laboratory for learning for those involved. Student organizations provide opportunities for leadership development, building lifelong friendships, sharpening interpersonal skills, and improving organization. The University Office of Leadership Engagement & Experiential Development (LEED) is the hub for all undergraduate and graduate student organizations on the Homewood Campus at JHU. A comprehensive list of all student organizations, inclusive of all nine schools at JHU can be found on Hopkins Groups.

Fraternity and Sorority Life
The University recognizes ten fraternities, fourteen sororities, and 2 co-educational professional fraternities which include approximately 25% of the student body. Fraternities and sororities has been a part of the university culture since 1877, when Beta Theta Pi fraternity became the first to form a chapter on campus. Sororities arrived at Hopkins in 1976. As with all Hopkins programs, discrimination on the basis of "marital status, pregnancy, race, color, ethnicity, national origin, age, disability, religion, sexual orientation, veteran status" is prohibited. JHU also has an anti–hazing policy and prohibits alcohol at recruitment activities. Hopkins does not permit "city–wide" chapters, and requires all members of a JHU recognized fraternity or sorority to be a JHU student.

As of spring 2021, 1,208 students were members of one of Hopkins' fraternities or sororities. The All–FSL Average GPA was 3.84, above the undergraduate average GPA. In spring 2010 the university was considering construction of a "fraternity row" of houses to consolidate the groups on campus.

All Johns Hopkins fraternities and sororities belong to one of three Councils: the Inter–Fraternity Council, the National Panhellenic Conference, and the Intercultural Greek Council which is a combination of the National Pan-Hellenic Council and the Multicultural Council. Additionally, four independent chapters report directly to Fraternity and Sorority Life a unit within Leadership Engagement & Experiential Development.

The Inter–Fraternity Council includes seven fraternities:
 ΑΕΠ – Alpha Epsilon Pi fraternity, Psi chapter founded 1936. Jewish interest.
 ΒΘΠ – Beta Theta Pi fraternity, Alpha Chi chapter founded 1877.
 ΦΔΘ – Phi Delta Theta fraternity, Maryland Delta chapter founded 2009.
 ΦΓΔ – Phi Gamma Delta fraternity, Beta Mu chapter founded 1891.
 ΦΚΨ – Phi Kappa Psi fraternity, Maryland Alpha chapter founded 1879.
 ΣΧ – Sigma Chi fraternity, Kappa Upsilon chapter founded 2004.
 ΣΦΕ – Sigma Phi Epsilon fraternity, Maryland Alpha chapter founded 1929.
The National Panhellenic Conference includes five sororities:
 ΑΦ – Alpha Phi sorority, Zeta Omicron chapter founded 1982.
 ΚΚΓ – Kappa Kappa Gamma sorority, Eta Epsilon chapter founded 1999.
 ΦΜ – Phi Mu sorority, Gamma Tau chapter founded 1982.
 ΠΒΦ – Pi Beta Phi sorority, Maryland Gamma chartered November 20, 2010.
 ΚΑΘ – Kappa Alpha Theta sorority, Zeta Chi chapter founded April 20, 1997, disbanded April 14, 2009, reorganized March 1, 2014.

The Intercultural Greek Council includes the following 10 organizations represented by the NPHC and the MGC.

The National Pan–Hellenic Council includes five historically African–American groups:
 ΔΣΘ – Delta Sigma Theta sorority, Mu Psi chapter chartered 1976.
 ΑΚΑ – Alpha Kappa Alpha sorority, Xi Tau chapter chartered 1976
 ΑΦΑ – Alpha Phi Alpha fraternity, Sigma Sigma chapter founded 1991.
 ΣΓΡ – Sigma Gamma Rho sorority, Rho Omega chapter founded 2009.
 ΚΑΨ – Kappa Alpha Psi fraternity, Rho Upsilon Chapter chartered 2019

The Multicultural Council includes five groups:

 αΚΔΦ – alpha Kappa Delta Phi sorority, associate chapter founded 1997. Asian–American interest.
 ΔΞΦ – Delta Xi Phi sorority, Lambda chapter founded 2003. Multicultural interest.
 ΛΠΧ – Latinas Promoviendo Comunidad Lambda Pi Chi sorority, Sigma chapter charted 2004
 ΛΥΛ – Lambda Upsilon Lambda fraternity, Phi chapter charted in 1995
 ΣΙΑ – Hermandad de Sigma Iota Alpha Incorporada, Beta Zeta chapter chartered 2017
 ΣΟΠ – Sigma Omicron Pi sorority, Lambda chapter founded 2002. Asian–American interest.

These four independent organization are recognized at the University under the Fraternity and Sorority Life a unit within Leadership Engagement & Experiential Development.

 IX Society, Johns Hopkins Chapter founded in 2009 primarily for women.
 ΔΦ – Delta Phi Fraternity (also known as St. Elmo's), Xi chapter founded in 1885
 ΑΚΨ – Alpha Kappa Psi business fraternity, Rho Psi chapter founded 2001.
 ΘΤ – Theta Tau engineering fraternity.

Kappa Alpha Theta, a National Panhellenic Conference (NPC) sorority, was disbanded by its national headquarters on April 14, 2009 after twelve years on campus. The removal was due to repeated risk management violations.  Theta was reorganized on March 1, 2014.

In March 2010, Johns Hopkins University officially opened for NPC extension. In May 2010, the University Panhellenic Council selected Pi Beta Phi, which opened in the fall of 2010.

Recruitment for Inter–Fraternity Council and Panhellenic Conference fraternities and sororities takes place during the spring semester for freshmen, though some groups recruit upperclassmen during the fall semester. All participants must have completed one semester and must be in good academic standing.

Many of the fraternities maintain houses off campus, but no sororities do. Baltimore City allows housing to be zoned specifically for use as a fraternity or sorority house, but in practice this zoning code has not been awarded for at least 50 years. Only Sigma Phi Epsilon's building has this zoning code due to its consistent ownership since the 1920s.

Honor and Professional Societies
The university is home to a number of professional fraternities, societies, and honor organizations.
 ΑΕΔ – Alpha Epsilon Delta pre-med honor society.
 ΑΚΨ – Alpha Kappa Psi business fraternity, Rho Psi chapter founded 2001.
 ΑΦΩ - Alpha Phi Omega National Service Fraternity, Kappa Mu chapter founded 1952
 ΒΒΒ – Beta Beta Beta biology honor society, Rho Phi chapter.
 ΛΕΜ – Lambda Epsilon Mu Latino pre-med honor society.
 ΝΡΨ – Nu Rho Psi neuroscience honor society.
 ΣΙΡ – Sigma Iota Rho international studies honor society.
 SHPE – Society of Hispanic Professional Engineers.
 ΤΒΠ - Tau Beta Pi The Engineering Honor Society.
 ΘΤ – Theta Tau engineering fraternity.

Student publications

Hopkins has many student publications.
 The Johns Hopkins News-Letter, founded in 1896, is one of the oldest continuously published weekly college newspapers in the nation with a press run of 5,200. The News-Letter won an Associated Collegiate Press Newspaper Pacemaker award for four–year, non–daily college newspapers in 2007.
 JHU Politik, founded in 2008 as the University's only bi-partisan political publication, produces a weekly magazine, The Politik Press.  The Politik Press contains student written op-eds and interviews with professors, professionals and experts in various political fields.  Every semester JHU Politik publishes a special issue to highlight diverse perspectives on singular topics. JHU Politik is also responsible for numerous speakers events on campus as well as more informal discussions and conversations that take place throughout the academic year.
 Epidemic Proportions is the university's public health research journal, designed to highlight JHU research and field work in public health. Combining research and scholarship, the journal seeks to capture the breadth and depth of the JHU undergraduate public health experience.
 Thoroughfare, Zeniada and j.mag are literary magazines. Prometheus is the undergraduate philosophy journal.
 Frame of Reference is an annual magazine that focuses on film and film culture.
 The New Diplomat is the multi–disciplinary international relations journal. Foundations is the undergraduate history journal.
 Américas is the Latin American Studies journal.
 Argot is the undergraduate anthropology journal.
 The Triple Helix is the university's journal to address issues concerning science, law and society.
 Perspectives is the official newsletter of the Black Student Union.
 The Black & Blue Jay is among the nation's oldest campus humor magazines. It was founded in 1920. According to The Johns Hopkins News–Letter, the magazine's name led the News–Letter to first use the moniker Blue Jays to refer to a Hopkins athletic team in 1923. While the magazine enjoyed popularity among students, it received repeated opposition from the university administration, reportedly for its vulgar humor. In October 1934, Dean Edward R. Berry removed financial support for the magazine; without funding, the magazine continued under the name The Blue Jay until Berry threatened to expel the editors in 1939. The magazine had a revival in 1984, and has appeared intermittently since then.
 The Hopkins Donkey was a political newspaper with a Democratic perspective on international, national and statewide political topics. It is now defunct.
 The Carrollton Record was a political newspaper with an American conservative perspective on campus and citywide politics.  Like the Hopkins Donkey, it is now defunct.

Student–Run businesses

Hopkins Student Enterprises (HSE) is a venture capital fund and umbrella organization with the goal of fostering innovation and facilitating resources and mentorship to student entrepreneurs. Current businesses that are in operation are as follows:
 Hopkins Consulting Agency (HCA)—Business and technology consulting company that prepares technology commercialization reports and business plans.
 Hopkins Student Movers (HSM)—Moving and storage company that serves JHU faculty, staff, and students and the broader Baltimore community.
 Hopkins Creative Design (HCD)—Full service graphic design company.
 The Blue Jay Cleaners - Student janitorial service for all on-campus and select off campus residence halls.
 The Complete Dorm Room - Dorm essential delivery service with over 130 items to choose from including everything from linens to printers.
 Hop and Shake - Health food kiosk located in the Ralph S. O' Connor Recreation center
 Hopkins: Printer ink delivery service for Homewood and surrounding Charles Village.
 Blue Jay Bay: Service for all Hopkins students to buy and sell gently used furniture.
 Blue Jay Boxes - Care package and cake delivery business for Charles Village and the surrounding neighborhoods.

Programming Boards
In addition to the many clubs that exist on campus, there are a set of groups that directly advised by the LEED office to run major activities and events on campus including:
 The HOP - The Hopkins Organization for Programming  is a group responsible for a continuous stream of on campus events. They also are known to co-sponsor other groups events to help bring other groups visions to life when as the HOP has more resources at their disposal then most groups.

Symposiums
Established in 1967, the Milton S. Eisenhower Symposium is a student-run lecture series at the Johns Hopkins University.

Established in 1998, the Foreign Affairs Symposium has hosted a speaker series for the student body, with past speakers including Edward Snowden, Gloria Steinem and Dr. Cornel West.

Established in 2018, the Osler Medical Symposium is a student-run speaker series that hopes to bridge the divide between those making decisions in medicine and those affected by these decisions. Notable past symposia include Dr. Leana Wen, former Baltimore City Commissioner of Health and current President of Planned Parenthood; Dr. Paul B. Rothman, Dean/CEO of Johns Hopkins Medicine; Dr. Joshua Sharfstein, former Principal Deputy Commissioner of the Food and Drug Administration and former Baltimore Commissioner of Health; Dr. Peter Beilenson, former Baltimore City Commissioner of Health and current Sacramento County Commissioner of Health; and Dr. Peter Agre, Nobel Prize Winner in Chemistry.

Additional student organizations
Since 1918, the Johns Hopkins University Barnstormers (originally known as the dramatics club) has been performing various works on campus. In their current set-up they put on five shows a year. Two MainStage productions (a fall play and a spring musical) as well as a spring cabaret, an intersession show and series of one-acts done performed by freshmen. They will be celebrating their 100th anniversary in the 2018-2019 school year.

Since 1972, the Johns Hopkins Outdoors Club, or JHOC, has organized weekend trips for students looking to experience the outdoors. Along with Outdoor Pursuits, an arm of the University's Rec Center, JHOC offers students the opportunity to participate in activities such as canoeing, kayaking, caving, and mountain biking.

The Johns Hopkins Student Government Association represents undergraduates in campus issues and projects. It is elected annually.

Blueprints for a new programming board called The Hopkins Organization for Programming ("The HOP") were drawn up during the summer and fall of 2006.

References

Johns Hopkins University
Johns Hopkins University
Johns Hopikins University


text:
Hans Jordan (Joinville,  - Joinville, ) foi um político brasileiro.

Casou com Rose Tamm Jordan e tiveram duas filhas.

Nas eleições de 1945 concorreu à Câmara Federal pelo PSD, recebendo 4.224 votos, ficando na posição de último suplente. Foi convocado deputado federal para a 38ª Legislatura (1947-1951), e tomou posse em 5 de fevereiro de 1946.

Deputados estaduais de Santa Catarina
Naturais de Joinville

redpajama_set_name: RedPajamaWikipedia


Over 125 people attended the Convention of the Construction, Maintenance and Allied Workers in Canada (CMAW) that was held in Richmond, British Columbia on 19-21 September.
BWI was represented by Dora Cervantes, Secretary-Treasurer and Chair of the BWI North American Women's Committee; Vasyl Andreyev, President of Profbud; and Nilton Freitas, Regional Representative of BWI Latin America and the Carribean.
The delegates elected Chris Wasilenchuk as new president of the union after the retirement of the previous president Jan Noster. Chris has been an active member of CMAW since 2003 and brings a host of experience to his new role, including his most recent position as Vice-President and Business Agent of CMAW Local 1995.
They also decided to establish bilateral relations with the Ukrainian union PROFBUD and to support the youth committee of the Building and Wood Workers' International by providing a contribution to BWI's work on youth leadership development.
"The BWI recognizes that the future of a strong and dynamic global trade union moment is dependent upon a movement that is inclusive and committed to organizing women workers, young workers, and migrant workers," said Vasyl Andreyev from the Ukrainian construction union PROFBUD and member of the BWI World Board who was invited as a guest to the Convention.


PARIS (Reuters) - Air France-KLM (AIRF.PA) posted strong quarterly earnings and an upbeat outlook despite rising fuel costs, as new Chief Executive Ben Smith prepares to seek pilots' approval for plans likely to include an expansion of the group's low-fare operations.
Revenue rose 4 percent to 7.55 billion euros(6.74 billion pounds) in the third quarter, the group said on Wednesday. Recurring operating profit fell 6.7 pct to 1.07 billion euros - still beating analysts' expectations, according to a Reuters poll.
Describing the results as a "solid performance", the new Air France-KLM boss pledged to "build an ambitious and innovative strategy" for the group in coming months.
As Air Canada second-in-command, Smith pushed through new labour deals, a global network expansion and the launch of separately branded low-fare operations which together cemented the carrier's rebound from a 2009 brush with bankruptcy.
Smith is due to open negotiations with Air France pilots' unions next week, the company said.


text:
Sofija Tyzenhauzaitė de Šuazel-Gufjė, född 1790, död 28 maj 1878, var en litauisk författare. Hon räknas som de första litauiska författarna av sitt kön.

Hon tillhörde den litauiska adeln och gifte sig med en fransk adelsman som tjänstgjorde i Napoleons armé under invasionen av Ryssland 1812. Hennes romaner beskriver det samtida livet för kvinnor inom den litauiska adeln. Efter makens död styrde hon ett tag ensam familjens gods. Hon tillbringade sitt senare liv i den franska kulturkretsen i Paris.

Källor

Kvinnor
Födda 1790
Avlidna 1878
Litauiska författare
Litauisk adel
Personer i Litauen under 1800-talet


Before applying, it's important to understand the 1/8 Rule, 2/65 Rule, 1/90 Rule, and bonus eligibility.
The Citi 1/8 means you can only submit one application every eight days, and additional applications will automatically be rejected.
For example, if you apply to two credit cards in one day, one of them will be automatically rejected. Wait for 8–9 days before trying to apply for the second card.
The second Citi rule to be aware of is 2/65: 2 applications every 65 days, any additional applications will automatically be rejected.
In practice, the applications should probably be more spaced out, but the dates are used to illustrate the rules.
Realistically, you probably don't want to apply to that many Citi cards in one year because of the bonus rules, which we'll cover later in the post.
Citi 1/90 only applies to business cards; you can submit one business card application every 90 days.
Business cards also count towards the 1/8 AND 2/65 rules. Business and personal cards are separate applications, unlike other banks.
Some people have been affected by Citi 1/60; one application every 60 days, but not everyone.
Since people have been able to get around this rule, it may only apply to those who don't have a banking relationship or a thin credit profile.
You can only have six hard inquiries in 6 months, or else you'll automatically be rejected. The hard inquiries are not Citi specific.
You can only get a bonus for a specific group of cards once every two years. The 2-year clock is not based off when you last applied for a card; it's also based on product changes and card cancellations.
Example: If you apply for the Hilton Honors in May 2017, you'll be eligible to receive another Hilton bonus in May 2019. However, if you product change the card in 2018, it resets the bonus clock, and you won't be eligible for another Hilton bonus until 2020.
The good thing about Citi is you can freely product change cards from different groups. If you have a Costco credit card, you can product change it to a Citi Double Cash card.
Apply to no annual fee + no sign-up bonus cards first, then go for the annual fee + high sign up bonus cards later.
The other benefit of this strategy is that you are building the foundation for your credit history.


text:
Amerikai amatőr ökölvívók listája
Demetrius Andrade
Michael Bennett
Tyrell Biggs
Riddick Bowe
Chris Byrd
Howard Davis
Andre Dirrell
George Foreman
Joe Frazier
Paul Gonzales
Kenneth Gould
Virgil Hill
Oscar De La Hoya
Jeff Lacy
Ray Leonard
Floyd Mayweather Jr.
Ray Mercer
Jerry Page
David Reid
Leon Spinks
Frank Tate
Henry Tillman
José Torres
Andre Ward
Raushee Warren
Rhoshii Wells
Pernell Whitaker
Ricardo Williams

Amerikai ökölvívók
Ökölvívólisták


text:
Q: conditional volatility plot in R - GARCH When I build a GARCH(1,1) with a skewed generalised error dist to model the innovations, with a linear trend using the fGarch library I get the following:
mod = lm(Jobs ~ Month, df)

library(fGarch)
fit1 = garchFit(~garch(1,1), cond.dist="sged",  data= mod$residuals ,trace=F)

Model summary output:
Title:
 GARCH Modelling

Call:
 garchFit(formula = ~garch(1, 1), data = yres1, cond.dist = "sged",
    trace = F)

Mean and Variance Equation:
 data ~ garch(1, 1)
<environment: 0x3a54c3a0>
 [data = yres1]

Conditional Distribution:
 sged

Coefficient(s):
         mu        omega       alpha1        beta1         skew        shape
-6.1535e-16   5.6885e-01   6.9053e-02   8.8967e-01   1.7432e+00   1.9838e+00

Std. Errors:
 based on Hessian

Error Analysis:
         Estimate  Std. Error  t value Pr(>|t|)
mu     -6.154e-16   4.964e-01    0.000    1.000
omega   5.688e-01   6.554e-01    0.868    0.385
alpha1  6.905e-02   5.550e-02    1.244    0.213
beta1   8.897e-01   6.919e-02   12.858  < 2e-16 ***
skew    1.743e+00   2.669e-01    6.532 6.51e-11 ***
shape   1.984e+00   3.916e-01    5.065 4.08e-07 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Log Likelihood:
 -329.9066    normalized:  -2.945595

Description:
 Tue Jun 09 10:09:50 2020 by user: ben-i

Standardised Residuals Tests:
                                Statistic p-Value
 Jarque-Bera Test   R    Chi^2  12.39209  0.002037473
 Shapiro-Wilk Test  R    W      0.9560595 0.001006272
 Ljung-Box Test     R    Q(10)  5.87567   0.8255984
 Ljung-Box Test     R    Q(15)  11.69413  0.7020176
 Ljung-Box Test     R    Q(20)  15.61049  0.7404756
 Ljung-Box Test     R^2  Q(10)  11.52942  0.3177839
 Ljung-Box Test     R^2  Q(15)  13.25381  0.5827007
 Ljung-Box Test     R^2  Q(20)  14.56133  0.800934
 LM Arch Test       R    TR^2   17.38531  0.1356689

Information Criterion Statistics:
     AIC      BIC      SIC     HQIC
5.998332 6.143966 5.992972 6.057420

Plot:
plot.ts(sigma(fit1), ylab="sigma(t)", col="blue")

first 14 values:
sigma(fit1)
5.239727 5.806760 5.837686 5.855970 5.722280 5.920535 5.747552 5.489287 5.274133 5.225501 7.195435 7.082418 6.744105 6.736294

And when I use the rugarch library I get the following:
library(rugarch)
spec = ugarchspec(variance.model=list(model="sGARCH", garchOrder=c(1,1)),
        mean.model=list(armaOrder=c(0,0), external.regressors = matrix(df$Month)),
        distribution.model="sged")
fit2 <- ugarchfit(data = df$Jobs, spec = spec)

Model summary output:
*---------------------------------*
*          GARCH Model Fit        *
*---------------------------------*

Conditional Variance Dynamics
-----------------------------------
GARCH Model : sGARCH(1,1)
Mean Model  : ARFIMA(0,0,0)
Distribution    : sged

Optimal Parameters
------------------------------------
        Estimate  Std. Error    t value Pr(>|t|)
mu     14.491863    0.756088  19.166899 0.000000
mxreg1 -0.087486    0.012902  -6.780849 0.000000
omega   0.000000    0.003364   0.000085 0.999932
alpha1  0.003441    0.004869   0.706637 0.479792
beta1   0.987222    0.007582 130.210874 0.000000
skew    2.140304    0.524488   4.080750 0.000045
shape   1.952177    0.418083   4.669351 0.000003

Robust Standard Errors:
        Estimate  Std. Error   t value Pr(>|t|)
mu     14.491863    1.358462 10.667844 0.000000
mxreg1 -0.087486    0.025644 -3.411523 0.000646
omega   0.000000    0.000437  0.000658 0.999475
alpha1  0.003441    0.007121  0.483161 0.628981
beta1   0.987222    0.011632 84.874789 0.000000
skew    2.140304    0.939204  2.278850 0.022676
shape   1.952177    0.701973  2.780984 0.005419

LogLikelihood : -329.0167

Information Criteria
------------------------------------

Akaike       6.0003
Bayes        6.1702
Shibata      5.9931
Hannan-Quinn 6.0692

Weighted Ljung-Box Test on Standardized Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.7412  0.3893
Lag[2*(p+q)+(p+q)-1][2]    0.8391  0.5535
Lag[4*(p+q)+(p+q)-1][5]    1.1372  0.8279
d.o.f=0
H0 : No serial correlation

Weighted Ljung-Box Test on Standardized Squared Residuals
------------------------------------
                        statistic p-value
Lag[1]                     0.2884  0.5912
Lag[2*(p+q)+(p+q)-1][5]    1.3548  0.7755
Lag[4*(p+q)+(p+q)-1][9]    6.3467  0.2603
d.o.f=2

Weighted ARCH LM Tests
------------------------------------
            Statistic Shape Scale P-Value
ARCH Lag[3]    0.2343 0.500 2.000  0.6284
ARCH Lag[5]    2.2926 1.440 1.667  0.4101
ARCH Lag[7]    3.7211 2.315 1.543  0.3886

Nyblom stability test
------------------------------------
Joint Statistic:  1.1102
Individual Statistics:
mu     0.16284
mxreg1 0.15200
omega  0.05662
alpha1 0.07337
beta1  0.06320
skew   0.06851
shape  0.12872

Asymptotic Critical Values (10% 5% 1%)
Joint Statistic:         1.69 1.9 2.35
Individual Statistic:    0.35 0.47 0.75

Sign Bias Test
------------------------------------

Adjusted Pearson Goodness-of-Fit Test:
------------------------------------
  group statistic p-value(g-1)
1    20     17.29       0.5705
2    30     22.46       0.8005
3    40     30.86       0.8208
4    50     35.32       0.9289

Plot:
plot.ts(sigma(fit2), ylab="sigma(t)", col="blue")

First 14 values:
volatility(fit2)
5.379336 5.373246 5.347906 5.322474 5.291569 5.273909 5.252040 5.218381 5.190804 5.162108 5.280149 5.256321 5.222694 5.202587

Why are the values for the unconditional variance so different? Are the models not identically specified?

A: The fitted model coefficients are considerably different (compare the estimates of $\alpha$ and $\beta$ across the models), hence the difference in fitted conditional standard deviations is unsurprising. The reason could be

*

*different solvers used for numerical optimization of the likelihood functions,

*different starting values,

among other. After your updates, it appears it was the starting values causing the discrepancy.
(A side issue is that the estimation sequence is not identical; in one case, the estimation is done in one step, in the other case in two. But given that the regressor is a linear time trend and the estimation of the slope will likely be about as precise in each case due to the superconsistency of estimators with linearly trending regressors, this does not seem to have the potential of causing such large differences between fitted $\sigma_t$s as observed in your case.)

redpajama_set_name: RedPajamaStackExchange


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Rio Grande, Ohio: The Alice Lloyd women's and men's cross country team traveled to Rio Grande, Ohio on Saturday, September 23rd to participate in the 47th annual Patty Forgey Invitational Cross Country Meet. The weather was beautiful and sunny but temperatures hovering near 90 degrees made the course quite challenging. A total of 44 men's runners and 37 women runner's from various colleges completed the race for the challenging 5K (3.1 mile) women's race and 8K (5 mile) men's race. The humid temperatures definitely had an effect on the runner's times and several of them were forced to drop out due to cramping.
In the women's meet, the Lady Eagles were paced by junior Brittney Buttrey who completed the challenging course in 11th place with a time of 24:02. Junior Rachel McCoy and sophomore Kacey Trout finished their runs with times of 25:08 and 26:29 respectively. Coming in fourth and fifth place were freshman Taylor Bullock (27:55) and junior Cheyenne Boggs in a time of 30:02. For the event, ALC finished in 4th place out of the 6 teams that participated.
In the men's race, the Eagles were paced by freshman Jacob VanHook who finished with a time of 35:52. Junior John Driskill and sophomore Dustin Blair were next in line with an identical time of 36:29 while freshmen Noah Smith completed the course in 50:15. For the event, six teams sent runners but the weather definitely took a toll as only three of them had enough runners complete the course to qualify as a full squad.
The next meet will be on Thursday, September 28th. Both squads will race at Winchester Kentucky in the Lykins Park Invitational. The race begins at 6 pm.


If you feel a case needs to be referred urgently, please telephone the surgery on 015395 67241 and tell us that you consider the case to be urgent.
Wherever possible we will try to make sure that you can speak to a Veterinary Surgeon directly if immediate advice is needed. Please send any available images with the client or by email and fax a referral submission form as soon as possible.
After hours please follow the instructions on the answering service.
Please contact us by telephone if you wish to discuss the case before arranging a referral. If you feel the case is routine, please fill out a referral submission form and fax or email it back, with a copy of the clinical history on 0845 862 8612.
We do not need a specific referral letter unless you wish to produce one.
Please let us know if you wish us to contact the owner directly once we have received your instructions, or if the client will contact us directly to arrange an appointment.
Please ensure we have a clinical history provided and completed an xray submission form If you have sent radiographs but have not heard back within three days, please telephone the surgery as we may have been trying to contact you.


text:
Q: Why won't my python code print? So I'm extremely new to programming, and I'm having troubles getting my code to print. The error it continues to give is:
TabError: inconsistent use of tabs and spaces in indentation (<wingdb_compile>, line 5

My code is:
def main ():

        name = ____________("Tell me your name please ")
        num = int(input("Enter a number "))
    print(name,"your number is", num)
    d1 = num // ______________
    num = num _________  100
    d2 = _________________
    d3 = _________________

    print("The number reversed is ", d3, d2, d1, ________________)

main()

So as I said, I'm so new it may be a silly typing error. It has blanks since my professor gave us this much to start, but since I've not actually had the class to learn it yet, I'm a bit stuck. I'm trying to get it to say:
Tell me your name please Debby
Enter a number 849
Debby your number is 849
The number reversed is 948

Line 5 is def main ():.

A: It is a type error, very inconsistent use of indentation, this will confuse python.
It should look more like this:
def main ():
    name = ____________("Tell me your name please ")
    num = int(input("Enter a number "))
    print(name,"your number is", num)
    d1 = num // ______________
    num = num _________  100
    d2 = _________________
    d3 = _________________
    print("The number reversed is ", d3, d2, d1, ________________)

main()

A general rule to follow is after a colon : the next line should indented with tab once more than the line above it. This gets especially important when using if/elif and try/except blocks within a function.
I've included some pseudo script just to show what I mean
def foo():
    for var in list:
        if var in bar:
            print "var is in bar"
        elif var in some_list:
            try:
                some_function(var)
            except:
                print "error"
                continue

Notice how after each line that ends with : the next line is indented again.

A: You are using tabs for the lines with less indentation, spaces for the others.
Don't do that. To avoid this problem, in Wing IDE adjust your indentation preferences to use only spaces, and use the indentation manager to repair this file.

A: print should be within the function, your indentation is inconsistent so python doesn't even know the print function is there. Also it should look more like print('this is what it prints')


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text:
\section{Introduction}

An important contribution of Kostant has been the definition of geometric quantization via the cohomology associated to the sheaf of sections of a chosen pre-quantum line bundle that are flat along a given polarization. This construction using real polarizations is an abstraction of K\"{a}hler quantization and has been used in connection to representation theory (see for instance \cite{GuSt}). Generalizations of this scheme considering non-degenerate singularities have also been obtained by Hamilton \cite{Ha}, Hamilton and Miranda \cite{HaMi}, and Solha \cite{Solha}.

A toric manifold is a symplectic manifold endowed with an effective Hamiltonian action of a torus whose rank is half of the dimension of the manifold. A theorem of Delzant \cite{Del} establishes a one-to-one correspondence between closed toric manifolds in dimension $2m$ and a class of polytopes (called Delzant polytopes) on $\CR^m$. The real geometric quantization of closed toric manifolds can be read from the Delzant polytope, as it was proved by Hamilton \cite{Ha} (generalizing previous results by \'{S}niatycki \cite{Sni} to the singular context): given a toric manifold, its real geometric quantization is determined by the sheer count of integral points inside (boundary points are excluded) its associated Delzant polytope.

Toric manifolds are central in the study of the geometry of symplectic manifolds and their symmetries, as well as their generalizations, like semitoric integrable systems \cite{pelayo4} or almost toric manifolds \cite{LeSy}, in which the rank of the torus is no longer half of the dimension of the manifold. Examples of almost toric manifolds are given by K3 surfaces, which are also of relevance in complex geometry. A semitoric integrable system (see for instance \cite{pelayo3, pelayo4}) is an integrable system admitting only non-degenerate singularities composed of elliptic and focus-focus types, but excluding any hyperbolic components.

As observed in \cite{Solha} the quantization of almost toric manifolds can be reduced to the computation of the contribution of a neighborhood of a Bohr--Sommerfeld focus-focus singular fiber by the use of factorization tools\footnote{Which behave following a K\"{u}nneth formula \cite{MiPr}, as a simple sheaf cohomology.}. This is because the geometric quantization of neighborhoods of Bohr--Sommerfeld fibers computes the geometric quantization of the whole manifold (by means of a standard Mayer--Vietoris sequence).

In this article we associate Kostant's model to focus-focus singularities and conclude its computation showing that the first cohomology group associated to the real geometric quantization of a small neighborhood of a focus-focus fiber of a 4-dimensional semitoric integrable system is trivial, but not the second cohomology group, which is infinite dimensional when the singular fiber is Bohr--Sommerfeld. This determines completely the geometric quantization when the real polarization has focus-focus fibers (the cohomology group in degree zero is trivial and had already been computed in \cite{Solha}) and thus closes up the problem of geometric quantization of integrable systems with non-degenerate singularities as initiated in \cite{Ha} and \cite{HaMi} for $4$-dimensional manifolds with no hyperbolic-hyperbolic fibers.

As a motivation for these results, we present K3 surfaces as an example of almost toric manifolds and analyze the effect of nodal trades \cite{LeSy} in their real quantization. Other models of quantization for K3 surfaces have been recently obtained by Castejón \cite{castejon} using the Berezin--Toepliz operators approach \cite{bms}. For this direction see also \cite{pelayo1}. We also analyse the examples of a spin-spin system, the spherical pendulum, and a spin-oscillator system.

\section{Main definitions}

\subsection{Singular Lagrangian fibrations}

The symplectic manifolds of interest to this article have a great deal of symmetry, and such symmetries are related to some particular classes of integrable systems: the ones admitting only non-degenerate singularities.

\begin{definition}An integrable system on a symplectic manifold $(M,\omega)$ of dimension $2m$ is a set of $m$ functions, $f_1,\dots,f_m\in C^\infty(M;\CR)$, satisfying
\begin{equation*}
\ud f_1\wedge\cdots\wedge\ud f_m\neq 0 \text{ over an open dense subset of $M$ and}
\end{equation*}
\begin{equation*}
\{f_j,f_k\}_\omega=0 \text{ for all $j,k$.}
\end{equation*}
\end{definition}

The \emph{Poisson bracket} is defined by $\{f,\cdot\}_\omega=X_f(\cdot)$, where $X_f$ is the unique vector field defined by the equation $\imath_{X_f}\omega+\ud f=0$, called the \emph{Hamiltonian vector field} of $f$.

The next definition refers to the critical set of an integrable system, i.e. the set of points where $\ud f_1\wedge\cdots\wedge\ud f_m$ vanishes.

\begin{definition}A critical point of rank $k_r=m-k_e-k_h-2k_f$ of an integrable system $(f_1,\dots,f_m):M\to\CR^m$ is a non-degenerate singular point of Williamson type $(k_e,k_h,k_f)$ if the quadratic parts of $f_1,\dots,f_m$ can be written as:
\begin{equation*}\begin{array}{lc}
\quad h_j=x_j \phantom{lmmmmmmmmm} \text{(regular)} & 1 \leq j \leq k_r \\
\quad h_j=x_j^2+y_j^2 \phantom{nmmmmmm} \text{(elliptic)}& k_r+1 \leq j \leq k_r+k_e \\
\quad h_j=x_jy_j \phantom{lnmmmmmmm} \text{(hyperbolic)} & k_r+k_e+1 \leq j \leq k_r+k_e+k_h \\
\left\{\begin{array}{l} h_j=x_j y_j+x_{j+1} y_{j+1} \\ h_{j+1}=x_j y_{j+1}-x_{j+1} y_j \end{array}\right. \text{(focus-focus)} & \begin{array}{c}
j=k_r+k_e+k_h+2l-1,\\ 1\leq l \leq k_f \end{array}
\end{array}
\end{equation*}in some Darboux local coordinates $(x_1,y_1,\dots,x_m,y_m)$.
\end{definition}

\begin{example}The stable equilibrium point in the simple pendulum is an elliptic singularity ($k_e=1$, $k_h=0$, and $k_f=0$), while the unstable one is of hyperbolic type ($k_e=0$, $k_h=1$, and $k_f=0$). The spherical pendulum has a stable equilibrium point which is a purely elliptic singularity ($k_e=2$, $k_h=0$, and $k_f=0$), the unstable equilibrium point is a focus-focus singularity ($k_e=0$, $k_h=0$, and $k_f=1$).
\end{example}

A singular fiber of an integrable system is said to be of \emph{Williamson type} $(k_e,k_h,k_f)$ if all of its singular points are non-degenerate singular points of that same Williamson type. Another terminology is also used in this article: in dimension 2 an \emph{elliptic fiber} and a \emph{hyperbolic fiber} are singular fibers of Williamson type $(1,0,0)$ and $(0,1,0)$, in dimension 4 a \emph{focus-focus fiber} is a singular fiber of Williamson type $(0,0,1)$.

When we refer to a \emph{foliation associated to an integrable system} we refer to the foliation described by the orbits of the Hamiltonian vector fields. When we refer to a \emph{fibration associated to an integrable system} we refer to the foliation defined by the fibers. Those two foliations do not necessarily coincide at the singular points. As it was proved by Eliasson \cite{eliasson,eliasson1} and Miranda \cite{Mi, miranda, MiZu} non-degenerate singularities are characterized by the fact that the foliation associated to an integrable system is equivalent to the foliation described by its quadratic part.

\begin{theorem} The foliation of the integrable system given by $f_1,\dots,f_m$ is locally equivalent, in a neighborhood of a non-degenerate singular point of Williamson type $(k_e,k_h,k_f)$, to the foliation of the integrable system given by:
\begin{equation*}\begin{array}{lc}
\quad h_j=x_j \phantom{lmmmmmmmmm} \text{(regular)} & 1 \leq j \leq k_r \\
\quad h_j=x_j^2+y_j^2 \phantom{nmmmmmm} \text{(elliptic)}& k_r+1 \leq j \leq k_r+k_e \\
\quad h_j=x_jy_j \phantom{lnmmmmmmm} \text{(hyperbolic)} & k_r+k_e+1 \leq j \leq k_r+k_e+k_h \\
\left\{\begin{array}{l} h_j=x_j y_j+x_{j+1} y_{j+1} \\ h_{j+1}=x_j y_{j+1}-x_{j+1} y_j \end{array}\right. \text{(focus-focus)} & \begin{array}{c}
j=k_r+k_e+k_h+2l-1,\\ 1\leq l \leq k_f \end{array}
\end{array}
\end{equation*}in some Darboux local system of coordinates $(x_1,y_1,\dots,x_m,y_m)$.
\end{theorem}

Let us define the notion of singular Lagrangian fibration.

\begin{definition}A singular Lagrangian fibration is a symplectic manifold $(M,\omega)$ of dimension $2m$ together with a surjective map $\mathcal{F}:M\to N$, where $N$ is a topological space of dimension $m$, such that for every point in $N$ there exist an open neighborhood $V\subset N$ and a homeomorphism $\chi:V\to U\subset\CR^m$ satisfying that $\chi\circ\mathcal{F}\big{|}_{\mathcal{F}^{-1}(V)}$ is an integrable system on $(\mathcal{F}^{-1}(V),\omega\big{|}_{\mathcal{F}^{-1}(V)})$.
\end{definition}

When the integrable systems in a singular Lagrangian fibration do not have singularities, one refers to it as a \emph{regular Lagrangian fibration}. The real geometric quantization of such manifolds were computed in \cite{Sni}, whereas (closed) \emph{locally toric manifolds} were considered in \cite{Ha} (see \cite{Solha} for the non-compact case); these symplectic manifolds are singular Lagrangian fibrations whose singularities are only of Williamson type $(k_e,0,0)$.

\emph{Almost toric manifolds} are singular Lagrangian fibrations admitting only singularities of Williamson type $(k_e,0,k_f)$. In particular, regular Lagrangian fibrations and locally toric manifolds (which include toric manifolds), are examples of almost toric manifolds; as well as the \emph{semitoric integrable systems} in dimension four, which are included in the almost toric manifolds whose bases are subsets of $\CR^2$. The semi-local and global classification of these symplectic manifolds have been the object of study of \cite{chaperon, LeSy, pelayo3, pelayo4, san}.

\subsection{Real geometric quantization}

Let $(M,\omega)$ be a symplectic manifold of dimension $2m$ whose de Rham class $[\omega]$ admits an integral lift. Such a symplectic manifold will be called \emph{pre-quantizable}.

\begin{definition}\label{prequantumdef}A pre-quantum line bundle for $(M,\omega)$ is a complex line bundle $L$ over $M$ with a connection $\nabla^{\omega}$ satisfying $curv(\nabla^{\omega})=-i\omega$.
\end{definition}

\begin{definition}\label{polarisationdef} A real polarization $\mathcal{P}$ is an integrable (in the Sussmann's sense) distribution of $TM$ whose leaves are generically Lagrangian. The complexification of $\mathcal{P}$ is denoted by $P$ and will be called polarization.
\end{definition}

The most relevant real polarization for this work is $\langle X_{f_1},...,X_{f_m}\rangle_{C^\infty(M;\CR)}$: the distribution of the Hamiltonian vector fields of an integrable system. The leaves of the associated (possibly singular) foliation are isotropic submanifolds and they are Lagrangian at points where the first integrals are functionally independent.

\begin{definition} Let $\mathcal{J}$ denote the sheaf of sections of a pre-quantum line bundle $L$ such that for each open set $V\subset M$ the set $\mathcal{J}(V)$ is the module (over the ring of smooth leafwise constant complex-valued functions of $V$) of sections $s\in L$ defined over $V$ satisfying $\nabla^{\omega}_X s=0$ for all vector fields $X$ in $P$ defined over $V$.
\end{definition}

\begin{definition} The quantization of $(M,\omega,L,\nabla^{\omega},P)$ is given by
\begin{equation*}
\mathcal{Q}(M)=\displaystyle\bigoplus_{n\geq 0}H^n(M;\mathcal{J}) \ ,
\end{equation*}where $H^n(M;\mathcal{J})$ are the sheaf cohomology groups associated to $\mathcal{J}$.
\end{definition}

The following definition plays a very important role in the computation of the cohomology groups appearing in geometric quantization:

\begin{definition}A leaf $\ell$ of $\mathcal{P}$ is Bohr--Sommerfeld if there exists a non-vanishing section $s:\ell\to L$ such that $\nabla^\omega_X s=0$ for any complex vector field $X$ in the polarization $P$ (restricted to $\ell$). Fibers that are a union of Bohr--Sommerfeld leaves are called Bohr--Sommerfeld fibers.
\end{definition}

\section{A motivating example: K3 surfaces}\label{KKK}

A K3 surface is an example of a total space of an almost toric manifold: it admits an almost toric fibration over the sphere with 24 focus-focus fibers. The base is a sphere with 24 marked points, and on the complement of these points one has a regular Lagrangian fibration with torus fibers.

A way to construct such an almost toric manifold, as done in \cite{LeSy} (cf. \cite{Gom}), is to consider two copies of a (symplectic and toric) blowup of the complex projective plane at 9 different points as toric manifolds, apply nodal trades to all of their elliptic-elliptic singular fibers, and take their symplectic sum along the symplectic tori corresponding to the preimage of the boundary of their respective bases (as almost toric fibrations). Starting with a pre-quantizable K3 surface, this construction (together with a gluing result described in subsection \ref{glue}) allows one to obtain a K3 surface with up to 24 Bohr--Sommerfeld focus-focus fibers.

Here is how the construction works.

\begin{itemize}
\item Starting with a complex projective plane, understood as a toric manifold and described here by its Delzant polytope \cite{Del}, one performs three blowups at different points, represented in their Delzant polytopes by cuts based on their three vertices (cf. \cite{KKP}), followed by another six blowups at different points, represented in their Delzant polytopes by cuts based on their six new vertices formed after the first three blowups: see figure \ref{figK3}.

\begin{figure}[h]
\centering
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\begin{picture}(11,9)
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\end{picture}
\begin{picture}(6,9)
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\put(2,1){\line(-1,1){1}}
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\end{picture}
\caption{$\CC P^2$, $\CC P^2\# 3\overline{\CC P}^2$, and $\CC P^2\# 9\overline{\CC P}^2$.}\label{figK3}
\end{figure}

\item Now, following \cite{LeSy}, one can perform nodal trades to all the vertices of the resulting Delzant polytope. In figure \ref{figK32} each nodal trade is being represented by a vector based at a vertex, and the monodromy around each of the resulting focus-focus fibers can be read from those vectors.

\begin{figure}[h]
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\caption{Nodal trades on $\CC P^2\# 9\overline{\CC P}^2$.}\label{figK32}
\end{figure}

\item The resulting manifold, $\CC P^2\# 9\overline{\CC P}^2$, is endowed with an almost toric fibration, and the preimage of the boundary of its base is a symplectic torus. Thus, one can consider two copies of this symplectic manifold and perform a symplectic sum along these tori (cf. \cite{Gom}), obtaining a K3 surface, $(\CC P^2\# 9\overline{\CC P}^2)\#_{T^2}(\CC P^2\# 9\overline{\CC P}^2)$; together with an almost toric fibration whose base is the sphere formed by gluing two copies of the previously constructed disk (with its twelve marked points) along their boundary (see figure \ref{figK33}).

\begin{figure}[h]
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\caption{K3 surface as a singular fiber bundle over the sphere.}\label{figK33}
\end{figure}

\end{itemize}

Before the nodal trades, the toric manifold $\CC P^2\# 9\overline{\CC P}^2$ of figure \ref{figK3} admits a pre-quantum line bundle such that all the integer lattice points belonging to its Delzant polytope are images of Bohr--Sommerfeld fibers \cite{GuSt,Solha}. Nodal trades produce a one parameter family of symplectomorphic manifolds (via nodal slides \cite{LeSy}), and the almost toric manifold constructed in figure \ref{figK32} is related by a symplectomorphism isotopic to the identity (the same is true for different nodal trades resulting in up to 12 focus-focus fibers outside the integer lattice).
Therefore, it inherits a pre-quantum line bundle whose Bohr--Sommerfeld fibers are still given by the integer lattice points in the base (which now includes up to 12 focus-focus fibers, depending on the size of the nodal trades).

When gluing two copies of those almost toric manifolds by a symplectic sum, one can glue the pre-quantum line bundles to obtain a pre-quantum line bundle on a K3 surface having any number (between 0 and 24) of focus-focus Bohr--Sommerfeld fibers. This last assertion is justifyied by applying lemma \ref{Franlemma} and corollary \ref{Francoro} (which can be found, together with their proofs, in subsection \ref{glue}).

\section{Preliminary results}

The first three subsections of this section collect results from the literature needed for the proof of the main theorems of this article, and they are included here for the convenience of the reader.

\subsection{Poincar\'{e} lemmata}

Given a pre-quantizable symplectic manifold $(M,\omega)$ with polarization $P$ and pre-quantum line bundle $(L,\nabla^{\omega})$, it is possible to construct a fine resolution for the sheaf of flat sections $\mathcal{J}$, even when $P$ has non-degenerate singularities \cite{JHR,Msolha2,Solha}.

Let $\Omega_P^n(M)$ denote the set of multi-linear maps
\begin{equation*}
\mathrm{Hom}_{C^\infty(M;\CC)}(\wedge^n_{C^\infty(M;\CC)}P; C^\infty(M;\CC)) \ ,
\end{equation*}usually called polarized $n$-forms, and
\begin{equation*}
S_P^n(L)=\Omega_P^n(M)\otimes_{C^\infty(M;\CC)}\Gamma(L) \ .
\end{equation*}Then, the set of \emph{line bundle valued polarized forms} is
\begin{equation*}
{S_P}^\bullet(L)=\displaystyle\bigoplus_{n\geq 0}S_P^n(L) \ .
\end{equation*}

Therefore, $\nabla=\nabla^{\omega}\big{|}_P:S^0_P(L)\to S^1_P(L)$, the restriction of the connection $\nabla^{\omega}$ to the polarization, extends to a derivation of degree $+1$ on the space of line bundle valued polarized forms: if $\alpha\in\Omega_P^n(M)$ and $s\in\Gamma(L)$,
\begin{equation*}
\ud^\nabla(\alpha\otimes s)=\ud_P\alpha\otimes s+(-1)^n\alpha\wedge\nabla s \ ,
\end{equation*}with the exterior derivative $\ud_P$ being the restriction of the de Rham differential to the directions of the polarization.

Since $\omega=i \ curv(\nabla^\omega)$ vanishes along $P$, $\ud^\nabla$ is a coboundary operator.

If $\mathcal{S}_P^n$ denotes the associated sheaf of $S_P^n(L)$, one can extend $\ud^\nabla$ to a homomorphism of sheaves, $\ud^\nabla:\mathcal{S}_P^n\to\mathcal{S}_P^{n+1}$. The sheaf $\mathcal{S}$ of sections of the line bundle $L$ is isomorphic to $\mathcal{S}_P^0$, and $\mathcal{J}$ is isomorphic to the kernel of $\nabla:\mathcal{S}\to\mathcal{S}_P^1$, understood as a map between sheaves. The associated complex
\begin{equation*}
0\longrightarrow\mathcal{J}\hookrightarrow\mathcal{S}\stackrel{\nabla}{\longrightarrow}\mathcal{S}_P^1\stackrel{\ud^\nabla}{\longrightarrow}\cdots\stackrel{\ud^\nabla}{\longrightarrow}\mathcal{S}_P^m\stackrel{\ud^\nabla}{\longrightarrow}0
\end{equation*}is called the \emph{Kostant complex}, and its cohomology is denoted by $H^\bullet({S_P}^\bullet(L))$.

\begin{theorem}\label{fineresolution1} The Kostant complex is a fine resolution for $\mathcal{J}$ when $\mathcal{P}$ is a subbundle of $TM$, or when $\mathcal{P}$ is induced by an integrable system whose moment map has only non-degenerate singularities. Therefore, each of its cohomology groups, $H^n({S_P}^\bullet(L))$, is isomorphic to $H^n(M;\mathcal{J})$.
\end{theorem}

This theorem is proved in \cite{JHR,Msolha2,Solha} by showing that Poincar\'{e} lemmata exist for the Kostant complex. One Poincar\'{e} lemma of particular importance to the present work is the following:

\begin{theorem}[Solha \cite{Solha}]\label{poincaresolha}The cohomology groups $H^n({S_P}^\bullet(L))$ vanish for all $n$ in any sufficiently small contractible open neighborhood of a focus-focus singularity.
\end{theorem}

\begin{remark}\label{metarema} The only property of $L$ being used here is the existence of flat connections along $P$; thus, the results here work if metaplectic correction is considered.
\end{remark}

\subsection{Mayer--Vietoris sequence}

Analogously to the de Rham cohomology case, there exists a Mayer--Vietoris sequence for $\mathcal{J}$. One can construct, for each pair of open subsets $V$ and $W$ of $M$, the injective homomorphism
\begin{equation*}
R_{V,W}:\mathcal{S}_P^n(V\cup W)\to\mathcal{S}_P^n(V)\oplus\mathcal{S}_P^n(W)
\end{equation*}defined by $R_{V,W}(\boldsymbol{\zeta})=\boldsymbol{\zeta}\big{|}_{V}\oplus\boldsymbol{\zeta}\big{|}_{W}$ and the surjective homomorphism
\begin{equation*}
R_{V,V\cap W}-R_{W,V\cap W}:\mathcal{S}_P^n(V)\oplus\mathcal{S}_P^n(W)\to\mathcal{S}_P^n(V\cap W)
\end{equation*}defined by $R_{V,V\cap W}-R_{W,V\cap W}(\boldsymbol{\alpha}\oplus\boldsymbol{\beta})=\boldsymbol{\alpha}\big{|}_{V\cap W}-\boldsymbol{\beta}\big{|}_{V\cap W}$. The injectivity of $R_{V,W}$ is due to the local identity property of the sheaves, while the surjectivity of $R_{V,V\cap W}-R_{W,V\cap W}$ comes from the existence of partitions of unity for $\mathcal{S}^n_P$.

Thanks to the gluing condition of the sheaves, the image of $R_{V,W}$ is equal to the kernel of $R_{V,V\cap W}-R_{W,V\cap W}$, and the long exact sequence associated to the short exact sequences
\begin{equation*}
0\rightarrow\mathcal{S}_P^n(V\cup W)\hookrightarrow\mathcal{S}_P^n(V)\oplus\mathcal{S}_P^n(W)\twoheadrightarrow\mathcal{S}_P^n(V\cap W)\rightarrow 0 \
\end{equation*}yields the Mayer--Vietoris sequence for the Kostant complex (or $\mathcal{J}$, since the Kostant complex is a resolution for $\mathcal{J}$).

\subsection{K\"{u}nneth formula}

The classical K\"{u}nneth formula also holds for the geometric quantization scheme \cite{MiPr}. Let $(M_1, \mathcal{P}_1)$ and $(M_2, \mathcal{P}_2)$ be a pair of pre-quantizable symplectic manifolds endowed with Lagrangian foliations. The natural Cartesian product for the foliations is Lagrangian with respect to the product symplectic structure. The induced sheaf of flat sections associated to the product foliation will be denoted $\mathcal{J}_{12}$. Note that we use the pre-quantum line bundle defined as pull-backs of the ones defined over $M_1$ and $M_2$.

\begin{theorem}[Miranda and Presas \cite{MiPr}] \label{thm:Kun}
There is an isomorphism
\begin{equation*}
H^n(M_1 \times M_2,\mathcal{J}_{12})\cong\bigoplus_{p+q=n}H^p(M_1, \mathcal{J}_1) \otimes H^q(M_2, \mathcal{J}_2) \ ,
\end{equation*}whenever $M_1$ admits a good cover, the geometric quantization associated to $(M_2,\mathcal{J}_2)$ has finite dimension and $M_2$ is a submanifold of a compact manifold.
\end{theorem}

As an illustration, and to anticipate some needed results, let us mention what is the geometric quantization for $M=T^*I\times (I_s\times S^1)$ with $\omega=\ud x_1\wedge\ud y_1+\ud x_2\wedge\ud y_2$, endowed with a trivial pre-quantum line bundle with connection $\nabla=\ud-i(x_1\ud y_1+x_2\ud y_2)$, and $\mathcal{P}$ generated by $\depp{}{y_1}$ and $\depp{}{y_2}$, where $x_1$ is the coordinate function along the fibers of $T^*I$, $y_1$ is the coordinate function along the open interval $I\subset\CR$, $x_2$ is the coordinate function along the open interval $I_s\subset\CR$, and $y_2$ is the periodic coordinate function along $S^1$.

\begin{proposition}\label{kunn}The geometric quantization of $M=T^*I\times (I_s\times S^1)$, with the extra structures described above, is given by
\begin{equation*}
H^1(T^*I\times (I_s\times S^1);\mathcal{J}_{12})\cong\bigoplus_ {j\in\{1,\dots,n\}}H^0(T^*I;\mathcal{J}_1)\cong\bigoplus_ {j\in\{1,\dots,n\}} C^\infty(\CR;\CC) \ ,
\end{equation*}where $n$ is the number of integers inside $I_s$.
\end{proposition}

The techniques from \cite{MiPr,Sni,Solha} provide isomorphisms from $H^1(T^*I\times (I_s\times S^1);\mathcal{J}_{12})$ to flat sections of trivial pre-quantum line bundles $L$ over $T^*I$, for a given open interval $I\subset\CR$. These flat sections are all of the form
\begin{equation*}
T^*I\ni (x,y)\mapsto h(x)\e^{ixy}s(x,y)\in L|_{(x,y)}\cong\CC
\end{equation*}where $h\in C^\infty(\CR;\CC)$ and $s\in\Gamma(L)$ is a unitary section of $L$ with potential $1$-form $-x\ud y$. For example, theorem \ref{thm:Kun} gives
\begin{equation*}
H^1(T^*I\times (I_s\times S^1);\mathcal{J}_{12})\cong H^0(T^*I;\mathcal{J}_1)\otimes H^1(I_s\times S^1;\mathcal{J}_{2})\cong C^\infty(\CR;\CC)\otimes\CC^n \ .
\end{equation*}

\section{Contribution from focus-focus singularities}

Let $V\subset M$ be an open neighborhood of a non-degenerate focus-focus fiber $\ell_f$ (compact or not) over which a Hamiltonian $S^1$-action is defined \cite{Zu2}. Note that a focus-focus fiber might have more than one singular point (also called a \emph{node}, or \emph{nodal point} \cite{LeSy}).

\begin{lemma}[Solha \cite{Solha}]\label{corosolha}
In the neighborhood of $\ell_f$ over which a Hamiltonian $S^1$-action is defined, there exists a neighborhood $V$ containing only $\ell_f$ as a Bohr--Sommerfeld fiber such that $H^0(V;\mathcal{J}\big{|}_V)=\{0\}$.
\end{lemma}

Therefore, without loss of generality, one can assume that $V$ contains no Bohr--Sommerfeld fiber, or only one if $\ell_f$ is itself Bohr--Sommerfeld. Such a neighborhood $V$ of a focus-focus fiber $\ell_f$ is called a \emph{saturated neighborhood} (since it is saturated by the orbits of the $S^1$-action).

\begin{theorem}\label{ffalive1}The geometric quantization of a saturated neighborhood of a focus-focus fiber with only one node is isomorphic to $C^\infty(\CR;\CC)$ if the focus-focus fiber is compact and Bohr--Sommerfeld, and zero otherwise.
\end{theorem}
\begin{proof}
Let $p\in\ell_f$ be the singular point of the focus-focus fiber, $W\subset V$ be a contractible open neighborhood of the singular point, and $V_0\subset V$ be an open (not connected when $\ell_f$ is not compact) neighborhood with $p\notin V_0$, together satisfying $V=V_0\cup W$ and $V_0\cap W=W^-\sqcup W^+$.

The neighborhood $V$ is the total space of a singular Lagrangian fibration over an open disk $D^2\cong\CR\times I_s$ (with $I_s\subset\CR$ an open interval representing the circle action direction), as well as $W$ (which is diffeomorphic to an open $4$-ball centered in the nodal point $p\in\ell_f$), while $W^-$, $W^+$, and $V_0$ are regular trivial Lagrangian fibrations. Indeed,
\begin{equation*}
V_0\cong (I_0\times S^1)\sqcup (I_{2\pi}\times S^1)\times D^2 \ ,
\end{equation*}with $I_0=(0,b^-)$, $I_{2\pi}=(a^+,2\pi)$, and $a^+>b^-$,
\begin{equation*}
W^-\cong(I^-\times S^1)\times D^2
\end{equation*}with $I^-=(a^-,b^-)$ and $a^->0$, and
\begin{equation*}
W^+\cong(I^+\times S^1)\times D^2
\end{equation*}with $I^+=(a^+,b^+)$ and $b^+<2\pi$ (see figure \ref{figI}). For a compact fiber $\ell_f$, one connects $I_0$ and $I_{2\pi}$ via $0\sim 2\pi$, yielding $V_0$ with only one connected component.

\begin{figure}[h]
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\end{picture}
\caption{Intervals along a focus-focus fiber with only one node.}\label{figI}
\end{figure}

Let us represent the trivial regular Lagrangian fibrations as a product of two cotangent bundles
\begin{equation*}
V_0\cong\left(T^*(I_0\sqcup I_{2\pi})\right)\times (I_s\times S^1) \ ,
\end{equation*}
\begin{equation*}
W^-\cong T^*I^-\times (I_s\times S^1) \ ,
\end{equation*}and
\begin{equation*}
W^+\cong T^*I^+\times (I_s\times S^1) \ .
\end{equation*}We do so in order to use lemma \ref{corosolha}, theorem \ref{poincaresolha}, and proposition \ref{kunn}, which give:
\begin{equation*}
H^0(V;\mathcal{J}\big{|}_V)=\{0\} \ ,
\end{equation*}
\begin{equation*}
H^0(W;\mathcal{J}\big{|}_{W})=H^1(W;\mathcal{J}\big{|}_{W})=H^2(W;\mathcal{J}\big{|}_{W})=\{0\} \ ,
\end{equation*}
\begin{equation*}
H^0(V_0;\mathcal{J}\big{|}_{V_0})=H^0(W^-;\mathcal{J}\big{|}_{W^-})=H^0(W^+;\mathcal{J}\big{|}_{W^+})=\{0\} \ ,
\end{equation*}
\begin{equation*}
H^2(V_0;\mathcal{J}\big{|}_{V_0})=H^2(W^-;\mathcal{J}\big{|}_{W^-})=H^2(W^+;\mathcal{J}\big{|}_{W^-})=\{0\} \ ,
\end{equation*}
\begin{equation*}
H^1(V_0;\mathcal{J}\big{|}_{V_0})\cong\left\{\begin{array}{l}
\{0\}\text{, if $\ell_f$ is not Bohr--Sommerfeld} \\
C^\infty(\CR;\CC)\oplus C^\infty(\CR;\CC)\text{, if $\ell_f$ is non-compact}\\
C^\infty(\CR;\CC)\text{, if $\ell_f$ is compact}
\end{array}\right. \ ,
\end{equation*}
\begin{equation*}
H^1(W^-;\mathcal{J}\big{|}_{W^-})\cong\left\{\begin{array}{ll}
C^\infty(\CR;\CC) & \text{, if $\ell_f$ is Bohr--Sommerfeld} \\
\{0\} & \text{, otherwise} \end{array}\right. \ ,
\end{equation*}and
\begin{equation*}
H^1(W^+;\mathcal{J}\big{|}_{W^+})\cong\left\{\begin{array}{ll}
C^\infty(\CR;\CC) & \text{, if $\ell_f$ is Bohr--Sommerfeld} \\
\{0\} & \text{, otherwise} \end{array}\right. \ .
\end{equation*}

Considering the open covering $V_0,W$ of $V$, one has the following exact sequence from the Mayer--Vietoris sequence:
\begin{multline*}
0\rightarrow H^0(V_0\cup W;\mathcal{J}\big{|}_{V_0\cup W}) \\
\rightarrow H^0(V_0;\mathcal{J}\big{|}_{V_0})\oplus H^0(W;\mathcal{J}\big{|}_W)\rightarrow H^0(V_0\cap W;\mathcal{J}\big{|}_{V_0\cap W}) \\
\rightarrow H^1(V_0\cup W;\mathcal{J}\big{|}_{V_0\cup W}) \\
\rightarrow H^1(V_0;\mathcal{J}\big{|}_{V_0})\oplus H^1(W;\mathcal{J}\big{|}_W)\rightarrow H^1(V_0\cap W;\mathcal{J}\big{|}_{V_0\cap W}) \\
\rightarrow H^2(V_0\cup W;\mathcal{J}\big{|}_{V_0\cup W}) \\
\rightarrow H^2(V_0;\mathcal{J}\big{|}_{V_0})\oplus H^2(W;\mathcal{J}\big{|}_W)\rightarrow H^2(V_0\cap W;\mathcal{J}\big{|}_{V_0\cap W}) \\
\rightarrow H^3(V_0\cup W;\mathcal{J}\big{|}_{V_0\cup W})\rightarrow\cdots
\end{multline*}Exploiting the dimension of $V$ (cohomology groups in degree higher than two vanish) and the fact that the cohomology groups in degree zero vanish, as well as in degree two for $V_0$, $W$, and $V_0\cap W$, one has
\begin{multline*}
0\rightarrow H^1(V_0\cup W;\mathcal{J}\big{|}_{V_0\cup W}) \\
\hookrightarrow H^1(V_0;\mathcal{J}\big{|}_{V_0})\oplus H^1(W;\mathcal{J}\big{|}_W)\rightarrow H^1(V_0\cap W;\mathcal{J}\big{|}_{V_0\cap W}) \\ \twoheadrightarrow H^2(V_0\cup W;\mathcal{J}\big{|}_{V_0\cup W})\rightarrow 0 \ .
\end{multline*}

Because $H^1(W;\mathcal{J}\big{|}_{W})=\{0\}$, the middle map
\begin{equation*}
H^1(V_0;\mathcal{J}\big{|}_{V_0})\oplus H^1(W;\mathcal{J}\big{|}_W)\longrightarrow H^1(V_0\cap W;\mathcal{J}\big{|}_{V_0\cap W})
\end{equation*}is injective. This can be seen by identifying the pertinent cohomology groups with $C^\infty(\CR;\CC)$ (see proposition \ref{kunn} and comments below it); thus, the map can be identified with
\begin{equation*}
C^\infty(\CR;\CC)\oplus\{0\}\ni h\oplus 0\mapsto h\oplus h\in C^\infty(\CR;\CC)\oplus C^\infty(\CR;\CC) \ ,
\end{equation*}when the fiber is compact and Bohr--Sommerfeld, and
\begin{equation*}
C^\infty(\CR;\CC)\oplus C^\infty(\CR;\CC)\oplus\{0\}\ni h_1\oplus h_2\oplus 0\mapsto h_1\oplus h_2\in C^\infty(\CR;\CC)\oplus C^\infty(\CR;\CC) \ ,
\end{equation*}when the fiber is Bohr--Sommerfeld but not compact. From the exactness of the sequence and using the first isomorphism theorem, this implies the following:
\begin{equation*}
H^1(V_0\cup W;\mathcal{J}\big{|}_{V_0\cup W})=\{0\}
\end{equation*}and
\begin{equation*}
H^2(V_0\cup W;\mathcal{J}\big{|}_{V_0\cup W})\cong\frac{H^1(V_0\cap W;\mathcal{J}\big{|}_{V_0\cap W})}{H^1(V_0;\mathcal{J}\big{|}_{V_0})} \ .
\end{equation*}

Thus, identifying $V_0\cup W=V$ and $V_0\cap W=W^-\sqcup W^+$, a nodal point on a compact Bohr--Sommerfeld focus-focus fiber provides an infinite dimensional contribution,
\begin{equation*}
H^1(V;\mathcal{J}\big{|}_V)=\{0\}
\end{equation*}and
\begin{equation*}
H^2(V;\mathcal{J}\big{|}_V)\cong\left\{\begin{array}{ll}
C^\infty(\CR;\CC) & \text{, if $\ell_f$ is compact and Bohr--Sommerfeld} \\
\{0\} & \text{, otherwise}
\end{array}\right. \ .
\end{equation*}
\end{proof}

It is possible to adapt the argument of the proof of theorem \ref{ffalive1} to include more nodal points to one focus-focus fiber.

\begin{theorem}\label{ffalive}The geometric quantization of a saturated neighborhood of a focus-focus fiber (with any number of nodes) vanishes if the singular fiber is not Bohr--Sommerfeld, or it is isomorphic to
\begin{equation*}
\bigoplus_{j\in\{1,\dots,n_f\}}C^\infty(\CR;\CC) \ ,
\end{equation*}with $n$ being the number of singular points in the Bohr--Sommerfeld focus-focus fiber, and $n_f=n$ if the focus-focus fiber is compact, or $n_f=n-1$ if the focus-focus fiber is non-compact.
\end{theorem}
\begin{proof}(Sketch)
Instead of one singular point we now have $n$ singularities $p_1,\dots,p_n\in\ell_f$ on the focus-focus fiber. As before we take $W_1,\dots,W_n\subset V$ contractible open neighborhoods of the singular points such that $W_j\cap W_k=\emptyset$ for $j\neq k$, $V_0\subset V$ an open (not connected, even if the fiber is compact) neighborhood satisfying $p_1,\dots,p_n\notin V_0$, as well as, $V=V_0\cup W_1\cup\cdots\cup W_n$, and $V_0\cap W_j=W^-_j\sqcup W^+_j$ for each $W_j$.

Therefore, one can apply the Mayer--Vietoris argument considering the open covering $V_0,V_n=W_1\cup\dots\cup W_n$ of $V$. In place of $W$ one has $V_n$ and since $V_0\cap V_n=\displaystyle\bigsqcup_{j\in\{1,\dots,n\}}W^-_j\sqcup W^+_j$:
\begin{equation*}
H^1(V_0;\mathcal{J}\big{|}_{V_0})\cong\left\{\begin{array}{cl}
\{0\} & \text{, if $\ell_f$ is not Bohr--Sommerfeld} \\
\displaystyle\bigoplus_{j\in\{1,\dots,n+1\}}C^\infty(\CR;\CC) & \text{, if $\ell_f$ is non-compact} \\
\displaystyle\bigoplus_{j\in\{1,\dots,n\}}C^\infty(\CR;\CC) & \text{, if $\ell_f$ is compact}
\end{array}\right. \
\end{equation*}and
\begin{multline*}
H^1(V_0\cap V_n;\mathcal{J}\big{|}_{V_0\cap V_n})\cong\displaystyle\bigoplus_{j\in\{1,\dots,n\}}H^1(W^-_j;\mathcal{J}\big{|}_{W^-_j})\oplus H^1(W^+_j;\mathcal{J}\big{|}_{W^+_j}) \\
\cong\left\{\begin{array}{cl}
\displaystyle\bigoplus_{j\in\{1,\dots,n\}}C^\infty(\CR;\CC)\oplus C^\infty(\CR;\CC) & \text{, if $\ell_f$ is Bohr--Sommerfeld} \\
\{0\} & \text{, otherwise}
\end{array}\right. \ .
\end{multline*}Finally, similarly to the case $n=1$, the map
\begin{equation*}
H^1(V_0;\mathcal{J}\big{|}_{V_0})\oplus H^1(V_n;\mathcal{J}\big{|}_{V_n})\rightarrow H^1(V_0\cap V_n;\mathcal{J}\big{|}_{V_0\cap V_n})
\end{equation*}is injective.
\end{proof}

\section{Semitoric systems and almost toric manifolds}

As for the quantization of Lagrangian fibrations and locally toric manifolds, quantization of neighborhoods of Bohr--Sommerfeld fibers computes the quantization of the whole manifold \cite{Solha} (cf. \cite{Ha} and \cite{Sni}). Consequently, mimicking the Lagrangian bundle and locally toric cases, theorem \ref{ffalive} together with the factorization tools from \cite{Solha} provide the following results.

\begin{theorem}\label{GQalmosttoric}For a $4$-dimensional closed almost toric manifold $M$, with $BS_r$ and $BS_{f}$ the images of the regular and focus-focus Bohr--Sommerfeld fibers on the base:
\begin{equation*}
\mathcal{Q}(M)\cong\left(\bigoplus_{p\in BS_r}\CC\right)\oplus\left(\bigoplus_{p\in BS_{f}}\oplus_{n(p)} C^\infty(\CR ;\CC)\right) \ ,
\end{equation*}with $n(p)$ the number of nodes on the fiber whose image is $p\in BS_{f}$.
\end{theorem}

Closed almost toric manifolds in dimension four were classified in \cite{LeSy}, and in order to obtain their real geometric quantization is enough to identify the image of the Bohr--Sommerfeld fibers at each of the seven possible bases, and then apply theorem \ref{GQalmosttoric}. The total number of regular Bohr--Sommerfeld fibers is determined by the symplectic volume of the almost toric manifold, and the number of focus-focus fibers can be read from table 1 in \cite{LeSy}. Via nodal slides is always possible to modify the real polarization to change the number of focus-focus fibers that are actually Bohr--Sommerfeld fibers; this is exemplified in subsection \ref{glue} for the K3 surface.

Semitoric sytems \cite{pelayo4} are a particular example of almost toric manifolds in dimension four, their bases are subsets of $\CR^2$. However, their total spaces need not to be closed symplectic manifolds; therefore, there is no upper bound on the number of Bohr--Sommerfeld fibers, although their image still form a countable set on the base. Their real geometric quantization is essentially the same as in the case of closed almost toric manifolds, the formula of theorem \ref{GQalmosttoric} holds, but the sets $BS_r$ and $BS_{f}$ can be countably infinite.

\section{Four dimensional examples}

Let us compute the geometric quantization for some specific examples in this last section.

\subsection{K3 surface}\label{glue}

As mentioned in the construction of a pre-quantizable K3 surface (section \ref{KKK}), one can obtain a K3 surface with up to 24 Bohr--Sommerfeld focus-focus fibers. In the particular example constructed in section \ref{KKK}, an application of theorem \ref{GQalmosttoric} yields
\begin{equation*}
\mathcal{Q}(K3)\cong\CC^{14}\oplus\displaystyle\bigoplus_{j\in\{1,\dots,24\}} C^\infty(\CR;\CC) \ .
\end{equation*}

The real geometric quantization of the K3 surface can be, then, drastically different from the K\"{a}hler case, which is always finite dimensional.

On the K3 surface, the dimension of the vector space of holomorphic sections for a given ample holomorphic line bundle $L$ equals $\frac{1}{2}c_1(L)^2+2$ (cf. \cite{Hu}), and the dimension of its K\"{a}hler quantization is exactly this number. Since the first Chern class of a pre-quantum line bundle $L$ is represented by the symplectic form $\omega$, it holds that
\begin{equation*}
c_1(L)^2=\displaystyle\int_{K3}\omega\wedge\omega \ .
\end{equation*}

In the particular example above $K3=(\CC P^2\# 9\overline{\CC P}^2)\#_{T^2}(\CC P^2\# 9\overline{\CC P}^2)$, and the symplectic volume of a symplectic sum is the sum of the symplectic volumes \cite{Gom}. Thus, the symplectic volume can be computed from the symplectic volume of each toric manifold (as nodal trades produce symplectomorphic manifolds \cite{LeSy}), which is simply two times the Euclidean volume of each Delzant polytope \cite{Gu} (up to a $(2\pi)^2$ factor due to different conventions); and the volume of the Delzant polytopes are $24$ in this case. Therefore, for the particular example computed above, the dimension of its K\"{a}hler quantization is
\begin{equation*}
\frac{1}{2}c_1(L)^2+2=\frac{1}{2}(2\cdot 24+2\cdot 24)+2=50 \ .
\end{equation*}

But even when the real geometric quantization is finite dimensional, for this symplectic $K3$ one would have $\mathcal{Q}(K3)\cong\CC^{38}$, which is still different from $\CC^{50}$. This difference is due to how real and K\"{a}hler quantization behave with respect to singular Bohr--Sommerfeld elliptic fibers of a toric manifold. In the real case those fibers (that lie on the the boundary of the Delzant polytope) do not contribute to geometric quantization \cite{Ha}, while they do contribute in the K\"{a}hler case. This means that the real geometric quantization has a simpler behavior under symplectic sum ($19+19=38$) than the K\"{a}hler quantization ($31+31=50+12$).

What is missing is to actually show how to glue pre-quantum line bundles when performing a symplectic sum. We begin by reviewing the symplectic sum construction, and we, then, keep track of this construction when considering pre-quantum line bundles in the picture.
\begin{lemma}[Gompf \cite{Gom}]\label{gomlemma}
Let $(M_j, \omega_j)$, $j=1,2$, be two symplectic manifolds. Assume that there are two codimension $2$ symplectic submanifolds $\Sigma_j \subset M_j$, a symplectomorphism $\Psi: \Sigma_1 \to \Sigma_2$ and a complex isomorphism identifying the symplectic normal bundle $\nu_1$ of $\Sigma_1$ and the dual symplectic normal bundle $\nu_2^*$ of $\Sigma_2$. Then, there is a symplectic structure on the fiber connected sum $M_1 \#_{\Psi} M_2$ of $M_1$ and $M_2$ along $\Sigma_2\simeq \Psi(\Sigma_1)$.
\end{lemma}

Denote by $U_j$ a small tubular neighborhood of $\Sigma_j$ and $U_j^*=U_j \setminus \Sigma_j$. Recall that Gompf's construction provides a symplectomorphism $\Phi:U_1^* \to U_2^*$ that takes the outer boundary of one domain to the inner boundary of the other one and vice versa. This is used as gluing morphism. Let us upgrade the previous Lemma.

\begin{lemma}\label{Franlemma}
Let $(M_j, \omega_j, L_j, \nabla_j)$, $j=1,2$, be two symplectic manifolds equipped with pre-quantum line bundles.
Under the hypotheses of lemma \ref{gomlemma}, assume moreover that $\Psi^* L_2 \simeq L_1$ (as topological complex line bundles), then the fiber connected sum $M_1 \#_{\Psi} M_2$ admits a pre-quantum line bundle $(L, \nabla)$ whose restriction to $M_j\setminus U_j$ coincides with $(L_j, \nabla_j)$
\end{lemma}

\begin{proof}
Since $U_j$ deform retracts to $\Sigma_j$, we have that $\Phi^* L_2 \simeq L_1$ (the topological isomorphism can be extended to $U_j$ and then restricted to $U_j^*$). Denote $(\tilde{L}_1, \tilde{\nabla}_1) = (\Phi^* L_2, \Phi^* \nabla_2)$. We have that over $U_1^*$ the two bundles also satisfy that $curv(\tilde{\nabla}_2) = -\Phi^* i\omega_2 =- i\omega_1 =curv(\nabla_1)$.

Denote by $(\bar{L}_1, \bar{\nabla}_1)$ the dual vector bundle of $(L_1, \nabla_1)$.The bundle $V= \tilde{L}_1 \otimes \bar L_1$, defined over $U_1^*$, is topologically trivial. Moreover, it is equipped with a flat connection $\nabla_V= \tilde{\nabla}_1 - \nabla_1$, therefore we have that $\tilde{\nabla}_1 = \nabla_1 + e^{if}$, for some smooth function $f: U_1^* \to \CR$.

Take a slightly smaller open neighborhood $V_1^* \subset U_1^*$ and the corresponding $\Phi(V_1^*)= V_2^* \subset U_2^*$. Define a function $g:U_1^* \to \CR$ such that $f|_{V_1^*}= g|_{V_1^*}$ and $g$ is compactly supported in $U_1^*$. Define a gauge equivalent connection $\nabla_1' = \nabla_1 +e^{ig}$. By construction, it coincides with $\nabla_1$ away from $U_1^*$ and it coincides with $\Phi^* \nabla_2$ over $V_1^*$. We conclude by gluing $M_1$ and $M_2$ over $V_2^*= \Phi(V_1^*)$.
\end{proof}

\begin{corollary}\label{Francoro}
With the same hypothesis, further assume that there exists a symplectomorphism $\psi:(M,\omega_1,L_1,\nabla_1)\to (M,\omega_2,L_2,\nabla_2)$ with $\psi$ isotopic to the identity ($M=M_1=M_2$), then for any $\Sigma$ we have that ${L_1}|_{\Sigma}$ and ${L_2}|_{\Sigma}$ are isomorphic as topological complex line bundles
\end{corollary}
Do note that $\psi(\Sigma)\neq \Sigma$ in general. If this were not the case, the statement and the gluing would be trivial.
\begin{proof}
Denote by $\psi_t$ the isotopy connecting $\psi_1=id$ with $\psi_2=\psi$. We define $L_t= (\psi_t)_* L_1$ and $\Sigma_t= \psi_t(L_1)$. Therefore, the bundles ${L_2}|_{\Sigma_2}$ and ${L_1}|_{\Sigma_1}$ are isomorphic. Now, since $\Sigma_2$ and $\Sigma_1$ are isotopic submanifolds then the bundles ${L_2}|_{\Sigma_2}$ and ${L_2}|_{\Sigma_1}$ are isomorphic.
\end{proof}

\subsection{Spin-spin system}

Another example (which may be considered as a toy model to the spin-spin system of \cite{SaZh}) is to consider the product of two spheres. As a toric manifold it is represented by its Delzant polytope (which is a square), and one can perform a nodal trade on one of its vertices: see figure \ref{figS2}.

\begin{figure}[h]
\centering
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\put(0,2){\circle*{0.2}}
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\put(0,2){\circle*{0.2}}
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\multiput(0,1)(1,0){3}{\circle*{0.1}}
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\put(0,0){\line(1,0){2}}
\put(2,0){\line(0,1){1.5}}
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\put(1,1){\circle{0.3}} \qbezier(2,1.5)(2,2)(1.5,2)
\put(0,2){\line(0,-1){2}}
\end{picture}
\caption{Nodal trade on $S^2\times S^2$.}\label{figS2}
\end{figure}

Considering the two spheres as unit spheres in $\CR^3$ (with the induced area form as the symplectic structures) parametrized by coordinate functions $(x_j,y_j,z_j)$, with $j=1,2$ and $x_j^2+y_j^2+z_j^2=1$, the toric system of figure \ref{figS2} is described by the first integrals $f_1=z_1$ and $f_2=z_2$, while the spin-spin system of \cite{SaZh} is descibed by the first integrals $\displaystyle f_1=\frac{z_1}{2}+\displaystyle\frac{x_1x_2+y_1y_2+z_1z_2}{2}$ and $f_2=z_1+z_2$, whose qualitative traits are represented by the semitoric system obtained from the above toric system via a nodal trade.

The resulting construction yields,
\begin{equation*}
\mathcal{Q}(S^2\times S^2)\cong C^\infty(\CR;\CC)
\end{equation*}for the semitoric system.

\subsection{Spherical pendulum and the spin-oscillator system}

Particular choices of the parameters on the system of a spherical pendulum (mass and length of the pendulum, as well as the gravitational acceleration) imply that the focus-focus singular fiber is Bohr--Sommerfeld. As a result, the eigenspace attached to the eigenvalue given by the energy of the classical unstable equilibrium state of a focus-focus singular point might be infinite dimensional.

The coupled classical spin and harmonic oscillator of \cite{pelayo2} (a classical version of the Jaynes--Cummings model \cite{JaCu}) also behaves in a similar fashion. While the spherical pendulum is described by an integrable system on the cotangent bundle of the sphere, the spin-oscillator system is described by a semitoric system on the trivial bundle $\CR^2\times S^2$ over the sphere, whose first integrals are $f_1=z+\frac{1}{2}(u^2+v^2)$ and $f_2=\frac{1}{2}(xu+yv)$, where $(u,v)$ are the coordinate functions on the fibers $\CR^2$ and $(x,y,z)$ satisfying $x^2+y^2+z^2=1$ are the coordinates on the unit sphere; the symplectic structure is the product one, with the one in the fibers being $\ud u\wedge\ud v$ and the one in the sphere being the induced area form from the Euclidean $\CR^ 3$.

This sort of degeneracy at energy values associated with classical unstable equilibrium states does not represent the physics of the quantum systems (cf. \cite{Condon,CuDu}), and it would be interesting to refine the definition of geometric quantization with singular real polarizations to mod out these infinite dimensional contributions to get a finite dimensional Hilbert space instead.

It is worth mentioning, though, that this infinite dimensional degeneracy on these examples is not generic. If the singular Lagrangian fibration is perturbed so the focus-focus fibers are no longer Bohr--Sommerfeld (e.g. by slightly changing the parameters of a spherical pendulum, by performing a nodal slide on almost toric manifolds, or by perturbing the connection in the pre-quantum line bundle), their real geometric quantisation ends up being finite dimensional.

\section{Higher dimensions}

We may iterate these techniques to obtain higher dimensional results. We would like to stress out that there is no complete symplectic topological classification of integrable systems in a neighbourhood of semitoric fibers in dimension greater than 4. However, by applying the Mayer--Vietoris argument used in the proof of theorem \ref{ffalive1} together with the K\"{u}nneth formulae from \cite{MiPr}, we conjecture:

\begin{conjecture}
Let $M$ is a $2m$-dimensional closed almost toric manifold with singularities of any corank and of Williamson type $(k_e, 0, k_f)$ with $k_f< 2$:
\begin{equation*}
\mathcal{Q}(M)\cong\left(\bigoplus_{p\in BS_r}\CC\right)\oplus\left(\bigoplus_{p\in BS_{f-r}}\oplus_{n(p)} C^\infty(\CR ;\CC)\right) \ ,
\end{equation*}where $BS_r$ and $BS_{f-r}$ are the images of the regular and focus-focus-regular (Williamson type $(0,0,1)$) Bohr--Sommerfeld fibers on the base, and $n(p)$ the number of nodes on the fiber whose image is $p\in BS_{f-r}$.
\end{conjecture}

The conjecture above  does not take into account singular Lagrangian fibrations with hyperbolic components but those can be considered using the results in  \cite{HaMi}:

\begin{conjecture}
For non-degenerate integrable systems with singularities of any corank and of Williamson type $(k_e, k_h, h_f)$ with $k_h< 2$, $k_f<2$, and $k_h+k_f\leq 1$ on a closed symplectic manifold:
\begin{equation*}
\mathcal{Q}(M)\cong\left(\bigoplus_{p\in BS_r}\CC\right)\oplus\left(\bigoplus_{p\in BS_{f-r}}\oplus_{n(p)} C^\infty(\CR ;\CC)\right)\oplus\left(\bigoplus_{p\in BS_{h-r}}\CC^{\CN}\oplus\CC^{\CN}\right) \ .
\end{equation*}
\end{conjecture}

Due to the infinite dimensional contributions coming from hyperbolic and focus-focus singularities, the analogous results containing mixed product of these components are not considered here, e.g. Williamson types $(0,2,0)$, $(0,1,1)$, $(0,0,2)$, or higher dimensional analogues of any codimension. In these examples a completion might be needed when considering the K\"{u}nneth formulae (see \cite{MiPr,kaup,grothendieck}).

redpajama_set_name: RedPajamaArXiv


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