title: Stony Brook University

text: 2002 saw the opening of the $22 million Kenneth P. LaValle Stadium and the inauguration of the massive Charles B. Wang Center dedicated to Asian and American culture, funded by a $50 million donation from Charles B. Wang. At the time, it was the largest private donation to a SUNY institution. In 2003, chemistry professor Paul Lauterbur received a Nobel Prize in Physiology or Medicine for his research and discovery of Nuclear Magnetic Resonance, which was instrumental in the development of NMR Imaging (MRI) while at Stony Brook. In 2005, the university bought the Flowerfield property adjacent to campus through eminent domain as land for the development of a Research and Development Park. Plans for a law school were in the talks but scrapped shortly after.


title: Stony Brook University

text: In 2009, president Shirley Strum Kenny stepped down, and in May, Dr. Samuel Stanley, Jr. was announced as Stony Brook's fifth president. The late 2000s saw the university receive historic philanthropic donations. Hedge funder Jim Simons made multiple multi-million donations, including a $25 million donation to the Stony Brook Foundation in 2006, a $60 million donation for the development of the Simons Center for Geometry and Physics in 2008, and a landmark $150 million donation to the university in 2011. Other major donations were provided by alumni Joe Nathan, Stuart Goldstein, and Glenn Dubin for major renovation of athletic facilities. In 2010 Dr. Stanley announced Project 50 Forward, a comprehensive plan for the development of the university in the next fifty years with a focus on "operational excellence, academic greatness, and building for the future."


title: Stony Brook University

text: In 2012, the $40 million Walter J. Hawrys Campus Recreation Center opened, soon followed by the on-campus Hilton Garden Inn in May 2013. Frey Hall, named after alumnus Dr. Robert Frey, opened in 2013 after undergoing renovations as the former Old Chemistry building. The Stony Brook University Arena underwent a $21 million overhaul, re-opening as the Island Federal Arena in 2014. In July 2015, a new $40.8 million Computer Science building opened, spanning . New dormitories, known as Chavez Hall and Tubman Hall, along with a new East Side Dining hall, opened in the fall of 2016. In January 2019, Stony Brook Medicine opened their $194 million cancer center to the public.


title: Stony Brook University

text: In 2018, Stony Brook received its best ranking yet in the U.S. News & World Report, at 80th in the United States.


title: Stony Brook University

text: President Stanley left Stony Brook effective August 1, 2019 to become the president of Michigan State University following the Larry Nassar scandal. Provost Michael A. Bernstein was named interim president in his place. On March 26, 2020, Maurie McInnis, the executive vice president and provost of the University of Texas was named the sixth president of Stony Brook, effective July 1, 2020.


title: Stony Brook University

text: During the COVID-19 pandemic, the United States Army constructed $50 million temporary field hospitals on the campus of Stony Brook University. However, the beds were dismantled in 2021 without ever being used. A $63.4 million renovation of the Stony Brook Union was completed during the pandemic and opened in 2020.


title: Stony Brook University

text: On January 5, 2022, New York governor Kathy Hochul officially designated Stony Brook University and the University at Buffalo as the two flagship universities of the State University of New York system. The announcement came with the dedication of a new $100 million multidisciplinary engineering building. On January 25, 2022, Stony Brook announced its departure from the America East Conference after 21 years to the Colonial Athletic Association, effective July 1, 2022.


title: Stony Brook University

text: In the 2022–23 edition of the U.S. News & World Report national college rankings, Stony Brook was ranked 77th, its highest-ever placement as well as the first time that Stony Brook held sole possession of the highest-ranked public university in New York.


title: Stony Brook University

text: The main campus is in the historic north shore hamlet of Stony Brook near the geographic midpoint of Long Island, approximately east of Manhattan and west of Montauk. Bounded to the north by New York State Route 25A (North Country Road) the campus is subdivided into "West Campus" and "East Campus" by the thoroughfare County Road 97 (Nicolls Road). The Ashley Schiff Forest Preserve separates the South Campus from West Campus. The Long Island Rail Road serves the community with the Stony Brook station situated along the northern edge of the campus.


title: Stony Brook University

text: The west campus is the center of the academic life of the university. It houses the majority of academic, athletic, and undergraduate student housing facilities while also being the original site of the university.


title: Stony Brook University

text: The modern campus is centered around the Academic Mall, which stretches for more than a quarter of a mile from the Simons Center for Geometry and Physics at the west end to the Administration Building at the east end. The Academic Mall includes the Student Activity Center, Frank Melville Jr. Memorial Library, Staller Center for the Arts, Humanities building, Psychology A & B, Harriman Hall, Frey Hall (previously known as Old Chemistry), the Earth and Space Sciences Building, Math Tower, and Physics building.


title: Stony Brook University

text: The Engineering Quad is located near the Academic Mall, and contains the Engineering, Light Engineering, Heavy Engineering, and Computing Center facilities. The Javits Lecture Center, Social and Behavioral Sciences Building, Computer Science building, New Computer Science building and Student Union facilities are also on the west campus. The Life Sciences complex, also on the west campus, consists of the Life Sciences Building, Laufer Center for Physical and Quantitative Biology, Centers for Molecular Medicine, Bioengineering building, and the Institute for Advanced Computational Science.


title: Stony Brook University

text: Among the latest additions to the campus are the Simons Center for Geometry and Physics, the new Walter J. Hawrys Campus Recreation Center, the Hilton Garden Inn hotel, Frey Hall, and a new Computer Science building. The Staller Center, which contains the largest movie screen in Long Island's Suffolk County, holds the annual Stony Brook Film Festival.


title: Stony Brook University

text: The athletic facilities are in the northwest quadrant of west campus, which include the Stony Brook Sports Complex, Island Federal Credit Union Arena, Kenneth P. LaValle Stadium, Joe Nathan Field, University Track, and University Field.


title: Stony Brook University

text: The South Campus is about half a mile south of the Academic Mall and separated from West Campus by the Ashley Schiff Forest Preserve. It is home to the School of Dental Medicine, the School of Marine and Atmospheric Sciences (SoMAS), the Cody Center for Autism and Developmental Disabilities, and the University Police headquarters.


title: Stony Brook University

text: The Research and Development Park is on Stony Brook Road, a mile from the center of campus. On November 3, 2005, the university announced it had formally acquired of the adjacent Flowerfield property, originally owned by the St. James Gyrodyne Company of America, through eminent domain, three years after the university had expressed its desire to acquire the property.


title: Stony Brook University

text: Stony Brook is using this property as a Research and Development Park, similar to other university-affiliated science parks around the country. The campus will ultimately house ten new buildings. The first building, the Center of Excellence in Wireless and Information Technology (CEWIT), was completed in October 2008. CEWIT houses the world's first (and currently the largest) immersive gigapixel facility in the world, called the Reality Deck. In 2019, Stony Brook University celebrated the opening of its astonishing new SMART Cluster in CEWIT, a dual use GPU Cluster for both machine learning and visualization. The SMART Cluster is also the first hardware-accelerated ray-tracing cluster for real-time cinematic quality rendering, allowing scientists, engineers and physicians to visualize huge amounts of data in a new way.. Construction for the Advanced Energy Research and Technology Center, designed by Flad Architects, commenced in the Summer of 2008 and is open as of spring 2010.


title: Stony Brook University

text: The East Campus is separated from the main campus by Nicolls Road (County Road 97). It is home to the Stony Brook University Hospital and the Health Science Center. Stony Brook University Hospital, completed in 1980, is Suffolk County's only tertiary hospital and Level 1 Trauma Center, and the only academic medical center in Suffolk County—larger also than any in Nassau County.


title: Stony Brook University

text: The hospital is the largest in Suffolk County, and the attached Health Sciences Center (HSC) and Basic Science Tower (BST) houses numerous laboratories, the School of Medicine (1972), the School of Nursing (1970), the School of Health Technology and Management (1970) and the School of Social Welfare (1971). The area also includes the Ambulatory Surgery Center and the Center for Outpatient Services.


title: Stony Brook University

text: Construction on the Medical and Research Translation (MART) building began in November 2013, and the 240,000 square foot building opened on November 1, 2018, with its $194 million cost funded by state grants and donations from Jim Simons. The MART building is the home to the Stony Brook Cancer Center. In November 2019, Stony Brook Medicine opened a four-story, $73 million expansion to the Stony Brook Children's Hospital.


title: Stony Brook University

text: Also in the east side of campus are the Chapin apartments, which provide housing for graduate students. The Long Island High Technology Incubator, one of the four business incubators of the university, is a short walk north of the hospital. The Long Island State Veterans Home serving the Long Island veteran community is in this part of campus.


title: Stony Brook University

text: In 2002, the university established a presence in Manhattan with the opening of Stony Brook Manhattan. The original site was at 401 Park Avenue South; a newer operation opened in late 2008 in the adjacent building on the third floor of 387 Park Avenue South. The university consolidated operations in 2011 to just the 3rd floor of 387 Park Avenue South, with a classroom entrance around the corner at 101 East 27th Street. The site allows Stony Brook to offer professional and graduate courses targeted towards students in New York City; undergraduate courses are held primarily during the summer and winter sessions. Conferences and special events take place throughout the year. In February 2017 however, the lease for this campus was terminated, and there are no classes offered at this location.


title: Stony Brook University

text: On March 24, 2006, the university completed the purchase of the Southampton College (on the east end of Long Island) property from Long Island University with the intent to develop it as a full college campus focusing on academic programs related to the environment and sustainability. Stony Brook expanded its original program, started in the fall of 2005, when it offered an undergraduate marine sciences program, with teaching and research facilities at the campus leased from Long Island University. An enrollment of about 2,000 students is expected within the next five years. Professor Martin Schoonen was appointed interim dean of Southampton campus on August 3, 2006, and conservationist Mary Pearl was appointed dean and vice president in March 2009.


title: Stony Brook University

text: On April 7, 2010, the university had suspended residential programs and transferred sustainability programs to the main campus. The change was prompted by severe state budget cuts. Although the Marine Sciences and Graduate Writing programs are still in session at Southampton, undergraduates were relocated to the main campus. As a result of the suspension of residential programs, all dining services and retail operations were suspended by the Faculty Student Association. The old LIU radio station and National Public Radio affiliate no longer operate on the campus.


title: Stony Brook University

text: In September 2011 Stony Brook Southampton began offering an undergraduate program called Semester by the Sea, where students attend undergraduate classes to study the Ocean or the Arts. Students studying the Ocean are immersed in marine topics that are enhanced with close proximity to the water, a fleet of research vessels and graduate research projects that are ongoing. Students studying the Arts are engaged in studies for filmmaking and creative writing. Both programs offer a Public Lecture Series.


title: Stony Brook University

text: As of 2015, the Stony Brook Southampton campus has shown growth, despite almost being closed down in 2010. Programs had been added back and the Board of Trustees of the State University of New York approved a long-awaited partnership agreement between Southampton and Stony Brook University hospitals. Enrollment increased to over 400, after being around 175, three years following the addition of new funding.


title: Stony Brook University

text: As of 2019, Stony Brook University operates the Stony Brook Southampton Hospital with plans to take ownership of more eastern Long Island hospitals.


title: Stony Brook University

text: Stony Brook University has operations in Incheon, South Korea as part of the Incheon Global Campus (IGC). The other schools involved include George Mason University, the University of Utah, Ghent University and the Fashion Institute of Technology.


title: Stony Brook University

text: In July 2011, Dr. Stanley announced that the Ministry of Education, Science and Technology in South Korea has approved the establishment of SUNY Korea as part of Songdo International Business District in Incheon. The campus was expected to begin academic programs in March 2012 with an enrollment of 200. As of 2021, the student body contained roughly 1,000 students from over 40 countries. Its president is Arthur H. Lee, who was appointed to the role effective January 1, 2022. Stony Brook University students are allowed to apply for an exchange program at SUNY Korea, while SUNY Korea students are required to take a year of classes at Stony Brook University.


title: Stony Brook University

text: Stony Brook University has four gallery spaces on campus. As was the desire of donor Paul W. Zuccaire, the Paul W. Zuccaire Gallery, formerly known as the University Art Gallery, showcases professional exhibitions as well as annual graduate and undergraduate student works. The Paul W. Zuccaire Gallery is in the Staller Center for the Arts.


title: Stony Brook University

text: Also on campus is the Latin American and Caribbean Studies Center's Art Gallery, which features works from Latino and Latin American artists as well as local artists who fall under that category. The SAC Art Gallery is a center for interactive and participatory art projects.


title: Stony Brook University

text: The Tabler Center for Arts, Culture, and Humanities includes an art gallery and blackbox theater performance space.


title: Stony Brook University

text: The Stony Brook University consists of a main campus in Stony Brook, and additional satellite campuses in Southampton and South Korea. The university is composed of twelve schools and colleges. By enrollment, the largest college or school is the College of Arts and Science.


title: Stony Brook University

text: The university is governed by the State University of New York board of trustees, a body of eighteen members which regulate all the individual units of the SUNY system. The trustees have the authority to appoint the president of each state-operated institution, grant all degree diplomas and certificates for the completion of studies at any state-operated campus, and regulation of admissions, tuition, curricula, and all other matters pertaining to the operation and administration of each state-operated campus. The president of Stony Brook is the principal executive officer of the university. The position was first held by John Francis Lee and is held by the sixth president in the institution's history, Maurie McInnis, who took office on July 1, 2020.


title: Stony Brook University

text: Stony Brook's financial endowment is managed by the Stony Brook Foundation. The foundation was established in 1965 as a not-for-profit corporation under the New York State Education Law. Chartered to collect and manage gifts from private and non-state resources to supplement the funding of the university and managed by a voluntary Board of Trustees. Donations can be made to a wide selection of funds which benefit different areas of the university. In 2012 the endowment was valued at approximately $125 million with total assets amounting to nearly $350 million and has fully recovered from the losses endured in the 2008 economic downturn. After a strong fundraising campaign led by Jim Simon's $150 million donation, the university amounted to more than $180 million in fundraising for the 2011-12 year and raised $200 million by March 2013. It is the second largest endowment among State University of New York university centers behind the University at Buffalo. However, the University's endowment remains far below the average of its Association of American Universities peers.


title: Stony Brook University

text: Housed in the Student Activity Center, the Undergraduate Student Government (USG) is the governing body representing the undergraduate students of the university. The main functions of USG involve regulation, funding, and recognizing official clubs and organizations of the university. Undergraduate students are obligated to pay a Student Activity Fee per semester which is then administered by the Undergraduate Student Government. USG manages the yearly Homecoming events, Roth Pond Regatta and the traditional end-of-the year Brookfest concert and a series of concerts and events branded as "Stony Brook Concerts" that occur throughout the academic year while also directly funding undergraduate organizations, clubs, and other student services. USG at Stony Brook has a long history going back to the founding of the Student Polity Association (Polity) in 1959. After the controversial de-certification of Polity by the administration in 2002, USG was founded in 2003.


title: Stony Brook University

text: Like USG, the Graduate Student Organization (GSO) is the governing body representing the graduate students of the university. The GSO advocates for graduate student interests to senior university administration, and is part of shared governance at the university. Graduate students pay a per-semester activity fee which is used to fund events and programs for the Graduate community. The GSO provides services for graduate students including funding for conferences, seminars, speaker events, travel to Brookhaven and Cold Spring Harbor Laboratories, funding for social events, departmental organizations, and student clubs. The GSO hosts social events, professional development programming, and free legal and tax clinics. Traditional events hosted by the GSO include graduate student orientation, Three Minute Thesis, an annual speaker event, and national conferences. The GSO co-manages the University Cafe. GSO executive board members partake in national advocacy, and work with organizations such as the National Association of Graduate-Professional Students (NAGPS) on advocating for higher education policy issues.


title: Stony Brook University

text: Stony Brook was one of ten national universities awarded a National Science Foundation recognition award in 1998 for their integration of research and education. Between 2005 & 2007, two Nobel Prizes were awarded to professors for their work conducted at Stony Brook. The university has an annual $4.65 billion economic impact on the region. Stony Brook co-manages Brookhaven National Laboratory through Brookhaven Science Associates, a 50-50 partnership with Battelle Memorial Institute. Stony Brook is also one of two public schools in New York to have a medical school and a dental school, the other being University at Buffalo, The State University of New York.


title: Stony Brook University

text: The university's health science and medical component, collectively referred to as Stony Brook Medicine, includes the Renaissance School of Medicine and the Schools of Dental Medicine, Nursing, Health Technology and Management, Pharmacy and Pharmaceutical Sciences, and Social Welfare, as well as the Hospital, major centers and institutes, programs, clinics and community-based healthcare settings, and the Long Island State Veterans Home.


title: Stony Brook University

text: For the 2021–22 academic year, annual undergraduate tuition and fees was $10,410 for in-state students and $28,080 for out-of-state students. Tuition alone cost $7,070 and $24,740, respectively, with $3,020 in fees. Average room and board cost is $14,884.


title: Stony Brook University

text: 78% of full-time enrolled freshmen received financial aid, with an average of $13,100 per student. 71% of full-time enrolled freshmen received a scholarship or aid from a federal or state government institution. 42% of full-time enrolled freshmen received loan aid from the federal government at an average of $5,000 per student.


title: Stony Brook University

text: In fall 2021, the university had an enrollment of 26,782 students: 18,010 undergraduate students and 8,772 graduate students.


title: Stony Brook University

text: Of all students, 23,072 (86 percent) are U.S. citizens or permanent residents representing all states of the United States and 3,536 (13 percent) are international students representing over a hundred countries around the world.


title: Stony Brook University

text: 21,103 students hail from New York state, accounting for 79 percent of the student body. 12,061 students (45 percent) of the student body reside in Nassau or Suffolk County on Long Island, while 6,766 students (25 percent) reside in New York City.


title: Stony Brook University

text: 2,276 students (8.5 percent) reside north of New York City, with Westchester County, Rockland County and Orange County as the three most common. 7.3 percent of students live out of state, most represented by New Jersey, California and Connecticut.


title: Stony Brook University

text: Half of Stony Brook's international students hail from China. An additional 18% come from India and 10% from South Korea. In 2021, there were more Asian American undergraduate students than White American students attending Stony Brook for the first time in the university's history.


title: Stony Brook University

text: In 2022, "U.S. News & World Report" ranked Stony Brook University 77th overall among national universities and 31st among public universities. In the same year, the QS US University Rankings ranked Stony Brook University as No. 39 in the US. In the same year, Stony Brook University was ranked the top public university in New York. This is the first time the university has done so without tying with another university.


title: Stony Brook University

text: In 2020, "The Wall Street Journal" ranked Stony Brook University tied with two others as the second-best public school in the Northeastern United States.


title: Stony Brook University

text: In 2015, "Kiplinger's Personal Finance" ranked Stony Brook 33rd best value among the country's public institutions for in-state students, and 26th for out-of-state students. In 2012, "The Wall Street Journal" ranked Stony Brook 8th among public universities sending students to elite graduate programs.


title: Stony Brook University

text: The School of Engineering is ranked 67th, the School of Social Work is ranked 71st, the School of Medicine is ranked 59th in Research and a 'Rank Not Published' in Primary Care nationally.


title: Stony Brook University

text: Nuclear Physics (categorized as a Physics specialty) ranked 4th; Geometry (categorized as a Mathematics specialty) ranked 4th; Clinical Psychology ranked 4th; Topology (categorized as a Mathematics specialty) ranked 11th; Physician Assistant program ranked 16th; Physics ranked 23rd; Midwifery ranked 23rd; Mathematics ranked 25th; Political Science ranked 29th; Earth Science ranked 34th; Materials Science (categorized as an Engineering specialty) ranked 37th; Psychology ranked 39th; Sociology ranked 40th; Computer Science ranked 40th; Occupational Therapy ranked 44th; Biological Sciences ranked 55th; Chemistry ranked 56th; English ranked 60th; Economics ranked 63rd; History ranked 63rd; Physical Therapy ranked 64th; and Fine Arts ranked 98th.


title: Stony Brook University

text: In 2017, the Shanghai Global Rankings of Academic Subjects ranked Stony Brook's Mathematics program 13th best worldwide.""


title: Stony Brook University

text: "College Factual"s 2015 survey ranked Stony Brook University's Applied Mathematics program as 3rd best in the United States.


title: Stony Brook University

text: In 2021, a team of students of the Journalism School won the Edward R. Murrow Awards for Excellence in Audio Feature Reporting.


title: Stony Brook University

text: The School of Marine and Atmospheric Sciences (SoMAS) is the SUNY center for marine and atmospheric research, education, and public service. More than 300 graduate and undergraduate students from 16 different nations work and study at SoMAS. The School's students study coastal oceanographic processes and atmospheric sciences. The Marine Sciences Research Center, the original institute for marine studies, was incorporated into the new School of Marine and Atmospheric Sciences (SOMAS) on June 15, 2007.


title: Stony Brook University

text: The university co-manages Brookhaven National Laboratory, which is affiliated with the United States Department of Energy. In the Physical Sciences, Mathematics and Engineering area, some of the research centers of Stony Brook University are the Institute for Mathematical Sciences, the Institute for Advanced Computational Science, and the C. N. Yang Institute for Theoretical Physics among others. In the biomedical sciences, Stony Brook houses the Center for Biotechnology and the Institute of Chemical Biology and Drug Discovery, among many others. In March 2008, the university received $60 million endowment from James Simons to establish the Simons Center for Geometry and Physics. The Louis and Beatrice Laufer Center for Physical and Quantitative Biology was established by a generous gift in 2008 from Dr. Henry Laufer.


title: Stony Brook University

text: In July 2007 Stony Brook won a grant from the Department of Defense to devise ways to prevent terrorists from corrupting computers, and another from the Department of Homeland Security to design a system to detect radiation without triggering false alarms.


title: Stony Brook University

text: The New York Center for Computational Sciences (NYCCS), formed in 2007, is a joint venture of Stony Brook University and Brookhaven National Laboratory. Its centerpiece is an 18-rack Blue Gene /L and 2 rack Blue Gene/P massively parallel supercomputer


title: Stony Brook University

text: based on the IBM system-on-chip technology, also known as New York Blue Gene supercomputer. In the June 2008 Top 500 supercomputer rankings New York Blue Gene/L was ranked 17th, and Blue Gene/P was ranked 75th. The total peak performance for both Blue Gene/L and Blue Gene/P consists 103.22 teraflops (trillion floating-point calculations per second).


title: Stony Brook University

text: In 2016, Stony Brook University placed second at the Long Island regional round of the New York State Business Plan Competition. New York Institute of Technology placed first with four teams qualifying for the state competition's final round, while Stony Brook University had three teams qualifying for the state competition's final round.


title: Stony Brook University

text: In 2018, Stony Brook University accepted 41.8%, or 15,800, of the 37,828 freshman applications it received.


title: Stony Brook University

text: Stony Brook has a wide variety of student-run organizations on campus, which include sororities and fraternities, and a count of almost 300 recognized student clubs and organizations. The Undergraduate Student Government at Stony Brook University is trusted with the responsibility of budgeting the undergraduate student activity fee which funds most student run organizations on campus. The Graduate Student Organization is responsible for budgeting the graduate student activity fee, and supplies a variety of funding opportunities, programming, student services, and funding for departmental and student organizations.


title: Stony Brook University

text: The oldest campus newspaper is "The Statesman", which was founded in 1957 when the university was in Oyster Bay. Other publications include the "Stony Brook Press", "Stony Brook Independent", "Blackworld", and the Asian American E-Zine. Stony Brook also has a campus-wide public radio station, WUSB, which serves most of Long Island and dedicates programming to Stony Brook athletics and other events on campus.


title: Stony Brook University

text: Stony Brook has the greatest number of students who live on campus out of any public school in New York. Once seen as a "suitcase school" of commuters, Stony Brook has undergone efforts to shed that negative label, creating a model that struggling schools such as Central Connecticut State University have tried to emulate. Around 83% of freshmen students are residents.


title: Stony Brook University

text: Incoming freshmen are welcomed to the university in August with First Night Out, a night of events taking place on the Friday in which they move in. Organized by Student Engagement and Activities, the night consists of a "Party on the Plaza" as well as various assorted activities. Wolfieland, an annual carnival, began in 2016 and takes place during September. Stony Brook University's annual Homecoming celebration, known as "Wolfstock", takes place in October and features numerous activities throughout the week, including Homecoming Hoopla & Carnival on Wednesday and the Seawolves Showcase talent show on Friday. Wolfstock culminates in Stony Brook's annual Homecoming football game on Saturday, which traditionally draws record-breaking crowds upwards of 12,000 people. The homecoming court is presented during halftime, dating back to 1984. Beginning in 2018, Stony Brook shifted to the gender-neutral title of 'royal' in lieu of 'king' and 'queen'.


title: Stony Brook University

text: Academic activity pauses weekly on Wednesdays from 1:00 to 2:30 p.m. in a period known as "Campus Life Time", which was established in February 1991. During Campus Life Time, no classes are scheduled and events take place at the Academic Mall, allowing students to take a break from their studies and come together for social activity.


title: Stony Brook University

text: First-year students are assigned a book written by a notable figure, who speaks to the entire freshman class on Commons Day in the fall semester. Commons Day speakers have included Supreme Court associate justice Sonia Sotomayor, author Janet Mock, designer Joshua Davis and journalist Charles M. Blow.


title: Stony Brook University

text: The Festival of Lights started in 2000 as an annual Stony Brook tradition that celebrates the numerous cultures and faiths which celebrate during the holiday season. The festival is preceded by "Light the Brook", a tree lighting ceremony at the Academic Mall. Midnight Breakfast takes place during the first Monday of Finals Week in both semesters as the dining halls open late to serve breakfast foods to students.


title: Stony Brook University

text: To celebrate Earth Day, Stony Brook holds a week-long Earthstock celebration in the week leading up, culminating in the Earthstock Festival. Numerous environment-themed events take place throughout the week, with the most known tradition being the Rubber Duck Race held at "The Brook" adjacent to the Administration building.


title: Stony Brook University

text: Starting in 1993, Strawberry Fest is held on the first Wednesday of May, where students and faculty gather at the Academic Mall to eat an array of strawberry-themed foods with live music and student performances. In 1998, Stony Brook began to hold Diversity Day during the same day as Strawberry Fest, planned and organized by the Office of Multicultural Affairs to highlight the diverse cultures which make up the university.


title: Stony Brook University

text: The yearly Roth Pond Regatta, held since 1989, attracts dozens of competitors and thousands of attendees, including students, faculty, staff, and alumni. The competition involves groups making boats out of cardboard and tape, with the challenge to get across the Roth Pond first without sinking.


title: Stony Brook University

text: Stony Brook holds two annual concerts – Back to the Brook during the fall semester and Brookfest during the spring semester. The inaugural Back to the Brook took place in 2012 and featured Reel Big Fish at the Staller Steps. Since then, Mac Miller, Lupe Fiasco, Walk the Moon, Fetty Wap and Post Malone have headlined Back to the Brook. 2018's Back to the Brook was infamously cancelled after only 24 tickets were sold for headliner Ashanti a week before the concert. 2019's Back to the Brook was controversially cancelled again. The historic Stony Brook concert series was revived in 2011 with Brookfest hosting headliners Bruno Mars and Janelle Monáe. Brookfest has since been headlined by Wiz Khalifa and Miguel, Ludacris and Grouplove, Childish Gambino and Diplo, Panic! at the Disco and Twenty One Pilots, Future and Cash Cash, DNCE and Joey Badass, 21 Savage and A Boogie wit da Hoodie, and ASAP Ferg and Aminé.


title: Stony Brook University

text: In the early 2010s, Stony Brook's Graduate Student Organization sponsored a concert series on the first Thursday of every month titled Stony Brooklyn, focused on exposing the student body to up-and-coming indie rock musicians from the New York area. This concert series included acts like The Antlers, Parquet Courts, Beach Fossils, The Drums, Ra Ra Riot, Das Racist and The Front Bottoms playing concerts on the Stony Brook campus. As interest in this scene declined, the series was shut down by 2017.


title: Stony Brook University

text: The Spirit of Stony Brook Marching Band was created in 2006 by Jerrold Stein, the Dean of Students at the time, and plays at athletic games and other events. The first public performance was at the September 2006 convocation. The band grew to 70 members the second year and added additional staff. The band first traveled to the America East men's basketball tournament in March 2007 and has done so regularly ever since. By July 2008, the band had reached 100 members, and by the mid-2010s had attracted over 200 members.


title: Stony Brook University

text: The Stony Brook marching band first participated in the NYC Columbus Day Parade in 2011, as well as appeared in an episode of "" and a commercial for the New York Lottery. The band performed on ESPN's "College GameDay" in 2017 when an episode was hosted in New York City's Times Square.


title: Stony Brook University

text: In 2021 during the COVID-19 pandemic, members of the Stony Brook marching band performed in the virtual halftime show for the College Football Playoff National Championship game between Alabama and Ohio State.


title: Stony Brook University

text: Stony Brook's current Director of Athletic Bands is Dr. Justin R. Stolarik, who previously was a director at the University of Wisconsin–Madison and the University of Oklahoma.


title: Stony Brook University

text: Stony Brook University's intercollegiate athletics teams, known as the Stony Brook Seawolves, compete in the National Collegiate Athletic Association (NCAA) at the Division I level and are members of the Colonial Athletic Association, beginning on July 1, 2022. The school's current Director of Athletics is Shawn Heilbron, who was hired in May 2014 after serving as Senior Associate Athletic Director for Development at Oregon State University.


title: Stony Brook University

text: The university's athletics teams were originally known as the Soundmen or the Baymen in the early 1950s when the campus was located in Oyster Bay. Their name was changed to the Warriors in 1960, and again to the Patriots and Lady Patriots in 1966. In 1994, as Stony Brook prepared to become a Division I program, the team nickname was changed again, this time to its current day incarnation, the Seawolves. The team's mascot is named Wolfie.


title: Stony Brook University

text: Beginning in 2019, Stony Brook announced a partnership with SNY to broadcast football, basketball and lacrosse games on the channel. Stony Brook games on WUSB were announced by Josh Caray, grandson of famed broadcaster Harry Caray and son of Skip Caray until his departure in 2019.


title: Stony Brook University

text: The Stony Brook Patriots participated at the Division III level until 1995, when they moved up to Division II with the ultimate goal of soon reaching Division I. On June 3, 1997, President Shirley Kenny announced that the Seawolves' entire athletics program would play at the Division I level beginning during the 1999–00 season. Stony Brook joined the America East Conference in 2001 until leaving in 2022 and also played women's tennis in the Missouri Valley Conference.


title: Stony Brook University

text: Stony Brook garnered national attention during their 2012 College World Series run. The Seawolves upset the LSU Tigers in a three-game series to win the Baton Rouge Super Regional and reach the College World Series in Omaha, the first America East team to do so. Coach Matt Senk was awarded the National College Baseball Writers Association's Coach of the Year award. Outfielder Travis Jankowski was drafted by the San Diego Padres in the first round of the 2012 MLB Draft.


title: Stony Brook University

text: After going winless in four consecutive America East Finals in men's basketball, the Seawolves earned their first bid to the NCAA tournament in 2016 by defeating the Vermont Catamounts 80–74. They lost to Kentucky in the first round of the NCAA Tournament by a score of 85–57.


title: Stony Brook University

text: Stony Brook has established itself as a dominant force in women's lacrosse. Since 2013, the Seawolves have finished in first place in the America East for eight straight seasons, making eight consecutive NCAA Division I Women's Lacrosse Championship tournaments. During the 2018 season, the Seawolves were ranked No. 1 nationally in all three major polls (IWLCA Coaches' Poll, Cascade/Inside Lacrosse, Nike/US Lacrosse) for at least ten weeks.


title: Stony Brook University

text: In recent years, Stony Brook's athletic facilities have undergone several additions and renovations. The Goldstein Family Student-Athlete Development Center opened in 2006 after a million-dollar donation by alumnus Stuart Goldstein. In 2011, Joe Nathan Field, dedicated to six-time MLB All-Star relief pitcher and Stony Brook alumnus Joe Nathan, opened after renovations to the former University Field were made possible by Nathan's $500,000 donation. Island Federal Arena, formerly known as the Stony Brook University Arena, opened in 2014 after a two-year, $21.1 million renovation. The Pritchard Gymnasium, current home of the volleyball team and former home of the men's and women's basketball teams, underwent a $1.5 million renovation in 2008. Alumnus Glenn Dubin donated $4.3 million to a strength and conditioning facility named the Dubin Family Athletic Performance Center, which opened in 2012. The Dubin family also pledged $5 million for a $10 million for the Dubin Family Indoor Training Center, which opened in 2020.


title: Stony Brook University

text: In 2013, Stony Brook University launched its own bike share system to provide a sustainable transportation alternative for students (Wolf Ride Bike Share). As of 2016, the university provides 8 stations and 63 bikes. Docking stations and bikes are supplied by PBSC Urban Solutions.


title: Stony Brook University

text: There is also a system of buses operated by the university. This system is accessible to anyone on the Stony Brook Campus at no charge.


title: Apple A14

text: The Apple A14 Bionic is a 64-bit ARMv8.5-A system on a chip (SoC), designed by Apple Inc. It appears in the fourth generation iPad Air and tenth generation iPad, as well as iPhone 12 Mini, iPhone 12, iPhone 12 Pro, and iPhone 12 Pro Max. Apple states that the central processing unit (CPU) performs up to 40% faster than the A12, while the graphics processing unit (GPU) is up to 30% faster than the A12. It also includes a 16-core neural engine and new machine learning matrix accelerators that perform twice and ten times as fast, respectively.


title: Apple A14

text: The Apple A14 Bionic features an Apple-designed 64-bit, six-core CPU, implementing ARMv8 with two high-performance cores called Firestorm and four energy-efficient cores called Icestorm.


title: Apple A14

text: The A14 integrates an Apple-designed four-core GPU with 30% faster graphics performance than the A12. The A14 includes dedicated neural network hardware that Apple calls a new 16-core Neural Engine. The Neural Engine can perform 11 trillion operations per second. In addition to the separate Neural Engine, the A14 CPU includes second-generation machine learning matrix scalar multiplication accelerators (which Apple calls AMX blocks). The A14 also includes a new image processor with improved computational photography capabilities.


title: Apple A14

text: A14 is manufactured by TSMC on their first-generation 5 nm fabrication process, N5. This makes the A14 the first commercially available product to be manufactured on a 5 nm process node. The transistor count has increased to 11.8 billion, a 38.8% increase from the A13's transistor count of 8.5 billion. According to Semianalysis, the die size of A14 processor is 88 mm, with a transistor density of 134 million transistors per mm. It is manufactured in a package on package (PoP) together with 4 GB of LPDDR4X memory in the iPhone 12 and 6 GB of LPDDR4X memory in the iPhone 12 Pro.


title: Apple A14

text: The A14 has video codec encoding support for HEVC and H.264. It has decoding support for HEVC, H.264, MPEG‑4 Part 2, and Motion JPEG.


title: Apple A14

text: The A14 would be later used as the basis for the M1 series of chips, used in various Macintosh and iPad models.


title: Ghost World (film)

text: Ghost World is a 2001 black comedy film directed by Terry Zwigoff and starring Thora Birch, Scarlett Johansson, Brad Renfro, Illeana Douglas and Steve Buscemi. Based on the 1993–97 comic book of the same name by Daniel Clowes, with a screenplay co-written by Clowes and Zwigoff, the story focuses on the lives of Enid (Birch) and Rebecca (Johansson), two teenage outsiders in an unnamed American city. They face a rift in their relationship as Enid takes interest in an older man named Seymour (Buscemi), and becomes determined to help his romantic life.


title: Ghost World (film)

text: The film debuted at the Seattle International Film Festival in 2001. It had little box office impact, but was critically acclaimed. It was nominated for the Academy Award for Best Adapted Screenplay and has become a cult film.


title: Ghost World (film)

text: Best friends Enid and Rebecca face the summer after their high school graduation, with no plans for their future other than to find jobs and live together. The girls are cynical social outcasts, but Rebecca is more popular with boys than Enid. Enid's diploma is withheld on the condition that she attend a remedial art class. Even though she is a talented artist, her art teacher, Roberta, believes that art must be socially meaningful and dismisses Enid's sketches as nothing more than "light entertainment".


title: Ghost World (film)

text: The girls see a personal ad in which a lonely, middle-aged man named Seymour asks a woman he met recently to contact him. Enid makes a prank phone call to Seymour, pretending to be the woman and inviting him to meet her at a diner. The two girls and their friend, Josh, secretly watch Seymour at the diner and make fun of him. Enid soon begins to feel sympathy for Seymour, and they follow him to his apartment building. Later they find him selling vintage records in a garage sale. Enid buys an old blues album from him, and they become friends. She decides to try to find women for him to date.


title: Ghost World (film)

text: Enid has meanwhile been attending her remedial art class, and she persuades Seymour to lend her an old poster depicting a grotesquely caricatured black man, which was once used as a promotional tool by Coon Chicken Inn, the fried chicken franchise now known as Cook's Chicken, where Seymour works in corporate. Enid presents the poster in class as a social comment about racism, and Roberta is so impressed with the concept that she offers Enid a scholarship to an art college.


title: Ghost World (film)

text: Seymour receives a phone call from Dana, the intended recipient of his personal ad. Enid encourages him to pursue a relationship with Dana, but she becomes unexpectedly jealous when he does so.


title: Ghost World (film)

text: Enid's and Rebecca's lives start to diverge. While Enid has been spending time with Seymour, Rebecca starts working at a coffee shop. Enid gets a job at a movie theater so she can afford to rent an apartment with Rebecca, but her cynical attitude and reluctance to upsell concessions get her fired on her first day. The girls argue, and Rebecca abandons the idea of living with Enid.


title: Ghost World (film)

text: When Enid's poster is displayed in an art show, school officials find it so offensive they force Roberta to give her a failing grade and revoke the scholarship. Enid turns to Seymour for solace, resulting in a drunken one-night stand. Seymour breaks up with Dana and is called to account at work when the Coon Chicken poster is publicized in a local newspaper. He unsuccessfully tries to contact Enid, only for Rebecca to tell him about Enid's prank phone call, describing the way they mocked him at the diner. Seymour is upset and goes to the convenience store where Josh works. Another customer ends up in a violent confrontation with Seymour, resulting in his being injured and hospitalized. Enid visits him in the hospital to apologize.


title: Ghost World (film)

text: After everything that has occurred Enid gives in to her childhood fantasy of running away from home and disappearing. She has seen an old man, Norman, continually waiting at an out-of-service bus stop for a bus that will never come. Finally, as Enid watches from across the street, Norman boards an out-of-service bus. The next day, while Seymour discusses the summer's events with his therapist, Enid returns to the bus stop and boards the out-of-service bus when it arrives.


title: Ghost World (film)

text: A post-credits scene shows an alternate version of Seymour's scene in the convenience store, in which he wins the fight and is not injured.


title: Ghost World (film)

text: The film was directed by Terry Zwigoff with cinematography by Affonso Beato. Zwigoff and "Ghost World" comic creator Daniel Clowes wrote the screenplay together. Years later, Clowes admitted that writing the screenplay came with a significant learning curve. He recalled, "I started by trying to transcribe the comic into Final Draft. I figured that’s how you do an adaptation. Then I tried throwing everything away and writing an entirely new story that was very different from the book. And I synthesized those two things into a final screenplay. The actual film itself is very different from the script we wrote. We ended up jettisoning the last twenty pages and rethought the whole thing as we were filming. It was really held together by hair and spit."


title: Ghost World (film)

text: Zwigoff and Clowes presented Beato with the task of making a comic book look to the movie. They asked for a fresh technique: earlier examples of the form such as "X-Men" and "Dick Tracy" were dismissed as literal-minded and "insulting" to the art form. According to Clowes, cameraman Beato "really took it to heart," carefully studying the style and color of the original comics. The final cut is just slightly oversaturated, purposefully redolent of "the way the modern world looks where everything is trying to get your attention at once."


title: Ghost World (film)

text: Zwigoff also added his individual vision to the adaptation, particularly in his capture and editing of languid, lingering shots, a technique derived from his experience as a documentarian. Another notable touch is his minimal use of extras in the film, making the city and its streets intentionally empty – Clowes notes approvingly, "It captures this weird feeling of alienation in the endless modern consumer culture."


title: Ghost World (film)

text: In a 2002 interview, Daniel Clowes and Terry Zwigoff were asked if the ending of the film adaptation was a metaphor for suicide. Daniel replied, "Yeah, it could be. It's hard to figure out why people have that response. The first time I heard that I said, 'What? You're out of your mind. What are you talking about?' But I've heard that hundreds of times". Zwigoff expanded on his views in a 2021 interview, saying: "Many interpreted it to mean Enid died by suicide [...] I personally thought of the ending as more positive: that she’s moving on with her life, that she had faith in herself".


title: Ghost World (film)

text: Birch, on the other hand, stated: "Honestly, it’s a sad film, to me... I have a very dark view of where that story is leading, unfortunately".


title: Ghost World (film)

text: Music in the film includes "Jaan Pehechan Ho" by Mohammed Rafi, a dance number choreographed by Herman Benjamin from the 1965 Bollywood musical "Gumnaam" which Enid watches and dances to early in the film, and "Devil Got My Woman" by Skip James (1931), as well as "Pickin' Cotton Blues" by the bar band, Blueshammer.


title: Ghost World (film)

text: There are songs by other artists mentioned in the film, including Lionel Belasco, which are reflective of the character Seymour, and of director Terry Zwigoff. Zwigoff is a collector of 78 RPM records, as portrayed by Seymour. Other tracks are by Vince Giordano, a musician who specializes in meticulous recreations of songs from old 78 RPM records.


title: Ghost World (film)

text: Referenced in the film is R. Crumb & His Cheap Suit Serenaders, a band that Zwigoff played in. Enid asks Seymour about the band's second album, "Chasin' Rainbows", and Seymour replies, "Nah, that one's not so great."


title: Ghost World (film)

text: Missing from the soundtrack album are "What Do I Get?" by Buzzcocks, which can be heard when Enid dresses up like a punk, and the song "A Smile and a Ribbon" by Patience and Prudence.


title: Ghost World (film)

text: "Ghost World" premiered on June 16, 2001 at the Seattle International Film Festival, to lower than average recognition by audiences, but admiration from critics. It was also screened at several film festivals worldwide including the Fantasia Festival in Montreal.


title: Ghost World (film)

text: Following the film's theatrical exhibition in the United States, "Ghost World" was released on VHS and DVD format via MGM Home Entertainment in early 2002. Additional features include deleted and alternative scenes, "Making of Ghost World" featurette, Gumnaam music video "Jaan Pehechaan Ho", and the original theatrical trailer. The film was released on Blu-ray on May 30, 2017 by The Criterion Collection, with a 4K transfer, interviews with the performers, and audio commentary.


title: Ghost World (film)

text: With a limited commercial theatrical run in the United States, "Ghost World"'s commercial success was minimal. The film was released on July 20, 2001 in five theaters grossing $98,791 on its opening weekend; it slowly expanded to more theaters, reaching a maximum of 128 by the end of the year. It went on to make $6.2 million in North America and $2.5 million in the rest of the world for a worldwide total of $8.7 million, just above its $7 million budget.


title: Ghost World (film)

text: "Ghost World" has a 93% approval rating on Rotten Tomatoes, based on 162 reviews, with an average rating of 7.8/10. The site's critical consensus read, "With acerbic wit, Terry Zwigoff fashions Daniel Clowes' graphic novel into an intelligent, comedic trip through deadpan teen angst." As of November 2020, the film had a score of 88 out of 100 on Metacritic, based on 31 reviews, indicating "universal acclaim."


title: Ghost World (film)

text: Roger Ebert gave the film four out of four stars and wrote: In his review for "The New York Times", A. O. Scott praised Thora Birch's performance as Enid: In his "Chicago Reader" review, Jonathan Rosenbaum wrote: However, Andrew Sarris of "The New York Observer" disliked the character of Enid: Kevin Thomas, in his review for the "Los Angeles Times", praised Steve Buscemi's portrayal of Seymour: "Time" magazine's Andrew D. Arnold wrote:


title: Ghost World (film)

text: Michael Dean of "The Comics Journal" addressed the concerns of comics fans head-on: "Those with higher expectations – and, certainly, "Ghost World" purists – are likely to experience at least a degree of disappointment. Some of the comic's air of aimless mystery has been paved over with the semblance of a Hollywood plot, and to that extent, the movie is a lesser work than the comic. But it's still a far better movie than we had a right to expect." According to Dean: "The injection of a relatively trite plot situation into "Ghost World"s more enigmatic stream of events is perhaps forgivable, since the film might otherwise never have been produced. Its greatest sin, the misappropriation of Enid's longing, is not so forgivable, though the overlap between Zwigoff's distaste for modernity and Enid's distrust of social acceptability makes it almost palatable. In any case, we want to forgive it, because so much is right about the movie."


title: Ghost World (film)

text: "Entertainment Weekly" gave the film an "A−" rating and Owen Gleiberman wrote, ""Ghost World" is a movie for anyone who ever felt imprisoned by life, but crazy about it anyway." In her review for the "LA Weekly", Manohla Dargis wrote, "If Zwigoff doesn't always make his movie move (he's overly faithful to the concept of the cartoon panel), he has a gift for connecting us to people who aren't obviously likable, then making us see the urgency of that connection." In "Sight & Sound", Leslie Felperin wrote, "Cannily, the main performers deliver most of their lines in slack monotones, all the better to set off the script's wit and balance the glistering cluster of varyingly deranged lesser characters." In his review for "The Guardian", Peter Bradshaw wrote, "It is an engaging account of the raw pain of adolescence: the fear of being trapped in a grown-up future and choosing the wrong grown-up identity, and of course the pain of love, which we all learn to anaesthetise with jobs and mundane worries." Several critics referred to the film as an art film.


title: Ghost World (film)

text: "Ghost World" topped MSN Movies' list of the "Top 10 Comic Book Movies", it was ranked number 3 out of 94 in Rotten Tomatoes' "Comix Worst to Best" countdown (where #1 was the best and #94 the worst), ranked 5th "Best" on IGN's "Best & Worst Comic-Book Movies", and "Empire" magazine ranked the film 19th in their "The 20 Greatest Comic Book Movies" list. It is considered a cult film. It was added to the Criterion Collection in 2017.


title: Vision (Marvel Cinematic Universe)

text: The Vision is a fictional character portrayed by Paul Bettany in the Marvel Cinematic Universe (MCU) media franchise, based on the Marvel Comics character of the same name. Vision is a vibranium-based male android (or "Synthezoid") created by Ultron to serve as his body. However, the body is stolen by the Avengers and brought to sentience by Tony Stark and Bruce Banner, who upload the core software of Stark's AI, J.A.R.V.I.S., into the body constructed by Ultron, with energy provided by Thor. Vision subsequently joins the Avengers, kills Ultron, and later develops a romantic relationship with his teammate, Wanda Maximoff. Most of Vision's powers including flight, matter phasing, and energy-based beam attacks, stem from the Mind Stone in his forehead, which also keeps him alive. Vision is killed by Thanos when he removes the Mind Stone to complete the Infinity Gauntlet in order to initiate the Blip. Years later, Vision's body is reactivated via Wanda's magic, albeit initially without his memories and with a different appearance. Additionally, Wanda recreates Vision when she magically forms a false reality in the town of Westview, New Jersey, but this version vanishes when she returns to reality.


title: Vision (Marvel Cinematic Universe)

text: Vision first appeared in "" (2015), and as of 2022, has appeared in three films. He has a lead role in the television series "WandaVision" (2021), and alternate versions from within the MCU multiverse appear in the animated series "What If...?" (2021).


title: Vision (Marvel Cinematic Universe)

text: A character named The Vision debuted in a short comic story in "Marvel Mystery Comics" #13 (Nov. 1940) as an alien law enforcement officer also known as Aarkus, continuing as a regular feature in that superhero anthology through issue #48 (Oct. 1943). In the late 1960s, editor Stan Lee and writer Roy Thomas decided to add a new team member to the superhero-team series "The Avengers". Thomas wanted to bring back the Golden Age alien Vision but Lee was set on introducing an android member. Thomas ultimately compromised by using a new, android Vision. The second Vision first appeared in "The Avengers" #57 (Oct. 1968). Thomas wanted the character to be white as befitting his ghostly name, but printing limitations of the time would have rendered him colorless, with un-inked paper where his skin should be. He settled on red as he did not want Vision to be green like the Hulk or blue like the Atlanteans. The character has been compared with Spock from "Star Trek", but Thomas said that he was barely aware of the TV series at the time. He acknowledged being influenced by the Adam Link character by Otto Binder, one of the first robots treated as a sympathetic character rather than as a mechanical tool.


title: Vision (Marvel Cinematic Universe)

text: In "The Avengers" #75 (April 1970), the Scarlet Witch is reintroduced to the team and soon becomes a love interest for the Vision. Thomas recounted, "I felt that a romance of some sort would help the character development in "The Avengers", and the Vision was a prime candidate because he appeared only in that mag... as did Wanda, for that matter. So they became a pair, for just such practical considerations. It would also, I felt, add to the development I was doing on the Vision's attempting to become ‘human.’"


title: Vision (Marvel Cinematic Universe)

text: Prior to the introduction of Vision in the MCU, Paul Bettany had voiced J.A.R.V.I.S., Tony Stark's A.I. companion in previous films. Bettany admitted that he had little idea of what the role was, even as he recorded it, simply doing it as a favor for Jon Favreau. He was cast again as the Vision, an android created by Ultron. Bettany stated that he was surprised when Whedon asked him if he wanted to be the Vision because once an actor has been cast as a particular character in the MCU, they usually are not cast as another. On what intrigued him about the Vision, Bettany said, "The thing that appealed to me is that this sort of nascent creature being born, being both omnipotent and totally naive, the sort of danger of that and complex nature of a thing being born that is that powerful and that created in a second and the choices he makes morally are really complex and interesting. They've really managed to maintain all of that". Bettany also stated that the Vision feels paternal and protective to a number of people in the film, particularly Wanda Maximoff, and has the ability to change his density. Bettany did wire work for the part. Whedon stated he wanted to include the Vision in a second "Avengers" film before he signed onto the first film. Vision's on-screen appearance was designed by Trent Claus and his team at Lola VFX, based on concepts by Ryan Meinerding. Bettany's make-up, which consisted of a mix of face paint and prosthetics, took two hours to apply with make-up artists Jeremy Woodhead and Nik Williams citing the correct hue of the Vision's skin as the hardest thing to figure out. Ultimately, however, the prosthetics and tracking dots were removed digitally and replaced with CG.


title: Vision (Marvel Cinematic Universe)

text: Discussing the development of the character in "", Bettany noted that because the Vision was only created in the previous film, "Age of Ultron", "you see my character get born... He must be both omnipotent and yet totally naive at the same time. And experiencing the world in real time and his place in it. Is he going to be a force of good or a force of evil?" Bettany also said he was interested in exploring "what it means to be human and what love is" with the character, as "The only way one can guarantee one's loyalty is love." This is exhibited in the connection Vision begins to form with Wanda Maximoff, with Bettany commenting, "They both have these new burgeoning powers that they don't understand ... I think he's worried that they're both dangerous. So he feels this real connection with her." As the Vision has the ability to create a projected disguise, he chooses to dress similarly to Howard Stark's attache, Edwin Jarvis. One review notes that "Vision is an android who gained consciousness and an affinity for love all at the same time, and so the latter is of the utmost importance to him".


title: Vision (Marvel Cinematic Universe)

text: In "WandaVision", Bettany portrays a new version of the character created by Wanda within her reality from the part of the Mind Stone that lives in her, who is the embodiment of her sadness, hope, and love. Given this, Bettany described this Vision as "decent and honorable". He was influenced by the performances of Dick Van Dyke and Hugh Laurie for this version. Bettany also plays the original character, referred to as "The Vision", who is reassembled and reactivated by S.W.O.R.D. (Sentient Weapon Observation and Response Division). That version has an all-white appearance similar to when the comic book character was resurrected with an all-white body and without his memories and emotions. Bettany differentiated the two versions by portraying The Vision as familiar and intimidating at the same time.


title: Vision (Marvel Cinematic Universe)

text: In 2015, J.A.R.V.I.S. is apparently destroyed by Ultron, but it is later revealed that he actually distributed his consciousness throughout the Internet, allowing his security protocols to delay Ultron's attempt to access Earth's nuclear weapon launch codes long enough for Tony Stark to work out what had happened. The Avengers capture a synthetic vibranium body created by Ultron for himself, powered by the Mind Stone, and Stark and Bruce Banner upload J.A.R.V.I.S. as the core software for the body. After a brief fight with other Avengers, who disagree with this effort, Thor uses Mjolnir's lightning to power its completion, creating the Vision. Thor explains that the gem in his forehead is one of the six Infinity Stones, the most powerful objects in existence. Having gained consciousness, Vision sides with the Avengers against Ultron as he wishes to protect life. He lifts and hands Thor Mjolnir, which has been enchanted so that only "worthy" individuals can lift it, convincing the Avengers that he can be trusted.


title: Vision (Marvel Cinematic Universe)

text: In Sokovia, Vision fights scores of Ultron's sentries, rescues Wanda Maximoff from the collapsing city center, and destroys Ultron himself. After Stark, Clint Barton, Thor, and Banner leave, Steve Rogers and Natasha Romanoff lead a new roster of Avengers consisting of Wanda, Sam Wilson, James Rhodes, and Vision.


title: Vision (Marvel Cinematic Universe)

text: In 2016, Vision joins Stark in signing the Sokovia Accords, giving the United Nations control over the Avengers, remarking that the group's existence invites challenge and conflict, which often results in catastrophe. The Accords include confining Maximoff, who accidentally killed Wakandan humanitarian aid workers during a confrontation with terrorists, to the Avengers Compound. He attempts to make her feel comfortable by cooking her paprikash and they start to form a romantic relationship. He overpowers Barton after he tries to break Wanda out on Rogers' behalf, until she interferes, telekinetically pushing Vision beneath the Compound. Unharmed, Vision flies to Germany to help stop Rogers' team. During the fight, he aims to disable Wilson with an energy beam, but accidentally shoots down and cripples Rhodes, as he was distracted while tending to an injured Maximoff.


title: Vision (Marvel Cinematic Universe)

text: In 2018, Vision, now in a romantic relationship with Maximoff, runs away with her to Scotland. However, the Mind Stone begins alerting Vision to an unknown threat before they are ambushed by Corvus Glaive and Proxima Midnight, two of Thanos' children. Vision is badly wounded by Glaive's namesake, which limits his powers. Glaive and Midnight attempt to remove Vision's Mind Stone, but Maximoff is able to hold them off until the pair are rescued by Rogers, Wilson, and Romanoff and return to the Avengers Compound. Learning that Thanos is after Vision's Stone, Vision volunteers to destroy it and kill himself, but the Avengers refuse and bring him to Wakanda, where Shuri attempts to safely remove the Stone without killing Vision, with Maximoff standing vigil over the operation.


title: Vision (Marvel Cinematic Universe)

text: When Thanos' children and their Outriders attack Wakanda, Wanda intervenes in the battle, leaving Shuri defenseless to Glaive, who infiltrated her lab. Vision fights Glaive, and is nearly defeated before being rescued by Rogers. Vision in turn saves Rogers, killing Glaive with his own weapon. Thanos arrives, and as the Avengers attempt to hold him off, Vision manages to convince Maximoff to destroy the Mind Stone, which she does, at the cost of his life. However, Thanos uses the Time Stone to reverse the action, allowing him to rip the Mind Stone out of Vision's head, killing him once more. Thanos initiates the Blip, and Vision's body is taken into S.W.O.R.D.'s custody.


title: Vision (Marvel Cinematic Universe)

text: In 2023, S.W.O.R.D. runs experiments on Vision's corpse at their headquarters. Maximoff, desiring to give him a burial, learns he is being experimented on, but is told that he is government property and cannot be released to her.


title: Vision (Marvel Cinematic Universe)

text: She travels to Westview, New Jersey, and in her grief creates a false reality to her liking, later called the "Hex", along with Hex Vision. Within the Hex, she and Hex Vision are married, live in a suburban neighborhood, and he works for Computational Services Inc. After Wanda becomes pregnant with twins, Hex Vision momentarily grows suspicious of their surroundings. Nonetheless, Hex Vision joins the neighborhood watch and Wanda gives birth to Billy and Tommy. When the twins grow up rapidly, he continues to question his and Wanda's life in Westview and uses his powers to learn that one of his co-workers is under mind control. He investigates further, and on the edge of town he finds residents frozen in place, including his neighbor Agnes. He discovers a hexagonal barrier and tries to leave, but begins to disintegrate before he can reach the nearby S.W.O.R.D. outpost while telling the S.W.O.R.D. agents and Darcy Lewis that the people inside need help. Maximoff expands the Hex, restoring Vision. After waking up, he finds Lewis in Westview and awakens her from her trance. While heading home with her to confront Maximoff, Lewis tells him about his past life and death.


title: Vision (Marvel Cinematic Universe)

text: Acting S.W.O.R.D. director Tyler Hayward uses Maximoff's powers to reactivate Vision, now with a white body and his memories erased, and deploys him into Westview. He attacks Maximoff, but Hex Vision rescues her and battles his double, during which they discuss the paradox of the Ship of Theseus. Hex Vision restores Vision's memories, and he flies away to process this information. Hex Vision learns that he is a memory of Wanda that was created through her powers and the Mind Stone, and says his final goodbyes before he fades from existence when Wanda removes the Hex.


title: Vision (Marvel Cinematic Universe)

text: Several alternate universe versions of Vision appear in the animated series "What If...?", with Bettany reprising his role while Ross Marquand voices a version of Ultron in Vision's body.


title: Vision (Marvel Cinematic Universe)

text: In an alternate 2018, a quantum virus is unleashed and infects several humans, turning them into zombies. The unaffected Vision keeps the zombified Maximoff contained at Camp Lehigh, where he experiments with the Mind Stone to find a cure and manages to cure Scott Lang and preserve his severed head in a jar. However, Vision is unable to cure Wanda, leading him to lure in individuals, such as T'Challa, and feed them to her. Eventually, after the surviving Avengers and their allies arrive at the base, Vision hands over the Mind Stone to Bruce Banner, killing himself in the process.


title: Vision (Marvel Cinematic Universe)

text: In an alternate 2015, Ultron successfully transfers his consciousness into Vision's body and goes on to kill most of the Avengers and eliminate all life in the universe after obtaining the other five Infinity Stones from Thanos. After learning about the Watcher and the existence of other realities, Ultron travels to the Watcher's observatory, from where he gains access to every timeline in the multiverse, intending to destroy each of them. However, he is foiled by the Watcher and the Guardians of the Multiverse, who upload Arnim Zola's mind into his body, allowing Zola to delete his consciousness. Although Zola, now controlling Vision's body, tries to fight Erik "Killmonger" Stevens for control over the Infinity Stones, they are both frozen in a pocket dimension by the Watcher and Doctor Strange Supreme, with the latter agreeing to guard them for the rest of eternity.


title: Vision (Marvel Cinematic Universe)

text: A review of the character for "The Hollywood Reporter" notes: "The comic book Vision employs a garish green-and-yellow costume, matched with a bright red face — a color scheme that may be a little over-the-top for the more subtly-hued Marvel Cinematic Universe — but even so, the mixture of purple, blue and grey is an unexpected, and unexpectedly bold, choice for Paul Bettany's character".


title: Vision (Marvel Cinematic Universe)

text: In terms of fashion while maintaining a civilian appearance, Vision attempts to emulate classic human style, including wearing an ascot tie.


title: Vision (Marvel Cinematic Universe)

text: Following the release of "WandaVision", Jen Chaney of "Vulture" stated, "Olsen and Bettany's characters were often treated like benchwarmers on an all-star team in the Avengers movies. Here, they really shine." The review further found that Bettany "slides easily into the role of the devoted, kinda square, goofball husband," and praised his physical comedy skills. Saim Cheeda of "CBR.com" praised Bettany's performance across the MCU, especially in "WandaVision", writing, "Paul Bettany has proved in "WandaVision" that he is the absolute perfect choice for playing Vision, nailing many of his characteristics." Tyler Pisapia of "Looper" inducted Bettany's portrayal of Vision through the MCU among his best performances overall, stating, "From his heroics — and ultimate death — in the Avengers franchise to his subsequent revival in WandaVision, Bettany's Vision has added a deeply soulful component to the MCU through his portrayal of a being whose synthetic structure doesn't deprive him of his essential humanity." "TVLine" named Bettany "Performer of the Week" for the week of January 16, 2021.


title: Definite matrix

text: In mathematics, a symmetric matrix formula_1 with real entries is positive-definite if the real number formula_2 is positive for every nonzero real column vector formula_3 where formula_4 is the transpose of More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is


title: Definite matrix

text: positive-definite if the real number formula_5 is positive for every nonzero complex column vector formula_3 where formula_7 denotes the conjugate transpose of formula_8


title: Definite matrix

text: Positive semi-definite matrices are defined similarly, except that the scalars formula_2 and formula_5 are required to be positive "or zero" (that is, nonnegative). Negative-definite and negative semi-definite matrices are defined analogously. A matrix that is not positive semi-definite and not negative semi-definite is sometimes called indefinite.


title: Definite matrix

text: A matrix is thus positive-definite if and only if it is the matrix of a positive-definite quadratic form or Hermitian form. In other words, a matrix is positive-definite if and only if it defines an inner product.


title: Definite matrix

text: Positive-definite and positive-semidefinite matrices can be characterized in many ways, which may explain the importance of the concept in various parts of mathematics. A matrix is positive-definite if and only if it satisfies any of the following equivalent conditions.


title: Definite matrix

text: A matrix is positive semi-definite if it satisfies similar equivalent conditions where "positive" is replaced by "nonnegative", "invertible matrix" is replaced by "matrix", and the word "leading" is removed.


title: Definite matrix

text: Positive-definite and positive-semidefinite real matrices are at the basis of convex optimization, since, given a function of several real variables that is twice differentiable, then if its Hessian matrix (matrix of its second partial derivatives) is positive-definite at a point , then the function is convex near , and, conversely, if the function is convex near , then the Hessian matrix is positive-semidefinite at .


title: Definite matrix

text: Some authors use more general definitions of definiteness, including some non-symmetric real matrices, or non-Hermitian complex ones.


title: Definite matrix

text: In the following definitions, formula_14 is the transpose of formula_15, formula_16 is the conjugate transpose of formula_15 and formula_18 denotes the "n"-dimensional zero-vector.


title: Definite matrix

text: An formula_19 symmetric real matrix formula_1 is said to be positive-definite if formula_21 for all non-zero formula_22 in formula_23. Formally,


title: Definite matrix

text: An formula_19 symmetric real matrix formula_1 is said to be positive-semidefinite or non-negative-definite if formula_26 for all formula_22 in formula_23. Formally,


title: Definite matrix

text: An formula_19 symmetric real matrix formula_1 is said to be negative-definite if formula_31 for all non-zero formula_22 in formula_23. Formally,


title: Definite matrix

text: An formula_19 symmetric real matrix formula_1 is said to be negative-semidefinite or non-positive-definite if formula_36 for all formula_37 in formula_23. Formally,


title: Definite matrix

text: An formula_19 symmetric real matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.


title: Definite matrix

text: The following definitions all involve the term formula_40. Notice that this is always a real number for any Hermitian square matrix formula_1.


title: Definite matrix

text: An formula_19 Hermitian complex matrix formula_1 is said to be positive-definite if formula_44 for all non-zero formula_22 in formula_46. Formally,


title: Definite matrix

text: An formula_19 Hermitian complex matrix formula_1 is said to be positive semi-definite or non-negative-definite if formula_49 for all formula_37 in formula_46. Formally,


title: Definite matrix

text: An formula_19 Hermitian complex matrix formula_1 is said to be negative-definite if formula_54 for all non-zero formula_22 in formula_46. Formally,


title: Definite matrix

text: An formula_19 Hermitian complex matrix formula_1 is said to be negative semi-definite or non-positive-definite if formula_59 for all formula_22 in formula_46. Formally,


title: Definite matrix

text: An formula_19 Hermitian complex matrix which is neither positive semidefinite nor negative semidefinite is called indefinite.


title: Definite matrix

text: Since every real matrix is also a complex matrix, the definitions of "definiteness" for the two classes must agree.


title: Definite matrix

text: For complex matrices, the most common definition says that "formula_1 is positive-definite if and only if formula_64 is real and positive for all non-zero "complex" column vectors formula_65". This condition implies that formula_1 is Hermitian (i.e. its transpose is equal to its conjugate). To see this, consider the matrices formula_67 and formula_68, so that formula_69 and formula_70. The matrices formula_71 and formula_11 are Hermitian, therefore formula_73 and formula_74 are individually real. If formula_64 is real, then formula_74 must be zero for all formula_65. Then formula_11 is the zero matrix and formula_79, proving that formula_1 is Hermitian.


title: Definite matrix

text: By this definition, a positive-definite "real" matrix formula_1 is Hermitian, hence symmetric; and formula_82 is positive for all non-zero "real" column vectors formula_83. However the last condition alone is not sufficient for formula_1 to be positive-definite. For example, if


title: Definite matrix

text: then for any real vector formula_83 with entries formula_87 and formula_88 we have formula_89, which is always positive if formula_65 is not zero. However, if formula_65 is the complex vector with entries formula_92 and formula_93, one gets


title: Definite matrix

text: On the other hand, for a "symmetric" real matrix formula_1, the condition "formula_97 for all nonzero real vectors formula_65" "does" imply that formula_1 is positive-definite in the complex sense.


title: Definite matrix

text: If a Hermitian matrix formula_1 is positive semi-definite, one sometimes writes formula_101 and if formula_1 is positive-definite one writes formula_103. To denote that formula_1 is negative semi-definite one writes formula_105 and to denote that formula_1 is negative-definite one writes formula_107.


title: Definite matrix

text: The notion comes from functional analysis where positive semidefinite matrices define positive operators. If two matrices formula_71 and formula_11 satisfy formula_110, we can define a non-strict partial order formula_111 that is reflexive, antisymmetric, and transitive; It is not a total order, however, as formula_112 in general may be indefinite.


title: Definite matrix

text: A common alternative notation is formula_113, formula_114, formula_115 and formula_116 for positive semi-definite and positive-definite, negative semi-definite and negative-definite matrices, respectively. This may be confusing, as sometimes nonnegative matrices (respectively, nonpositive matrices) are also denoted in this way.


title: Definite matrix

text: Let formula_1 be an formula_19 Hermitian matrix (this includes real symmetric matrices). All eigenvalues of formula_1 are real, and their sign characterize its definiteness:


title: Definite matrix

text: Let formula_125 be an eigendecomposition of formula_1, where formula_127 is a unitary complex matrix whose columns comprise an orthonormal basis of eigenvectors of formula_1, and formula_129 is a "real" diagonal matrix whose main diagonal contains the corresponding eigenvalues. The matrix formula_1 may be regarded as a diagonal matrix formula_129 that has been re-expressed in coordinates of the (eigenvectors) basis formula_127. Put differently, applying formula_1 to some vector , giving , is the same as changing the basis to the eigenvector coordinate system using , giving , applying the stretching transformation to the result, giving , and then changing the basis back using , giving .


title: Definite matrix

text: With this in mind, the one-to-one change of variable formula_134 shows that formula_64 is real and positive for any complex vector formula_65 if and only if formula_137 is real and positive for any formula_138; in other words, if formula_129 is positive definite. For a diagonal matrix, this is true only if each element of the main diagonal—that is, every eigenvalue of formula_1—is positive. Since the spectral theorem guarantees all eigenvalues of a Hermitian matrix to be real, the positivity of eigenvalues can be checked using Descartes' rule of alternating signs when the characteristic polynomial of a real, symmetric matrix formula_1 is available.


title: Definite matrix

text: formula_1 is positive definite if and only if such a decomposition exists with formula_11 invertible.


title: Definite matrix

text: More generally, formula_1 is positive semidefinite with rank formula_153 if and only if a decomposition exists with a formula_154 matrix formula_11 of full row rank (i.e. of rank formula_153).


title: Definite matrix

text: If moreover formula_1 is positive definite, then the eigenvalues are (strictly) positive, so formula_162 is invertible, and hence formula_160 is invertible as well.


title: Definite matrix

text: If formula_1 has rank formula_153, then it has exactly formula_153 positive eigenvalues and the others are zero, hence in formula_160 all but formula_153 rows are all zeroed.


title: Definite matrix

text: The columns formula_172 of formula_11 can be seen as vectors in the complex or real vector space formula_174, respectively.


title: Definite matrix

text: Then the entries of formula_1 are inner products (that is dot products, in the real case) of these vectors


title: Definite matrix

text: In other words, a Hermitian matrix formula_1 is positive semidefinite if and only if it is the Gram matrix of some vectors formula_172.


title: Definite matrix

text: In general, the rank of the Gram matrix of vectors formula_172 equals the dimension of the space spanned by these vectors.


title: Definite matrix

text: if formula_157 for some formula_154 matrix formula_11 and if formula_183 is any unitary formula_184 matrix (meaning formula_185),


title: Definite matrix

text: However, this is the only way in which two decompositions can differ: the decomposition is unique up to unitary transformations.


title: Definite matrix

text: More formally, if formula_71 is a formula_154 matrix and formula_11 is a formula_191 matrix such that formula_192,


title: Definite matrix

text: then there is a formula_193 matrix formula_183 with orthonormal columns (meaning formula_195) such that formula_196.


title: Definite matrix

text: let the columns of formula_71 and formula_11 be the vectors formula_201 and formula_172 in formula_174.


title: Definite matrix

text: A real unitary matrix is an orthogonal matrix, which describes a rigid transformation (an isometry of Euclidean space formula_174) preserving the 0 point (i.e. rotations and reflections, without translations).


title: Definite matrix

text: Therefore, the dot products formula_205 and formula_206 are equal if and only if some rigid transformation of formula_174 transforms the vectors formula_201 to formula_172 (and 0 to 0).


title: Definite matrix

text: A matrix formula_1 is positive semidefinite if and only if there is a positive semidefinite matrix formula_11 (in particular formula_11 is Hermitian, so formula_213) satisfying formula_214. This matrix formula_11 is unique, is called the "non-negative square root" of formula_1, and is denoted with formula_217.


title: Definite matrix

text: When formula_1 is positive definite, so is formula_219, hence it is also called the "positive square root" of formula_1.


title: Definite matrix

text: Some authors use the name "square root" and formula_219 for any such decomposition, or specifically for the Cholesky decomposition,


title: Definite matrix

text: A positive semidefinite matrix formula_1 can be written as formula_227, where formula_228 is lower triangular with non-negative diagonal (equivalently formula_221 where formula_230 is upper triangular); this is the Cholesky decomposition.


title: Definite matrix

text: If formula_1 is positive definite, then the diagonal of formula_228 is positive and the Cholesky decomposition is unique. Conversely if formula_228 is lower triangular with nonnegative diagonal then formula_228 is positive semidefinite.


title: Definite matrix

text: A closely related decomposition is the LDL decomposition, formula_235, where formula_129 is diagonal and formula_228 is lower unitriangular.


title: Definite matrix

text: Let formula_1 be an formula_19 real symmetric matrix, and let formula_240 be the "unit ball" defined by formula_1. Then we have the following


title: Definite matrix

text: Let formula_1 be an formula_19 Hermitian matrix. The following properties are equivalent to formula_1 being positive definite:


title: Definite matrix

text: The (purely) quadratic form associated with a real formula_19 matrix formula_1 is the function formula_291 such that formula_292 for all formula_37. formula_1 can be assumed symmetric by replacing it with formula_295.


title: Definite matrix

text: A symmetric matrix formula_1 is positive definite if and only if its quadratic form is a strictly convex function.


title: Definite matrix

text: More generally, any quadratic function from formula_297 to formula_298 can be written as formula_299 where formula_1 is a symmetric formula_19 matrix, formula_88 is a real formula_303-vector, and formula_304 a real constant. In the formula_305 case, this is a parabola, and just like in the formula_305 case, we have


title: Definite matrix

text: Theorem: This quadratic function is strictly convex, and hence has a unique finite global minimum, if and only if formula_1 is positive definite.


title: Definite matrix

text: Proof: If formula_1 is positive definite, then the function is strictly convex. Its gradient is zero at the unique point of formula_309, which must be the global minimum since the function is strictly convex. If formula_1 is not positive definite, then there exists some vector formula_311 such that formula_312, so the function formula_313 is a line or a downward parabola, thus not strictly convex and not having a global minimum.


title: Definite matrix

text: One symmetric matrix and another matrix that is both symmetric and positive definite can be simultaneously diagonalized. This is so although simultaneous diagonalization is not necessarily performed with a similarity transformation. This result does not extend to the case of three or more matrices. In this section we write for the real case. Extension to the complex case is immediate.


title: Definite matrix

text: Let formula_1 be a symmetric and formula_258 a symmetric and positive definite matrix. Write the generalized eigenvalue equation as formula_316 where we impose that formula_37 be normalized, i.e. formula_318. Now we use Cholesky decomposition to write the inverse of formula_258 as formula_320. Multiplying by formula_183 and letting formula_322, we get formula_323, which can be rewritten as formula_324 where formula_325. Manipulation now yields formula_326 where formula_327 is a matrix having as columns the generalized eigenvectors and formula_328 is a diagonal matrix of the generalized eigenvalues. Now premultiplication with formula_329 gives the final result: formula_330 and formula_331, but note that this is no longer an orthogonal diagonalization with respect to the inner product where formula_325. In fact, we diagonalized formula_1 with respect to the inner product induced by formula_258.


title: Definite matrix

text: Note that this result does not contradict what is said on simultaneous diagonalization in the article Diagonalizable matrix, which refers to simultaneous diagonalization by a similarity transformation. Our result here is more akin to a simultaneous diagonalization of two quadratic forms, and is useful for optimization of one form under conditions on the other.


title: Definite matrix

text: For arbitrary square matrices formula_1, formula_258 we write formula_337 if formula_338 i.e., formula_339 is positive semi-definite. This defines a partial ordering on the set of all square matrices. One can similarly define a strict partial ordering formula_340. The ordering is called the Loewner order.


title: Definite matrix

text: Every positive definite matrix is invertible and its inverse is also positive definite. If formula_341 then formula_342. Moreover, by the min-max theorem, the "k"th largest eigenvalue of formula_1 is greater than the "k"th largest eigenvalue of formula_258.


title: Definite matrix

text: If formula_1 is positive definite and formula_346 is a real number, then formula_347 is positive definite.


title: Definite matrix

text: The diagonal entries formula_369 of a positive-semidefinite matrix are real and non-negative. As a consequence the trace, formula_370. Furthermore, since every principal sub-matrix (in particular, 2-by-2) is positive semidefinite,


title: Definite matrix

text: An formula_19 Hermitian matrix formula_1 is positive definite if it satisfies the following trace inequalities:


title: Definite matrix

text: Another important result is that for any formula_1 and formula_258 positive-semidefinite matrices, formula_379


title: Definite matrix

text: If formula_380, although formula_362 is not necessary positive semidefinite, the Hadamard product is, formula_382 (this result is often called the Schur product theorem).


title: Definite matrix

text: Regarding the Hadamard product of two positive semidefinite matrices formula_383, formula_384, there are two notable inequalities:


title: Definite matrix

text: If formula_380, although formula_362 is not necessary positive semidefinite, the Kronecker product formula_389.


title: Definite matrix

text: If formula_380, although formula_362 is not necessary positive semidefinite, the Frobenius inner product formula_392 (Lancaster–Tismenetsky, "The Theory of Matrices", p. 218).


title: Definite matrix

text: The set of positive semidefinite symmetric matrices is convex. That is, if formula_1 and formula_258 are positive semidefinite, then for any formula_395 between 0 and 1, formula_396 is also positive semidefinite. For any vector formula_15:


title: Definite matrix

text: This property guarantees that semidefinite programming problems converge to a globally optimal solution.


title: Definite matrix

text: The positive-definiteness of a matrix formula_71 expresses that the angle formula_400 between any vector formula_15 and its image formula_402 is always formula_403:


title: Definite matrix

text: A Hermitian matrix is positive semidefinite if and only if all of its principal minors are nonnegative. It is however not enough to consider the leading principal minors only, as is checked on the diagonal matrix with entries 0 and −1.


title: Definite matrix

text: where each block is formula_19. By applying the positivity condition, it immediately follows that formula_71 and formula_129 are hermitian, and formula_430.


title: Definite matrix

text: A similar argument can be applied to formula_129, and thus we conclude that both formula_71 and formula_129 must be positive definite. The argument can be extended to show that any principal submatrix of formula_1 is itself positive definite.


title: Definite matrix

text: Converse results can be proved with stronger conditions on the blocks, for instance using the Schur complement.


title: Definite matrix

text: A general quadratic form formula_439 on formula_303 real variables formula_441 can always be written as formula_442 where formula_22 is the column vector with those variables, and formula_1 is a symmetric real matrix. Therefore, the matrix being positive definite means that formula_445 has a unique minimum (zero) when formula_22 is zero, and is strictly positive for any other formula_22.


title: Definite matrix

text: More generally, a twice-differentiable real function formula_445 on formula_303 real variables has local minimum at arguments formula_441 if its gradient is zero and its Hessian (the matrix of all second derivatives) is positive semi-definite at that point. Similar statements can be made for negative definite and semi-definite matrices.


title: Definite matrix

text: In statistics, the covariance matrix of a multivariate probability distribution is always positive semi-definite; and it is positive definite unless one variable is an exact linear function of the others. Conversely, every positive semi-definite matrix is the covariance matrix of some multivariate distribution.


title: Definite matrix

text: The definition of positive definite can be generalized by designating any complex matrix formula_1 (e.g. real non-symmetric) as positive definite if formula_452 for all non-zero complex vectors formula_65, where formula_454 denotes the real part of a complex number formula_304. Only the Hermitian part formula_456 determines whether the matrix is positive definite, and is assessed in the narrower sense above. Similarly, if formula_15 and formula_1 are real, we have formula_459 for all real nonzero vectors formula_15 if and only if the symmetric part formula_461 is positive definite in the narrower sense. It is immediately clear that formula_462is insensitive to transposition of "M".


title: Definite matrix

text: Consequently, a non-symmetric real matrix with only positive eigenvalues does not need to be positive definite. For example, the matrix formula_463 has positive eigenvalues yet is not positive definite; in particular a negative value of formula_464 is obtained with the choice formula_465 (which is the eigenvector associated with the negative eigenvalue of the symmetric part of


title: Definite matrix

text: In summary, the distinguishing feature between the real and complex case is that, a bounded positive operator on a complex Hilbert space is necessarily Hermitian, or self adjoint. The general claim can be argued using the polarization identity. That is no longer true in the real case.


title: Definite matrix

text: Fourier's law of heat conduction, giving heat flux formula_466 in terms of the temperature gradient formula_467 is written for anisotropic media as formula_468, in which formula_469 is the symmetric thermal conductivity matrix. The negative is inserted in Fourier's law to reflect the expectation that heat will always flow from hot to cold. In other words, since the temperature gradient formula_470 always points from cold to hot, the heat flux formula_466 is expected to have a negative inner product with formula_470 so that formula_473. Substituting Fourier's law then gives this expectation as formula_474, implying that the conductivity matrix should be positive definite.


title: Bloomberg Terminal

text: The Bloomberg Terminal is a computer software system provided by the financial data vendor Bloomberg L.P. that enables professionals in the financial service sector and other industries to access Bloomberg Professional Services through which users can monitor and analyze real-time financial market data and place trades on the electronic trading platform. It was developed by employees working for businessman Michael Bloomberg. The system also provides news, price quotes, and messaging across its proprietary secure network. It is well known among the financial community for its black interface, which has become a recognizable trait of the service. The first version of the terminal was released in December 1982.


title: Bloomberg Terminal

text: Most large financial firms have subscriptions to Bloomberg Professional Services. Many exchanges charge their own additional fees for access to real time price feeds across the terminal. The same applies to various news organizations.


title: Bloomberg Terminal

text: All Bloomberg Terminals are leased in two-year cycles (in the late 1990s and early 2000s, three-year contracts were an option), with leases originally based on how many displays were connected to each terminal (this predated the move to Windows-based application). Most Bloomberg setups have between two and six displays. It is available for an annual fee of $20,000 per user ($24,000 per year for the small number of firms that use only one terminal). As of 2022, there were 325,000 Bloomberg Terminal subscribers worldwide.


title: Bloomberg Terminal

text: In 1981, Michael Bloomberg was fired from Salomon Brothers. He was given no severance package, but owned $10 million worth of equity as a partner at the firm. Using this money, Bloomberg, having designed in-house computerized financial systems for Salomon, set up a data services company named Innovative Market Systems (IMS) based on his belief that Wall Street would pay a premium for high-quality business information, delivered instantaneously on computer terminals in a variety of usable formats. The company sold customized computer terminals that delivered real-time market data, financial calculations and other analytics to Wall Street firms. At first, the machine was called the Market Master terminal, but later became known as the Bloomberg Terminal or simply "The Bloomberg". The terminal was released to market in December 1982. Merrill Lynch became the company's first customer, purchasing a 30% stake in IMS for $30 million in exchange for a five-year restriction on marketing the terminals to Merrill Lynch's competitors. In 1984, Merrill Lynch released IMS from the restriction.


title: Bloomberg Terminal

text: In 1990, the Bloomberg keyboard was released with a trackball and built-in voice-chat features. In 1991, the first color edition of the terminal was released.


title: Bloomberg Terminal

text: Michael Bloomberg stepped away from working on the terminal in 2001 to run for New York City mayor, but returned to lead the project in 2014. Starting in 2012, Bloomberg Terminal had a greater annual revenue than Thomson Reuters, the company that founded the market data business.