text:
\begin{document}

\title{Now that we're together*: Biography of the Chilean Collective of Women Mathematicians.}

\author{Mar\'{\i}a Isabel Cortez and Andrea Vera-Gajardo}
\address{Facultad de Matem\'aticas, Pontificia Universidad Cat\'olica de Chile.
 Instituto de Matem\'aticas, Universidad de Valpara\'{\i}so.} \email{maria.cortez@mat.uc.cl,
andreaveragajardo@gmail.com}
\thanks{Both authors were supported by proyecto ANID PIA Anillo SOC180025.}
\thanks{* "Now that we’re together, now that we’re seen" (\textit{Ahora que estamos juntas, ahora que s\'{\i} nos ven}) is a slogan that has been widely used both in the “Ni una menos” (“Not a single woman less”) movement in Argentina, and in the movement born during the so-called Feminist May of 2018 in Chile. }
  \keywords{Chilean women in math organization, Latin American women in science organization, Chilean math community}

\footnotesize{This is submitted to an upcoming book aimed at a general audience. Below is the second version of the draft.}

\begin{abstract} In this article we give an account of the history of the Chilean collective of women
mathematicians. Unlike the AWM, this collective is not a membership organization, but rather a group that was spontaneously armed with common interests, and whose number of members varies between 15 and 5 members, depending on the moment. We begin by describing the context of the mathematical community
in Chile and the process of forming the Collective, together with the first objectives
we set ourselves. Then we continue with an analysis of some reasons that support
the need to create a group formed by women mathematicians and also the choice of
horizontality and autonomy as structural pillars of our organization. On the
other hand, we refer to the main activities that we have carried out and provide an
overview of the women in mathematics organizations that exist in Latin America.
Finally, we conclude by discussing the research project to which some of the
members of the collective are dedicated nowadays in 2020 (in particular the authors of this
article).
\end{abstract}

 \maketitle{}

\section*{The context of the mathematical community in Chile.}
\label{sec:1}

According to the International Mathematical Union, the modern development of mathematics in Chile began in 1930 with the arrival of Carlos Grandjot \cite{DeLaPena2014}, a German mathematician who played a key role training the first Chilean mathematicians \cite{Gutierrez2002}. On the other hand, the first local articles published in international journals were written in the 1940s by Roberto Frucht, another German mathematician based in Chile, who worked at the Universidad Técnica Federico Santa María (UTFSM) until the end of his professional career \cite{Mercado2020}. It is worth noting that in 1949 Frucht published "Graphs of degree three with a given abstract group" in the \textit{Canadian Journal of Mathematics}, the same year in which women’s right to vote in presidential elections was approved in Chile.

By 1970, Chilean mathematicians had already graduated from PhD programs abroad. The first was Jaime Michelow, who obtained his PhD from the University of Washington in 1962. Nevertheless, as the study commissioned by the National Commission of Science and Technology (CONICYT) points out  \cite{Harding1970}, by 1970 there was still no "systematic and planned mathematical research, nor high-level researchers to initiate it" \cite[p. 2]{Harding1970}, and there was also "a lack of human resources in terms of quantity, capacity and adequate knowledge, for university teaching and postgraduate studies" \cite[p. 2]{Harding1970}. This reality stood in stark contrast to other Latin American countries, such as Brazil and Argentina. The case of Brazil is emblematic, since by the 1950s it had already trained mathematicians who would later gain international recognition through their guest lectures at the International Congress of Mathematicians \cite{DeLaPena2014}. On the other hand, Argentina had established the Argentine Mathematical Union (UMA) in 1936, which was created with the aim of "promoting the evident progress of mathematical research in Argentina"  \cite{UMA2020}.

In Chile, institutional development for the training of mathematicians began in 1962 with the creation of the first degree in Mathematics at the Universidad de Concepción \cite{Harding1970}. This was followed in 1965 by degrees in Mathematics at both the Universidad Católica and Universidad de Chile, as well as a degree in Mathematical Engineering from the Faculty of Physical and Mathematical Sciences (FCFM) at the latter institution \cite{Harding1970}. In 1967, the Universidad Técnica del Estado (UTE)—now the Universidad de Santiago de Chile (USACH)—created the country's first postgraduate program in mathematics. This corresponded to the Academic Mathematics Degree (LAM) program, which awarded a Master's degree in Mathematics. The first PhD in Mathematics offered in Chile was awarded in 1975 by the Universidad Católica (UC) to Pablo Salzberg.\footnote{Private communication with Irene Mikenberg} The second one was awarded in 1978 by the same program to Irene Mikenberg, who became the first woman to obtain her PhD in mathematics in Chile. By the end of the 1970s, Chile had 47 PhDs in mathematics working in the country, of which five were women: Carmen Cortázar (UC), Irene Mikenberg (UC), Lidia Consiglieri (U. Católica de Valparaíso), María Angélica Astaburuaga (UTE) and Nancy Lacourly (FCFM U. de Chile)  \cite{CONICYT1978}. It is important to highlight Inés Harding, since she played a very relevant role in the mathematical community of the time. Harding read further postgraduates courses leading to a PhD degree at the University of Moscow in the 1960s  \cite{CONICYT1978}, working as a professor \cite{Pedemonte2017} at UTE until 1981.\footnote{Private communication with Rafael Labarca.}  She became one of the pillars of the LAM program \cite{Calderon2003}, managing the program and recruiting students, while also co-authoring  groundbreaking textbooks from those years, such as "Elementos de Computación" (Computing Elements) in 1973 \cite{Michellow1973}. She passed away in Paris in 1996.

One of the many consequences of the 1973 coup d'état was the loss of many students and academics who were forced to leave the country \cite{SOMACHI2014}, thus delaying the consolidation of the local mathematical community. This situation slowly began to turn around in the early 1980s as several mathematicians who had obtained their PhDs abroad started returning to Chile. Founded in 1975, the Chilean Mathematical Society (SOMACHI) was instrumental in drawing these mathematicians back home as well as promoting mathematical activity at the time \cite{SOMACHI2014}.

A milestone that shaped the way science is done in Chile was the creation of the National Fund for Scientific and Technological Development (FONDECYT) in 1981, which finances basic scientific research in Chile through "open competition for public grants" \cite{FONDECYT2020}. Even today, being awarded a FONDECYT project is considered a decisive milestone in a researcher's career. In most cases, opportunities for national and international collaboration, the possibility to train students, and even the academic workload allocated to each person depend on a research project being awarded one of these grants.

Between 1982 and 1990, FONDECYT funded 144 math projects,\footnote{FONDECYT provided us with data for all the FONDECYT mathematics projects between 1983 and 2019.} of which 13 were awarded to women. These women included Carmen Cortázar, Marta García-Huidobro and Rubí Rodríguez—recognized scholars who remain scientifically active to this day— and Cecilia Yarur, who has recently retired after leading a successful career as a researcher and teacher.

After the return to democracy in 1990, the development of Chilean mathematics started to gain momentum. Not only did the number of researchers and publications grow, but also the number of international cooperation schemes \cite{SOMACHI2014}. The number of FONDECYT projects in mathematics awarded in the 1990s grew by more than 80\% compared to the previous decade, although the ratio of women among the researchers behind these projects remained largely unchanged.

There are currently more than 350 mathematics researchers working in Chile  \cite{SOMACHI2014}. According to data collected from 11 mathematics departments in the country, approximately 20\% of the scholars who hold a PhD in mathematics or related fields are women \cite{Cortez2016}. Nevertheless, a review of the statistics of FONDECYT mathematics projects shows that grants awarded to women between 2011 and 2020 \footnote{The numbers about FONDECYT projects since 2001 include both Regular and Initiation calls. Regular calls are oriented to senior researchers whereas Initiation calls are for junior researchers.}  did not surpass 15\%—an even lower rate than the 2001-2010 period, at 17\%. This suggests that women in our community are even more underrepresented in leading positions, \footnote{Being awarded a FONDECYT grant is considered a sort of symbolic credential of prestige for the beneficiary researcher.}and this situation does not seem to have improved over time. To further emphasize this situation, it is worth mentioning that in the "FONDECYT regular" \footnote{This is the fund oriented to senior researchers.} call for 2020 there were no  projects awarded for women in the area of mathematics, something not seen since 1990.   At the school level, the picture is no more promising, as can be seen from the gender gap in performance in mathematics, present since primary school \cite{Radovic18}.

There seems to be a consensus within our community on the need to increase female participation in the discipline. What is less clear, however, is how to do this; and even more so, whether that is indeed the problem or rather a symptom of something that goes much deeper. Clémence Perronnet clarifies this problem by stating that institutions expect to receive "individualizing and psychologizing" explanations for inequalities in science, which leads to hiding and concealing "the collective and structural dimension" of these inequalities \cite{Perronnet2020}. In other words, the lack of women in mathematics is not a problem of motivation of individuals (women in this case), but rather the consequence of practices that are usually incompatible with the way in which those individuals are socialized.

\medskip

In the following lines we present the process of forming the Chilean Collective of Women Mathematicians and we refer to the main activities that we have carried out.   We also describe  other  mathematics organizations  in Latin America that support girls' and women's participations in mathematics.

\section*{How the Collective began.}
\label{sec:2}

  After one of the roundtable discussions at the International Congress of Women in Mathematics 2014 in Seoul, several colleagues from Chile who attended met to discuss the issues presented at the conference. The general consensus that day was that our experience had been very different from that of the speakers, in the sense that our parents had not prevented us from studying mathematics, nor had we endured explicit complaints from our husbands or partners about the profession being unladylike. Yet we were familiar with the experience of being the "only woman" in our workplace, and we perceived that treading through a heavily male-dominated space had consequences that were not easy to identify and describe, given how subtle they generally were. Trying to understand this phenomenon triggered our need to meet as a group for further discussion; this group would later become the Chilean Collective of Women Mathematicians.

Our discussion group was structured around weekly meetings in the Council Room of the Mathematics Department in the Science Faculty at Universidad de Chile. Javiera Barrera, M. Isabel Cortez, Leslie Jiménez, Monica Musso, Adriana Piazza, Anita Rojas, Mariel Sáez and Andrea Vera participated in these first meetings. This space was used to share our experiences, as well as the literature that we came across. We think it was very liberating for all of us to realize that we shared the same experiences as many other women, some of which had even been documented. Shortly after we created our discussion group, we received an invitation from USACH students to speak about women in mathematics at an activity organized for the anniversary of the Mathematical Engineering degree. This was a great opportunity to present the issues we discussed every week to a wider audience. Instead of preparing a simple slideshow with biographies of female mathematicians, we decided that it would be much more interesting to shine a light on the situation of women and mathematics in Chile in quantitative terms, with the aim of starting a conversation. The activity was titled "Panel Discussion: Women and Mathematics" (Conversatorio: mujeres y matemáticas), and was held on November 4, 2014, in a nearly full auditorium with capacity for 100 people. The audience included mathematical engineering students (mostly male), some mathematical colleagues, and several social science researchers specializing in gender issues. After our brief presentation, the discussion with the audience lasted almost two hours. Students' stories related to their experience and difficulties of studying in a place where there are mostly men. Some colleagues were uncomfortable with questions about the Math Olympiad that arose during the conversation, while colleagues from the social sciences summarized and organized the discussion within certain theoretical frameworks. What took place on that day made us realize that there was a great need to address certain issues. We agreed that meeting once a week to discuss these matters was not enough. That is when we decided to come together under the umbrella of the Chilean Collective of Women Mathematicians, as a way of making gender issues in mathematics and our future efforts visible. Having the experience of other women's mathematical associations in different countries as a reference—for example,
the Association for Women in Mathematics (AWM)—was crucial to creating the Collective.

It is important to mention that the collective is not the only initiative carried out by  women in mathematics in Chile. Some of these examples are MATEA\footnote{https://matea.cl/hypatia/}  and Colectivo Resistencia Notheriana\footnote{https://fiestadelaciencia.cl/colectivo-resistencia-noetheriana-universidad-de-chile/}.  The first one focus on the organization of summer maths camps for high school girls, and the second one aims to ensure inclusion in the teaching and dissemination of mathematics, from an interdisciplinary perspective. One of the founders of Resistencia Notheriana is also a member of the Collective. Besides mathematics, it is important to highlight the Red de Investigadoras\footnote{https://redinvestigadoras.cl} (Network of Researchers), which is an association that promotes gender equality in research in all areas of knowledge.  A mathematician who is part of the Collective of Women in Mathematics was its first president.

\section*{Organizational characteristics of the Collective.}
\label{sec:3}

Unlike other organizations, the objectives of our Collective were not entirely clear upon its establishment. In fact, the main reason why we decided to band together was the need to be together. We needed to share our experiences with other women mathematicians and confirm that, despite the differences in our lives and academic careers, the sole condition of being women mathematicians in Chile was a shared experience that united all of us. Every one of us had many questions. In time, we realized the need to share them with other women mathematicians, so we could begin to understand them and seek answers. The decision to form an all-women's collective can be explained through Julieta Kirkwood's words: "Only women have the problem of asking themselves what their condition means. Womanhood is otherness (...)" \cite{Kirkwood1987}.

It is important to say that unlike the AWM, this Collective is not -yet- a membership organization, but rather a group that was spontaneously conformed with common interests, and whose number of members varies between 15 and 5 members, depending on the moment and the particular initiative that is being carried out.

We were very clear from the start about the way in which we made our decisions and the structure that we wanted for our organization. We all agreed that our organization was to be horizontal; in other words, that its organizational structure should be flat, without hierarchies, resulting in democratic decision making which means most of the time by consensus and in some other times by simple majority. As stated by Pérez-Arrau, Espejo, Mandiola, Ríos and Toro \cite{Perez-Arrau2020}, "groups lacking allocated hierarchical leadership may develop different forms of collective leadership." Much like these authors, our notion of collective leadership is a relationship that is built among us, one that is nourished by taking advantage of our different paths and experiences, and that gives way to collective decision making. On the other hand, since our respective workloads are different and dynamic, our time availability for the Collective also varies. Therefore, we could also talk about a kind of "occasional leadership," in which depending on the time available and what each individual seeks, members may from time to time take a leading role to organize specific activities. Still, there are certainly many initiatives that have been possible thanks to the commitment and dedication of everyone involved. We will talk about this in detail in the next section.

Although our work takes place within academic institutions, we decided from the onset to be totally independent, both organizationally and financially. That is to say, we do not want to depend on any institutions, including the Mathematical Society of Chile. The purpose of this decision was to remain free to discuss whatever we believe to be important, all the while protecting our capacity for independent thought. However, from time to time, especially when organizing large events, we resort to some financing and contingent sponsorship. For example, to organize the II Meeting of Women in Mathematics in Latin America, held in 2018 in Valdivia, Chile, we received funding from various institutions such as: International Mathematical Union's Committee for Women in Mathematics (CWM), Conicyt Chile, our home universities, and some local research grants in mathematics.

Today we can say that these decisions we made for our organization—horizontality and independence—turned out to make our Collective unique compared to similar organizations across Latin America. The establishment of the Collective, however, has been questioned by some colleagues in the local mathematical community. During our first year, we sensed that few colleagues understood the need for and relevance of an organization devoted to issues about gender and mathematics. We even saw considerable resistance from some colleagues to our organization. Moreover, our decision to be horizontal and independent also brought about challenges and greater efforts to achieve credibility within the local community. This is definitely something we have in common with AWM and its history, sharing the same feelings expressed in \cite{Greenwald2015} when referring to the decision made by AWM to be autonomous from AMS and MAA: "the advantages of forming an independent association outweighed the challenges of doing so."

Another principle that we share with AWM is the desire to foster a cultural change within the mathematics community, so that young people who are new to the field will experience a less hostile environment than those who came before them. We also seek to highlight and put a spotlight on the life and work of women throughout Latin America who have devoted themselves to mathematics. In fact, one of our permanent activities is a photo exhibition of women mathematicians from Latin America and Europe, which we will explore in the following section.

\section*{Joys and challenges: some activities organized by the Collective.}

The Maths and Gender Panel Discussion (Conversatorio Matemáticas y Género) held on November 27, 2015 in Pucón, has been one of the most complex activities we have organized so far, mainly because of the high amount of exposure it meant for us. This discussion was held as part of SOMACHI's annual meeting and it was probably the first time that gender issues in mathematics were addressed at such a massive local scientific event. Participants included Javiera Barrera, M. Isabel Cortez, Adriana Piazza, Anita Rojas and Mariel Sáez, in addition to social science researcher Marcela Mandiola, who we invited as a guest speaker after having met her at the first discussion group at USACH. Organizing this event was far from easy. In fact, a dispute about which names to include on the list of sponsors in the promotional poster even led to a rift within the Collective. Yet the activity carried on in spite of it all, generating enormous expectation among the local mathematical community. On the day of the event, during lunchtime, we realized that the congress participants (mostly men) felt a kind of "moral duty" to attend our discussion, and that not doing so was somehow considered socially reprehensible. And indeed, when the event took place, everyone at SOMACHI was there, including foreigners who could hardly speak any Spanish, which was the language in which the discussion took place. The activity consisted of two lectures: in the first one, M. Isabel Cortez provided a quantitative overview of women mathematicians in Chile. The second lecture was given by Marcela Mandiola, dealing with her research paper on gender in academia \cite{Mandiola19}.  In her presentation she explained how the different practices in academia where hierarchically gendered, where research and teaching appear as masculinized and  feminized respectively. Attendees found these conclusions to be very controversial, triggering a discussion that lasted for hours. Some colleagues felt under attack, others kept making jokes in mathematical language, and some others expected to hear concrete solutions. The conversation ended very late, so we all went to have dinner together. The discussion went on all through the evening and until the end of the congress. For us, the greatest achievement of this activity was to have introduced the topic to our colleagues.

In 2017 Mariel Sáez suggested the possibility of bringing to Chile the travelling photographic exhibition "Women of mathematics throughout Europe: a gallery of portraits," created by mathematician Sylvie Paycha and photographer Noel Matoff and presented for the first time in 2016 at the European Congress of Mathematics. As a Collective we approved of the idea but decided to modify the original exhibition to also include portraits of women mathematicians from Latin America. Thus, the Mathematics Portraits Exhibition was born, whose first version took place between March 8 and 28, 2018, at the GAM Cultural Center in Santiago, Chile. This was a major project that involved many people and received funding from different sources. The first stages were managed by CONICYT's EXPLORA program. The interviews were conducted by Paula Arenas and the photographs were taken by Cristian Translaviña, Nicolás Sanhueza and Dario Cuellar.  We chose to portray diversity in mathematics from multiple perspectives: regional (not only mathematics in the capital), institutional (from different universities), age and also in terms of their main occupation, taking into account that we were also interested in showcasing outstanding women who excel not only in their research, but also in terms of teaching or outreach initiatives. This has been the largest activity we have organized for a more general target audience. The exhibition has continued to travel around Chile and Latin America (Uruguay), currently thanks to the funding provided by a research project involving M. Isabel Cortez, Mariel Sáez and Andrea Vera. Thanks to this exhibition Mariel Sáez edited a book called "Retratos de Matemáticas" (Mathematical Portraits)  \cite{Saez2018}, whose printed version will be published soon.

Organizing and running the Second Meeting of Latin American Women in Mathematics has been undoubtedly our most important event, and probably the one we are most proud of. It all started at the meeting dubbed "Women in Mathematics in Latin America: Barriers, Advancements and New Perspectives," held in August 2016 at the Casa Matemática de Oaxaca (BIRS), Mexico. M. Isabel Cortez attended this meeting and suggested Chile as the venue for the next such encounter. The project was kicked off with an application to the IMU's Committee for Women in Mathematics (CWM) 2017 call for proposals, which we fortunately won. Even though the awarded amount was not enough for the event, it still meant a lot to us because it encouraged us to shape our expectations and venture forth to get more funding, form a scientific committee, start inviting the first speakers, etc. The organizing committee was made up of  Javiera Barrera, M. Isabel Cortez, M. Isabel del Río, Adriana Piazza, Salomé Martínez, Mariel Sáez and Andrea Vera. We decided to hold the meeting in the last week of January 2018, which is summer in the Southern Hemisphere and right before the holiday season in Chile. The place: Valdivia, a beautiful city in the South of Chile, bordering Patagonia \footnote{The most important street in Valdivia is called Ram\'on Picarte, in recognition of the father of the first Chilean mathematician (of the same name) having international recognition. \cite{GutierrezGutierrez16}} . While preparing for this meeting, we once again encountered less-than-supportive reactions from some colleagues who did not see the need or the relevance of holding a meeting on Women in Mathematics. It would seem that questioning certain practices within our community is seen as a threat by some.

It was clear to us that in order to deal with the issue of women in the sciences, it is necessary to draw on knowledge and expertise from other areas such as the social sciences and philosophy. Therefore, the event was roughly divided as follows: 50\% of the time was spent on lectures and posters on mathematics (presented by women), 25\% on lectures in the social sciences and/or philosophy and the rest of the time was devoted to a leadership workshop and a roundtable entitled "Mujeres en ambientes masculinizados” (Women in male-driven environments), which was open to the general public. Also, during the whole week and in parallel to the above-mentioned activities, we set up an exhibition called “Iluminación Matemática” (Mathematical Illumination), organized by Mónica Canales.
The fact that we included the participation of social scientists in this meeting was undoubtedly a great decision. These points of view gave new insights to our discussions, which led us to approach the issue of women in science/mathematics from a structural perspective, allowing us to get a taste of interdisciplinary collaboration and its advantages.

 \section*{Women's mathematical organizations in other Latin American countries.}

Talking about the Chilean Collective of Women Mathematicians also involves our colleagues and allies in other Latin American countries, since to some extent we consider them part of our organization. Indeed,  some of the activities where the Collective has been involved were organized  jointly with these organizations. For example, the panel of Gender and Mathematics of the Mathematical Congress of the Americas, held in  2017 in Montreal, was organized  together with the Mexico's organization described below. The  next conference of Women in Mathematics in Latin America (to be held next year in Colombia), is another example  of international collaboration.

Between 2013 and 2018, several organizations of women mathematicians (and scientists in general) emerged throughout Latin America. Many of these organizations, in addition to the Collective in Chile, constitute what we have named the "Network of Women in Mathematics in Latin America and the Caribbean" (Red de Mujeres Matemáticas en América Latina y El Caribe), which began to take shape during 2016 and was consolidated in 2018 at the International Congress of Mathematicians (ICM) Rio de Janeiro-Brazil.

From a historical point of view, it should be noted that in 2018 the "WM2: World Meeting for Women in Mathematics" took place as a satellite event of the ICM in Rio de Janeiro, Brazil. This probably contributed to the creation of some of the organizations that we will mention in the following paragraphs. In this same line, we also think it is relevant to mention that in May 2018 the so-called "Feminist May" broke out in Chile. It started out as a university protest movement to complain about the lack of protocols to prevent and sanction cases of sexual harassment and abuse in higher education institutions. The movement gained a lot of traction and finally became a national and citizen-wide protest united under the slogan "Non-Sexist Education." Much has been written about this social movement, see for example \cite{TFS19} and \cite{Zeran18}, or \cite{Dessi18} for an article in english.

Next, we provide a review of the women's mathematical organizations in Latin America that we are currently aware of. We particularly refer to their organizational characteristics. To do this, we relied on the report by Araujo-Pardo \& Vera-Gajardo (2019) and updated the information according to conversations we had with colleagues in Peru, Argentina and Brazil.\footnote{Roxana López Cruz (Peru), Gabriela Ovando (Argentina) and Christina Brech (Brazil)}
In Argentina there is the Gender Commission of the Argentine Mathematical Union (UMA), founded in 2018. At first it was made up only of women, but now it is a mixed group of 5 people, including a female coordinator. They also have representatives in different parts of the country. The committee sees itself as a coordinating entity rather than a resolving body. It works as a gender advisor to the UMA and proposes its own specific initiatives.

In Colombia there is the Commission on Equity and Gender of the Colombian Mathematical Society \cite{CEGC2017}, created in 2017. It is currently a mixed organization made up of six members, with a female chairperson. One if its objectives is to encourage participation in mathematics by women and minorities in Colombia.

Mexico’s organization is the oldest in Latin America. It is called the Commission on Equity and Gender of the Mexican Mathematical Society  \cite{CEG2020}, founded in 2013. It is a mixed group of three people, one of whom acts as Coordinator. Its purpose is to promote the inclusion of underrepresented groups, particularly women, in the country's mathematical endeavors.

The case of Brazil is slightly different. Because it is such a large and diverse country, there are different organizations and networks. Some of them include the Gender Commission \cite{CDG2020} created in 2018 as a joint effort between the Brazilian Mathematical Society and the Brazilian Society for Applied and Computational Mathematics. On the other hand, some Brazilian women mathematicians created the platform "Mathematics, a feminine noun" \cite{MSF2020}, which aims to promote actions being developed in different parts of the world, especially in Brazil, in order to increase women's participation in mathematics. We should underscore that these are just some of the many networks that exist in Brazil.

In Peru, the "Group of Peruvian Women in Mathematical Sciences" \cite{MPM2016}  was formed in 2016. This is a network of more than 100 members, which has an organizing committee, a list of representatives in different parts of the country and financially dependent on the Peruvian Society of Applied Mathematics and Computer Science. The main—although not the only—activity performed by this group is to coordinate and organize the Sofia Kovalevskaia Prize in Peru, awarded in many countries by the eponymous Fund.

\section*{The present and the future: paths taken by the Collective’s members.}

In 2020, the members of the Collective continue to carry out some activities together and others on their own, focusing on topics related to mathematics and gender. For example, we continue to regularly display the exhibition "Retratos de Matemáticas" (\textit{Mathematical Portraits}) in different cities and contexts, as well as attending most of the activities in which we are invited to present. We are also in permanent touch with our partners in the "Network of Women in Mathematics in Latin America and the Caribbean."

Javiera Barrera and Adriana Piazza generally participate in a few national activities to spread scientific knowledge, such as the "Feria Ingeniosas." Similarly, the Collective is one of the organizations involved in the global coordination of the "12 de Mayo Celebration of Women in Mathematics initiative honoring Maryam Mirzakhani on her birthday.", represented by Andrea Vera. Javiera Barrera is also part of the Equity and Gender Committee of the Faculty of Engineering and Sciences at her home institution.

Since 2018, M. Isabel Cortez, Mariel Sáez and Andrea Vera have been part of an interdisciplinary research team in the "Anillo Matemáticas y Género" (Mathematics and Gender “Anillo” Grant)  \cite{Anillo2020}. The team is also integrated by Jeanne Hersant, Marcela Mandiola and Tania de Armas, who are all social science researchers. This is a three-year project funded by the Chilean National Agency for Research and Development. The main objective of this research is to understand and analyze the situation of women in mathematics in Chile from a gender perspective, told through the stories and career paths of those who are part of it. To achieve this, we have separated it into three specific goals giving rise to four stages of research. These goals are as follows:
\begin{enumerate}
\item To gain awareness of the scientific field of mathematics in Chile from a gender perspective, as well as its subsequent analysis.
\item To understand and analyze the subjectivities of female mathematics scholars from their interpretations
and self-interpretations regarding the academic field of mathematics, as well as in relation to their career paths.
\item To identify the common career pattern of women researchers in mathematics, that is, the different stages they must go through to reach the status of researcher; in addition, to analyze the barriers and difficulties that women researchers in mathematics must overcome to become researchers, in relation to internal hierarchies and prestige ladders.
\end{enumerate}

Regarding the future, in these months (November, December 2020) we are preparing a workshop that will have the objective of re-founding the Collective. Specifically, in January 2021 we will hold an open day to women in mathematics in Chile, who wish to be part of the Collective. Additionally, one of our plans is to create, within the Collective, a chapter for students.

 On the other hand, together with AWM and some Mathematician colleagues from Latin America, we are organizing a virtual bilingual (English-Spanish) math outreach program.

Undoubtedly,  since the formation of the Collective, our careers have been strongly influenced by issues that touch upon gender and mathematics. Without a doubt, one of the lessons learned from belonging to a Collective of Women Mathematicians is the certainty that any change we want to promote requires coordination with others; that is, collaboration.

\section*{Acknowledgement}   We would like to thank Hernán Henríquez, Rafael Labarca and Irene Mikenberg for providing us valuable information regarding the history of mathematics in Chile. We would also like to thank FONDECYT for providing us the complete database concerning FONDECYT grants in mathematics.


TITLE: Compactness, Hausdorffness and a Descending Sequence of Closed Sets
QUESTION [1 upvotes]: Still studying topology and recently stumbled on Exercise 41/Chapter 11  in Royden and Fitzpatrick's Real Analysis (4th edition). Thing is, I feel my proof is incorrect because it doesn't use the fact that $X$ is Hausdorff. I'll enunciate propositions and definitions at the end of the proof.
Let $(X, \mathcal{T})$ be a compact, Hausdorff topological space and $\{F_n\}_{n = 1}^\infty$ be a descending collection of closed subsets of $X$. Let $O$ be a neighborhood of the intersection $\cap_{n = 1}^\infty F_n$. Show there is an index $N$ such that $F_n \subseteq O$ for $n \geq N$.
We start with the fact that $O$ is a neighborhood of the intersection, that is, $\cap_{n = 1}^\infty F_n \subseteq O$, where $O \in \mathcal{T}$. This implies that:
\begin{equation}
(X - O) \subseteq (X -  \cap_{n = 1}^\infty F_n) = \cup_{n = 1}^\infty (X - F_n) \tag{1}
\end{equation}
where $X - O$ is closed, as the complement of an open set, and $\{(X - F_n)\}_{n = 1}^\infty$ an open cover of $X - O$. By proposition 1, $X - O$ is compact. I'm assuming my mistake lies in the next step. Using Claim 1 there exists a finite subcover $\{(X - F_{n_k})\}_{k = 1}^m \;$ of $X - O$, i.e.:
\begin{equation}
(X - O) \subseteq \cup_{k = 1}^m(X - F_{n_k}) \tag{2}
\end{equation}
let $t = max\{n_1, ..., n_m\}$. We argue that
\begin{equation}
\cup_{i  =  t + 1}^\infty \, F_{i} \cap (X - O) = \varnothing  \tag{3}
\end{equation}
By contradiction, assume there is $w \in \cup_{i  =  t + 1}^\infty \, F_{i} \cap (X - O) \neq \varnothing$. Since $w \in (X - O)$, Because of $(2)$, there is $ k\in \{1, ..., m\}$ s.t. $w \in (X - F_{n_k})$. Also, there is $i \in \mathbb{N} - \{1, ..., t\}$ s.t. $ w \in F_i$. Because the sequence is descending and $i > n_k$ it has to be the case that $F_i \subseteq F_{n_k}$, meaning $w \in F_{n_k}$, a contradiction. Hence $(3)$ holds.
Finally, by $(3)$, $\cup_{i  =  t + 1}^\infty \, F_{i} \subseteq O$. Now, if we define $N = t + 1$ then for arbitrary $n \geq N$, $F_n \subseteq \cup_{i  =  t + 1}^\infty \, F_{i} \subseteq O$, proving the exercise.

Definition$_1$: A topological space $(X, \mathcal{T})$ is said to be compact provided every open conver of $X$ has a finite subcover. A subset $K$ of $X$ is called compact provided $K$, considere as a topological space $(K, \mathcal{S})$, where $\mathcal{S}$ is the inherited topology from $X$, is compact.
Definition$_2$: For a subset $K$ of a topological space $(X, \mathcal{T})$, we call a subset $A$ of $X$ a neighborhood of $K$ if $A$ is open and $K \subseteq A$.
Proposition 1: A closed subset $K$ of a compact topological space $(X, \mathcal{T})$ is compact. (Page 234 - Royden's Real Analysis)
Claim 1: If $K$ is a compact subset of the topological space $(X, \mathcal{T})$, then every convering of $K$ by a collection of open subsets of $X$ has a finite subcover.

REPLY [1 votes]: It is indeed the case that $X$ need not be Hausdorf at all, and your proof shows this correctly; I'll recap it in shorter form:
If $\bigcap_n F_n \subseteq O$, where $O$ is open, then $\{X-F_n \mid n \in \Bbb N \} \cup \{O\}$ is an open cover of $X$ (if $p$ is in no $X-F_n$, it's in all $F_n$, so in $O$) and there is a finite subcover $\{X-F_{n_1}, X-F_{n_2},\ldots, X-F_{n_k}\} \cup \{O\}$ where $n_1 < n_2 < \ldots n_k$, say, and then $F_{n_k} \subseteq O$ by the nestedness (if $p \in F_{n_k}$, then $p \notin X-F_{n_i}$ for $i \le k$ and so must be in $O$).


TITLE: Is the zero element and zero vector the same thing in the context of Linear Independence?
QUESTION [0 upvotes]: Say..
A finite set of elements from a linear space are linearly independent if:

Their trivial linear combination is equal to the zero element
Their trivial linear combination is equal to the zero vector

I am not sure if statements 1 and 2 say the same thing.
I know that the zero element has the properties; for elements x and y of a linear space :
x + zero element = x
x + y = zero element
And I believe a zero vector would be an n tuple with only zero elements: (0,0,...,0)
Right now I believe they are the same thing in terms of properties that they have when talking about linear spaces and I think it is just different terminology.
But I also know from an example of a field with 2 elements (even and odd) that, even is the zero element in the field and it is not an n tuple with only zero elements.
As even + even = even and even + odd = odd

REPLY [2 votes]: Both your definitions are wrong. It should be, if the set is $\{v_1,\ldots,v_k\}$,

$$\sum_{i=1}^k \lambda_iv_k=0 \;\Rightarrow\;\text{all $\lambda_i=0$.}$$

You are correct that the zero vector in $\mathbb{R}^n$ has exactly $n$ entries that are equal to the real number $0$.
Finally, the terms zero element and zero vector are often used interchangeably in this context.

REPLY [1 votes]: The most conscise definition of vector space is:

an abelian group on which a field acts

The zero vector of such a vector space is nothing else but the zero (or neutral) element of said abelian group.
Specifically, in the vector space $F^n$ of $n$-tuples of elements of the field $F$, the zero vector is also the tuple with all entries zero, $(0,0,\ldots,0)$.
Noe however, that we always also have the ground field in the context of vector spaces. The additive group of that filed of course also has a zero element, which (unless we consider the field as vector space over itself) is not referred to as zero vector.


TITLE: Prove area of a quadrilateral is $\frac14[4m^2n^2-(b^2+d^2-a^2-c^2)^2]^{\frac12}$
QUESTION [2 upvotes]: Someone asked me this question which I am really stuck at, any help is appreciated.

If $a,b,c,d$ are the sides of a  quadrilateral and $m,n$ are diagonals of the quadrilateral, then prove that area of the quadrilateral is $$\frac14[4m^2n^2-(b^2+d^2-a^2-c^2)^2]^{\frac12}$$

I do not recall anything like this from my school days. I need some hints to get started on this, I do not remember high school geometry formulas, may be one of them could help here.
It works fine for squares and recatngles, so I assume it is a valid formula.

REPLY [4 votes]: For convex $\square ABCD$, we can proceed as follows:

Let the diagonals meet a $P$, which subdivides the diagonals into segments of length $m_1$, $m_2$, $n_2$, $n_2$; and let the diagonals make an angle of $\theta$ at $P$ as shown. Then we can write:
$$\begin{align}
|\square ABCD| &= |\triangle APB| + |\triangle BPC| + |\triangle CPD| + |\triangle DPA| \\[4pt]
&= \frac12 \sin\theta \;\left( m_1 n_1 + m_2 n_1 + m_2 n_2 + m_1 n_2 \right) = \frac12 (m_1 + m_2)(n_1+n_2) \sin\theta \\[4pt]
&= \frac12 m n \sin\theta \\[4pt]
\therefore 4\;|\square ABCD| &= 2 m n \sin\theta \tag{$\star$}
\end{align}$$
Applying the Law of Cosines in each triangle gives:
$$\begin{align}
a^2 &= m_1^2 + n_1^2 - 2 m_1 n_1 \cos\theta \qquad\qquad
b^2 = m_2^2 + n_1^2 + 2 m_2 n_1 \cos\theta \\
c^2 &= m_2^2 + n_2^2 - 2 m_2 n_2 \cos\theta \qquad\qquad
d^2 = m_1^2 + n_2^2 + 2 m_1 n_2 \cos\theta
\end{align}$$
so that
$$a^2 - b^2 + c^2 - d^2 = -2 m n \cos \theta$$
Finally, recalling $(\star)$,
$$\begin{align}
4 m^2 n^2 = 4 m^2 n^2\left( \sin^2\theta + \cos^2 \theta \right) = 16\;|\square ABCD|^2 + \left( a^2 - b^2 + c^2 - d^2 \right)^2
\end{align}$$
which gives the desired formula. $\square$


text:
\begin{document}
\maketitle

\begin{abstract}
We establish a Schn$\ddot{\text{u}}$rer's convergence result and then apply it to obtain the existence of  solutions on the second boundary value problem for a family of special Lagrangian equations.
\end{abstract}

\let\thefootnote\relax\footnote{
2010 \textit{Mathematics Subject Classification}. Primary 53C44; Secondary 53A10.

\textit{Keywords and phrases}. parabolic type special Lagrangian equation; special Lagrangian  diffeomorphism; special Lagrangian  graph.}
\section{Introduction}
Let $\Omega$ be a bounded domain with smooth boundary in $\mathbb{R}^{n}$ and $\mathcal{S}$ be the open connected subset of $\mathbb{S}_{n}$ where
$$  \mathbb{S}_{n}=\{n\times n\,\, \text{real\,\,symmetric\,\,matrix}\}.$$
Given $u_{0}: \Omega\rightarrow\mathbb{R}$,
we consider nonlinear parabolic equation with second boundary condition:
\begin{equation}\label{e1.1}
\left\{ \begin{aligned}u_{t}-F(D^{2}u,Du,x)&=0,
&  x\in \Omega,\,\, t>0, \\
h(Du,x)&=0,& x\in \partial\Omega,\,\, t>0,\\
u&=u_{0},& x\in \Omega,\,\, t=0,
\end{aligned} \right.
\end{equation}
where $F: \mathcal{S}\times \mathbb{R}^{n}\times\Omega\rightarrow \mathbb{R}$ is $C^{2+\alpha}$ for some $0<\alpha<1$ and
satisfies
\begin{equation}\label{e1.2}
A<B\Rightarrow F(A,p,x)<F(B,p,x).
\end{equation}
In other words, (\ref{e1.2}) means that $F$ is non-decreasing with respect to the matrix variable and we say that
$F$ is strictly parabolic. Here $h:\mathbb{R}^{n}\times\partial\Omega\rightarrow \mathbb{R}$ is $C^{1}$ and satisfies
\begin{equation}\label{e1.3}
\sum_{k=1}^{n}h_{p_{k}}(\cdot,x)\nu_{k}>0,
\end{equation}
 where $\nu=(\nu_{1},\nu_{2}, \cdots,\nu_{n})$ is the unit inward normal vector of $\partial\Omega$.
 (\ref{e1.3}) shows that $h(Du,x)=0$ is  oblique.

Throughout the following,  Einstein's convention of
 summation over repeated indices will be adopted.
Denote $$u_{i}=\dfrac{\partial u}{\partial x_{i}},
u_{ij}=\dfrac{\partial^{2}u}{\partial x_{i}\partial x_{j}},
u_{ijk}=\dfrac{\partial^{3}u}{\partial x_{i}\partial x_{j}\partial
x_{k}}, \cdots $$ and
$$[u^{ij}]=[ u_{ij}]^{-1},\,\,\,
F^{ij}(D^{2}u)=\frac{\partial F}{\partial u_{ij}},\,\,\,h_{p_{k}}(Du)=\frac{\partial h}{\partial u_{k}}, \,\,\, \Omega_{T}=\Omega\times(0,T). $$

In this paper, we are concerned with the convergence of nonlinear parabolic equation \eqref{e1.1} to translating solution under some a-priori estimates as $t\rightarrow \infty$.
The translating solutions are intimately related to the solutions of minimal Lagrangian diffeomorphism problem. In general,
evolution equations often have special solutions called solitons which keep their shape during the evolution. For examples, two very important classes of solitons
in mean curvature flow are self-shrinker and translating solutions which evolve by a homothety or a translation respectively.  Translating solutions are interesting examples
of the evolution equations since they are precise solutions in the sense that their evolution is known, which is very hard to determine
in general.
Our main result concerning the asymptotic behavior of nonlinear parabolic equation \eqref{e1.1}
under certain assumptions on a-priori estimates can be summarized as follows.

\begin{theorem}\label{t1.1}
For any $T>0$, we assume that  $u\in C^{4+\alpha,\frac{4+\alpha}{2}}(\bar{\Omega}_{T})$
be a unique solution of the nonlinear parabolic equation (\ref{e1.1}) which satisfy
\begin{equation}\label{e1.4}
\|u_{t}(\cdot,t)\|_{C(\bar{\Omega})}+\|Du(\cdot,t)\|_{C(\bar{\Omega})}+\|D^{2}u(\cdot,t)\|_{C(\bar{\Omega})}\leq C_{1},
\end{equation}
\begin{equation}\label{e1.40}
\|D^{2}u(\cdot,t)\|_{C^{\alpha}(\bar{D})}\leq C_{2},\quad \forall D\subset\subset\Omega,
\end{equation}
\begin{equation}\label{e1.5}
\sum_{k=1}^{n}h_{p_{k}}(Du(\cdot,t),x)\nu_{k}\geq \frac{1}{C_{3}},
\end{equation}
where the positive constants $C_{1}$, $C_{2}$ and $C_{3}$  are independent of ~ $t\geq 1$.
Then  $u(\cdot,t)$ converges to a function $u^{\infty}(x,t)=\tilde{u}^\infty(x)+C_{\infty}\cdot t$
in $C^{1+\zeta}(\bar{\Omega})\cap C^{4+\alpha'}(\bar{D})$ as $t\rightarrow\infty$
  for any $D\subset\subset\Omega$, $\zeta<1$ and $\alpha'<\alpha$, that is
 $$\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\qquad
  \lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{4+\alpha'}(\bar{D})}=0.$$
And $\tilde{u}^{\infty}(x)\in C^{2}(\bar{\Omega})$ is a solution of
\begin{equation}\label{e1.6}
\left\{ \begin{aligned}F(D^{2}u,Du,x)&=C_{\infty},
&  x\in \Omega, \\
h(Du,x)&=0, &x\in\partial\Omega.
\end{aligned} \right.
\end{equation}
The constant $C_{\infty}$ depends only on $\Omega$ and $F$. The solution to (\ref{e1.6}) is unique up to additions of constants.
\end{theorem}
\begin{rem}\label{r1.1}
By Evans-Krylov theorem for parabolic equations \cite{W},
we can replace (\ref{e1.40}) by  $F(A,p,x)$ being concave with respect to the variable $A$.
\end{rem}

We first recall some results concerning the convergence of solutions to translating solutions in history.
Let $\Omega$ be a smooth strictly convex domain in $\mathbb{R}^{n}$. Schn$\ddot{\text{u}}$rer \cite{OC} studied a class of curvature flow in $\mathbb{R}^{n+1}$:
\begin{equation}\label{e1.7}
\left\{ \begin{aligned}\dot{X}&=-(\ln F-\ln f)\nu, \\
\nu(M)&=\nu(M_{0}),\\
M\mid_{t=0}&=M_{0},
\end{aligned} \right.
\end{equation}
where $X$ denote the embedding vector of a smooth strictly convex hypersurface with boundary, $M=\text{graph}\, u\mid_{\Omega},$
$u: \bar{\Omega}\rightarrow \mathbb{R}$, and $\dot{X}$ denote the total time derivative. By a given smooth positive function
$f:\bar{\Omega}\rightarrow \mathbb{R}$ and a curvature function $F$, Schn$\ddot{\text{u}}$rer
 transformed  the curvature flow (\ref{e1.7}) into some sort of (\ref{e1.1})
and then obtained the estimates (\ref{e1.4}), (\ref{e1.5}). Finally he proved that the initial value problem (\ref{e1.7})
admits a convex solution $M(t)=\text{graph}\, u(t)\mid_{\Omega}$ that exists for all times $t\geq 0$ and converges smoothly
to a translating solution  $M^{\infty}=\text{graph}\, u^{\infty}\mid_{\Omega}$ of the flow (\ref{e1.7}), that is, there exists
$v^{\infty}\in\mathbb{R}$ such that $$u^{\infty}(x,t)= u^{\infty}(x,0)+v^{\infty}\cdot t.$$

A similar convergence result for graphic mean curvature flow with Neumann boundary condition in arbitrary dimension
was studied by a recent work of Ma-Wang-Wei \cite{MWW}.
They studied nonparametric surfaces over strictly convex bounded domains in $\mathbb{R}^n$
which are evolving by the mean curvature flow with Neumann boundary value
\begin{equation*}
\left\{ \begin{aligned}u_t &= (\delta_{ij}-\frac{u_i u_j}{1+|Du|^2})u_{ij} \quad \text{in} \qquad \Omega\times (0,\infty), \\
u_{\nu} & = \varphi(x) \quad \text{on} \qquad \partial \Omega\times (0,\infty),\\
 u (x,0)& = u_{0}(x),
\end{aligned} \right.
\end{equation*}
where $\varphi(x)$ and $u_0(x)$ are smooth functions satisfying $u_{0,\nu}=\varphi(x)$ on $\partial \Omega$.
They proved a convergence theorem that the solution of the above mean curvature flow with Neumann boundary value converges
to a translating solution which moves at a constant speed up to a translation $\lambda t+ w $ in arbitrary dimensions,
where $(\lambda, w)$ are suitable solution to
\begin{equation*}
\left\{ \begin{aligned}  (\delta_{ij}-\frac{u_i u_j}{1+|Du|^2})u_{ij} &= \lambda \quad \text{in} \qquad \Omega, \\
u_{\nu} & = \varphi(x) \quad \text{on} \qquad \partial \Omega.
\end{aligned} \right.
\end{equation*}

Our work was also inspired by the reading of the  papers of  Altschuler-Wu \cite{SL}, and Kitagawa \cite{JK}
where the translating solutions were obtained.
In this paper we show Schn$\ddot{\text{u}}$rer's convergence result to more general case
and discuss the application of this convergence result
in the study of minimal Lagrangian diffeomorphism problem. For the proof of Theorem \ref{t1.1},
we  borrow the ideas from \cite{OC}, \cite{JK} and \cite{SL}.

The rest part of this paper is organized as follows. We give a simple case which is the convergence of the solution
to one-dimensional heat equation to illustrate the convergence in Theorem \ref{t1.1}.
We collect several preliminary results in section 3.  In section 4, we give the proof of Theorem \ref{t1.1}.
In section 5, we exhibit the applications of  Theorem \ref{t1.1} to the study of  a family of special Lagrangian graphs.

\section{A simple case}
In this section, we first give a simple case to illustrate the convergence result in Theorem \ref{t1.1}.
We consider the convergence of the solutions to the following heat equation on the interval $[0,1]$.
 \begin{equation}\label{heat}
\left\{ \begin{aligned}u_t &= u_{xx} \quad \text{for}\quad x \in [0,1],\quad t>0, \\
u_x|_{x=0} & = a ,  \quad  u_x|_{x=1} = b  ,\\
 u (x,0)& = u_{0}(x),
\end{aligned} \right.
\end{equation}
where $a$ and $b$ are two constants and $u_0(x)$ is a smooth function on the interval $[0,1]$. In physical meaning,
the model characterizes the heat conduction problem of a fine iron rod in the interval [0,1]
with heat exchanges on the two interval endpoints $0$ and $1$.
By principle of superposition in classical linear PDE theory,
we know that the solution of equation \eqref{heat} can be decomposed into three parts
\begin{equation*}
u(x,t)=V(x)+Z(t)+w(x,t),
\end{equation*}
 where $V(x)$ , $Z(t)$ and  $w(x,t)$ are solutions of the following equations respectively:
 \begin{equation}\label{heat1}
\left\{ \begin{aligned}V_{xx} &= b-a \quad \text{for}\quad x\in [0,1], \\
V_x|_{x=0} & = a ,  \quad  V_x|_{x=1} = b  ,
\end{aligned} \right.
\end{equation}

 \begin{equation}\label{heat2}
\left\{ \begin{aligned}Z_{t} &= b-a \quad \text{for}\quad t>0, \\
& Z|_{t=0} = 0,
\end{aligned} \right.
\end{equation}
and
 \begin{equation}\label{heat3}
\left\{ \begin{aligned}w_t &= w_{xx} \quad \text{for}\quad x\in [0,1],\quad t>0, \\
w_x|_{x=0} & = 0 ,  \quad  w_x|_{x=1} = 0  ,\\
 w (x,0)& = u_{0}(x)-(\frac{b-a}{2}x^2+ax).
\end{aligned} \right.
\end{equation}
By solving the equations \eqref{heat1} and \eqref{heat2}, we obtain that
\begin{equation}\label{sol1}
V(x)=\frac{b-a}{2}x^2+ax
\end{equation}
 and
 \begin{equation}\label{sol2}
 Z(t)=(b-a)t.
\end{equation}
Then it is left to solve the equation \eqref{heat3}. To that end,
we use the methods of separation of variables. We suppose the equation \eqref{heat3} has
the solutions in the following form
\begin{equation}\label{sep}
w(x,t)=X(x)T(t).
\end{equation}
Substituting \eqref{sep} into equation \eqref{heat3}, we obtain:
 \begin{equation}\label{heat4}
\frac{X''(x)}{X(x)}=\frac{T'(t)}{T(t)}=-\lambda,
\end{equation}
where $\lambda$ is a constant which will be determined later.
From equation \eqref{heat4}, we need to solve
\begin{equation}\label{heat5}
\left\{ \begin{aligned} & X''(x)+\lambda X(x) = 0 ,\quad \text{for}\quad x\in [0,1], \\
&X_x|_{x=0}= 0 ,  \quad  X_x|_{x=1} = 0  ,
\end{aligned} \right.
\end{equation}
and
\begin{equation}\label{heat6}
T'(t)+\lambda T(t) = 0 ,\quad \text{for}\quad t>0.
\end{equation}

For equation \eqref{heat5}, the eigenvalues $\lambda_n$ and corresponding eigenfunctions are $n^2\pi^2$ and $X_n(x)=C_n cos n\pi x$
for $n=0,1,2, \cdot\cdot\cdot$. For $\lambda_n$, the solutions of equation \eqref{heat6} are
 $T_n(t) = D_n e^{-n^2\pi^2 t}$ for $n=0,1,2, \cdot\cdot\cdot$. From the method of separation of variables, we know that the solution of
 equation \eqref{heat3} has the form of Fourier series:
 \begin{equation}\label{sol3}
 w(x,t)=\sum_{n=0}^{\infty} C_n e^{-n^2\pi^2 t}cos n\pi x.
 \end{equation}
 Using the orthogonality of the eigenfunctions and initial value of equation \eqref{heat3},
 we know $w(x,0)=u_0(x)-(\frac{b-a}{2}x^2+ax)$ and hence we can
 determine the coefficient constants as follows:
 \begin{equation*}
 C_n = \int_{0}^1 [u_0(x)-(\frac{b-a}{2}x^2+ax)]cos n\pi x dx,
 \end{equation*}
 for $n=0,1,2, \cdot\cdot\cdot$.
 Combining \eqref{sol1}, \eqref{sol2} and \eqref{sol3}, we know the solution to the equation \eqref{heat} is
 \begin{equation*}
 u(x,t)= (b-a)t+\frac{b-a}{2}x^2+ax + \sum_{n=0}^{\infty} C_n e^{-n^2\pi^2 t}cos n\pi x.
 \end{equation*}
 If we denote the speed constant $C_{\infty}=b-a$, Then the solution can be reformulated as
 \begin{equation*}
 u(x,t)-C_{\infty} t = \frac{b-a}{2}x^2+ax + \sum_{n=0}^{\infty} C_n e^{-n^2\pi^2 t}cos n\pi x.
 \end{equation*}
 By using the exponential decay of the exponential functions,
 we deduce that the function $u(x,t)-C_{\infty} t$
 is asymptotically approaching $\tilde{u}(x) \triangleq V(x) = \frac{b-a}{2}x^2+ax  $ up to a constant $C_0$
 and the limit function $\tilde{u}(x) $ satisfies
  \begin{equation*}
\left\{ \begin{aligned}V_{xx} &= b-a \triangleq C_{\infty} \quad \text{for}\quad x\in [0,1], \\
V_x|_{x=0} & = a ,  \quad  V_x|_{x=1} = b.
\end{aligned} \right.
\end{equation*}
Therefore we have completed the proof that the solution of the heat equation \eqref{heat} $u(x,t)$ converges to a function $u^{\infty}(x,t)=\tilde{u}^\infty(x)+C_{\infty}\cdot t$
as $t\rightarrow\infty$.

\section{Preliminary results}
In this section, we collect several preliminary results which will be used to prove
Theorem \ref{t1.1} in the next section.

\begin{lemma}\label{l2.1}
For any $T>0$, we assume that $a^{ij}\in C(\bar{\Omega}_{T})$, $b^{i}\in C(\bar{\Omega}_{T})$ and $[a^{ij}]|_{\bar{\Omega}_{T}}>0$.
If $w\in C^{2,1}(\bar{\Omega}_{T})$ satisfies
\begin{equation}\label{e2.1}
\left\{ \begin{aligned}w_{t}-a^{ij}w_{ij}-b^{i}w_{i}&=0,
&  x\in \Omega,\,\, t>0, \\
\beta^{k}w_{k}&=0,& x\in \partial\Omega,\,\, t>0,\\
\end{aligned} \right.
\end{equation}
where $\beta=(\beta^{1},\beta^{2},\cdots,\beta^{n})$ is a uniformly strictly oblique vector field,
that is, $\beta^{k}\nu_{k}\geq \frac{1}{C}$. Then
\begin{equation*}
osc_{w}(t)\triangleq \max_{x\in \bar{\Omega}}w(x,t)-\min_{x\in \bar{\Omega}}w(x,t)
\end{equation*}
is a strictly decreasing function or $w$ is a constant
\end{lemma}
\begin{proof}
Suppose $w$ is not a constant, then we can prove that $\min_{x \in \Omega}w(x,t)$ is a strictly increasing function and
$\max_{x \in \Omega}w(x,t)$ is a strictly decreasing function with respect to $t$.
In fact, since $w(x,t)$ is a solution of the parabolic equation \eqref{e2.1},
then for any $t>t_0$, by using the strong maximum principles \cite{W}, we obtain
\begin{equation*}
w(x,t)>\min_{(x,t)\in \partial \Omega\times[t_0,T)\cup \Omega\times \{t_0\}}w(x,t),\,\, \forall x\in \Omega,\,\, t_0<t<T.
\end{equation*}
By the boundary condition of equation \eqref{e2.1} and the Hopf Lemma, we know that $w(x,t)$
can not attain its minimum on the side boundary $\partial \Omega\times [t_0,T)$, then we deduce
 \begin{equation*}
w(x,t)>\min_{x\in \Omega}w(x,t_0),\,\, \forall x\in \Omega,\,\, t>t_0.
\end{equation*}
Taking the minimum of $x\in \Omega$, we deduce that
\begin{equation}\label{e2.1a}
\min_{x\in \Omega} w(x,t)>\min_{x\in \Omega}w(x,t_0),\,\, \forall t>t_0.
\end{equation}
Similarly, we have
\begin{equation}\label{e2.1b}
\max_{x\in \Omega} w(x,t)<\max_{x\in \Omega}w(x,t_0),\,\, \forall t>t_0.
\end{equation}
Combining \eqref{e2.1a} and \eqref{e2.1b}, we deduce that for $t>t_0$,
\begin{equation*}
osc_{w}(t)\triangleq \max_{x\in \bar{\Omega}}w(x,t)-\min_{x\in \bar{\Omega}}w(x,t)
<\max_{x\in \bar{\Omega}}w(x,t_0)-\min_{x\in \bar{\Omega}}w(x,t_0)=osc_{w}(t_0),
\end{equation*}
which shows that $osc_{w}(t)$ is a strictly decreasing function.
\end{proof}

\begin{lemma}\label{l2.2}
For any sequence $\{t_{n}\} \,\, (t_{n}\rightarrow +\infty)$ and for $x_{0}\in \bar{\Omega}$ and $t_{0}>0$  fixed, there exists a subsequence of $\{t_{n}\}$ (again denoted by itself), such that
\begin{equation*}
u(x,t+t_{n})-u(x_{0},t_{n}) \quad \text{and}\quad u(x,t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})
\end{equation*}
 converge to two functions which we denote by $u^{\infty}$ and $u^{t_{0},\infty}$ respectively  in the following sense,
\begin{equation}\label{e2.2}
\begin{aligned}
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{n})-u(x_{0},t_{n})-u^{\infty}(x,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\\
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})-u^{t_{0},\infty}(x,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\\
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{n})-u(x_{0},t_{n})-u^{\infty}(x,t)\|_{C^{2}(\bar{D})}=0,\\
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})-u^{t_{0},\infty}(x,t)\|_{C^{2}(\bar{D})}=0,\\
 \end{aligned}
\end{equation}
where $D\subset\subset\Omega$, $0<\zeta<1$.
\end{lemma}
\begin{proof}
By making use of the intermediate value theorem, we obtain
\begin{equation*}
\begin{aligned}
&|u(x,t+t_{n})-u(x_{0},t_{n})|\\
=&|u(x,t+t_{n})-u(x_{0},t+t_{n})+u(x_{0},t+t_{n})-u(x_{0},t_{n})|\\
\leq &|u(x,t+t_{n})-u(x_{0},t+t_{n})|+|u(x_{0},t+t_{n})-u(x_{0},t_{n})| \\
\leq &diam(\Omega)\|Du\|_{C(\bar{\Omega})}+t\|u_{t}(\cdot,t)\|_{C(\bar{\Omega})},
 \end{aligned}
\end{equation*}
\begin{equation*}
\begin{aligned}
&|u(x,t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})|\\
=&|u(x,t+t_{0}+t_{n})-u(x_{0},t+t_{0}+t_{n})+u(x_{0},t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})|\\
\leq &|u(x,t+t_{0}+t_{n})-u(x_{0},t+t_{0}+t_{n})|+|u(x_{0},t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})| \\
\leq &diam(\Omega)\|Du\|_{C(\bar{\Omega})}+t\|u_{t}(\cdot,t)\|_{C(\bar{\Omega})}.
 \end{aligned}
\end{equation*}
Combining the above two inequality with (\ref{e1.4}), we have
\begin{equation}\label{e2.3}
\begin{aligned}
\|u(\cdot,t+t_{n})-u(x_{0},t_{n})\|_{C^{2}(\bar{\Omega})}\leq C,\\
\|u(\cdot,t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})\|_{C^{2}(\bar{\Omega})}\leq C,
 \end{aligned}
\end{equation}
where the constant $C$ is independent of $t_{n}$.

By $Arzel\grave{a}-Ascoli$ theorem, there exist functions $u^{\infty}$ and $u^{t_{0},\infty}$, such that
\begin{equation}\label{e2.4}
\begin{aligned}
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{n})-u(x_{0},t_{n})-u^{\infty}(x,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\\
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})-u^{t_{0},\infty}(x,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\\
 \end{aligned}
\end{equation}
where  $0<\zeta<1$.

By intermediate Schauder estimates for parabolic
equations (cf. Lemma 14.6  and Proposition 4.25 in \cite{GM}) and (\ref{e1.4}), for any $D\subset\subset \Omega$, we have
\begin{equation}\label{e2.40}
\begin{aligned}
\sup_{t\geq 1}\|D^{3}u(\cdot,t)\|_{C(\bar{D})}\leq C,
\end{aligned}
\end{equation}
where $C$ is the constant depending only on  the known data and dist$(\partial \Omega, \partial D)$. Using (\ref{e2.3}),
we obtain
\begin{equation*}
\begin{aligned}
\|u(\cdot,t+t_{n})-u(x_{0},t_{n})\|_{C^{3}(\bar{D})}\leq C,\\
\|u(\cdot,t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})\|_{C^{3}(\bar{D})}\leq C,
 \end{aligned}
\end{equation*}
where the constant $C$ is independent of $t_{n}$. By $Arzel\grave{a}-Ascoli$ theorem again, we get
\begin{equation}\label{e2.5}
\begin{aligned}
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{n})-u(x_{0},t_{n})-u^{\infty}(x,t)\|_{C^{2}(\bar{D})}=0,\\
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n})-u^{t_{0},\infty}(x,t)\|_{C^{2}(\bar{D})}=0.\\
 \end{aligned}
\end{equation}
Putting (\ref{e2.4}) and (\ref{e2.5}) together, we obtain the desired results.
\end{proof}
Define
\begin{equation*}
osc_{f}(t)\triangleq \max_{x\in \bar{\Omega}}f(x,t)-\min_{x\in \bar{\Omega}}f(x,t),
\end{equation*}
for any $f\in C(\bar{\Omega}\times [0,+\infty))$.  We fix the positive constant $t_0$ and write
$$w(x,t)=u(x,t)-u(x,t+t_0).$$
Then we have
\begin{equation}\label{e2.6}
\begin{aligned}
\dot{w}&=F(D^2u(x,t), Du(x,t),x)-F(D^2u(x,t+t_{0}), Du(x,t+t_{0}),x)\\
&=\int_0^1\frac{d}{ds}F(sD^2u(x,t)+(1-s)D^2u(x,t+t_{0}),sDu(x,t)+(1-s)Du(x,t+t_{0}),x)ds\\
&=\big(\int_0^1 \nabla_{r_{ij}}F(sD^2u(x,t)+(1-s)D^2u(x,t+t_{0}),sDu(x,t)+(1-s)Du(x,t+t_{0}),x)ds\big)w_{ij}\\
&+\big(\int_0^1 \nabla_{p_{i}}F(sD^2u(x,t)+(1-s)D^2u(x,t+t_{0}),sDu(x,t)+(1-s)Du(x,t+t_{0}),x)ds\big)w_{i}\\
&=a^{ij}w_{ij}+b^{i}w_{i},\qquad x\in\Omega, \quad t>0,
\end{aligned}
\end{equation}
where we denote
\begin{equation*}
\begin{aligned}
a^{ij}&=\int_0^1 \nabla_{r_{ij}}F(sD^2u(x,t)+(1-s)D^2u(x,t+t_{0}),sDu(x,t)+(1-s)Du(x,t+t_{0}),x)ds,\\
b^{i}&= \int_0^1 \nabla_{p_{i}}F(sD^2u(x,t)+(1-s)D^2u(x,t+t_{0}),sDu(x,t)+(1-s)Du(x,t+t_{0}),x)ds.
\end{aligned}
\end{equation*}
Calculating in a similar way, we know $w$ satisfies the following boundary condition,
\begin{equation}\label{e2.7}
\begin{aligned}
0&=h(Du(x,t),x)-h(Du(x,t+t_{0}),x)\\
&=\int_0^1\frac{d}{ds}h(sDu(x,t)+(1-s)Du(x,t+t_{0}),x)ds\\
&=\beta^{i}w_{i},\qquad x\in\partial\Omega, \quad t>0,
\end{aligned}
\end{equation}
where
\begin{equation*}
\begin{aligned}
\beta^{i}&= \int_0^1 \nabla_{p_{i}}h(sDu(x,t)+(1-s)Du(x,t+t_{0}),x)ds.
\end{aligned}
\end{equation*}
By (\ref{e1.4}) and (\ref{e1.5}), there exists $N\in Z^{+}$ depending only on the property of $h(\cdot,x)$ in $\partial\Omega$, such that if $0<t_{0}\leq\frac{1}{N}$, then  $a^{ij}, b^{i}, $ and $\beta^{i}$ satisfy the conditions in Lemma \ref{l2.1}.

\begin{lemma}\label{l2.3}
$u^{\infty}$ and $u^{t_{0},\infty}$ satisfy the following evolution equation with second boundary condition:
\begin{equation}\label{e2.8}
\left\{ \begin{aligned}u_{t}-F(D^{2}u,Du,x)&=0,
&  x\in \Omega,\,\, t>0, \\
h(Du,x)&=0,& x\in \partial\Omega,\,\, t>0,\\
\end{aligned} \right.
\end{equation}
and
\begin{equation*}
osc_{[u^{\infty}-u^{t_{0},\infty}]}(t)
\end{equation*}
is a strictly decreasing function or $u^{\infty}-u^{t_{0},\infty}$ is constant. In addition, $u^{\infty}$ and $u^{t_{0},\infty}$  satisfy the estimate (\ref{e1.4}),(\ref{e1.40}) and (\ref{e1.5}).
\end{lemma}
\begin{proof}
We first point out that $u^{\infty}$ satisfy the evolution equation with second boundary condition.
In fact, if we denote $w_n(x,t) = u(x,t+t_{n})-u(x_{0},t_{n})$, then it satisfies equation \eqref{e2.8}.
For any $x \in \Omega$, we can choose an open subset $D$ with compact support in $\Omega$ such that $x \in D\subset \bar{D}\subset \Omega$.
By \eqref{e2.2} in Lemma \ref{l2.2}, we know $w_n$ and $F(D^2w_n,Dw_n,x)$ uniformly converge to $u^{\infty}$ and $F(D^2u^{\infty},u^{\infty},x)$ in $D$
respectively as $n\rightarrow \infty$.

 By letting $n\rightarrow \infty$ in $(w_n)_t = F(D^2w_n,Dw_n,x)$, we obtain
 \begin{equation*}
 \begin{split}
& u^{\infty}(x,t)-u^{\infty}(x,t_0)=\lim_{n\rightarrow \infty} ( w_n(x,t)-w_n(x,t_o))\\
= & \lim_{n\rightarrow \infty} \int_{t_o}^t  (w_n)_s ds =\lim_{n\rightarrow \infty} \int_{t_o}^t  F(D^2 w_n, D w_n,x) ds\\
= &  \int_{t_o}^t \lim_{n\rightarrow \infty} F(D^2 w_n, D w_n,x) ds = \int_{t_o}^t  F(D^2 u^{\infty}, D u^{\infty},x) ds .
 \end{split}
 \end{equation*}
Taking derivative with respect to the variable, we deduce that $u^{\infty}_t= F(D^2 u^{\infty}, D u^{\infty},x)$.
By uniform convergence of $w_n(x,t)$ to $u^{\infty}(x,t)$ in $C^{1,\zeta}(\bar{\Omega})$ as $n\rightarrow \infty$
 in \eqref{e2.2}, we easily derive that $h(D u^{\infty},x)=0$ on $\partial \Omega$. Hence $u^{\infty}$ satisfies equation \eqref{e2.8}.
Similarly, $u^{t_0,\infty}$ also satisfies equation \eqref{e2.8}.

If we set  $$w(x,t)=u^{\infty}(x,t)-u^{t_{0},\infty}(x,t),$$
then by \eqref{e2.6} and \eqref{e2.7}, we know $w(x,t)$ satisfies
\begin{equation*}
\left\{ \begin{aligned}w_{t}-a^{ij}w_{ij}-b^{i}w_{i}&=0,
&  x\in \Omega,\,\, t>0, \\
\beta^{k}w_{k}&=0,& x\in \partial\Omega,\,\, t>0.\\
\end{aligned} \right.
\end{equation*}
By Lemma \ref{l2.1}, we deduce that $osc_{[u^{\infty}-u^{t_{0},\infty}]}(t)$ is a strictly decreasing function
or $u^{\infty}-u^{t_{0},\infty}$ is constant.
Furthermore, since $u\in C^{3+\alpha,\frac{3+\alpha}{2}}(\bar{\Omega}_{T})$,
by using \eqref{e2.40}, \eqref{e2.5} and  $Arzel\grave{a}-Ascoli$ theorem,
we know that  $u^{\infty}$ and $u^{t_{0},\infty}$  satisfy the estimate (\ref{e1.4}),(\ref{e1.40}) and (\ref{e1.5}).
\end{proof}

\begin{lemma}\label{l2.4}
If we take any  positive constant $0<t_0\leq \frac{1}{N}$ for some large $N>0$  depending only on the property of $h(\cdot,x)$ in $\partial\Omega$ and write $$w(x,t)=u(x,t)-u(x,t+t_0),$$
then we have
$\lim_{t\rightarrow+\infty}osc_{w}(t)=0$.
\end{lemma}
\begin{proof}
By Lemma \ref{l2.1}, $\lim_{t\rightarrow+\infty}osc_{w}(t)$ exists.

If $\lim_{t\rightarrow+\infty}osc_{w}(t)=\epsilon>0.$
Then $w$ is not a constant function. For any sequence $\{t_{n}\} \,\, (t_{n}\rightarrow +\infty)$
and for $x_{0}\in \bar{\Omega}$ fixed, let
\begin{equation*}
\begin{aligned}
\varphi_{n}(x,t)&=u(x,t+t_{n})-u(x_{0},t_{n}),\\
\varphi_{t_{0},n}(x,t)&=u(x,t+t_{0}+t_{n})-u(x_{0},t_{0}+t_{n}).
\end{aligned}
\end{equation*}
It is easy to check that,
\begin{equation}\label{e2.9}
osc_{[\varphi_{n}-\varphi_{t_{0},n}]}(t)=osc_{w}(t+t_{n}).
\end{equation}
There exists a subsequence of $\{t_{n}\}$ (again denoted by itself), such that
\begin{equation*}
\lim_{n\rightarrow+\infty}osc_{[\varphi_{n}-\varphi_{t_{0},n}]}(t)=\max_{\bar{\Omega}}(u^{\infty}-u^{t_{0},\infty})-\min_{\bar{\Omega}}(u^{\infty}-u^{t_{0},\infty}),
\end{equation*}
where we use Lemma \ref{l2.3}. On the other hand, by the assumption of the beginning ,
\begin{equation*}
\lim_{n\rightarrow+\infty}osc_{w}(t+t_{n})=\epsilon.
\end{equation*}
Then by (\ref{e2.9}), we obtain
\begin{equation*}
\begin{aligned}
osc_{[u^{\infty}-u^{t_{0},\infty}]}(t)&=\max_{\bar{\Omega}}(u^{\infty}-u^{t_{0},\infty})(t)-\min_{\bar{\Omega}}(u^{\infty}-u^{t_{0},\infty})(t)\\
&=\epsilon.
\end{aligned}
\end{equation*}
Since $ u^{\infty}$ and $u^{t_{0},\infty}$ satisfy equation \eqref{e2.8}, then we know $u^* = u^{\infty}-u^{t_{0},\infty}$
satisfies the uniformly  parabolic equation
\begin{equation*}
\left\{ \begin{aligned}u^*_{t}-a^{ij}u^*_{ij}-b^{i}u^*_{i}&=0,
&  \quad \text{in} \quad \Omega\times (-\infty,+\infty), \\
\beta^{k} u^*_{k}&=0,& \quad \text{on} \quad\partial\Omega\times (-\infty,+\infty),\\
\end{aligned} \right.
\end{equation*}
where
\begin{equation*}
\begin{aligned}
a^{ij}&=\int_0^1 \nabla_{r_{ij}}F(sD^2u^{\infty}(x,t)+(1-s)D^2u^{t_0,\infty}(x,t),sDu^{\infty}(x,t)+(1-s)Du^{t_0,\infty}(x,t),x)ds,\\
b^{i}&= \int_0^1 \nabla_{p_{i}}F(sD^2u^{\infty}(x,t)+(1-s)D^2u^{t_0,\infty}(x,t),sDu^{\infty}(x,t)+(1-s)Du^{t_0,\infty}(x,t),x)ds.
\end{aligned}
\end{equation*}

By the strong maximum principle and Hopf's Lemma,
we know $u^*= u^{\infty}-u^{t_{0},\infty}$ is a constant and hence $osc_{[u^{\infty}-u^{t_{0},\infty}]} = 0$.
This makes a contradiction to
$osc_{[u^{\infty}-u^{t_{0},\infty}]}(t) = \epsilon$ and we get the desired results.
\end{proof}
As in the proof of Lemma \ref{l2.2}, we have the following result:
\begin{lemma}\label{l2.5}
For any sequence $\{t_{n}\} \,\, (t_{n}\rightarrow +\infty)$ and for $x_{0}\in \bar{\Omega}$ fixed,
there exists a subsequence of $\{t_{n}\}$ (again denoted by itself), such that
\begin{equation*}
u(x,t+t_{n})-u(x_{0},t_{n})
\end{equation*}
 converge to   $u^{0}(x,t)$   in the following sense,
\begin{equation}\label{e2.10}
\begin{aligned}
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{n})-u(x_{0},t_{n})-u^{0}(x,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\\
 &\lim_{n\rightarrow+\infty}\|u(x,t+t_{n})-u(x_{0},t_{n})-u^{0}(x,t)\|_{C^{2}(\bar{D})}=0,\\
 \end{aligned}
\end{equation}
where $D\subset\subset\Omega$, $0<\zeta<1$. In addition,   $u^{0}(x,t)$ satisfies (\ref{e2.8}), (\ref{e1.4}),(\ref{e1.40}) and (\ref{e1.5}).
\end{lemma}
\begin{lemma}\label{l2.6}
Let $t_{0}$ be some positive constant as in Lemma \ref{l2.4}.
There exists some constant $v^{\infty}$ such that
\begin{equation}\label{e2.11}
u^{0}(x,t+t_{0})=u^{0}(x,t)+v^{\infty}\cdot t_{0}\quad \text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\}.
\end{equation}
\end{lemma}
\begin{proof}
By  making use of Lemma \ref{l2.5}, we have
\begin{equation}\label{e2.12}
\begin{aligned}
 &\lim_{n\rightarrow+\infty}u(x,t+t_{n})-u(x_{0},t_{n})=u^{0}(x,t),\\
 &\lim_{n\rightarrow+\infty}u(x,t+t_{0}+t_{n})-u(x_{0},t_{n})=u^{0}(x,t+t_{0})\\
 \end{aligned}
\end{equation}
for any  $(x,t)\in \bar{\Omega}\times \{t|t\geq 0\}$. On the other hand,
using Lemma \ref{l2.4} we conclude that
as $t\rightarrow+\infty$, $u(x,t)-u(x,t+t_{0})$ converges to some constant uniformly in $\bar{\Omega}$. We denote the process by
\begin{equation*}
\lim_{t\rightarrow+\infty}(u(x,t)-u(x,t+t_{0}))=-v^{\infty}\cdot t_{0}.
\end{equation*}
Replacing $t$ by $t+t_{n}$ in the above, we obtain
\begin{equation}\label{e2.13}
\lim_{n\rightarrow+\infty}(u(x,t+t_{n})-u(x,t+t_{n}+t_{0}))=-v^{\infty}\cdot t_{0}.
\end{equation}
Combining (\ref{e2.12}) with (\ref{e2.13}),  we get
\begin{equation*}
\begin{aligned}
u^{0}(x,t+t_{0})-u^{0}(x,t)&=\lim_{n\rightarrow+\infty}(u(x,t+t_{0}+t_{n})-u(x_{0},t_{n}))\\
&-\lim_{n\rightarrow+\infty}(u(x,t+t_{n})-u(x_{0},t_{n}))\\
&=\lim_{n\rightarrow+\infty}(u(x,t+t_{n}+t_{0})-u(x,t+t_{n}))\\
&=v^{\infty}\cdot t_{0}.
\end{aligned}
\end{equation*}
\end{proof}
Let
\begin{equation*}
P=\{p_{i}|p_{0}<p_{1}<\cdots,i=0,1,\cdots\}
\end{equation*}
be the set of all positive prime numbers.

By Lemma \ref{l2.5}, $u^{0}(x,t)$ satisfies (\ref{e2.8}), (\ref{e1.4}), (\ref{e1.40}) and (\ref{e1.5}).
If we replace $(u, t_{1})$ by $(u^{0}, t_{0})$  and repeat the argument as in the proof of Lemma \ref{l2.6},
 we have
\begin{lemma}\label{l2.7}
There exist $N\in Z^{+}$  depending only on the property of $h(\cdot,x)$ in $\partial\Omega$
and the function $u^{1}(x,t)$ satisfying (\ref{e2.8}), (\ref{e1.4}), (\ref{e1.40}) and (\ref{e1.5}), such that we obtain
\begin{equation}\label{e2.14}
u^{1}(x,t+t_{1})=u^{1}(x,t)+\tilde{v}^{\infty}\cdot t_{1}\quad \text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\},
\end{equation}
\begin{equation}\label{e2.15}
u^{1}(x,t+t_{0})=u^{1}(x,t)+v^{\infty}\cdot t_{0}\quad \text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\},
\end{equation}
where $t_{0}=\frac{1}{Np_{0}}, t_{1}=\frac{1}{Np_{1}}, \tilde{v}^{\infty}=v^{\infty}$.
\end{lemma}
\begin{proof}
As Lemma \ref{l2.5},  there exist sequence $\{t_{n}\} \,\, (t_{n}\rightarrow +\infty)$ for  $x_{0}\in \bar{\Omega}$ to be  fixed and $u^{1}(x,t)$, such that
\begin{equation}\label{e2.16}
\begin{aligned}
 &\lim_{n\rightarrow+\infty}\|u^{0}(x,t+t_{n})-u^{0}(x_{0},t_{n})-u^{1}(x,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\\
 &\lim_{n\rightarrow+\infty}\|u^{0}(x,t+t_{n})-u^{0}(x_{0},t_{n})-u^{1}(x,t)\|_{C^{2}(\bar{D})}=0,\\
 \end{aligned}
\end{equation}
where $\forall t\geq 0, D\subset\subset\Omega$, $0<\zeta<1$,  In addition,   $u^{1}(x,t)$ satisfies (\ref{e2.8}), (\ref{e1.4}) and (\ref{e1.5}).
By (\ref{e2.16}), we also have
\begin{equation}\label{e2.17}
\begin{aligned}
 &\lim_{n\rightarrow+\infty}\|u^{0}(x,t+t_{0}+t_{n})-u^{0}(x_{0},t_{n})-u^{1}(x,t+t_{0})\|_{C^{1+\zeta}(\bar{\Omega})}=0,\\
 &\lim_{n\rightarrow+\infty}\|u^{0}(x,t+t_{0}+t_{n})-u^{0}(x_{0},t_{n})-u^{1}(x,t+t_{0})\|_{C^{2}(\bar{D})}=0.\\
 \end{aligned}
\end{equation}
Combining (\ref{e2.16}) with (\ref{e2.17}), we obtain
\begin{equation}\label{e2.18}
\begin{aligned}
u^{1}(x,t+t_{0})-u^{1}(x,t)&=\lim_{n\rightarrow+\infty}(u^{0}(x,t+t_{0}+t_{n})-u^{0}(x_{0},t_{n}))\\
&-\lim_{n\rightarrow+\infty}(u^{0}(x,t+t_{n})-u^{0}(x_{0},t_{n}))\\
&=\lim_{n\rightarrow+\infty}(u^{0}(x,t+t_{0}+t_{n})-u^{0}(x,t+t_{n}))\\
&=v^{\infty}\cdot t_{0},\quad \text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\},
\end{aligned}
\end{equation}
where we use Lemma \ref{l2.6}.

On the other hand, for any  positive constant $0<t_{1}\leq \frac{1}{N}$ for some large $N>0$  depending only
on the property of $h(\cdot,x)$ in $\partial\Omega$, as in the proof of Lemma \ref{l2.6}, there exists some constant $\tilde{v}^{\infty}$ such that
\begin{equation*}
\lim_{t\rightarrow+\infty}(u^{0}(x,t)-u^{0}(x,t+t_{1}))=-\tilde{v}^{\infty}\cdot t_{1}, \quad \text{uniformly}\,\,\, \text{in}\,\,\, \bar{\Omega}.
\end{equation*}
Replacing $t$ by $t+t_{n}$ in the above, we obtain
\begin{equation*}
\lim_{n\rightarrow+\infty}(u^{0}(x,t+t_{n})-u^{0}(x,t+t_{n}+t_{1}))=-\tilde{v}^{\infty}\cdot t_{1}.
\end{equation*}
Then we obtain
\begin{equation}\label{e2.19}
\begin{aligned}
\tilde{v}^{\infty}\cdot t_{1}&=\lim_{n\rightarrow+\infty}(u^{0}(x,t+t_{n}+t_{1})-u^{0}(x,t+t_{n}))\\
&=\lim_{n\rightarrow+\infty}(u^{0}(x,t+t_{n}+t_{1})-u^{0}(x_{0},t_{n})\\
&+u^{0}(x_{0},t_{n})-u^{0}(x,t+t_{n}))\\
&=\lim_{n\rightarrow+\infty}(u^{0}(x,t+t_{n}+t_{1})-u^{0}(x_{0},t_{n}))\\
&+\lim_{n\rightarrow+\infty}(u^{0}(x_{0},t_{n})-u^{0}(x,t+t_{n}))\\
&=u^{1}(x,t+t_{1})-u^{1}(x,t)\quad \text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\},
\end{aligned}
\end{equation}
where we use (\ref{e2.16}).

By (\ref{e2.18}) and (\ref{e2.19}), for any $m_{0}, m_{1}\in Z^{+}$, we get
\begin{equation}\label{e2.20}
\begin{aligned}
u^{1}(x,t+m_{0}t_{0})-u^{1}(x,t)&=v^{\infty}\cdot m_{0}t_{0},\\
u^{1}(x,t+m_{1}t_{1})-u^{1}(x,t)&=\tilde{v}^{\infty}\cdot m_{1}t_{1},
\end{aligned}
\end{equation}
for any $(x,t)\in \bar{\Omega}\times \{t|t\geq 0\}$. By the conditions of the choice according to $t_{0}$ and $t_{1}$,
there exist $N\in Z^{+}$  depending only on the property of $h(\cdot,x)$ in $\partial\Omega$ such that we can take
$t_{0}=\frac{1}{Np_{0}}$ and $t_{1}=\frac{1}{Np_{1}}$ respectively. Letting $m_{0}=Np_{0}, m_{1}=Np_{1}, t=0$, then by (\ref{e2.20}) we obtain
\begin{equation*}
\begin{aligned}
u^{1}(x,1)-u^{1}(x,0)&=v^{\infty},\\
u^{1}(x,1)-u^{1}(x,0)&=\tilde{v}^{\infty}.
\end{aligned}
\end{equation*}
So $\tilde{v}^{\infty}=v^{\infty}.$
\end{proof}
By the proof of Lemma \ref{l2.7}, we deduce that
\begin{lemma}\label{l2.8}
There exist $N\in Z^{+}$  depending only on the property of $h(\cdot,x)$ in $\partial\Omega$
and the function $u^{i}(x,t)(i=0,1,\cdots)$ satisfying (\ref{e2.8}), (\ref{e1.4}), (\ref{e1.40})and (\ref{e1.5}), such that we obtain
\begin{equation}\label{e2.21}
\begin{aligned}
&u^{i}(x,t+\tau_{k})=u^{i}(x,t)+v^{\infty}\cdot \tau_{k}(k=0,1,\cdots,i)\\
&\text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\},
\end{aligned}
\end{equation}
where $\tau_{k}=\frac{1}{Np_{k}}$ and $v^{\infty}$ is some constant.
\end{lemma}
\begin{lemma}\label{l2.9}
There exist $u^{\infty}$ satisfying (\ref{e2.8}), (\ref{e1.4}), (\ref{e1.40})and (\ref{e1.5}), such that we obtain
\begin{equation}\label{e2.22}
\begin{aligned}
&u^{\infty}(x,t+\tau)=u^{\infty}(x,t)+v^{\infty}\cdot \tau\\
&\text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\},\,\,\tau\geq 0,
\end{aligned}
\end{equation}
where  $v^{\infty}$ is some constant.
\end{lemma}
\begin{proof}
By $Arzel\grave{a}-Ascoli$ theorem, we consider a diagonal sequence of \{i\} (again denoted by itself), such that
\begin{equation}\label{e2.23}
\begin{aligned}
 &\lim_{i\rightarrow+\infty}\|u^{i}(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\quad \forall\, t\geq 0,\\
 &\lim_{i\rightarrow+\infty}\|u^{i}(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{2}(\bar{D})}=0,\quad\forall\, t\geq 0,\\
 \end{aligned}
\end{equation}
for some function $u^{\infty}$ satisfies (\ref{e2.8}), (\ref{e1.4}), (\ref{e1.40})and (\ref{e1.5}).

By (\ref{e2.21}) in Lemma \ref{l2.8}, we get
\begin{equation*}
\begin{aligned}
&u^{\infty}(x,t+\tau_{k})=u^{\infty}(x,t)+v^{\infty}\cdot \tau_{k}~(k=0,1,\cdots,)\\
&\text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\}.
\end{aligned}
\end{equation*}
It is easy to see that
\begin{equation*}
\begin{aligned}
&u^{\infty}(x,t+m\tau_{k})=u^{\infty}(x,t)+v^{\infty}\cdot m\tau_{k}~(k=0,1,\cdots,)\\
&\text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\},\,\,m\in Z^{+}.
\end{aligned}
\end{equation*}
Then
\begin{equation*}
\begin{aligned}
&u^{\infty}(x,t+q)=u^{\infty}(x,t)+v^{\infty}\cdot q~(k=0,1,\cdots,)\\
&\text{for} \quad \text{any}\quad (x,t)\in \bar{\Omega}\times \{t|t\geq 0\},\,\,q\in Q^{+},
\end{aligned}
\end{equation*}
where $ Q^{+}=\{q\mid q=\frac{p_{i}}{p_{j}},\,\,p_{i}, p_{j}\in P\}.$
By the denseness of $Q^{+}$ in $\{\tau\geq 0\}$, we obtain the desired results.
\end{proof}
\begin{lemma}\label{l2.10}
\begin{equation*}
\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C(\bar{\Omega})}=0.
\end{equation*}
\end{lemma}
\begin{proof}
 Let
\begin{equation*}
W(x,t)=u(x,t)-u^{\infty}(x,t).
\end{equation*}
If we replace $u(x,t+t_{0})$ by $u^{\infty}(x,t)$ and repeat the arguments in  Lemma \ref{l2.4}, then we have the desired results.
\end{proof}

\section{Proof of Theorem \ref{t1.1}}
In this section, we give the proof of Theorem \ref{t1.1} by using the preliminary results which we establish in section 3.

We first give an estimates which will be used in the following proof.
 By intermediate Schauder estimates for parabolic
equations (cf. Lemma 14.6  and Proposition 4.25 in \cite{GM}), for any $D\subset\subset \Omega$, we have
\begin{equation}\label{e4.209}
\begin{aligned}
&\sup_{t\geq 1}\|D^{3}u(\cdot,t)\|_{C(\bar{D})}+\sup_{t\geq 1}\|D^{4}u(\cdot,t)\|_{C(\bar{D})}\\
&+\sup_{x_{i}\in D, t_{i}\geq 1}\frac{|D^{4}u(x_{1}, t_{1})-D^{4}u(x_{2}, t_{2})}{\max\{|x_{1}-x_{2}|^{\alpha},|t_{1}-t_{2}|^{\frac{\alpha}{2}}\}}\\
&\leq C_{4},
\end{aligned}
\end{equation}
where $ C_{4}$ is a constant depending on the known data and dist$(\partial \Omega, \partial D)$.

Now fixing some positive $t_0$ and writing $$w(x,t)=u(x,t)-u(x,t+t_0),$$ then we see that $w(x,t)$ satisfies equation \eqref{e2.1}.
By \eqref{e1.4} and \eqref{e1.40}, we know the equation \eqref{e2.1} are uniformly parabolic.
Since $h_{p_k}(x,Du(x,t))\nu_k\geq C>0$ for some $C$ uniform in $t$ and $x$.
Therefore we can choose $t_0$ small enough to ensure that $\alpha^k\nu_k\geq \frac{C}{2}>0$
and we see that $w$ satisfies a linear, uniformly oblique boundary condition.

By Lemma \ref{l2.1} to Lemma \ref{l2.9} in Section 2, we know the uniform parabolicity of equation \eqref{e2.1} ensures that
we can obtain a translating solution of the same regularity as $u$ : $u^{\infty}(x,t)=\tilde{u}^\infty(x)+C_{\infty}\cdot t$
for some constant $C_{\infty}$ which satisfies equation \eqref{e1.1}.
By Lemma \ref{l2.10},
 $u^{\infty}(x,t)$ satisfies
\begin{equation}\label{e3.1}
\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C(\bar{\Omega})}=0.
\end{equation}
By a result of Maz'ya-Shaposhnikova \cite{MS} on page 143 , For any $0<\alpha<1$, we have the following interpolation inequality
\begin{equation*}
|D u(x)|^{\alpha+1}\leq \frac{n(n+1)^{\alpha+1}(\alpha+1)^{\alpha+1}}{(n+\alpha)(n+\alpha+1)\alpha^{\alpha}}
(M^0(x))^{\alpha}\sup_y\frac{|D u(y)-D u(x)|}{|y-x|^{\alpha}},
\end{equation*}
where $M^0 u(x)=\sup_{\tau>0} \frac{1}{2\tau} |\int_{x-\tau}^{x+\tau} sign(x-y) u(y) dy |$ is the maximal operator.
It easily yields that for any $0<\alpha<1$,
\begin{equation*}
\|D u\|_{C(\bar{\Omega})}\leq \frac{n^{\frac{1}{\alpha+1}}(n+1)(\alpha+1)}{(n+\alpha)^{\frac{1}{\alpha+1}}(n+\alpha+1)^{\frac{1}{\alpha+1}}\alpha^{\frac{\alpha}{\alpha+1}}}
\|u\|_{C(\bar{\Omega})}^{\frac{\alpha}{\alpha+1}}\|D u \|^{\frac{1}{\alpha+1}}_{C^{\alpha}(\bar{\Omega})}.
\end{equation*}
Therefore by replacing $u(\cdot)$ with $u(\cdot,t)-u^{\infty}(\cdot,t)$ in the above inequality, we obtain
\begin{equation}\label{e3.2}
\begin{aligned}
& \|D(u(\cdot,t)-u^{\infty}(\cdot,t))\|_{C(\bar{\Omega})}\\
\leq & C(n,\alpha) \|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C(\bar{\Omega})}^{\frac{\alpha}{\alpha+1}}\|D (u(\cdot,t) - u^{\infty}(\cdot,t)) \|^{\frac{1}{\alpha+1}}_{C^{\alpha}(\bar{\Omega})}\\
\leq &  C(n,\alpha) \|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C(\bar{\Omega})}^{\frac{\alpha}{\alpha+1}}\|D^2 ( u(\cdot,t)- u^{\infty}(\cdot,t)) \|^{\frac{1}{\alpha+1}}_{C {(\bar{\Omega})}}.
\end{aligned}
\end{equation}

 From equation \eqref{e3.2} and the estimates \eqref{e1.40}, we obtain
\begin{equation}\label{e3.3}
\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{1}(\bar{\Omega})}=0.
\end{equation}
By \eqref{e1.4}, the interpolation inequalities  and (\ref{e4.209}), we get
\begin{equation*}\label{e3.4}
\begin{aligned}
&\|D^{2}(u(\cdot,t)-u^{\infty}(\cdot,t))\|^{2}_{C(\bar{D})}\\
&\leq c(D)\|D(u(\cdot,t)-u^{\infty}(\cdot,t))\|_{C(\bar{D})}(\|D^{3}(u(\cdot,t)-u^{\infty}(\cdot,t))\|_{C(\bar{D})}\\
&+\|D^{2}(u(\cdot,t)-u^{\infty}(\cdot,t))\|_{C(\bar{D})})\\
&\leq c(D)\|D(u(\cdot,t)-u^{\infty}(\cdot,t))\|_{C(\bar{D})}(2C_{4}+2C_{1}).
\end{aligned}
\end{equation*}
Using (\ref{e3.3}),  we also obtain
\begin{equation}\label{e3.5}
\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{2}(\bar{D})}=0.
\end{equation}
Repeating the above procedure and using (\ref{e4.209}),  we have
\begin{equation}\label{e3.6}
\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{4+\alpha'}(\bar{D})}=0.
\end{equation}
By \eqref{e1.4} and (\ref{e3.5}), we get
$$\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0.$$
Therefore by using the equation \eqref{e1.1} and letting $t\rightarrow\infty$, we obtain
 \begin{equation*}
C_{\infty} =\frac{\partial u^{\infty}(x,t)}{\partial t}= F(D^2u^{\infty}(x,t),Du^{\infty}(x,t),x)=F(D^2u^{\infty}(x),Du^{\infty}(x),x),
 \end{equation*}
 $$0=\lim_{t\rightarrow\infty}h(Du(x,t),x)=\lim_{t\rightarrow\infty}h(Du^{\infty}(x,t),x)=h(Du^{\infty}(x),x).$$
Then the proof of Theorem \ref{t1.1} is completed.
\qed

\section{Application of Theorem \ref{t1.1} to the convergence of special Lagrangian graphs}

In this section, we will show the applications of  Theorem \ref{t1.1} to study the existence of a family of special Lagrangian graphs.
We use the parabolic framework which consider the corresponding parabolic flows and show that
the flows converge to special Lagrangian graphs by using the convergence result.
The key steps are to verify the a-priori estimates \eqref{e1.4}, \eqref{e1.40} and the strict obliqueness estimates
of the boundary condition \eqref{e1.5}.

Elliptic methods have been successfully used to study the existence of fully-nonlinear equations with second boundary value conditions.
Elliptic Monge-Amp$\grave{e}$re equations and Hessian equations with Neumann boundary and oblique boundary conditions are solved in
\cite{P} and \cite{JU} respectively. In \cite{SM}, Brendle and Warren used the continuity method
to solve fully nonlinear elliptic equations with second boundary condition
where they obtained the existence and uniqueness of a special Lagrangian graph.

Let $\Omega$, $\tilde{\Omega}$ be two uniformly convex bounded
domains with smooth boundary in  $\mathbb{R}^{n}$  and  $a=\cot\tau$, $b=\sqrt{|\cot^{2}\tau-1|}$ for $\tau \in (0,\frac{\pi}{4}) \cup (\frac{\pi}{4},\frac{\pi}{2})$.
Here we consider the minimal Lagrangian diffeomorphism problem \cite{RS} which  is equivalent to
the following fully nonlinear elliptic equations with second boundary condition:
\begin{equation}\label{e4.1}
\left\{ \begin{aligned}F_{\tau}(D^{2}u)&=c,
&  x\in \Omega, \\
Du(\Omega)&=\tilde{\Omega},
\end{aligned} \right.
\end{equation}
where
$$F_{\tau}(A)=\left\{ \begin{aligned}
&\sum_{i}\ln\lambda_{i} , \quad
\qquad\quad\qquad\qquad\quad\quad\qquad\quad  \tau=0, \\
& \sum_{i}\ln(\frac{\lambda_{i}+a-b}{\lambda_{i}+a+b}) \qquad\qquad\quad\quad\qquad\quad 0<\tau<\frac{\pi}{4},\\
& -\sum_{i}\frac{1}{1+\lambda_{i}}, \qquad \qquad\qquad \qquad\qquad \qquad\tau=\frac{\pi}{4},\\
& \sum_{i}\arctan(\frac{\lambda_{i}+a-b}{\lambda_{i}+a+b}), \qquad \qquad\qquad \quad\frac{\pi}{4}<\tau<\frac{\pi}{2},\\
& \sum_{i}\arctan\lambda_{i}, \,\,\quad \qquad\qquad \qquad\qquad \qquad\tau=\frac{\pi}{2},
\end{aligned} \right.$$
$\lambda_i ~(1\leq i\leq n)$ are the eigenvalues of the Hessian $D^2 u$ and $Du$ are diffeomorphism from $\Omega$ to $\tilde{\Omega}$.
The existence and uniqueness of these problems have been obtained by elliptic methods in \cite{SM}, \cite{HRO} and \cite{MW}.
As in \cite{HR} and \cite{HRY}, we consider the corresponding parabolic type special Lagrangian equations and use parabolic methods to solve problem \eqref{e4.1}.
We settle the longtime existence and convergence of smooth solutions
for the following second boundary value problem to parabolic type special Lagrangian equations
\begin{equation}\label{e4.2}
\left\{ \begin{aligned}\frac{\partial u}{\partial t}&=F_{\tau}(D^{2}u),
& t>0,\quad x\in \Omega, \\
Du(\Omega)&=\tilde{\Omega}, &t>0,\qquad\qquad\\
 u&=u_{0}, & t=0,\quad x\in \Omega.
\end{aligned} \right.
\end{equation}

We will use Theorem \ref{t1.1} to prove the solutions of the
above special Lagrangian equations converging to those of problem \eqref{e4.1}. We consider different cases with respect to $\tau$.

In \cite{OK}, Schn$\ddot{\text{u}}$rer and Smoczyk studied the second boundary value problems for the Hessian and Gauss curvature flows.
They showed the convergence of the Hessian and Gauss curvature flow to the prescribed Gauss curvature equations.
We can use Theorem \ref{t1.1} to reprove part of their results.
For simplicity, we only give the proof of the special case corresponding to $\tau=0$ in equation \eqref{e4.1} and \eqref{e4.2}.
\begin{Example}
Assume that $\Omega$, $\tilde{\Omega}$ are bounded, uniformly convex domains with smooth boundary in $\mathbb{R}^{n}$, $0<\alpha<1$.
  Then for any given initial function $u_{0}\in C^{2+\alpha}(\bar{\Omega})$
  which is uniformly convex and satisfies $Du_{0}(\Omega)=\tilde{\Omega}$,  the  strictly convex solution of the following problem
  \begin{equation}\label{e4.3}
\left\{ \begin{aligned}\frac{\partial u}{\partial t}&= \ln \det D^2 u ,
& t>0,\quad x\in \Omega, \\
Du(\Omega)& = \tilde{\Omega}, &t>0,\qquad\qquad\\
 u& = u_{0}, & t=0,\quad x\in \Omega.
\end{aligned} \right.
\end{equation}
   exists for all $t\geq 0$ and $u(\cdot,t)$ converges to a function $u^{\infty}(x,t)=\tilde{u}^\infty(x)+C_{\infty}\cdot t$ in $C^{1+\zeta}(\bar{\Omega})\cap C^{4+\alpha}(\bar{D})$ as $t\rightarrow\infty$
  for any $D\subset\subset\Omega$, $\zeta<1$, that is,
 $$\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\qquad
  \lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{4+\alpha}(\bar{D})}=0.$$
And $\tilde{u}^{\infty}(x)\in C^{\infty}(\bar{\Omega})$ is a solution of
\begin{equation}\label{e4.4}
\left\{ \begin{aligned} \ln \det D^2 u & = C_{\infty},
&  x\in \Omega, \\
Du(\Omega)&=\tilde{\Omega}.
\end{aligned} \right.
\end{equation}
The constant $C_{\infty}$ depends only on $\Omega$, $\tilde{\Omega}$ and $F$. The solution to (\ref{e4.4}) is unique up to additions of constants.
\end{Example}
\begin{proof}
From section $7$ and section $9$ in \cite{OK}, Schn$\ddot{\text{u}}$rer and Smoczyk had established estimates \eqref{e1.4}.
Since the parabolic operator is concave with with respect to the $D^2 u$ variables, by Remark \ref{r1.1}, we know \eqref{e1.40} holds.
By Lemma 8.1 in section 8 of \cite{OK}, we get the strict obliqueness estimates \eqref{e1.5}. Therefore, by Theorem \ref{t1.1},
we obtain the desired convergence results.
\end{proof}

In \cite{SM}, Brendle and Warren obtained the existence and uniqueness of  special Lagrangian graph for the special case $\tau=\frac{\pi}{2}$. In the following example,
we give an alternative proof which shows the applications of Theorem \ref{t1.1} to the existence of solutions for the equations (\ref{e4.1}) when $\tau=\frac{\pi}{2}$.

\begin{Example}
Assume that $\Omega$, $\tilde{\Omega}$ are bounded, uniformly convex domains with smooth boundary in $\mathbb{R}^{n}$, $0<\alpha<1$.
  Then for any given initial function $u_{0}\in C^{2+\alpha}(\bar{\Omega})$
  which is   uniformly convex and satisfies $Du_{0}(\Omega)=\tilde{\Omega}$,  the  strictly convex solution of the following problem
  \begin{equation}\label{e4.5}
\left\{ \begin{aligned}\frac{\partial u}{\partial t}&= \sum_{i=1}^n arctan \lambda_i,
& t>0,\quad x\in \Omega, \\
Du(\Omega)& = \tilde{\Omega}, &t>0,\qquad\qquad\\
 u& = u_{0}, & t=0,\quad x\in \Omega.
\end{aligned} \right.
\end{equation}
   exists for all $t\geq 0$ and $u(\cdot,t)$ converges to a function $u^{\infty}(x,t)=\tilde{u}^\infty(x)+C_{\infty}\cdot t$ in $C^{1+\zeta}(\bar{\Omega})\cap C^{4+\alpha}(\bar{D})$ as $t\rightarrow\infty$
  for any $D\subset\subset\Omega$, $\zeta<1$, that is,
 $$\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\qquad
  \lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{4+\alpha}(\bar{D})}=0.$$
And $\tilde{u}^{\infty}(x)\in C^{\infty}(\bar{\Omega})$ is a solution of
\begin{equation}\label{e4.6}
\left\{ \begin{aligned}\sum_{i=1}^n arctan \lambda_i & = C_{\infty},
&  x\in \Omega, \\
Du(\Omega)&=\tilde{\Omega}.
\end{aligned} \right.
\end{equation}
The constant $C_{\infty}$ depends only on $\Omega$, $\tilde{\Omega}$ and $F$. The solution to (\ref{e1.3}) is unique up to additions of constants.
\end{Example}
\begin{proof}
By Lemma 3.1, Lemma 3.11 in \cite{HR} and boundary condition, we see that the estimates \eqref{e1.4} holds.
Since the parabolic operator is concave with respect to the $D^2 u$ variables, by Remark \ref{r1.1},
we obtain the estimate \eqref{e1.40}.
By Lemma 3.4 in \cite{HR}, we get the strict obliqueness estimates. By Theorem \ref{t1.1}, we obtain the desired results.
\end{proof}

Our next example is to show the existence of the special Lagrangian graphs for the general cases $0<\tau<\frac{\pi}{2}$.
We use Theorem \ref{e1.1} to show the convergence of the
corresponding parabolic flows to the translating solutions.
\begin{Example}
Assume that $\Omega$, $\tilde{\Omega}$ are bounded, uniformly convex domains with smooth boundary in $\mathbb{R}^{n}$, $0<\alpha<1$, $0<\tau<\frac{\pi}{2}$.
  Then for any given initial function $u_{0}\in C^{2+\alpha}(\bar{\Omega})$
  which is  uniformly convex and satisfies $Du_{0}(\Omega)=\tilde{\Omega}$,  the  strictly convex solution of the following problem
  \begin{equation}\label{e4.7}
\left\{ \begin{aligned}\frac{\partial u}{\partial t}&= F_{\tau}(D^{2}u),
& t>0,\quad x\in \Omega, \\
Du(\Omega)& = \tilde{\Omega}, &t>0,\qquad\qquad\\
 u& = u_{0}, & t=0,\quad x\in \Omega.
\end{aligned} \right.
\end{equation}
   exists for all $t\geq 0$ and $u(\cdot,t)$ converges to a function $u^{\infty}(x,t)=\tilde{u}^\infty(x)+C_{\infty}\cdot t$ in $C^{1+\zeta}(\bar{\Omega})\cap C^{4+\alpha}(\bar{D})$ as $t\rightarrow\infty$
  for any $D\subset\subset\Omega$, $\zeta<1$, that is,
 $$\lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{1+\zeta}(\bar{\Omega})}=0,\qquad
  \lim_{t\rightarrow+\infty}\|u(\cdot,t)-u^{\infty}(\cdot,t)\|_{C^{4+\alpha}(\bar{D})}=0.$$
And $\tilde{u}^{\infty}(x)\in C^{\infty}(\bar{\Omega})$ is a solution of
\begin{equation}\label{e4.8}
\left\{ \begin{aligned}F_{\tau}(D^{2}u) & = C_{\infty},
&  x\in \Omega, \\
Du(\Omega)&=\tilde{\Omega}.
\end{aligned} \right.
\end{equation}
The constant $C_{\infty}$ depends only on $\Omega$, $\tilde{\Omega}$ and $F$. The solution to (\ref{e1.3}) is unique up to additions of constants.
\end{Example}
\begin{proof}
In recent papers \cite{HRY} and \cite{CHY}, the authors study second boundary value problems for a class of fully nonlinear flows with general parabolic
operators.
By Lemma 3.1, Lemma 3.11 in \cite{HRY} and boundary condition, we know the priori estimates \eqref{e1.4} holds.
Since the parabolic operator is concave with respect to the $D^2 u$ variables, by Remark \ref{r1.1}, we see that the estimate \eqref{e1.40} holds.
By Lemma 3.4 in \cite{HRY}, we know the strict obliqueness estimates hold. By Theorem \ref{t1.1}, we obtain the desired convergence results.
\end{proof}

\vspace{5mm}

config: arxiv


TITLE: Combinatorics problem arising from physics
QUESTION [2 upvotes]: I'm currently studying general mechanics where the following problem came up:
Assume we have the space $\Gamma = \mathbb{R}^6$ which we are dividing in small cells $C_i$ . Let $f(\vec{x}, \vec{v})$ be a probability distribution on our $\Gamma$. The amount of elements in a cell $C_i$ is (not exactly) given by: $N_i = \int_{C_i} f(\vec{x}, \vec{v}) d^3x d^3v$.
Since it doesn't matter in my problem how my elements are numbered, as long as there are the same amount of particles in each cell, I'm interested in finding out how many possibilities for one configuration of particles in cells there are.
Let $N$ be the total amount of particles i.e. $N = \sum N_i$. My initial guess is, that there are $N!$ different possibilities because there are $N$ slots. So the first particle has $N$ possible places, the second one has $N-1$ possible places $\implies$ $N \cdot (N-1) \cdot ... \cdot 1$ = $N!$
But this isn't correct yet, because if I have a cell that has, lets say, $2$ elements, there are $2!$ possible ways for the elements to get labeled which still give the same overall state, which means I have to divide by $2!$. This makes me say the correct result would be $\frac{N!}{N_1! \cdot N_2! \cdot ... \cdot N_k!}$ ($k$ = amount of cells), which I know is the right solution but I'd like to know it with some more mathematical rigor.
Anybody care to help me?
Cheers!

REPLY [1 votes]: I don't know this way has the rigour you want, but here goes:
You have $N$ total particles, $N_1$ go to the first cell. The number of ways to achieve this is
$${N\choose N_1} = \frac{N!}{(N-N_1)!N_1!}$$
You are left with $N-N_1$ remaining particles, and now $N_2$ of these go to the second cell. The number of ways this can be done is
$${N-N_1\choose N_2} = \frac{(N-N_1)!}{(N-N_1-N_2)!N_2!}$$
Continue this way till the end; the overall number of ways to do this exercise is
$${N\choose N_1}{N-N_1\choose N_2} \cdots {N-N_1-N_2\cdots -N_{k-1} \choose N_k}= \frac{N!}{N_1!N_2!\cdots N_k!}  $$
All the factors other than $N!$ and $N_i!$ get cancelled due to successive multiplication.


TITLE: Is $:A: \; \;= A - \left<0\right|A\left|0\right>$ a correct definition of normal ordering?
QUESTION [3 upvotes]: My course notes say that normal ordering is defined as
$$:A: \;\; = A - \left< 0\right| A \left| 0\right>.\tag{1}$$
This works for $A = aa^\dagger$ and all already normal ordered expressions.
When $A = a a^\dagger a$, though, or anything that is not normal ordered but has at least one annihilation operator furthest right, the second term is immediately $0$ and the expression returned is simply $A$, which is not normal ordered in this case.
$$
\begin{align*}
:aa^\dagger a: \; \; &= a a^\dagger a  - \left<0\right| a a^\dagger a \left| 0 \right> \\
&= a a^\dagger a - 0 \\
&= a a^\dagger a
\end{align*}\tag{2}
$$
$A = a a^\dagger a^\dagger$ also doesn't work.
Have I misunderstood something, or are my notes incorrect?

REPLY [2 votes]: OP is right:

Eq. (1) is strictly speaking meaningless since the argument for the normal ordering on the lhs. should be a function/symbol, not an operator.  See my Phys.SE answer here for a necessary condition.
Normal ordering takes a symbol/function into an operator with creation (annihilation) operators to the left (right), respectively. It does not take an operator into an operator, cf. e.g. my Phys.SE answer here.


TITLE: Is a density function integrable?
QUESTION [2 upvotes]: Let $X \in \mathbb{R}$ be a continuous random variable with density function $f$ (i.e. $f(x)\geq 0$ and $\int f(x)dx=1$). Does this mean that $\int |f(x)|dx < \infty$ i.e. $f \in L^1$?
(The reason I ask this question is because I know the characteristic function of $X$ always exists and I wanted to check that this is because $f \in L^1$.)

REPLY [5 votes]: The probability density function is non-negative. Hence, $|f(x)|=f(x)$ and
$$
\int |f(x)|\mathrm dx=\int f(x)\mathrm dx=1.
$$
It follows that $f\in L^1$.