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Thinking Beyond Borders' team is comprised of exceptional educators who are deeply committed to providing transformational learning and growth through our gap year programs and impact fellowships. Collectively, we are determined not just to help each of our students find the purpose and direction they need to build a fulfilling life, but also to help them understand how they can create meaningful impact on critical global and local issues.
US Staff
Thinking Beyond Borders is fortunate to have a talented and committed USA staff to offer support to students, family, and the Program Leaders. Their tireless work helps to ensure the program safety, quality, and success for TBB's gap year students.
Kelly Sanders, Director of Programs
Kelly serves as the Director of Programs for Thinking Beyond Borders. In this role she manages field staff, collaborates with our in country partners, and empowers staff to create a meaningful learning space for students rooted in TBB's pedagogy and values.
Kelly graduated from North Carolina State University in 2011 with a B.A. in International Studies/International Relations. During her time at NC State, she had the opportunity to study abroad for a semester in Santander, Spain. This began a love for living abroad and learning Spanish. After graduation, she completed two years of AmeriCorps service in Chapel Hill, NC working with newly resettled refugees with a local literacy center. Her AmeriCorps experience helped facilitate a cultural exchange within the United States, and inspired a new career path in International Education.
In 2014, Kelly moved to Guatemala City to work with a Guatemalan women's cooperative, Unidas Para Vivir Mejor, as the English Program Coordinator. She created English curriculum for a primary school and provided training for English teachers. She completed her M.A. in International Education Policy at the University of Maryland in 2017. As a Graduate Coordinator with the office of Leadership and Community Service Learning at UMD, she worked with undergraduate students to help facilitate their leadership in various service-learning programs. This role helped Kelly realize her passion is working more closely with students than in policy research. Kelly joined the TBB family in 2017 and led Latin America Semester programs and then the Global Gap Year in 2018. TBB was a life changing career move and allowed the opportunity to practice authenticity, vulnerability, cross cultural experiences, and self-care in impactful ways side-by-side with students. She is thrilled to be stepping to the Director of Program role to continue to grow in this unique space.
Isaac Galef-Brown, Director of Outreach and Program Development
Isaac serves as TBB's Director of Outreach and Program Development. In this position, he works in student recruitment and the admissions process, represents TBB at school and college fairs and manages alumni engagement and the long-term development of programs. Driven by his background as an educator, he is excited to explore new ways to make programming accessible to all students, to continue growing the TBB community and to creating humanizing, cross-cultural educational opportunities.
After completing his undergraduate degree at the College of Wooster, Isaac returned to his hometown of Cambridge, MA. Needing to find some direction at that critical post-college juncture, he joined City Year, an Americorps Program which placed him in a public elementary school in Boston. Thus began Isaac's career in education! Over the course of the next four years, he taught in several different special education settings and in a Spanish-English Dual-Language classroom. He observed differences in learning styles, teaching approaches, and school cultures, and disparities in resource availability, teacher training and classroom support that led him to a deeper understanding of how education systems can be profoundly inequitable. These realizations provided newfound perspectives on his own experience growing up attending urban public schools, but he also began to recognize the tremendous potential within education that, on a micro-scale, could be the catalyst for personal connection, and on a macro-scale could be a vehicle for something as broad and necessary as social change.
Having learned alongside some amazing teachers, and bolstered by a Master's Degree in Elementary and Special Education from Wheelock College, Isaac switched his focus in 2017 to International Education to realign his teaching experience with his love of travel and experiential learning and signed on with Thinking Beyond Borders. As a Program Leader, he led two Latin America Semester Programs and a Global Gap Year Program for TBB. The most meaningful part of the work for him has been building authentic, trusting relationships with his students which continue after graduation. He is also passionate about engaging with people around the world in ways that attempt to break down cross-cultural barriers, inherent assumptions and power dynamics and instead seek greater understanding and empathy.
Cara Ferragamo Murray, Vice President Sales & Marketing
As the Vice President, Sales & Marketing for Thinking Beyond Borders and Impact Global Education, Cara leads the branding, marketing, and acquisition strategies for the gap programs. With over 25 years of experience in brand management, relationship marketing, and partnership alliances, Cara is committed to providing young adults with experiences and activities that will prepare them for a global society, and equip them with the tools they need to become the next generation of social impact leaders.
Gary Robinson, Psychological Consultant
Gary has dedicated himself for over 30 years to helping college-aged students to maximize their potential and overcome barriers to learning. His clinical specialties have included stress management, mood and anxiety disorders and life coaching/mentoring. He graduated summa cum laude from the University of Pittsburgh with a Bachelor's degree in Social Work.
After completing his graduate work in Counseling Psychology at the State University of New York at Albany, Gary was hired to work for the Semester at Sea program, mentoring students as they visited eleven countries on this around-the- world shipboard voyage of discovery. For Gary, Semester at Sea cemented his interest in intercultural learning and he found applications for this interest in his subsequent positions including a stint as a Foreign Student Advisor at Keystone College in Pennsylvania.
Gary has served in a variety of professional counseling roles in several university and college settings and currently holds the post of Director of Counseling at Hartwick College in Oneonta, New York. At Hartwick, he helps students during study abroad experiences to negotiate the numerous challenges they face as they confront cultural environments different than their own. He is licensed as a mental health counselor in two states and is board certified nationally. In 2016 he was appointed for a 3-year term to the International Committee of the American Counseling Association.
Gary finds his consulting work with TBB to be a perfect environment to ply his skills by training, advising, and supporting TBB's Program Leaders in their work with students. As an advocate for cultural immersion and a true believer in the life-changing impact overseas travel and education can have on young adults, Gary has found no better program than TBB for gap year students. Strongly influenced by his work with TBB, Gary recently joined with two other licensed therapists to form P3 Mental Health Advisors. "P3" now assists a growing list of Gap Year and Experiential Education programs both domestically and abroad.
Program Leaders
Thinking Beyond Borders' Program Leaders bring an outstanding array of skills, knowledge, and experience to their work educating TBB students. Strong program management skills, diverse educational backgrounds, impressive teaching experiences, and extensive international travel make each of these individuals a powerful addition to our programs for gap year students. All Program Leaders are certified in Wilderness First Responder training and each TBB program provides (at minimum) one Master's level educator.
Airon Whitt, Program Leader
Airon is originally from southern Mississippi, and comes from a family of artists. She chose to study Sustainable Tourism and Development Management from the University of Hawaii. After graduating, she stayed in Hawaii for a total of 7 years working for some of the biggest hotels on Waikiki Beach… and learning how hard it is to be a surfer.
After Hawaii, Airon joined the Peace Corps in Costa Rica where she worked in small, rural tourism development and making arts and crafts with the senior citizen group of her community. She then went on to get a dual Master's degree in Business Administration and International Policy and Development from the Middlebury Institute of International Studies in Monterey, California.
Airon has worked for several international education programs and has led over 350 students throughout Mexico, the Caribbean, and Central America. She is currently completing her final practicum in Malaysian Borneo working with small, remote, jungle ecolodges. In her free time, she enjoys a good beach day, hiking to places with killer views, exploring new cultures, and teaching the world's travelers about the importance of responsible and sustainable tourism.
Kandis Pogoda, Program Leader
Growing up in Southern California, Kandis attributes her initial curiosity to learn more about the world from her diverse surroundings. From living blocks away from Little India, that still to this day is not little, to attending elementary school where her classes were sporadically taught in Spanglish, Kandis always sought more exposure to anything unfamiliar to her.
Kandis attended California State University, Long Beach (CSULB) where she earned her Bachelor of Arts in Sociology and Spanish. Though initially disenchanted with the idea of staying home for university, CSULB became her diving board into the world. From her first semester, Kandis became involved with international student learning community on campus then studied abroad with Semester at Sea on two consecutive summer voyages through Central America and the Mediterranean. This was Kandis's first taste of the world and when she returned to California, she knew she was far from done.
After completing her B.A., Kandis served as a Health Educator for the Peace Corps in rural Belize, primarily focusing on the promotion of family planning and education and prevention of non-communicable diseases. Additionally, through living with host families and frequently translating for foreign medics, Kandis solidified her fluency in Spanish.
Shortly after returning from Belize, Kandis rejoined CSULB as a staff member. She began working for the Education Abroad Office where she worked as an advisor for exchange programs in Asia and Australia while simultaneously studying for her Master's in Linguistics. Now, with her M.A. in hand and 29 international countries under her belt, Kandis decided to expand her skillset as an educator and lead students on international programs. Combining her enthusiasm to work more directly with students as well as her alignment with the philosophies and pedagogies of TBB, Kandis cannot be more excited to join the team and co-lead the upcoming GGY program!
Joselin Hernandez, Program Leader
A Nicaraguan native, Joselin is an avid traveler and international leader with more than eight years experience in the field of international education. She has run programs ranging in length from two weeks to one year, worked directly with students on the ground, as well as in managerial roles leading diverse teams of local and international staff.
Joselin holds a bachelor's degree in Management of Tourist Enterprises from UNAN-Leon, graduating as an honors student. Joselin continued her studies in Salzburg, Austria, where she graduated with distinction and with a Master's Degree in Teacher´s Training and Tourism Management. Joselin speaks Spanish, English, French and German fluently and has lived, worked and traveled in 18 different countries in Europe and Latin America.
Joselin believes in the transformational power of travel, and has observed as well as experienced firsthand how exposure to new cultures can open up new perspectives on life, as well as develop self-awareness and leadership skills. She has dedicated her career to facilitating life-changing travel experiences for youth delivering workshops, leading excursions, and engaging students in social, environmental and cultural discussions. Joselin lived in Ecuador for 3 years, which impacted the way she saw the world, and has informed her teaching methodology. This experience motivated her to learn about natural construction and ancestral healing practices from the South American native peoples as a way to interact with mother earth and one another in a more conscious and holistic way.
Joselin also facilitates alternative dance forms, such as the dance of the 4 elements: Earth, Fire, Wind and Water to improve one's mind, body and soul connection, supporting individuals to achieve a more balanced life.
She is very excited to work closely with our TBB students this coming spring.
Jamie Hackbarth, Program Leader
Jamie identifies as a global citizen. Her roots were raised in the heartland of Columbus, Ohio and her seed has been planted worldwide from Spain to rural Peru, and most recently Colorado. Jamie's worldview was shaped by her own impactful experiences including assisting at an orphanage in Honduras, studying the impacts of urban development throughout Central America, and diving into a language immersion program in Spain.
Jamie's passion to make a lasting impact inspired her to serve as a Peace Corps volunteer in rural Peru for 2+ Years, focused on education and development to enhance community health and economic outcomes for youth and adults. She led a peer-mentorship program for 25+ youth and served as an environmental and financial advisor to several community residents. In addition to this, Jamie taught yoga and meditation in the community.
Jamie has spent the past two years leading rural economic development programs for the State of Colorado. This challenging yet rewarding work has taught her that many of the development challenges found abroad are directly transferable to the issues faced in rural America. Jamie is a certified small business consultant serving as a mentor and educator to entrepreneurs and small business owners. During her tenure with the State of Colorado, Jamie continued to lead student groups on community development trips to Colombia and Peru.
Jamie has a B.A. in Strategic Communications with specialties in marketing and Spanish. This has led her on an enthralling journey, gaining experience in public relations, youth empowerment and education, community development, public health, business planning, and investment. She is a mother-nature advocate, outdoor enthusiast, life long learner, and an avid traveler. Jamie believes in the power of giving back to the community and is thrilled to be apart of the Thinking Beyond Borders family!
As a team, the founders of Thinking Beyond Borders endeavored to create educational programs for gap year students that would develop the skills and capacities shared by great agents of social change. Although TBB's founders no longer work directly with the organization, their combined experience and expertise made it possible to launch ambitious programs and build Thinking Beyond Borders to what it is today.
Robin Pendoley, Founder & President
All roads in Robin's life lead to co-founding Thinking Beyond Borders. Born and raised near San Francisco, he discovered at an early age that youth can affect the disparities and challenges of our world. Robin also learned the need to truly understand a problem and oneself before proactive changes can be made. He sees Thinking Beyond Borders as the actualization of those lessons.
Robin equipped himself by earning a B.A. in International Development Studies from UCLA and a Masters in Education from the Harvard Graduate School of Education. He has lived and worked abroad extensively, both as a student and educator. Robin committed eight years to public schools in Los Angeles and Boston. He served as a teacher, administrator, coach, and college access program administrator.
Robin's blog posts on education and social change have been featured on Forbes, Ashoka, and Innovation Excellence. He also served for three years as the Co-Director of the Colloquium for the Common Good at Opportunity Collaboration, a global poverty conference for sector leaders. Today, he helps shape the US gap year market as both the Co-Director of the USA Gap Year Fairs and as a Board Member for the American Gap Association.
Chris Stakich, Founder
Chris has lived with a passion for community service and international travel almost his entire life. While growing up in Cleveland, Ohio, Chris was the president of his high school community service organization, worked for the United Way, and traveled to the Dominican Republic during his senior year to help build a hospital. During his four years at Harvard University, Chris grasped the importance of perspective and dedicated valuable time to experiencing the world. He studied abroad in Australia and traveled throughout Europe while earning an economics degree, playing on the Crimson varsity football team, and holding a part time job. After graduating college, Chris accepted a position as the Business Development Director for Rustic Pathways, an international education, community service, and adventure travel company. In 2007 Chris co-founded Thinking Beyond Borders with Robin Pendoley and Sandy Cooper. Thinking Beyond Borders is a cultivated dream for Chris, integrating education and international travel, providing service to communities in need, and mindfully developing future leaders who embody broad perspective and global compassion. Currently, Chris serves as the Chairman of the Board for Thinking Beyond Borders and also as the President of Operations for Rustic Pathways.
Sandy Cooper, Founder
Sandy's deep determination to contribute proactively to society began in middle school. After volunteering at a nursing home in her hometown of Gloucester, Massachusetts, she worked with the YMCA through high school and college to develop children's self-esteem and social skills. While earning an honors degree from Skidmore College, Sandy studied abroad and taught conversational English at La Universidad Autonoma in Madrid, Spain.
Following her growing passion for educating and traveling, Sandy spent two years as a Peace Corps Volunteer in the Republic of Kiribati. As a teacher, Sandy designed culturally appropriate curriculum to teach language and math skills to junior secondary school students. Her students' performances on national exams surpassed all previous classes.
With new understanding for the needs of developing communities and the importance of informed leadership, Sandy became a Team Leader for AmeriCorps*NCCC upon her return to the US. Through her diligent and creative guidance, Sandy's team of 9 Corps Members completed various service projects, including the supervision of 1500 volunteers in gutting the homes of hundreds of Hurricane Katrina victims.
Managing her team challenged Sandy to hone her conflict resolution and business organization skills. She also holds a certificate in Nonprofit Management through American Humanics. These many years of volunteerism, community service leadership, and formal study both abroad and within the United States endow Sandy with a focused and powerful vision for Thinking Beyond Borders.
Thinking Beyond Borders works with a wide array of partners abroad and in the Fall of 2018, officially partnered with the three international education organizations listed below to form Impact Global Education (IGE), an international travel and education conglomerate. These partners are crucial to ensuring the quality, educational outreach, and recruitment of exceptional students for TBB's gap year programs.
Winterline
Winterline is a unique and innovative global gap, semester and summer program organization, offering high school graduates and young adults the opportunity to equip themselves with the global skills and competencies necessary to not only succeed, but to lead.
Think Impact
Think Impact is a leading Social Entrepreneur organization offering programming to students, educators and working professionals. Their mission is to start and grow cross-cultural social enterprises built on an asset-based approach and immersive collaboration.
Cross-Cultural Solutions
Cross-Cultural Solutions has has placed over 35,000 volunteers in its sites around the world over the last two decades. CCS believes that through meaningful volunteer service we can build relationships and connections that are essential components to increased tolerance and global awareness.
The Thinking Beyond Borders gap year curriculum for all of its programs has been developed and refined with the support of a highly qualified group of advisors. With broad ranging expertise, these individuals have brought insight with regard to curricular content, delivery, and assessment. Their work in collaboration with the Thinking Beyond Borders staff ensures that each student has a powerful learning experience during the course of the program.
Eleanor Duckworth, Curriculum Advisor
Eleanor Duckworth is a professor at the Harvard Graduate School of Education and the author of several books including The Having of Wonderful Ideas: And Other Essays on Teaching and Learning and "Tell Me More": Listening to Learners Explain. A former student and translator of Jean Piaget, she grounds her work in Piaget and Inhelder's insights into the nature and development of understanding and in their research method, which she has developed as a teaching/research approach, Critical Exploration in the Classroom. She seeks to bring a Freirean approach to any classroom, valuing the learners' experience and insights. Her interest is in the experiences of teaching and learning of people of all ages, both in and out of schools. Duckworth is a former elementary school teacher and has worked in curriculum development, teacher education, and program evaluation in the United States, Europe, Latin America, Africa, Asia, and her native Canada. She is a coordinator for Cambridge United for Justice with Peace, and is a performing modern dancer.
Amanda Frost, Curriculum Advisor
Once Mandy Frost's children were raised, which she realizes now is never a completed job, she needed to make full use of the dynamic changes happening in South Africa after years of feeling powerless. Her experience in various bookkeeping and labour management jobs gave her the admin skills needed to develop an NGO. Her passion for human rights and human development, defined by the Montessori education method, gave her the faith and sensitivity needed to work in a highly politicized milieu.
Mandy was appointed, in 2003, as coordinator of what was intended to be a planning body monitoring local HIV & AIDS projects and identifying needs. Three months of local networking revealed that there were no projects to monitor, but the needs were clear. She got clearance from the NGO governing body and started initiating and implementing projects that were in line with the national strategic plan for HIV in South Africa. Two of their biggest projects, Bitou Community Home-based Care/Hospice and Invicta House Womens' Shelter, are firmly established and integrated with other services. The Plettaid Foundation employs 35 people, has achieved membership of Hospice Palliative Care Association of SA and has a dream to build and run a grassroots hospice to support home-based care.
Sam Mitchell, Curriculum Advisor
Sam Mitchell passed away in the Fall of 2011. His belief in, contributions to, and support of TBB were fundamental to the formation of our programs. He was truly a friend. In celebration of his life and work, Thinking Beyond Borders created the Sam Mitchell Scholarship, awarded to a deserving TBB student each year.
Sam Mitchell received his PhD in Asian History at the University of Hawaii, where he was an East-West Center fellow. While undertaking language training in Hindi, Urdu, and Nepali, he lived in Pakistan for one year, Nepal for three and India for four. Sam lived in Yunnan Province, China for 16 years. Dr. Mitchell served as Academic Director of the School for International Training Nepal program from 1990-93 and was a visiting assistant professor of East Asian History at Western Washington University during the 2001-2002 academic year. He helped to curate an exhibition of his photographs, as well as clothing, cultural artifacts, and music from the ethnic minority groups of Northwest Yunnan entitled, "Yunnan: Enchanting Region of Ethnic Diversity" on display at the East-West Center and the University of Hawaii during February and March of 2004. Sam and his wife, Lu Yuan, created and initiated the Chinese Culture and Ethnic Minorities Program for what is now SIT Study Abroad, based in Brattleboro Vermont, in 1994. They resided in Kunming ever since.
Under the auspices of the Center for Cultural Learning and Development Sam facilitated cultural and Traditional Chinese Medicine programs for various universities, colleges from throughout the United States, including the gap year program, Thinking Beyond Borders. He was an avid hiker, snorkeler and photographer.
Joshua Muldavin, Curriculum Advisor
Dr. Joshua Muldavin has a long and distinguished record as a researcher and teacher concerning his interests in globalization, China, Japan, and Asia, policy analysis, rural development, international aid, agriculture and food, climate change, political economy, and political ecology. He has completed thirty years of field research, primarily in rural China, as well as in Japan, Nepal, India, Russia, Hungary, northern Europe, Cuba, and Mexico.
He was Chair and Director of International Development Studies at UCLA from 1994-2001. He was a Resident Research Fellow with the East West Center's Program on Environment in 2003, Visiting Scientist at the International Center for Integrated Mountain Research and Development (ICIMOD) in Kathmandu from 2006-2008, Visiting Professor at China Agricultural University from 2007-2008, Visiting Scholar at Sophia University in Tokyo 2006-2010, and is currently a Visiting Scholar at the China Academy of Social Sciences in Beijing, as well as the Executive Director of the Action 2030 Institute, a small policy think tank.
While at UCLA he won the Distinguished Teaching Award, and was also given the Eby Award as the best teacher of the year. He is a frequent contributor to international news reporting on China and Asia, including for the BBC, International Herald Tribune, South China Morning Post, Boston Globe, Los Angeles Times, New York Times, and the Wall Street Journal. At the annual meetings of the AAG in Washington, D.C., he was given the Award for Media Achievement of the Association of American Geographers for 2010 for his long-term efforts to bridge the gap between academic research and public awareness on key contemporary global issues. He is currently Professor of Geography at Sarah Lawrence College.
Manda Parikh, Curriculum Advisor
Dr. Manda Parikh earned a Masters in Economics at Gujarat University and a Ph.D. in Economics at M.S. University, Baroda. Her career pursuits reflect her deep commitment to the people and communities in and around the city of Ahmedabad, India. Currently, Dr. Parikh is the Director of the Rural Development and Management Institute at Ahmedabad, specializing in rural area research, training, and extension; the Director of Pranava Industrial Services Pvt. Ltd., a professional consultancy firm specializing in market, financial, and pre-investment studies with international operations; and the Director of International Programs for the Indian Association of Experiment in International Living. Additionally, she serves as a Trustee of the Indian Society For Population Education, the Indian Society for Community Education, Yuvak Vikas Trust, and Green Serve India Ltd. Through these diverse professional opportunities, Dr. Parikh works to empower urban and rural communities with education, sound and just economic practice, and conscious engagement. She draws inspiration from the work and writing of Mohandas K. Gandhi and his vision of a diverse, peaceful, and sustainable India.
Sherman Teichman, Curriculum Advisor
Sherman Teichman is the inaugural Executive Director of the Institute for Global Leadership at Tufts University. The programs under his direction include Education for Public Inquiry and International Citizenship (EPIIC), a rigorous interdisciplinary program for analysis of global issues and active citizenship; the Tufts Initiative for Leadership and International Perspective centered in Hong Kong and Beijing; [EXPOSURE], a photojournalism, documentary studies and human rights initiative and a range of global immersive education student initiated research projects.
A lecturer in the arts and social sciences at Tufts since 1984, Mr. Teichman was a former fellow and lecturer at the Institute of Politics, the Kennedy School of Government, Harvard University and a faculty member at Boston University and Emerson College. As a journalist, he was a Peabody Award-winning foreign policy analyst for National Public Radio, WBUR, in Boston and a social science editor of the Boston Review. An adviser on counterterrorism and long-range strategic planning for the Israeli government, Mr. Teichman was also active with Amnesty International in the former Soviet Union and Central America. He was educated at the United States Naval Academy, The Johns Hopkins University, The Hebrew University of Jerusalem and the University of Chicago's Committees on International Relations and Social Thought.
An Olympic saber fencer, Mr. Teichman is currently the Tufts University saber fencing coach. He previously coached fencing at Boston University, the University of Chicago and Harvard University. TBB is fortunate to work with Mr. Teichman and the Institute for Global Leadership at Tufts University. With similar core values and program goals, it is an organizational relationship with great potential for future developments.


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Tea can be easily brewed and enjoyed at home.
Q. I like to drink tea, but I'm tired of having the same flavor all the time. Is there any way to liven up my favorite drink?
A. Tea is not just tea anymore. The selection is immense, including brew-your-own flavored traditional teas blended with fruit or spices, flavored herbal teas or fruity bottled cold beverages.
There are several ways to liven up your tea. You can either choose a brand that has additional flavor to it (try one with a peppermint flavor during the holidays or maybe a hint of vanilla for something a little smoother).
If you like your tea slightly bitter or with a bite to it, stick to the herbal teas, such as Chai or various green teas. If you like the aroma of an herbal tea but don't like how strong it is, try brewing it with fewer tea leaves or take the bag out sooner. You can also mellow out your beverage by adding a little milk and sugar to your tea. You can also add a slice of lemon to your herbal tea, adding a little extra vitamin C in with all the antioxidants already present in many teas.
Whether you want to grow your own tea at home, learn about the many different types of tea, or just try something new during tea time, enjoying tea can be beneficial for both your mind and body. It's a great way to relax and warm you from the inside out on a cold day and it is also a great alternative to coffee if you want a little caffeine.


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View source for The hot spots conjecture
← The hot spots conjecture
The hotspots conjecture can be expressed in simple English as: : Suppose a flat piece of metal, represented by a two-dimensional bounded connected domain, is given an initial heat distribution which then flows throughout the metal. Assuming the metal is insulated (i.e. no heat escapes from the piece of metal), then given enough time, the hottest point on the metal will lie on its boundary. In mathematical terms, we consider a two-dimensional bounded connected domain D and let u(x,t) (the heat at point x at time t) satisfy the heat equation with Neumann boundary conditions. We then conjecture that : For sufficiently large t > 0, u(x,t) achieves its maximum on the boundary of D This conjecture has been proven for some domains and proven to be false for others. In particular it has been proven to be true for obtuse and right triangles, but the case of an acute triangle remains open. The proposal is that we prove the Hot Spots conjecture for acute triangles! Note: strictly speaking, the conjecture is only believed to hold for ''generic'' solutions to the heat equation. As such, the conjecture is then equivalent to the assertion that the generic eigenvectors of the second eigenvalue of the Laplacian attain their maximum on the boundary. A stronger version of the conjecture asserts that # For all non-equilateral acute triangles, the second Neumann eigenvalue is simple; and # The second Neumann eigenfunction attains its extrema only at the boundary of the triangle. (In fact, it appears numerically that for acute triangles, the second eigenfunction only attains its maximum on the vertices of the longest side.) == Threads == * [http://www.math.missouri.edu/~evanslc/Polymath/ Initial web page for proposal] * [http://www.math.missouri.edu/~evanslc/Polymath/Polymath.pdf A more mathematical description of the problem] * [http://polymathprojects.org/2012/06/03/polymath-proposal-the-hot-spots-conjecture-for-acute-triangles First research thread] (June 3, 2012) '''Inactive''' * [http://polymathprojects.org/2012/06/09/polymath7-discussion-thread/ Current discussion thread] (June 9, 2012) '''Active''' * [http://polymathprojects.org/2012/06/12/polymath7-research-thread-1-the-hot-spots-conjecture/ Research thread 1] (June 12, 2012) '''Inactive''' * [http://polymathprojects.org/2012/06/15/polymath7-research-threads-2-the-hot-spots-conjecture/ Research thread 2] (June 15, 2012) '''Active''' == Basic concepts == * [http://www.math.missouri.edu/~evanslc/Polymath/BasicBessel Basic properties of Bessel functions] * Rayleigh quotients * Finite element methods * Separation of variables == Possible approaches == === Combinatorial approach === * [http://www.math.missouri.edu/~evanslc/Polymath/Combinatorics A writeup of a combinatorial approach] === Sturm comparison approach === How about approximating the eigenfunction by polynomials? The Neumann boundary conditions already imply that the corners are critical points. Perhaps starting with that observation, for sufficiently low degree, the geometry of the triangle implies that one of them must be a max/min. In an ideal world, a 2d version of the Sturm comparison theorem (if one exists) could then show that this feature of the polynomial approximation remains true for the actual eigenfunction. In my understanding, Sturm-type results allow you to establish some qualitative property for solutions of an ODE (a 1d elliptic problem) provided that the desired property holds for a nearby solution of a nearby ODE. The property usually considered is the presence of a zero (node) of an eigenfunction in some interval. I think a similar result holds for critical points of eigenfunctions. Say you could establish that the second eigenfunction of the Laplacian on a right angled triangle satisfies the desired maximum principle by showing that its only critical points are at the vertices (so that whatever the maximum is, it would have to be at one of these points). Then appealing to a comparison theorem might be able to show that the same property holds for almost-right acute triangles. Mind you, this is all speculation at this point… Ah, so maybe a Sturm-type result can make rigorous the statement: "the second eigenfunction depends continuously on the domain"? By physical considerations, this statement ought to be true. But then it seems this would only work to prove the conjecture for triangles close to right triangles? As a side note, if such a continuity statement were to hold then the hot spots conjecture for acute triangles must be true by the following *non-math* proof: If it weren't true then by continuity there would be an open set of angles where it failed to hold. But after simulating many triplets of angles the conjecture always holds. C'mon, What are the odds we missed that open set? :-) It appears that Sturm's classical work has far reaching generalizations, as described for instance in this monograph: Kurt Kreith, Oscillation Theory LNM-324, (Springer, 1973). In particular, Chapter 3 features some comparison theorems for solutions to elliptic equations. === Subdivision === Maybe ideas based on self-similarity of the whole triangle to its 4 pieces can help (i.e. modeling the whole triangle as its scaled copies + the heat contact). Then without going into the graph approximation (which looks fruitful anyway), one can see some properties. One advantage in the graph case is that after dividing and dividing the triangles you get to the graph G_1 which is simply a tree of four nodes and there I think the theorem shouldn't be too hard to prove. And from there there might be an argument by induction. In the continuous case, no matter how many times you subdivide the triangle, after zooming it it is still the same triangle. On the other hand, in the continuous case, each of the sub-triangles is truly the same as the larger triangle. Does anyone know of good examples where self-similarity techniques are used to solve a problem? In either case, there is the issue that while the biggest triangle has Neumann boundaries, the interior subtriangles have non-Neumann boundaries… Something that is true and uses that the whole triangle is the union of the 4 congruent pieces is the following. From each eigenfunction of the triangle with eigenvalue lambda we can build another eigenfunction with eigenvalue 4 lambda by scaling the triangle to half length, and then using even reflection through the edges. Note that this eigenfunction will have an interior maximum since every point on the boundary has a corresponding point inside (on the boundary of the triangle in the middle). I'm skeptical this observation could be of any use. Hmm… but, apart from the case of an equilateral triangle, when you divide a triangle into four sub-triangles, I don't believe that the sub-triangles are reflections of one another. === Brownian motion === Suppose I start off a Brownian motion at time 0 at some point <math>x</math> in the triangle, and the Brownian motion reflects off the sides of the triangles. At time <math>t</math> the probability density for the Brownian motion to be at location <math>y</math> is given by the solution <math>v(t,y)</math> of the heat equation with <math>v(0,y)=\delta(x-y)</math> and Neumann boundary conditions. So if I can show that the Brownian motion is eventually more likely to be in a corner (per unit area) than anywhere else, that will gives us the result. Here's a way to approach it: Suppose I fix <math>x</math>. I ask how many ways are there for Brownian motion to get from <math>x</math> to <math>y</math> in time <math>t</math>. For <math>t</math> large, If <math>y</math> is in a corner I might imagine that there are more ways of Brownian motion to get to <math>y</math> than otherwise, because the Brownian motion has all the ways it would in free space, but also a ton of new ways that involve bouncing off the walls. Perhaps you can using some argument showing that there are more paths into some corners than anywhere else in the triangle. (Perhaps, using the Strong Markov Property and Coupling?) This approach is also discussed at http://www.math.missouri.edu/~evanslc/Polymath/Polymath.pdf, and a similar approach was pursued in [BB1999]. Important tools in this method are the ''synchronous coupling'' and ''mirror coupling'' of two Brownian motions. == Computational strategies == Consider the region in the plane <math> \mathcal{R} :=\{ (\alpha, \beta) \vert \frac{\pi}{2}\leq \alpha+\beta <\pi, 0<\alpha, \beta \leq \frac{\pi}{2} \} </math> Acute triangles will be characterized by angles <math> (\alpha, \beta, \pi-\alpha -\beta)</math>, where <math> (\alpha, \beta) \in \mathcal{R}. </math> So we can identify acute triangles with points in <math> \mathcal{R} </math> and ask: for each such triangle, where do the extrema of the 2nd Neumann eigenfunction of the Laplacian go? Note that the line <math> \alpha = \beta </math> in <math>\mathcal{R}</math> gives us the isoceles triangles. Triangles corresponding to <math>(\alpha, \beta)</math> will have the same (upto similarity) eigenfunction as <math> (\beta, \alpha) </math> So we need to examine triangles corresponding to the region <math> \mathcal{R}_1 :=\{ (\alpha, \beta) \vert \frac{\pi}{2}\leq \alpha+\beta <\pi, 0<\alpha\leq \beta \leq \frac{\pi}{2} \} </math> The points <math> (\frac{\pi-\epsilon}{2}, \frac{\pi-\epsilon}{2}) </math> for <math>\epsilon>0</math> small have been examined in the section on nearly degenerate triangles. The point <math> (\frac{\pi}{3},\frac{\pi}{3})</math> has been examined in the section on equilateral triangles. If we can numerically demonstrate for a dense subset <math> \mathcal{J} </math> of <math>\mathcal{R}_1</math> that the conjecture holds, then the suggestion is to use continuity arguments to conclude the result for all triangles in the region. More precisely, if on <math>\mathcal{J} </math> the extrema of the eigenfunctions are well away from other stationary points, and if the second derivatives are bounded away from zero, then the location of the extrema cannot shift dramatically as the angles in the triangle shift. For each point in <math> \mathcal{J}, </math> we would construct an approximation <math> u_J </math> to the desired eigenfunction to high accuracy, and locate the extrema. If one is using a Ritz strategy, these calculations can be made more efficient. Note that any acute triangle can be mapped to a right-angled one. The integrals involved need only be computed on the right triangle, with the pullbacks carefully accounted for. === Reformulation on reference (right) triangle=== http://www.math.sfu.ca/~nigam/polymath-figures/transform.jpg Let <math>\Omega,\hat{\Omega}</math> be the acute triangle and the reference triangle, respectively. The acute triangle has corners <math> (0,0), (a_1,0), (a_2,b_2)</math> and the reference triangle has corners <math>(0,0), (1,0), (0,1)</math> Let <math> F:\hat{\Omega} \rightarrow \Omega</math> be the affine mapping <math> F(\hat\mathbf{x}) = B \hat{\mathbf{x}} = \mathbf{x}.</math> The matrix <math> B </math> is given by <math> \left(\begin{array}{cc} a_1 & a_2\\ 0 & b_2\end{array} \right) </math> Then if <math> p</math> is a scalar function on <math>\Omega </math> the corresponding function <math> \hat{p} </math> on the reference triangle <math> \hat{\Omega}</math> is given by <math> \hat{p} = p\circ F, \,\mbox{and} \, (B^{-1})^T \hat{\nabla}\hat{p} = (\nabla p) \circ F </math> The problem of locating the second eigenpair of the Neumann Laplacian on <math> \Omega </math> has two formulations, and a corresponding reformulation on the reference triangle. '''Formulation 1:''' Find <math> (u,\lambda) \in H^1(\Omega) \times \mathbb{R}^+ </math> so that <math> -\Delta u = \lambda u\,\mbox{in}\,\Omega,\,\, \frac{\partial u}{\partial n} =0\,\,\mbox{on}\,\partial\Omega,\, \int_\Omega u dV =0.</math> '''Formulation 1 on reference triangle:''' Find <math> (\hat{u}, \lambda) </math> so that <math> -\nabla \cdot (M \nabla \hat{u}) = \lambda \hat{u}\,\,\,\mbox{in}\,\hat{\Omega}, M\nabla \hat{u} \cdot \hat{n} =0\,\,\mbox{in}\,\partial\hat{\Omega},\, \int_{\hat{\Omega}} \hat{u} (det(B)) dV = a_1b_2\int_{\hat{\Omega}} \hat{u} dV= 0.</math> Here the matrix <math> M = B^{-1} (B^{-1})^T = \frac{1}{a_1^2b_2^2} \left(\begin{array}{cc} a_2^2+b_2^2 & -a_1a_2\\ -a_1a_2 & a_1^2\end{array} \right) </math> '''Formulation 2:''' Find the minimizer of the functional <math> \mathcal{I}_\Omega := \frac{ \int_\Omega |\nabla u|^2 dV}{ \int_\Omega | u|^2 dV} </math> over the set of non-zero functions in <math> H^1(\Omega) </math> with zero mean <math> \int_\Omega u dV =0.</math> '''Formulation 2 on reference triangle:''' Find the minimizer of the functional <math> \mathcal{I}_{\hat{\Omega}} := \frac{ \int_{\hat{\Omega}} \hat{\nabla} \hat{u}^T M \hat{\nabla} \hat{u} (det(B))dV}{ \int_{\hat{\Omega}} | \hat{u}|^2 (det B) dV} = \frac{ \int_{\hat{\Omega}} \hat{\nabla} \hat{u}^T M \hat{\nabla} \hat{u} dV}{ \int_{\hat{\Omega}} | \hat{u}|^2 dV}</math> over the set of non-zero functions with zero mean <math> \int_{\hat{\Omega}} \hat{u} (det B) dV =0 \Rightarrow \int_{\hat{\Omega}} \hat{u} dV=0 .</math> === Discrete approximation of 2nd eigenfunctions=== We're interested in numerical approximation of the 2nd Neumann eigenfunctions for triangles characterized by angles in the domain BDH in http://polymathprojects.files.wordpress.com/2012/06/triangle21.png Each point in this domain will require one eigenvalue calculation as follows. ====Physical domain specification ==== To fix notation, we're considering the domain given in http://www.math.sfu.ca/~nigam/polymath-figures/sine.jpg We've fixed one corner at the origin (Corner 1), another at (1,0) (Corner 2), and specified the interior angles <math> \alpha,\beta,\gamma</math>. We first need to calculate <math>(a_2,b_2), </math> the coordinates of the third corner. This is done using the law of cosines: <math> A=\frac{\sin(\alpha)}{\sin(\gamma)}, B= \frac{\sin(\beta)}{\sin(\gamma)}, \Rightarrow a_2=\frac{1}{2}(1- (\frac{\sin(\alpha)}{\sin(\gamma)})^2 + (\frac{\sin(\beta)}{\sin(\gamma)})^2), b_2=a_2 \tan(\alpha) </math> The area of this triangle is <math>b_2</math>. Call this triangle <math>\Omega</math> as above. ====Approximation algorithm specification ==== Step 0. We pick a point in parameter space, <math> (\alpha, \beta, \gamma) </math> and locate the associated physical domain, <math> \Omega.</math> We want to solve the eigenvalue problem on <math> H^1(\Omega).</math> Step 1. We first discretize the physical domain <math> \Omega</math> by a triangular mesh <math> \Tau_h = \cup_{i=1}^N \tau_i</math> where the individual triangles <math> \tau_j</math> have sizes parametrized by a parameter <math>h>0</math>, where <math> h \rightarrow 0</math> as <math> N </math> increases. Step 2. We now introduce a finite dimensional subspace <math> V_h \subset H^1(\Omega), V_h=span\{\phi_i\}_{i=1}^M </math> consisting of finite element basis functions <math> \phi_i</math> on <math> \Tau_h.</math> We have used both deg= 1 and deg= 2 finite elements, where <math> V_h:=\{ w_h \subset H^1(\Omega) \vert w_h \vert_{\tau_i}= \mbox{polynomial of degree}\, \leq deg ,\mbox{for each triangle} \,\, \tau_i \in \Tau_h\} </math> Step 3. We want to solve the discrete analog of Formulation 1: find <math> v_h =\sum_{i=1}^M c_i \phi_i \in V_h, \lambda_h \in \mathbb{R}^+ </math> to solve the generalized eigenvalue problem <math> A_h C = \lambda_h B_h C, \qquad \int_\Omega v_h =0</math> where <math> C=(c_1,c_2,...c_m)^T, (A_h)_{ij} = \int_\Omega\nabla \phi_i \cdot \nabla \phi_j, B_h = \int_\Omega \phi_i \phi_j </math> The triangulation of each domain is specified using Matlab. The quadratures are performed in Matlab, using over 200 points on each triangle <math> \tau_i.</math> The generalized discrete eigenvalue problem is solved using Matlab's eigs command, which in turn calls the ARPACK Arnoldi iteration with shift <math> \sigma=11</math>. The iteration runs until a prescribed tolerance of 1e-8 is reached. Step 4. We repeat Step 1-3 with smaller and smaller h, until we have achieved a tolerance for the desired eigenvalue calculation on this triangle. At the finest level, use a mesh with <math> h<1/4500.</math> Step 5. We repeat Steps 1-4 for a variety of points in parameter space. Results are reported in the 'Numerics' section below. == Special cases == === Isosceles triangles === :'''Proposition 1''' Let ABC be an acute isosceles triangle with AB=AC, and let u be a second eigenfunction which is symmetric with respect to reflection around the axis AD of symmetry (D being the midpoint of BC). Then the extrema of u only occur at the vertices A,B,C. '''Proof''' Being symmetric, u is also a Neumann eigenfunction of the right-angled triangle ADB of mean zero. It must be the second eigenfunction of that triangle, since any eigenfunction of lower energy can be reflected to give a lower energy eigenfunction of ABC. By Theorem 1 of [AB2004], the eigenfunction on this triangle ADB (which is a Lip domain) is simple. By the argument used to prove Theorem 3.3 of [BB2000], after a change of sign, the eigenfunction u on ADB is non-decreasing in the AD and DB directions. As such, it cannot attain an extremum anywhere on ADB except at A and B (and hence, by symmetry, u cannot attain an extremum on ABC except at the vertices) unless it is constant and extremal either on a segment of AD touching A, or a segment of BD touching B. But either case will ensure that u is zero (this can be seen for instance by Bessel function expansion at the vertex where this constancy occurs; one can presumably also use standard unique continuation results). This proves the claim. [] Note that from Proposition 2.3 of [BB2000] that if we are in the thin case <math>AB/BC > 2j_0/\pi \approx 1.53096</math>, where j_0 is the first zero of the Bessel function J_0, then all second eigenfunctions are symmetric (because there are no anti-symmetric eigenfunctions, and one can decompose any eigenspace into symmetric and antisymmetric functions when there is an axis of symmetry such as AD). This gives the conjecture for isosceles triangles with apex angle less than about 38 degrees. This was strengthened in [LS2009] to show that any subequilateral triangle (apex angle less than 60 degrees) has a symmetric second eigenfunction, which is simple by Theorem 1 of [AB2004] (or by observing that the nodal line, being symmetric, cannot pass through A). We thus have: :'''Corollary 2''' The hot spots conjecture is valid for isosceles triangles ABC with AB=BC and <math>\angle BAC \leq \pi/3= 60^\circ</math>. For super-equilateral triangles, it is a result of [LS2009] that the second eigenfunctions are anti-symmetric, and can thus be viewed as the lowest-energy Dirichlet-Neumann eigenfunctions on ADB with Dirichlet boundary conditions on AD and Neumann conditions on AB, BD. Since lowest-energy Dirichlet-Neumann eigenfunctions must have only one sign (either from the Courant nodal theorem or by minimising the Rayleigh-Ritz quotient), this implies that they are simple (otherwise one could take a linear combination of two lowest-energy eigenfunctions to obtain a signed eigenfunction). Now consider the following ''synchronously coupled Brownian motion'' on ADB. Start with two points x, y in the interior of ADB. We then define coupled Brownian motions <math>X_t, Y_t</math> initialised as <math>X_0=x, Y_0=y</math> and solving the Brownian motion equation :<math>dX_t = dB_t; dY_t = dB_t</math> with the ''same'' Brownian forcing term <math>dB_t</math>, so long as one is inside the triangle ADB, but with reflection whenever one hits the Neumann walls AB and BD and with absorption (i.e. the motion terminates) whenever one hits the Dirichlet wall AD. (The corners are almost surely avoided, so one does not need to define what happens there.) Formal details of this construction can be found for instance in [BB2000] or [AB2004]. (Alternatively, one can take parallel Brownian motions on the whole plane starting from x,y, and fold them over into the triangle ADB, cutting them off when they hit AD. Let n be the outward normal to AB. Let us say that y is ''to the right'' of x if either y=x, or the ray from x to y is oriented between the normal n and the ray from A to D. :'''Lemma 3''' If y is to the right of x, then <math>Y_t</math> is to the right of <math>X_t</math> as long as both paths survive. (In particular, <math>X_t</math> will hit the absorbing boundary AD first.) '''Proof''' (This is the argument from [BB2000], expressed somewhat informally; some more details are [http://www.math.missouri.edu/~evanslc/Polymath/MixedTriangle here].) As long as <math>X_t, Y_t</math> stay away from the boundary of the triangle, they move in parallel, and the claim is clearly preserved. If one of <math>X_t,Y_t</math> hits the absorbing boundary AD, there is nothing left to prove. If <math>X_t,Y_t</math> ever collide, then they will agree for all subsequent times and we are again done. As <math>Y_t</math> is to the right of <math>X_t</math>, <math>X_t</math> cannot hit AB or DB until <math>Y_t</math> does - but when that happens, the ray joining <math>X_t</math> to <math>Y_t</math> moves either towards the ray DB or the ray AB, and either such operation preserves the property of <math>Y_t</math> being to the right of <math>X_t</math>. [] :'''Corollary 4''' If we consider any solution u to the heat equation on ADB with the Dirichlet-Neumann data which is non-decreasing in the directions n and AD, then it will remain non-decreasing in these directions for all time. '''Proof''' This is immediate from Lemma 3 and the Brownian motion interpretation of the heat equation. But one can also establish it via a vector-valued maximum principle argument, as follows. At the initial time zero, we see from hypothesis that u is non-negative at time zero, and thus non-negative for all future times by the scalar maximal principle. Also, at time zero, that the gradient <math>\nabla u(0,x)</math> lies in the sector S through the origin with axes oriented in the directions DB, AB. The claim is then that the gradient stays in S for all time. For technical reasons (which are the same technical reasons that come up every time one tries to prove a maximum principle) it is convenient to first show that for any <math>\varepsilon>0</math>, the gradient stays in the <math>\varepsilon(t+1)</math>-neighborhood <math>S_{\varepsilon(t+1)}</math> of S for all time <math>t \geq 0</math>, since the claim then follows by sending <math>\varepsilon</math> to zero. If the claim failed, then there must be a first time t and a point x in ADB where <math>\nabla u(t,x)</math> first touches the boundary of <math>S_{\varepsilon(t+1)}</math> at some value v. (It should be possible to show that u is C^1 in space continuously in time, which is enough regularity to justify the above claim.) This time cannot be 0, since the boundary stays away from S. The point x cannot be on the Dirichlet boundary AD, since at that point the gradient must lie on the ray from the origin oriented in the DB direction (because of the non-negativity of u), and thus lies in S and cannot equal v. If x is in the interior of ADB, then from construction we see that <math>\partial_t \nabla u(t,x)</math> points strictly outwards from v in <math>S_{\varepsilon(t+1)}</math>, but that <math>\Delta \nabla u(t,x)</math> points inwards or tangentially, which is a contradiction since the heat equation implies that <math>\partial_t \nabla u= \Delta\nabla u</math>. If x is the corner D, then <math>\nabla u = 0</math>, a contradiction. If x lies on the interior of DB, then <math>v=\nabla u(t,x)</math> must be parallel to DB while remaining on the boundary of <math>\partial S_{\varepsilon(t+1)}</math>, and v is thus the point of <math>\partial S_{\varepsilon(t+1)}</math> that is distance <math>\varepsilon(t+1)</math> from the origin in the BD direction. One can then argue as before (after performing a reflection if necessary) to show that <math>\Delta \nabla u(t,x)</math> points inwards or tangentially while <math>\partial_t \nabla u(t,x)</math> points outwards, again a contradiction. Similarly if x is on the interior of AB. This covers all cases, and the claim follows. [] Picking generic data obeying the hypotheses of Corollary 4 to isolate the lowest energy eigenfunction, we conclude that (up to sign) the lowest energy eigenfunction is also non-decreasing in the directions n, AD, and thus obtains its non-zero extremum at the vertex B, and vanishes only at the Dirichlet boundary AD, giving the hot spots conjecture for super-equilateral triangles. === Equilateral triangles === It is convenient to work in the plane <math>\{ (x,y,z): x+y+z=0\}</math>, and with the triangle with vertices (0,0,0), (1,0,-1), (1,-1,0). A function on this triangle with Neumann data can be extended symmetrically to the entire plane, in such a way that it becomes periodic with respect to translation by (2,-1,-1), (-1,2,-1), and (-1,-1,2). In particular, it has a Fourier series arising from functions of the form <math>\exp(2 \pi i (ax+by+cz)/3 )</math> for integers a,b,c summing to 0; also, the Fourier coefficients need to be symmetric with respect to the reflections <math>(a,b,c) \mapsto (-a,-c,-b), (-c,-b,-a), (-b,-a,-c)</math>. To minimise the Rayleigh quotient, we see that (a,b,c) should only live among the six frequencies (1,0,-1), (1,-1,0), (0,1,-1), (-1,1,0), (0,-1,1), (-1,0,1), leading to the complex solution :<math> \exp( 2\pi i (y-z)/3 ) + \exp( 2\pi i (z-x)/3 ) + \exp( 2\pi i (x-y)/3 )</math> and its complex conjugate as basis vectors for the two-dimensional eigenspace. (See [McC2011], [McC2002] for more discussion.) All real second eigenfunctions are projections of this complex eigenfunction. With this normalization, the second eigenvalue is <math>8\pi^2/9</math>. After this, the next eigenvalue comes from the complex solution :<math> \exp( 2\pi i (2x-y-z)/3 ) + \exp( 2\pi i (2y-z-x)/3 ) + \exp( 2\pi i (2z-x-y)/3 )</math> and its complex conjugate, which have eigenvalue <math>8\pi^2/3</math>. An alternate (but slightly less symmetric) description is given in [LS2009], where the triangle is now in the plane with vertices (0,0), (1,0), <math>(1/2,\sqrt{3}/2)</math>. Now the second eivengalue is <math>16 \pi^2/9</math>, with eigenfunctions :<math>2 (\cos(\pi (2x-1)/3) + \cos(2\pi y/\sqrt{3})) \sin(\pi(2x-1)/3)</math> (anti-symmetric around x=1/2) and :<math>\cos(2\pi(2x-1)/3) - 2 \cos(\pi(2x-1)/3) \cos(2\pi y/\sqrt{3})</math> (symmetric around x=1/2). The complex second eigenfunction appears to map the equilateral triangle to a slightly concave version of that triangle, so that the extreme points are always on the corners. Has anyone done numerical work yet to estimate how sensitive the eigenvalue degeneracy lifting is to perturbations of a starting equilateral triangle? Any proof will have to address the issue that the third eigenvalue may be arbitrarily close to the second and has an eigenfunction with extremes at a different pair of corners. Here is some preliminary data. As I am not sure how to simulate the true eigenfunctions of the triangle, the following is for the graph whose eigenvectors should roughly approximate the true eigenfunctions: For the case that a=b=c=1 (Equilateral Triangle): HotSpotsAny(64,1,1,1,2) yields the corresponding eigenvalue -0.001070825047269 HotSpotsAny(64,1,1,1,3) yields the corresponding eigenvalue -0.001070825047269 HotSpotsAny(64,1,1,1,4) yields the corresponding eigenvalue -0.003211901603854 We see that indeed the eigenvalue -0.001070825047269 has multiplicity two. Perturbing a slightly we have that for a=1.1, b = 1, c = 1 (Isosceles Triangle where the odd angle is larger than the other two): HotSpotsAny(64,1.1,1,1,2) yields the corresponding eigenvalue -0.001078552707489 HotSpotsAny(64,1.1,1,1,3) yields the corresponding eigenvalue -0.001131412869938 Whereas for a=.9, b = 1, c = 1 (Isosceles Triangle where the odd angle is smaller than the other two): HotSpotsAny(64,.9,1,1,2) yields the corresponding eigenvalue -0.001004876221957 HotSpotsAny(64,.9,1,1,3) yields the corresponding eigenvalue -0.001062028119964 In either case we see that the third eigenvalue is perturbed away from the second. What strikes me as interesting is how different the outcome is in increasing a by 0.1 versus decreasing a by 0.1. In the former case, the second eigenvalue barely changes whereas in the latter case the second eigenvalue changes quite a bit. I imagine this ties into the heuristic "sharp corners insulate heat" — reducing a to .9 produces a sharper corner leading to more heat insulation and a much smaller (in absolute value) second eigenvalue, whereas increasing a to 1.1 makes that corner less sharp but the other two corners are still relatively sharp so the heat insulation isn't affected as much. Just a guess though… It looks like the degeneracy lifting might be linear. It's the ratio of the second and third eigenvalues that matters more, which is close to the same in either case, and approximately half as far from 1 as the perturbation. Using (9/10, 1, 1) should be the same as using (1, 10/9, 10/9) and then scaling all eigenvalues by 90%. === Obtuse triangles === The obtuse case was settled in [BB1999] using a reflected Brownian motion approach. The "hot" and "cold" spots are located at the most distant vertices. === Right-angled triangles === The right-angled case of the conjecture is known, for instance it follows from the more general results in [AB2002] on lip domains (domains between two Lipschitz graphs of constant at most 1). If ABC is a right angled triangle with right angle at B, then the second eigenfunction is simple, and (after changing sign if necessary) is monotone non-decreasing in the AB, BC directions. === Isosceles right-angled triangles === Consider the triangle with vertices (0,0), (1,0), (0,1). A Neumann function on this triangle can be reflected into a function on the entire plane which is periodic with periods (0,2), (2,0) and is symmetric with respect to reflection around the axes x=0, y=0, x=y. As such, the eigenfunctions can be easily computed by Fourier series; the second eigenfunction is simple, and takes the form <math>\cos(\pi x) + \cos(\pi y)</math> (up to multiplication by constants). === 30-60-90 triangle === Another triangle which can be computed explicitly by reflection is the 30-60-90 triangle, as this can be reflected into the equilateral triangle (which in turn can be reflected into the entire plane). In particular in the plane <math>\{ (x,y,z): x+y+z=0\}</math>, with the 30-60-90 triangle with vertices (0,0,0), (1,-1,0), (1,-1/2,-1/2), there is a simple second eigenfunction <math>\cos(\pi(x-y)/3) + \cos(\pi(y-z)/3) + \cos(\pi(z-w)/3)</math>. === Thin sectors === Consider the sector <math>\Omega_0:=\{ (r \cos \theta, r \sin \theta): 0 \leq r \leq R; 0 \leq \theta \leq \alpha \}</math> with <math>\alpha</math> small. By separation of variables, one can use an eigenbasis of the form <math>u(r,\theta) = u(r) \cos \frac{\pi k \theta}{\alpha}</math> for <math>k=0,1,2,\ldots</math>. For any non-zero k, the smallest eigenvalue of that angular wave number is the minimiser of the Rayleigh quotient <math>\int_0^R (u_r^2 + \frac{\pi^2 k^2}{\alpha^2} u^2)\ r dr / \int_0^R u^2\ r dr</math>; the <math>k=0</math> case is similar but with the additional constraint <math>\int_0^R u\ r dr = 0</math>. This already reveals that the second eigenvalue will occur only at either k=0 or k=1, and <math>\alpha</math> small enough it can only occur <math>k=0</math> (because the <math>k=1</math> least eigenvalue blows up like <math>1/\alpha^2</math> as <math>\alpha \to 0</math>, while the <math>k=0</math> eigenvalue is constant). Once <math>\alpha</math> is small enough that we are in the k=0 regime (i.e. the second eigenfunction is radial), the role of <math>\alpha</math> is no longer relevant, and the eigenfunction equation becomes the Bessel equation <math>u_{rr} + \frac{1}{r} u_r + \lambda u = 0</math>, which has solutions <math>J_0(\sqrt{\lambda} r)</math> and <math>Y_0(\sqrt{\lambda} r)</math>. The <math>Y_0</math> solution is too singular at the origin to actually be in the domain of the Neumann Laplacian, so the eigenfunctions are just <math>J_0(\sqrt{\lambda} r)</math>, with <math>\sqrt{\lambda} R</math> being a zero of <math>J'_0</math>. The second eigenfunction then occurs when <math>\sqrt{\lambda} R</math> is the first non-trivial zero of <math>J'_0</math> (<math>3.8317\ldots</math>, according to Wolfram alpha). This is a function with a maximum at the origin and a minimum at the circular arc, consistent with the hot spots conjecture. In Lemma 5.1 of [LS2009], this calculation is established for sectors of angle <math>\alpha \leq \pi/2.68 \approx 67^\circ</math>. After this point, the k=1 mode dominates instead. === Thin not-quite-sectors === From here, I was thinking along the following lines. The wedge can be turned into a nearly degenerate triangle <math> \Omega_1</math> by joining the two rays by the chord connecting the points <math>(R,0)</math> and <math>(R,\epsilon)</math>. Now suppose there is a map <em>F</em> such that the thin side of the triangle can be written parametrically as <math>(R+F(\theta),\theta)</math>. We know precisely the 2nd Neumann eigenfunction and eigenvalue on the thin wedge <math> \Omega_0</math>. We hypothesize this eigenfunction on the thin triangle is maximized at the same corner. We consider the situation on intermediate domains <math>\Omega_\delta</math> which are described as <math> \{(r(\theta),\theta)\vert 0<r<R+\delta F(\theta), 0<\theta<\epsilon\}</math> We can expand the Neumann eigenfunction in this intermediate region in terms of a series in powers of <math>\delta</math> as <math> u_\delta(r,\theta) = \sum_{n=0}^\infty \delta^n v_n(r,\theta)</math>. The first term in the sequence is just in terms of <math>J_0</math>. I think one can compute the higher order terms easily, and that ''these will satisfy a related, but not identical, problem on the fixed wedge'' <math> \Omega_0</math>. If the map <math> F </math> is smooth (analytic?) I think such a series must be in fact analytic; and that while <math> \delta=1</math> may be outside the disk of analyticity, it is still in the domain of analyticity. This will enable us to get the desired lower bounds on the eigenvalue. In fact, I think the analytic continuation literature may have some pointers. I'll try to add references tomorrow. However, if the mapping is not smooth this approach may fail. In this situation, one may yet be able to get a good lower bound on the true eigenvalue on the triangle. If <math> F</math> is not smooth, there will be a smooth map <math> G </math> which will not parametrize the chord, but some closeby curvilinear segment. The behaviour of the eigenfunction on the resulting domain can be obtained by the process described above. I think it is important is to get away from the wedge (where the tangents to the arc are perpendicular to the sides of the sector, rendering it 'morally right-angled') in some smooth fashion to a geometry where the tangent to the curve <math> (R+\delta F(\theta), \theta)</math> is ''not'' perpendicular to the sides. === Thin triangles === Main article: [[thin triangles]] === General domains === A counterexample to the hot spots conjecture for a domain with two holes was established in [BW1999]. A further counterexample in [BB2000] shows that both the maximum and minimum may be attained in the interior. On the other hand, the conjecture was established for convex domains with two axes of symmetry in [JN2000]. == Simplicity of eigenvalue == It is possible to show simplicity of the eigenvalue for non-equilateral triangles by getting a sufficiently good upper bound on the second eigenvalue <math>\lambda_2</math>, and a sufficiently good lower bound on the sum <math>\lambda_2+\lambda_3</math> of the second and third eigenvalues. The upper bound on <math>\lambda_2</math> can be obtained by using suitable test functions for the Rayleigh quotient, for instance by taking second eigenfunctions of reference triangles (e.g. equilateral, isosceles right-angle, or 30-60-90) and affinely transforming them to the triangle of interest. Lower bounds on <math>\lambda_2+\lambda_3</math> can be obtained by lower bounding quadratic forms such as <math>\int_{\Omega} |\nabla u_1|^2 + |\nabla u_2|^2</math> for orthonormal mean zero <math>u_1,u_2</math>, again by comparing with reference triangles where explicit bounds are known. An example of this method appears at http://pages.uoregon.edu/siudeja/simple.pdf == The nodal line == Courant's nodal line theorem tells us that the nodal set <math>\{u=0\}</math> of the second eigenfunction divides the domain T into at most two regions. Since u is mean zero, it must change sign, and so in fact it divides the domain into exactly two regions. It is in fact known that the nodal line is a continuous curve (ref?). It is not a closed loop, because if it enclosed a region D, then the first Dirichlet eigenvalue of D would be equal to the second Neumann eigenvalue of T. But the first Dirichlet eigenvalue of D exceeds the first Dirichlet eigenvalue of T, and a classical inequality of Polya asserts that the second Neumann eigenvalue of T is less than or equal to the first Dirichlet eigenvalue of T, giving a contradiction. Thus the nodal line must instead hit the boundary of T at two points. A variant of this argument shows that the two boundary points of the nodal line cannot lie on the same edge of the triangle T. For if this were the case, we could reflect around this edge and obtain a Neumann eigenfunction on a kite Q whose nodal line encloses a domain D. As before, the first Dirichlet eigenvalue of D equals the second Neumann eigenvalue of T, which is one of the Neumann eigenvalues of Q. On the other hand, as Q is convex, the results of [Fr1995] imply that the first Dirichlet eigenvalue of Q is greater than or equal to the third Neumann eigenvalue of Q. So we need to show that the unfolded version of the second Neumann eigenfunction of T is the second or third Neumann eigenfunction of Q. If this were not the case, then the second and third Neumann eigenfunctions of Q must be antisymmetric instead of symmetric. But then one of these eigenfunctions must change sign on T (as they will be orthogonal to each other), and thus have at least four nodal sign domains, contradicting the Courant nodal theorem. The nodal line of an equilateral triangle passes through a vertex. If the triangle is perturbed away from equilateral in an asymmetric fashion, numerics suggest the nodal line moves rapidly away from the vertex. http://www.math.sfu.ca/~nigam/polymath-figures/nearly-equilateral-1.jpg == Regularity and stability theory == In [CZ1994] some bounds for the Neumann heat kernels <math>P_t(x,y)</math> on a general domain are given; in particular, one can bound this kernel by a multiple of the Euclidean heat kernel for t small enough. Using this bound, [BP2008] showed uniform stability of the second eigenfunction (in the uniform topology) and eigenvalue with respect to uniform perturbations of the domain. In a sector <math>\{ (r \cos \theta, r \sin \theta): r > 0; 0 \leq \theta \leq \alpha \}</math>, solutions to the eigenfunction equation <math>\Delta u = -\lambda u</math> with Neumann data can be computed using separation of variables in polar coordinates as :<math> u(r,\theta) = \sum_{k=0}^\infty u^{(k)}(r,\theta)</math> where :<math> u^{(k)}(r,\theta) = c_k J_{\pi k / \alpha}(\sqrt{\lambda} r) \cos( \frac{\pi k}{\alpha} \theta ),</math> <math>c_k</math> are real coefficients, and <math>J_\beta</math> are the usual Bessel functions :<math>J_\beta(r) = \frac{1}{2^{\beta-1} \Gamma(\beta+\frac{1}{2}) \sqrt{\pi}} r^\beta \int_0^1 (1-t^2)^{\beta-1/2} \cos(rt)\ dt</math>. One consequence of this particular representation of the Bessel functions is that we observe the pointwise bounds :<math> c_\beta \leq J_\beta(r)/r^\beta \leq C c_\beta</math> for all <math>0<r<1</math> (say) and <math>\beta \geq 0</math>, and some constants <math>c_\beta, C</math>. Among other things, this gives pointwise bounds of the form :<math> u^{(k)}(r,\theta) = O( (r/r_0)^{\pi k/\alpha} (\frac{1}{\alpha} \int_0^\alpha |u(r_0,\theta)|^2\ d\theta)^{1/2} )</math> whenever <math>\sqrt{\lambda}r, \sqrt{\lambda} r_0 < 1</math>. If u is locally in L^2, this gives bounds near the corner of the sector of the form :<math> u^{(k)}(r,\theta) = O( (r/r_0)^{\pi k/\alpha} )</math> uniformly in k. Among other things, this implies for acute sectors <math>\alpha<\pi/2</math> that <math>\nabla^2 u</math> is bounded (and even Holder continuous) all the way up to the corner. In particular, Neumann eigenfunctions on acute triangles are <math>C^{2,\varepsilon}</math> all the way up to the boundary. The same analysis shows that third derivatives <math>\nabla^3 u</math> blow up by at worst <math>O(|x-x_0|^{-1+\varepsilon}</math> as one approaches an acute vertex <math>x_0</math>. Away from the corners, u is smooth, as can be seen by reflection arguments and elliptic regularity. On an acute triangle T, suppose we normalise the Neumann eigenfunction u as :<math>\displaystyle \int_T |u|^2 = |T|.</math> (E-0) Integration by parts, using the Neumann condition to eliminate boundary terms, then yields :<math>\int_T |\nabla u|^2 = \lambda |T|.</math> (E-1) One can carefully justify this integration by parts by cutting out the corners, replacing them with small circular arcs, applying Stokes' theorem, then sending the arcs to zero. A second integration by parts (still using the Neumann condition to eliminate boundary terms) gives :<math>\int_T |\nabla^2 u|^2 = \lambda^2 |T|.</math> (E-2) It turns out that the <math>C^{2,\varepsilon}</math> regularity of u, combined with the <math>O(|x-x_0|^{-1+\varepsilon})</math> decay of the third derivative, allows one to integrate by parts one final time to obtain :<math>\int_T |\nabla^3 u|^2 = \lambda^3 |T|,</math> (E-3) because the effect of the boundary terms at the corner circular arcs still goes to zero. Note that the boundary terms still vanish, because they involve products of pairs of derivatives of u, one of which contains an odd number of normal derivatives, which then vanishes on the boundary by symmetry. Among other things, this gives good C^2 bounds for a thin acute triangle ABC, with A near (0,0) and B,C within <math>O(\delta)</math> of (1,0) for some small <math>\delta = |BC|</math>. It should be the case here that <math>\lambda</math> is comparable to 1. Assuming this, we see from (E-0)-(E-3) that the first three derivatives of u have average size O(1) on T. The Bessel function analysis around A then already gives C^2 bounds of O(1) on the half of the triangle that is closer to A than to BC. On the other end, one can use the Poincare inequality to conclude that the average value of <math>|u|+ |\nabla u|+ |\nabla^2 u|</math> on the <math>O(\delta)</math>-neighbourhood of BC is also O(1), which by Bessel function analysis around B or C should then give C^2 bounds of O(1) on u in (a slightly smaller version of) this neighbourhood. Putting this all together, I think one has uniform C^2 bounds on all of T that are independent of delta. == Eigenvalue bounds == In [LS2009] it is shown that for any triangle of diameter D, the quantity <math>\lambda_2 D^2</math> is minimised for degenerate acute isosceles triangles (or for sufficiently acute sectors). In [LS2009] it is shown that for a triangle of perimeter P, <math>\lambda_2 P^2</math> is maximised at the equilateral triangle: :<math>14.7 \approx j_{1,1}^{2} \leq \lambda_2 D^2</math> and :<math>\lambda_2 (P/3)^2 \leq \frac{16\pi^2}{9} \approx 17.5.</math> In [Ch1975] the general upper bound :<math>\lambda_2 D^2 \leq 4j_{0,1}^2 \approx 23.1</math> for all convex domains is established; it becomes sharp for degenerate obtuse triangles. Here is a simple eigenvalue comparison theorem: if <math>T: D \to TD</math> is a linear transformation, then :<math>\|T\|_{op}^{-2} \lambda_k(D) \leq \lambda_k(TD) \leq \|T^{-1}\|_{op}^2 \lambda_k(D)</math> for each k. This is because of the Courant-Fisher minimax characterisation of <math>\lambda_k(D)</math> as the supremum of the infimum of the Rayleigh-Ritz quotient <math>\int_D |\nabla u|^2/ \int_D |u|^2</math> over all codimension k subspaces of <math>L^2(D)</math>, and because any candidate <math>u \in L^2(D)</math> for the Rayleigh-Ritz quotient on D can be transformed into a candidate <math>u \circ T^{-1} \in L^2(TD)</math> for the Rayleigh-Ritz quotient on TD, and vice versa. More sophisticated comparison theorems involving comparison of one triangle against two reference triangles appear in [LS2009]. One corollary of this theorem is that if one has a spectral gap <math>\lambda_2(D) < \lambda_3(D)</math> for some triangle D, then this spectral gap persists for all nearby triangles TD, as long as T has condition number less than <math>(\lambda_3(D)/\lambda_2(D))^{1/2}</math>. In [BP2008] it is also shown under the hypothesis of simple eigenvalues that the eigenfunctions vary uniformly with respect to small perturbations. == Miscellaneous remarks == If the sides of the triangle have lengths A, B, and C, currently you are using edge weights a=A, b=B, and c=C. This solves a different heat equation, when you pass to the limit, than the one you want. The assumption that each small triangle is at a uniform temperature gives an extra boost to the overall heat conduction rate in those directions along which the small triangles are longer, so in the limit you are modeling heat conduction in a medium with anisotropic thermal properties. It is just as though you started with a material of extremely high thermal conductivity and then sliced it in three different directions, with three different spacings, to insert thin strips of insulating material of a constant thickness. Based on some edge cases, I suspect that the correct formulas for edge weights to model an isotropic material in the limit are as follows: <math>a = 1/(-A^2+B^2+C^2)</math>, <math>b=1/(A^2-B^2+C^2)</math>, <math>c=1/(A^2+B^2-C^2)</math>. Interestingly enough, these formulas are only defined (with positive weights) in the case of acute triangles, which suggests that this approach, if it works, may not provide an independent proof of the known cases. Nadirashvili [Na1986] has shown that the multiplicity of the second eigenvalue is at most 2; a simpler proof was given in [BB1999]. == Software == * [http://www.math.missouri.edu/~evanslc/Polymath/HotSpotsAny.m A MATLAB program used to generate and display the eigenvectors of the graphs G_n]. The input format is HotSpotsAny(n,a,b,c,e), where a,b,c are the edge weights, n is the number of rows in the graph, and e determines which eigenvector of the graph Laplacian is displayed. (Note that in the proposal write up the graph G_n has 2^n rows so if you wanted to simulate the Fiedler vector for G_6 with edge weights a=1,b=2,c=3 then you would type HotSpotsAny(64,1,2,3,2)). For large n, the eigenvectors of the graph Laplacian should approximate the eigenfunctions of the Neumann Laplacian of the corresponding triangle… so the program can be used to roughly simulate the true eigenfunctions as well. == Numerics == Using a finite element method, here are the eigenfunctions of the Neumann Laplacian for a couple of acute triangles: * http://people.math.sfu.ca/~nigam/polymath-figures/isoceles1.jpg * http://people.math.sfu.ca/~nigam/polymath-figures/scaleneeig.jpg The domain is subdivided into a finite number N of smaller triangles. On each, I assumed the eigenfunction can be represented by a linear polynomial. Across interfaces, continuity is enforced. I used Matlab, and implemented a first order conforming finite element method. In other words, I obtained an approximation from a finite-dimensional subspace consisting of piece-wise linear polynomials. The approximation is found by considering the eigenvalue problem recast in variational form. This strategy reduces the eigenvalue question to that of finding the second eigenfunction of a finite-dimensional matrix, which in turn is done using an iterative method. As N becomes large, my approximations should converge to the true desired eigenfunction. This follows from standard arguments in numerical analysis. Additionally, I've computed a couple of approximate solutions to the heat equation in acute triangles. The initial condition is chosen to have an interior 'bump', and I wanted to see where this bump moved. Again, I've plotted the contour lines of the solutions as well, and one can see the bump both smoothing out, and migrating to the sharper corners: * http://people.math.sfu.ca/~nigam/polymath-figures/isoceles.avi * http://people.math.sfu.ca/~nigam/polymath-figures/isoceles2.avi * http://people.math.sfu.ca/~nigam/polymath-figures/scalene.avi I think in the neighbourhood of the corner with interior angle <math>\frac{\pi}{\alpha}</math>, the asymptotic behaviour of the (nonconstant) eigenfunctions should be of the form <math>r^\alpha \cos(\alpha \theta) + o(r^{\alpha})</math>, where r is the radial distance from the corner. Next, if one had an unbounded wedge of interior angle <math>\frac{\pi}{\alpha}</math>, the (non-constant) eigenfunctions of the Neumann Laplacian would be given in terms of <math>P_n(r,\theta) = J_{\alpha, n} (\sqrt(\lambda) r) cos(\alpha n \theta), n=0,1,2\ldots</math>. The <math>J_{\alpha,n}</math> are Bessel functions of the first kind. The spectrum for this problem is continuous. When we consider the Neumann problem on a bounded wedge, the spectrum becomes discrete. This makes me think approximating the second eigenfunction of the Neumann Laplacian in terms of a linear combination of such <math>P_n(r,\theta)</math> could be illuminating. This idea is not new, and a numerical strategy along these lines for the Dirichlet problem was described in [FHM1967]. Betcke and Trefethen have a nice recent paper [BT2005] on an improved variant of this 'method of particular solutions'. ===Results of finite element+Arnoldi iteration calculations === Here are some graphics, to explore the parameter region (BDH) above. The calculations are performed using FEM code developed by Joseph Coyle. To enable visualization, data is plotted as functions of the (alpha,beta). We taking a rectangular grid oriented with the sides BD and DH, with 25 steps in each direction. So there are (25)^5 grid points. Each parameter <math> (alpha,beta,gamma) </math> yields a triangle <math>\Omega</math>. We fix one side to be of unit length. For details, please see the section on Computational Strategies above. For each triangle, the second Neumann eigenvalue and third eigenvalue (first and second non-zero Neumann eigenvalue) is computed. We also kept track of where max|u| occurs, where u is the second eigenfunction. This is because numerically I can get either u or -u. A plot of the 2nd Neumann eigenvalue as we range through parameter space is here: http://www.math.sfu.ca/~nigam/polymath-figures/Lambda2.jpg A plot of the 3rdd Neumann eigenvalue as we range through parameter space is here: http://www.math.sfu.ca/~nigam/polymath-figures/Lambda3.jpg A plot of the spectral gap, \lambda_3-\lambda_2 multiplied by the area of the triangle \Omega as we range through parameter space is here: http://www.math.sfu.ca/~nigam/polymath-figures/SpectralGap.jpg One sees that the eigenvalues vary smoothly in parameter space, and that the spectral gap is largest for acute triangles without particular symmetries. For each triangle, we also kept track of the physical location of max|u|. If it went to the corner (0,0), I allocated a value of 1; if it went to (1,0) I allocated a value of 2, and if it went to the third corner, we allocated 3. If the maximum was not reported to be at a corner, we put a value of 0. http://www.math.sfu.ca/~nigam/polymath-figures/LocationOfAbsMax.jpg shows the result. Note that we obtain some values of 0 inside parameter space. ''This cannot be interpreted to mean the conjecture fails''. Rather, this is a signal that eigenfunction is likely flattening out near a corner, and that the numerical values at points near the corner are very close. == Bibliography == * [AF2012] Antunes, Pedro R. S; Freitas, Pedro, "[http://msvlab.hre.ntou.edu.tw/AntunesFreitas Numerical optimization of low eigenvalues of the Dirichlet and Neumann Laplacians]" to appear in Journal of Optimization Theory and Applications doi:10.1007/s10957-011-9983-3 (2012). * [AB2002] R. Atar and K. Burdzy (2002), [http://webee.technion.ac.il/people/atar/lip.pdf On Neumann eigenfunctions in lip domains], JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 17, Number 2, Pages 243--265. * [BO2005] I. Babuska, J. Osborn, [http://www.sciencedirect.com/science/article/pii/S1570865905800420 Eigenvalue problems], Handbook of Numerical Analysis Volume 2, 1991, Pages 641–787, Finite Element Methods (Part 1) * [Ba2000] R. Bañuelos, [http://www.math.purdue.edu/~banuelos/Papers/hotarticle.pdf In search of hot spots], Math PUrview, Summer 2000. * [BB1999] Rodrigo Bañuelos and Krzysztof Burdzy. [https://digital.lib.washington.edu/dspace/bitstream/handle/1773/2195/paper60.pdf?sequence=1 On the "hot spots" conjecture of J. Rauch]. J. Funct. Anal., 164(1):1–33, 1999. * [BP2008] Bañuelos, Rodrigo, Pang, Michael M. H., [http://eudml.org/doc/130789;jsessionid=080D9E5423278BA5ACFC818847CA97FE Stability and approximations of eigenvalues and eigenfunctions for the Neumann Laplacian. I.] Electron. J. Differential Equations 2008, No. 145, 13 pp. * [BH2010] A. Barnett, A. Hassell, [http://arxiv.org/abs/1107.2172 Estimates on Neumann eigenfunctions at the boundary, and the "Method of Particular Solutions" for computing them], Conference paper from a talk given at the International Conference on Spectral Geometry, Dartmouth College, July 2010. See also the slides at http://www.math.dartmouth.edu/~specgeom/Hassell.pdf * [BB2000] Richard F. Bass and Krzysztof Burdzy, [http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.dmj/1092403814 Fiber Brownian motion and the "hot spots" problem], Duke Math. J. Volume 105, Number 1 (2000), 25-58. * [Bu2002] K. Burdzy, [http://arxiv.org/pdf/math/0203017.pdf On nodal lines of Neumann eigenfunctions]. * [BW1999] K. Burdzy, W. Werner, [http://www.jstor.org/stable/121027 A counterexample to the hot spots conjecture]. Ann. of Math. (2) 149 (1999), no. 1, 309–317. * [BT2005] Timo Betcke, Lloyd N. Trefethen, [http://eprints.ma.man.ac.uk/589/01/MPSfinal.pdf Reviving the Method of Particular Solutions], Society for Industrial and Applied Mathematics Vol. 47,No . 3,pp . 469–491 * [CaZh1994] R.A. Carmona, W.I. Zheng, [http://www.sciencedirect.com/science/article/pii/S0022123684710858 Reflecting Brownian Motions and Comparison Theorems for Neumann Heat Kernels], Journal of Functional Analysis, Volume 123, Issue 1, July 1994, Pages 109-128 * [Ch1975] S.Y. Cheng, Eigenvalue comparison theorems and its geometric applications, Math. Z. 143 (3) (1975) 289–297. * [Ch1997] Fan R. K. Chung. Spectral graph theory, volume 92 of CBMS Regional Conference Series in Mathematics. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1997. * [Fi1973] Miroslav Fiedler. Algebraic connectivity of graphs. Czechoslovak Math. J., 23(98):298– 305, 1973. * [Fi1975] Miroslav Fiedler. A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory. Czechoslovak Math. J., 25(100)(4):619–633, 1975. * [FHM1967] L. Fox, P. Henrici and C. Moler, [http://www.jstor.org/stable/10.2307/2949737 Approximations and Bounds for Eigenvalues of Elliptic Operators], SIAM Journal on Numerical Analysis , Vol. 4, No. 1 (Mar., 1967), pp. 89-102. * [Fr1995] L. Friedlander, [http://www.jstor.org/stable/2375044 Remarks on Dirichlet and Neumann Eigenvalues], American Journal of Mathematics Vol. 117, No. 1 (Feb., 1995), pp. 257-262 * [G1985] P. Grisvard, Elliptic problems in non-smooth domains, Pitman 1985. * [HR2002] E. Hernandez, R. Rodriguez, [http://www.ams.org/journals/mcom/2003-72-243/S0025-5718-02-01467-9/S0025-5718-02-01467-9.pdf Finite element approximation of spectral problems with Neumann boundary conditions on curved domains], MATHEMATICS OF COMPUTATION Volume 72, Number 243, Pages 1099--1115. * [JN2000] D. Jerison, N. Nadirashvili, [http://www.ams.org/journals/jams/2000-13-04/S0894-0347-00-00346-5/S0894-0347-00-00346-5.pdf The "Hot Spots" Conjecture for Domains with Two Axes of Symmetry], Journal of the American Mathematical Society Vol. 13, No. 4 (Oct., 2000), pp. 741-772 * [LS2009] R. Laugesen, B. Siudeja, [http://arxiv.org/abs/0907.1552 Minimizing Neumann fundamental tones of triangles: an optimal Poincare inequality], Journal of Differential Equations, Volume 249, Issue 1, 1 July 2010, Pages 118–135. * [LS2009b] R. Laugesen, B. Siudeja, [http://dx.doi.org/10.1016/j.jde.2010.02.020 Maximizing Neumann fundamental tones of triangles], 22 pages. Journal of Mathematical Physics, 50:112903, 2009. * [McC2002] Brian J. McCartin, [ftp://ftp.math.ethz.ch/EMIS/journals/HOA/MPE/a223.html Eigenstructure of the equilateral triangle, Part II: The Neumann problem], Mathematical Problems in Engineering", Vol. 8 , Issue 6, Pages 517-539. * [McC2011] Brian J. McCartin, [http://www.m-hikari.com/mccartin-3.pdf Laplacian structure of the equilateral triangle], 2011. * [MP1968] C. B. Moler, L. E. Payne, [http://www.jstor.org/stable/2949550 Bounds for Eigenvalues and Eigenvectors of Symmetric Operators], SIAM Journal on Numerical Analysis Vol. 5, No. 1, Mar., 1968. * [Na1986] N.S. Nadirashvili, On the multiplicity of the eigenvalues of the Neumann problem, Soviet Mathematics, Doklady, 33, 281–282 (1986). * [PW1960] L. E. Payne and H. F. Weinberger. [http://www.springerlink.com/content/96378j462l111tw6/ An optimal Poincare inequality for convex domains]. Arch. Rational Mech. Anal. 5 (1960), 286–292.
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Consternation of a second run-way that will serve as alternative taxi way for aircraft in the event of future maintenance work on the first runway has began at the Akwa- Ibom State- Owned airport.
Speaker of the State House of Assembly, Rt. Hon. Onofiok Luke who disclosed this to newsmen at the Port Harcourt International Airport, Omagwa, weekend on his way to Abuja said everything is being done under the able leadership of Governor Udom Emmanuel to expand the airport.
Luke described the Akwa-Ibom airport at Uyo, as the best State-owned airport in the country, pointing out that the government is ensuring that there is no hitch in operations, especially in the run-way area.
"We would not like what happened in Abuja recently to repeat in Akwa- Ibom where the entire airport was closed for some months because of repair and maintenance work on the run-way.
"Because there was no alternative taxi-way, that was why the entice airport went out of use, and the Kaduna airport become the alternative then,'' he said.
On other infrastructural facilities at the airport, the speaker said that government is still talking with private investors for repairs and maintenance of infrastructure to see how they can get the facilities to be managed.
According to him, the landing service at the airport is quite fantastic as every other thing that is required are being put in place as the airport is expanding.
Luke however, commended the efforts of Governor Emmanuel, whom he described as a captain, adding that the governor is truly ensuring that nothing is lacking at the airport, which truly will be a testimony that the airport is the best State-owned airport in the country so far.


Many medical schools are associated with a free clinic. Students who speak and learn Spanish can help fill this gap.
AMA J Ethics. 2019;21(3):E232-238. doi: 10.1001/amajethics.2019.232.
Although undocumented immigrants contribute to the pool of available organs and to the US tax base, they are not eligible for organ transplantation in most US states.
AMA J Ethics. 2019;21(1):E17-25. doi: 10.1001/amajethics.2019.17.
Trafficking-specific ICD-10-CM codes account for physical, social, and psychological dimensions of trafficked patients' experiences. Data collected by clinicians can also motivate improvements in health policy, resource allocation, and prevention.
AMA J Ethics. 2018;20(12):E1143-1151. doi: 10.1001/amajethics.2018.1143.
Physicians should be aware of the powerful impact celebrities' cancer narratives can have on patients' experiences of their illnesses and treatment decisions. Partnering with celebrities is one strategy for delivering evidence-based health information and messaging to the public.
AMA J Ethics. 2018;20(11):E1075-1081. doi: 10.1001/amajethics.2018.1075.
Should NASA Collect Astronauts' Genetic Information for Occupational Surveillance and Research?
Though the National Aeronautics and Space Administration's collection of disaggregated genetic data for occupational surveillance and research raises numerous privacy concerns, the Genetic Information Nondiscrimination Act of 2008 allows genetic information to be used to develop personal pharmaceuticals.
AMA J Ethics. 2018;20(9):E849-856. doi: 10.1001/amajethics.2018.849.
Gene editing with CRISPR/Cas9 raises concerns about equitable access to therapies that could limit research participation by minority group members. These concerns can be addressed through public education, transparency, and stakeholder partnerships.
AMA J Ethics. 2018;20(9):E826-833. doi: 10.1001/amajethics.2018.826.


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Stuart Deignan takes the reins at tech consultancy Globant
IT consultancy Globant has a new Managing Director for its UK operation: former consultant Stuart Deignan.
Founded in 2003 by a quartet of co-founders in Buenos Aires, Globant is an IT firm that specialises in digital consulting, technology, software development and emerging technologies. The company was Latin America's first IT consultancy to go public in the US in 2014, and today has more than 16,000 employees across offices in nearly twenty countries.
Stuart Deignan has been active in the world of IT consulting for some two decades, and for him, "Globant is one of the most exciting tech organisations today," not just because of the firm's scale and capabilities, but also because the firm helps its clients deliver "meaningful and transformative" impact.
Now taking the reins of Globant's UK wing, Deignan has been tasked with spearheading the company's nationwide expansion strategy. "I plan to replicate Globant's explosive growth across LATAM and EMEA here in the UK," he said.
Deignan will leverage a more than twenty-five year career, most of which in the consulting industry. Following a stint at BP, he joined PA Consulting, where he spent eight years focused on clients in the energy sector. Deignan went on to work for CGI, latterly as a Vice President and global leader in its Oil & Gas vertical, and Wipro and ThoughtWorks, where he was Chief Growth Officer and a member of the Board.
"We are delighted to have Stuart on board and to benefit from his experience in organisational change. He will play a key role as we expand our footprint in the UK – a gateway to Europe and beyond," said Federico Pienovi, General Manager of Globant's Europe & Asia division.
The appointment comes shortly after Globant bolstered its cloud capabilities and headcount in the United Kingdom with the acquisition of CloudShift (70 professionals).
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