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Defense Information Systems Agency

The Defense Information Systems Agency (DISA), known as the Defense Communications Agency (DCA) until 1991, is a United States Department of Defense (DoD) combat support agency composed of military, federal civilians, and contractors. DISA provides information technology (IT) and communications support to the President, Vice President, Secretary of Defense, the military services, the combatant commands, and any individual or system contributing to the defense of the United States.

Headquarters
From January to July 2011, DISA relocated more than 4,500 military and civilian employees and supporting onsite contractors, 700 workstation suites with 11,000 pieces of IT equipment, and 58,000 square feet of lab equipment in accordance with the Base Realignment and Closure legislation of 2005. The relocation to Fort George G. Meade, MD., consolidated DISA headquarters elements that were housed in multiple locations in Arlington and Falls Church, Va. In April 2011, DISA held a ribbon cutting ceremony officially opening the new headquarters complex.

Services
DISA offers the following services:

Command and Control
Command and Control (C2) systems provide the U.S. military commander with the information to make effective decisions and provide the warfighter the capability to access the information necessary to complete their mission. The C2 portfolio contains the Global Command and Control System - Joint (GCCS-J), Multinational Information Sharing (MNIS), Joint Planning and Execution Services (JPES), and Personal Digital Assistant 184 (PDA-184).

PDA-184
The DISA PDA-184 software application implements a fully compliant MIL-STD-188-184A data controller. It provides reliable data communications over noisy, low bandwidth channels to support chat, file transfer, and email services via an easy-to-use graphical user interface. The application supports multiple bands and waveforms, including integrated waveform (IW), legacy ultra-high frequency (UHF) satellite communications (SATCOM), UHF line of sight (LOS), very high frequency (VHF), high frequency (HF), and extremely high frequency (EHF) SATCOM. PDA-184 is provided without charge to all combatant commands (CCMDs) / services / agencies (CC/S/A).

CENTRIXS
CENTRIXS is the Combatant Commander's network for coalition. CENTRIXS is designed to be a global, interoperable, interconnected, inexpensive, and easy-to use system to share intelligence and operations information through reliable communications connectivity, data manipulation, and automated processes. The CENTRIXS environment is a combination of network and applications services. CENTRIXS provides a secured exchange of intelligence and operational information through reliable communication networks There are 40+ CENTRIXS networks/communities of interest (COIs) providing selected centralized services including: Active Directory/DNS Roots, VoIP, WSUS and Anti-Virus Definitions, and at least 80 countries plus NATO nations participate in the various CENTRIXS networks/COIs.

Pegasus/Griffin
Pegasus/Griffin is a multinationally-developed, managed and resourced collection of networks and services that provides information sharing among the 5-Eyes national classified (SECRET level) networks and C2 systems. Griffin enables participating nations to plan, implement and execute multinational planning and operations from the strategic to tactical headquarters level. It permits users to share SECRET REL information from their national C2 system workstations.

CFBLNET
The Combined Federated Battle Laboratories Network (CFBLNet) is a coalition RDT&E (Research, Development, Test, and Evaluation) environment with Combined Communications-Electronics Board (CCEB) and the North Atlantic Treaty Organization (NATO) and other charter nations/organizations. CFBLNet leverages existing CCEB, NATO, and other national laboratories and test beds to support a wide range of coalition ISTAR (Intelligence, surveillance, target acquisition, and reconnaissance) experimentation and interoperability testing. As a combined environment and network, members have equal say in its use and management, yet specific initiatives may be configured among any number of participants. CFBLNet members respect sovereign and intellectual property rights of activities conducted on the network. CFBLNet is primarily a fee for service activity. and system injector.

APAN
All Partners Access Network (APAN): A web 2.0 service that combines the benefits of unstructured collaboration (wikis, blogs, forums) and structured collaboration (file sharing, calendar) with the personalization of social networking. The existing functionality is maintained at USPACOM Pacific Warfighting Center (PWC). Non-dot-mil; Gateway to DoD.

Computing
DISA's computing services portfolio includes mainframe hosting, application monitoring, and server hosting and virtualization. DISA manages all the partner data, hardware components, software, and labor.

Contracting
DISA purchases telecommunications and information technology (IT) products and services for the U.S. military using a variety of contract vehicles.

Enterprise Engineering
Enterprise Engineering refers to the Global Information Grid (a.k.a. the GIG). DISA plans, designs, constructs, and analyzes the effectiveness of the U.S. military's cyberspace and establishes the technological standards to make the GIG secure and reliable. The enterprise engineering portfolio includes the Joint Communication Simulation System (JCSS), GIG Technical Guidance for Information Technology Standards, and Interoperability Enhancement Process/iSmart (IEP/iSmart).

Enterprise Services
Enterprise services provided by DISA to its mission partners fall under three categories: Applications, Infrastructure, and Identity and Access Management.

Applications
 * Global Video Services (GVS)
 * DoD Enterprise Email (DEE)
 * DoD Enterprise Portal Service (DEPS)
 * Defense Collaboration Services (DCS)
 * FORGE.MIL
 * Defense Messaging Service
 * Strategic Knowledge Integration Web (SKIWeb)
 * Automated Time, Attendance, and Production System (ATAAPS)
 * Enterprise Search

Infrastructure
 * Rapid Access Computing Environment (RACE)
 * Global Content Delivery Service (GCDS)
 * Enterprise Service Monitoring
 * Enterprise Messaging
 * Enterprise Service Bus (ESB)
 * Data Services Environment (DSE)
 * Hosting Services
 * Secure Technology Application eXecution (STAX)

'''Identity and Access Management
 * DoD Visitor
 * Enterprise Directory Services (EDS)
 * Enterprise Attribute Application Forest (EASF) / Identity Synchronization Services (IDSS)
 * Open Source Attribute-Based Access Control (ABAC)

Information Assurance
DISA's Information Assurance services serve the purpose of:

 * Making data ubiquitously accessible while simultaneously restricting access
 * Promoting the safe sharing of information
 * Preventing attacks by having network protections in place

Network Services
The Defense Information Systems Network (DISN) is a worldwide-protected telecommunications network that enables the exchange of information in an interoperable and global space, partitioned by security demands, transmission requirements, and geographic needs of targeted end-user communities.

Nowadays, DISA maintains the following network services, to support diverse telecommunication requirements for organizations focused on, but not limited to, the Department of Defense (DoD):

Data: Voice:
 * SBU IP Data (formerly known as NIPRNet)
 * Secret IP Data (formerly known as SIPRNet)
 * TS/SCI IP Data (formerly known as JWICS)
 * Secret Test & Evaluation IP Data (formerly known as DISN-LES)
 * Private IP Service
 * SBU Voice (over VoIP and DSN networks)
 * VoSIP (VoIP over the secure SIPRNet)
 * TS/SCI Voice (formerly known as JWICS Voice)
 * Multilevel Secure Voice (formerly known as DRSN)
 * DISA Enterprise CVVoIP

Video:
 * Global Video Services (GVS)
 * TS/SCI Videoconferencing (formerly known as JWICS Videoconferencing)

Messaging:
 * Organizational Messaging Service (formerly known as DMS)

Wireless: Satellite: Providing access to DISN through
 * Enhanced Mobile Satellite Services (EMSS)
 * Secure Mobile Environment - Portable Electronic Device (SME-PED)
 * Commercial Satellite (COMSATCOM)
 * INMARSAT

Spectrum Management
Through the Defense Spectrum Organization (DSO), DISA provides commanders direct operational support, including electromagnetic battlespace planning, deconfliction, and joint spectrum interference resolution. DSO services include:
 * Spectrum Management Operational Support to the Combatant Commands
 * Spectrum-related Applied Engineering
 * Spectrum Technology Testbed Initiative (STTI)
 * Joint Spectrum Center Ordnance Electromagnetic Environmental Effects Risk Assessment Database (JOERAD)
 * Electromagnetic Environmental Effects (E3) and Spectrum Supportability Training and Awareness
 * Global Electromagnetic Spectrum Information System (GEMSIS)
 * Joint Spectrum Data Repository (JSDR)
 * Integrated Spectrum Desktop (ISD)
 * Spectrum XXI (SXXI)
 * End to End Spectrum Supportability (E2ESS): (Stepstone and Host Nation Spectrum Worldwide Database Online (HNSWDO))

Testing
DISA's Joint Interoperability Test Command (JITC) provides testing and joint certification for the net-centric systems employed by U.S. armed forces.

1960s: The Defense Communications Agency
DCA was established May 12, 1960, with the primary mission of operational control and management of the Defense Communications System (DCS).

The initial headquarters for 34 DCA members was Wake Hall, one of a complex of three buildings (which included Midway Hall and Guam Hall) on the site where the parking lot of the Robert F. Kennedy Stadium in Washington, D.C., stands today. Navy Rear Admiral William D. Irvin became the first DCA director in July. In September, Rear Admiral Irvin moved his staff to office space in Building 12 at the Naval Services Center, 701 Courthouse Road, Arlington, Va., the site of the U.S. Navy's old Radio Arlington Station.

DCA's first major tasks were to identify the DCS elements and develop an implementation and management plan. The DCS was essentially a collection of communications systems turned over by the military departments with considerable restrictions. Key among these responsibilities was the establishment of three common-user, defense-wide networks that would be known as the Automatic Voice Network (AUTOVON), the Automatic Digital Network (AUTODIN), and the Automatic Secure Voice Communications Network (AUTOSEVOCOM). For each, DCA sought to determine its overall system configuration and prepare the technical specifications necessary for the equipment for switching centers, interconnecting transmission media, and subscriber terminals.

With the arrival of the space-age, DCA was designated as the “strong focal point” for development, integration, and operation of the space and ground elements of a number of satellite-based communications initiatives. The most important of these would be the DCA-managed Defense Satellite Communications System (DSCS).

The Cuban Missile Crisis of October 1962 showed the need for direct, timely, and private communications between the leaders of the world's two superpowers, the United States and the Soviet Union. A duplex cable circuit (later augmented by a satellite hookup) between the two capitals known as the Moscow–Washington hotline or "Red Telephone", became operational August 30, 1963. Program management and engineering for the “Hotline” was assigned to DCA. The system continues intact today with direct links to more than 40 foreign leaders. Another direct result of the Cuban Missile Crisis was the creation of the Worldwide Military Command and Control System (WWMCCS) to enable national command authorities to exercise effective command and control of their widely dispersed forces.

While DCA dealt with the communication crises of the Cold War, a “hot war” was waging in Southeast Asia. America's commitment to South Vietnam led to the creation of a DCA Southeast Asia Region unit in 1964. DCA developed a plan to integrate the region's communication systems into a single modern network. The system would extend the commercial-quality communications provided by satellites and cables to the battlefield.

1970s
DCA assumed responsibility for the Minimum Essential Emergency Communications Network (MEECN), a subsystem of WWMCCS, in December 1971. The MEECN was developed to assure the timely receipt of emergency action messages by worldwide U.S. nuclear forces under nuclear attack by the Soviet Union. DCA served as the MEECN system engineer and provided the broad engineering necessary to ensure a more survivable future network with compatible, interoperable, and secure subsystems.

In the late 1960s and early 1970s DCA, through its NMCS Technical Support Directorate, had responsibility for the technical support, planning, and system engineering for the four command centers then comprising the NMCS. These were the NMCC and the National Emergency Airborne Command Post (these funded by the USAF), the Alternate National Military Command Center (funded by the Army), and the National Emergency Command Post Afloat (funded by the Navy, and deactivated in the late 1960s). DCA performed the requirements collection and analysis, system engineering, funding estimation, subsystem project plan development, and establishment of the initial physical architecture for the expansion of the NMCC that was implemented by the Air Force in the 1970s.

A DoD directive issued in the early 1970s appointed DCA as the system architect for all defense satellite communications. A major new DCA headquarters staff directorate, the Military Satellite Communications (MILSATCOM) System Office, was created to discharge the new role. As the system architect, DCA coordinated all defense satellite communications planning and programs to avoid duplication and ensure communications interoperability among the diverse systems serving the complete spectrum of defense needs.

1980s
The momentum of major improvements in national security telecommunications accelerated rapidly in the 1980s. Along with the unprecedented peacetime military build-up under the Reagan Administration came the proliferation of government-owned and government-leased networks and a high emphasis on interoperability among the military services. The pace of technological advancement brought with it new opportunities for system improvements.

The desire for interoperability in military communications did not originate in the 1980s. The need for communications systems that talked to each other within an individual military service and among the services together went back to the needs generated by the global proportions of WWII. Indeed, it was the lack of interoperability that drove the Eisenhower administration to seek one organization to pull together the services’ disparate systems to speak with one voice – that organization was DCA. But interoperability still had yet to be achieved by the 1980s.

In April 1986, the assistant secretary of defense for command and control, communications and intelligence proposed the consolidation of DCA and the Joint Tactical Command, Control, and Communications Agency (JTC3A) in view of the “climate within DoD of streamlining and reducing overhead functions.” The Joint Staff endorsed the proposal because it also provided some operational efficiency.

In January 1987, the secretary of defense approved the consolidation of DCA and JTC3A. A year later, DCA absorbed the Tri-Service Tactical Communications Joint Test Element and JTC3A Joint Operability Test Facility. DCA consolidated these organizations into a new organization in 1989, establishing the Joint Interoperability Test Command (JITC) at Fort Huachuca, Arizona. JITC provided the facility for DoD and private-sector interoperability compliance testing and certification.

In October 1989, the deputy secretary of defense established a DoD Corporate Information Management (CIM) Initiative to identify and implement management efficiencies in DoD information systems. DCA was given responsibility for implementing the CIM initiative, and its mission was expanded to include information support to the JCS and Office of the Secretary of Defense, tactical information system standards and interoperability, and White House information systems.

1990s
In 1990 and 1991, during Operations Desert Shield and Desert Storm, a team of planners, engineers, and operators from DCA's Defense Network Systems Organization (DNSO) assisted in the design of a semi-fixed telecommunications system, the Southwest Asia Telecommunications (SATS) for use in support of the theater commander's operations. SATS included satellite, microwave, copper cable, and fiber optic links; Defense Data Network packet-switching nodes; Defense Switched Network (DSN) multi-function voice switches; and technical control facilities. At their peak, these systems included more than 100 satellite links.

On June 25, 1991, DCA underwent a major reorganization and was renamed the Defense Information Systems Agency (DISA) to reflect its expanded role in implementing the DoD's CIM (Corporate Information Management) initiative and to clearly identify DISA as a combat support agency. DISA established the Center for Information Management to provide technical and program execution assistance to the assistant secretary of defense (C3I) and technical products and services to DoD and military components.

DISA's role in DoD information management continued to expand with implementation of several Defense Management Report Decisions (DMRD ), most notably DMRD 918, in September 1992. DMRD 918 created the Defense Information Infrastructure (DII) and directed DISA to manage and consolidate the Services’ and DoD's information processing centers into 16 mega-centers. During the 1990s, DISA fielded new systems to support the combatant commands. The Global Command and Control System (GCCS) and the Joint Chiefs’ C4I (Command, Control, Communications, Computers, and Intelligence) for the Warrior, and the Defense Message System were among the critical systems. GCCS was developed to replace WWMCCS, which had been in existence since the early 1960s.

2000s
With the new century, DISA faced even greater challenges as a DoD service provider. Preserving radio spectrum, information assurance, ensuring interoperability, and establishing secure wireless links were just some of the tasks performed by the agency. Perhaps the most significant achievement of the agency in 2001 was its immediate response in the aftermath of the attacks of Sept. 11, 2001. DISA justified $300 million in supplemental funds to support the Global War on Terrorism by providing critical communications paths and command and control enhancements for warfighters.

In the 18 months between September 2001 and April 2003, DISA supported the exponential use and increased capacity of information systems. The Defense Switched Network (DSN) infrastructure increased 400 percent. The Secret Internet Protocol (IP) Data Service (formerly known as the SIPRNet) capacity increased 292 percent. Sensitive but Unclassified Internet Protocol (IP) Data Service (formerly known as NIPRNet) capacity increased 509 percent. The Defense Video System Global (Secure) increased 1,150 percent. Satellite bandwidth increased 800 percent. The Enhanced Mobile Satellite Service (EMSS) capacity increased 300 percent, and usage increased 3,000 percent. EMSS allowed Special Operations forces to even call in air strikes from horseback in Afghanistan by permitting instantaneous communications in areas without any infrastructure whatsoever.

For Operation Iraqi Freedom in 2003, DISA provided 30 times more bandwidth to a 45-percent smaller force than in Operation Desert Storm in 1991. DISA facilitated multiple enhancements to the nation's preeminent joint command-and-control system and provided a real-time battle space picture.

After the previous consolidation of 194 data-processing centers in the 1990s into 16 computing mega-centers, DISA further reduced the number of mega-centers from 16 to six. Starting in 2003, DISA managed the six-year, $326 million effort to completely modernize presidential communications — the largest such initiative in the 61-year history of the White House Communications Agency. The “Pioneer Program” transformed presidential communications by employing net-centric concepts to put voice, video, and data at the president's fingertips on an around-the-clock basis.

The Global Information Grid Bandwidth Expansion (GIG-BE) Program was a major DoD net-centric transformational initiative executed by DISA. The $877 million program was the largest DoD information technology transport structure ever built. GIG-BE created a ubiquitous "bandwidth-available" environment to improve national security intelligence, surveillance, reconnaissance, information assurance, and command and control at locations worldwide. On Dec. 20, 2005, the GIG-BE program achieved the milestone of full operational capability at all of the almost 100 Joint Staff-approved sites.

2010s
DISA's 50 years of service as the Defense Communications Agency and later the Defense Information Systems Agency was recognized May 12, 2010, during an anniversary celebration at Seven Skyline Place, Falls Church, Va. Army LTG Carroll F. Pollett, the DISA director at the time, led the celebration of the agency's storied past.

From 2008 through 2010, DISA worked directly with the commander, United States Central Command (USCENTCOM), to design and implement a high-capacity, strategic communication network into an active Theater of Operations, ensuring reliable communications for intra-theater mission partners and to national leadership. Prior to this installation, the coalition forces in Afghanistan were dependent on satellite communications and tactical microwave links, which had limited bandwidth capacity and induced significant delay.

DISA operated and defended the Global Information Grid (GIG), providing information capabilities with a reach from the White House to forces at the tactical edge. DISA supported execution of military operations while simultaneously supporting peacekeeping, humanitarian assistance, and disaster-relief missions in multiple theaters around the globe. During a 4-month period in 2011, DISA provided support to an unprecedented six simultaneous operations: Operation NEW DAWN in Iraq; Operation ENDURING FREEDOM in Afghanistan; Operation UNIFIED RESPONSE in Haiti; Operation ODYSSEY DAWN and NATO Operation UNIFIED PROTECTOR in Libya, Operation TOMODACHI in Japan; and DISA's global cyber operations in support of United States Cyber Command (USCYBERCOM).

2020s
Computer systems controlled by DISA were hacked in the summer of 2019, and was reported in February 2020, exposing the personal data of about 200,000 people including names and social security numbers.

The US Army is seeking to reduce its NIPRNet footprint by 2023 and its SIPRnet footprint by 2024.

Military Unit Decorations
The unit awards below are issued to Headquarters, Defense Information Systems Agency and subordinate units unless otherwise noted.

Members of military-specific units (i.e., Army, Navy, Marine Corps, and Air Force) assigned or attached, even if they are under the operational control, are not eligible for the Joint Meritorious Unit Award (JMUA), but retain eligibility for unit awards from their respective military service department.

Members must have been permanently assigned or attached by official orders to the agency for 30 or more days during the designated award period. Service members individually assigned or attached (including temporary duty or temporary additional duty) status is determined by official orders that specify the agency as the duty unit. Some military services allow the “temporary-wear” of past unit citations when assigned to the organization. The JMUA is an exception to this rule and is not authorized for temporary wear.

title: Defense Information Systems Agency


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United States Department of Justice

The United States Department of Justice (DOJ), also known as the Justice Department, is a federal executive department of the United States government tasked with the enforcement of federal law and administration of justice in the United States. It is equivalent to the justice or interior ministries of other countries. The department is headed by the U.S. attorney general, who reports directly to the president of the United States and is a member of the president's Cabinet. The current attorney general is Merrick Garland, who has served since March 2021.

The Justice Department contains most of the United States' federal law enforcement agencies, including the Federal Bureau of Investigation, the U.S. Marshals Service, the Bureau of Alcohol, Tobacco, Firearms and Explosives, the Drug Enforcement Administration, and the Federal Bureau of Prisons. The department also has eight divisions of lawyers who represent the U.S. federal government in litigation: the Criminal, Civil, Antitrust, Tax, Civil Rights, Environment and Natural Resources, National Security, and Justice Management Divisions. The department also includes the U.S. Attorneys' Offices for each of the 94 U.S. federal judicial districts.

The U.S. Congress created the Justice Department in 1870 during the presidency of Ulysses S. Grant. The Justice Department's functions originally date to 1789, when Congress created the office of the Attorney General.

History
The office of the attorney general was established by the Judiciary Act of 1789 as a part-time job for one person, but grew with the bureaucracy. At one time, the attorney general gave legal advice to the U.S. Congress, as well as the president; however, in 1819, the attorney general began advising Congress alone to ensure a manageable workload. Until 1853, the salary of the attorney general was set by statute at less than the amount paid to other Cabinet members. Early attorneys general supplemented their salaries by running private law practices, often arguing cases before the courts as attorneys for paying litigants. The lightness of the office is exemplified by Edward Bates (1793–1869), Attorney General under Abraham Lincoln (1861 to 1864). Bates had only a small operation, with a staff of six. The main function was to generate legal opinions at the request of Lincoln and cabinet members, and handle occasional cases before the Supreme Court. Lincoln's cabinet was full of experienced lawyers who seldom felt the need to ask for his opinions. Bates had no authority over the US Attorneys around the country. The federal court system was handled by the Interior Department; the Treasury handled claims. Most of the opinions turned out by Bates's office were of minor importance. Lincoln gave him no special assignments and did not seek his advice on Supreme Court appointments. Bates did have an opportunity to comment on general policy as a cabinet member with a strong political base, but he seldom spoke up.

Following unsuccessful efforts in 1830 and 1846 to make attorney general a full-time job, in 1867, the U.S. House Committee on the Judiciary, led by Congressman William Lawrence, conducted an inquiry into the creation of a "law department" headed by the attorney general and also composed of the various department solicitors and United States attorneys. On February 19, 1868, Lawrence introduced a bill in Congress to create the Department of Justice. President Ulysses S. Grant signed the bill into law on June 22, 1870.

Grant appointed Amos T. Akerman as attorney general and Benjamin H. Bristow as America's first solicitor general the same week that Congress created the Department of Justice. The Department's immediate function was to preserve civil rights. It set about fighting against domestic terrorist groups who had been using both violence and litigation to oppose the 13th, 14th, and 15th Amendments to the Constitution.

Both Akerman and Bristow used the Department of Justice to vigorously prosecute Ku Klux Klan members in the early 1870s. In the first few years of Grant's first term in office, there were 1000 indictments against Klan members, with over 550 convictions from the Department of Justice. By 1871, there were 3000 indictments and 600 convictions, with most only serving brief sentences, while the ringleaders were imprisoned for up to five years in the federal penitentiary in Albany, New York. The result was a dramatic decrease in violence in the South. Akerman gave credit to Grant and told a friend that no one was "better" or "stronger" than Grant when it came to prosecuting terrorists. George H. Williams, who succeeded Akerman in December 1871, continued to prosecute the Klan throughout 1872 until the spring of 1873, during Grant's second term in office. Williams then placed a moratorium on Klan prosecutions partially because the Justice Department, inundated by cases involving the Klan, did not have the manpower to continue prosecutions.

The "Act to Establish the Department of Justice" drastically increased the attorney general's responsibilities to include the supervision of all United States attorneys, formerly under the Department of the Interior, the prosecution of all federal crimes, and the representation of the United States in all court actions, barring the use of private attorneys by the federal government. The law also created the office of Solicitor General to supervise and conduct government litigation in the Supreme Court of the United States.

With the passage of the Interstate Commerce Act in 1887, the federal government took on some law enforcement responsibilities, and the Department of Justice was tasked with performing these.

In 1884, control of federal prisons was transferred to the new department, from the Department of the Interior. New facilities were built, including the penitentiary at Leavenworth in 1895, and a facility for women located in West Virginia, at Alderson was established in 1924.

In 1933, President Franklin D. Roosevelt issued an executive order which gave the Department of Justice responsibility for the "functions of prosecuting in the courts of the United States claims and demands by, and offsenses [sic] against, the Government of the United States, and of defending claims and demands against the Government, and of supervising the work of United States attorneys, marshals, and clerks in connection therewith, now exercised by any agency or officer..."

Headquarters
The U.S. Department of Justice building was completed in 1935 from a design by Milton Bennett Medary. Upon Medary's death in 1929, the other partners of his Philadelphia firm Zantzinger, Borie and Medary took over the project. On a lot bordered by Constitution and Pennsylvania Avenues and Ninth and Tenth Streets, Northwest, it holds over 1000000 ft2 of space.

Various efforts, none entirely successful, have been made to determine the original intended meaning of the Latin motto appearing on the Department of Justice seal, Qui Pro Domina Justitia Sequitur (literally "Who For Lady Justice Strives"). It is not even known exactly when the original version of the DOJ seal itself was adopted, or when the motto first appeared on the seal. The most authoritative opinion of the DOJ suggests that the motto refers to the Attorney General (and thus, by extension, to the Department of Justice) "who prosecutes on behalf of justice (or the Lady Justice)".

The motto's conception of the prosecutor (or government attorney) as being the servant of justice itself finds concrete expression in a similarly-ordered English-language inscription ("THE UNITED STATES WINS ITS POINT WHENEVER JUSTICE IS DONE ITS CITIZENS IN THE COURTS") in the above-door paneling in the ceremonial rotunda anteroom just outside the Attorney General's office in the Department of Justice Main Building in Washington, D.C. The building was renamed in honor of former Attorney General Robert F. Kennedy in 2001. It is sometimes referred to as "Main Justice".

Leadership offices

 * Office of the Attorney General
 * Office of the Deputy Attorney General
 * Office of the Associate Attorney General
 * Office of the United States Principal Deputy Associate Attorney General
 * Office of the Solicitor General of the United States

Divisions
The Justice Department also had a War Division during World War II. It was created in 1942 and disestablished in 1945.

Law enforcement agencies
Several federal law enforcement agencies are administered by the Department of Justice:
 * United States Marshals Service (USMS) – The office of U.S. Marshal was established by the Judiciary Act of 1789. The U.S. Marshals Service was established as an agency in 1969, and it was elevated to full bureau status under the Justice Department in 1974.
 * Federal Bureau of Investigation (FBI) – On July 26, 1908, a small investigative force was created within the Justice Department under Attorney General Charles Bonaparte. The following year, this force was officially named the Bureau of Investigation by Attorney General George W. Wickersham. In 1935, the bureau adopted its current name.
 * Federal Bureau of Prisons (BOP) – the Three Prisons Act of 1891 created the federal prison system. Congress created the Federal Bureau of Prisons in 1930 by Pub. L. No. 71–218, 46 Stat. 325, signed into law by President Hoover on May 14, 1930.
 * National Institute of Corrections (NIC) – Founded in 1974, the National Institute of Corrections is organized under the Federal Bureau of Prisons and has a legislatively mandated mission to assist state and local correctional institutions, and to manage the American Federal Prison System by keeping records of inmates.
 * Bureau of Alcohol, Tobacco, Firearms and Explosives (ATF) – Except for a brief period during Prohibition, ATF's predecessor bureaus were part of the Department of the Treasury for more than two hundred years. ATF was first established by Department of Treasury Order No. 221, effective July 1, 1972; this order "transferred the functions, powers, and duties arising under laws relating to alcohol, tobacco, firearms, and explosives from the Internal Revenue Service to ATF. In 2003, under the terms of the Homeland Security Act, ATF was split into two agencies – the new Bureau of Alcohol, Tobacco, Firearms and Explosives (ATF) was transferred to the Department of Justice, while the Alcohol and Tobacco Tax and Trade Bureau (TTB) was retained by the Department of the Treasury.
 * Drug Enforcement Administration (DEA) – Created in 1973 as part of the war on drugs, the DEA was formed from various previously existing law enforcement agencies that were parts of either the Department of Justice, Department of the Treasury or the Food and Drug Administration. The DEA enforces the Controlled Substances Act and also interdicts foreign drug trafficking.
 * Office of the Inspector General (OIG) – The Office of Inspector General performs basic internal auditing functions, and has the power to make arrests and prosecute members of the Department of Justice who are found to be in violation of laws regulating conduct of government officials.

Offices

 * Executive Office for Immigration Review (EOIR)
 * Executive Office for U.S. Attorneys (EOUSA)
 * Executive Office of the United States Trustee (EOUST)
 * Office of Attorney Recruitment and Management (OARM)
 * Office of the Chief Information Officer
 * In May 2014, the department appointed Joseph Klimavicz as CIO. Klimavicz succeeds Kevin Deeley, who served as acting CIO since November 2013 when the previous office holder, Luke McCormack, left to take the CIO post at the Department of Homeland Security.
 * Office of Dispute Resolution
 * Office of the Federal Detention Trustee (OFDT)
 * Office of the Executive Secretariat
 * Office of Immigration Litigation
 * Office of Information Policy
 * Office of Intelligence Policy and Review (OIPR)
 * Office of Justice Programs (OJP)
 * Bureau of Justice Assistance (BJA)
 * Bureau of Justice Statistics (BJS)
 * National Institute of Justice (NIJ)
 * Office of Juvenile Justice and Delinquency Prevention (OJJDP)
 * Office for Victims of Crime (OVC)
 * Sex Offender Sentencing, Monitoring, Apprehending, Registering, and Tracking Office (SMART)
 * Office of the Police Corps and Law Enforcement Education
 * Office of Legal Counsel (OLC)
 * Office of Legal Policy (OLP)
 * Office of Legislative Affairs
 * Office of the Pardon Attorney
 * Office of Privacy and Civil Liberties (OPCL)
 * Office of Professional Responsibility (OPR)
 * Office of Public Affairs
 * Office on Sexual Violence and Crimes against Children
 * Office of Tribal Justice
 * Office on Violence Against Women (OVW)
 * Professional Responsibility Advisory Office (PRAO)
 * United States Attorneys Offices
 * United States Trustees Offices
 * Office of Community Oriented Policing Services (COPS)
 * Community Relations Service (CRS)

Other offices and programs

 * Foreign Claims Settlement Commission of the United States
 * INTERPOL, U.S. National Central Bureau
 * National Drug Intelligence Center
 * Obscenity Prosecution Task Force
 * United States Parole Commission

In March 2003, the United States Immigration and Naturalization Service was abolished and its functions transferred to the United States Department of Homeland Security. The Executive Office for Immigration Review and the Board of Immigration Appeals, which review decisions made by government officials under Immigration and Nationality law, remain under jurisdiction of the Department of Justice. Similarly the Office of Domestic Preparedness left the Justice Department for the Department of Homeland Security, but only for executive purposes. The Office of Domestic Preparedness is still centralized within the Department of Justice, since its personnel are still officially employed within the Department of Justice.

In 2003, the Department of Justice created LifeAndLiberty.gov, a website that supported the USA PATRIOT Act. It was criticized by government watchdog groups for its alleged violation of U.S. Code Title 18 Section 1913, which forbids money appropriated by Congress to be used to lobby in favor of any law, actual or proposed. The website has since been taken offline.

On October 5, 2021, U.S. Deputy Attorney General Lisa Monaco has announced the formation of a "Cryptocurrency Enforcement Team" during the Aspen Cyber Summit.

Finances and budget
In 2015, the Justice Department's budget was as follows:

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: United States Department of Justice


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Partial differential equation

In mathematics, a partial differential equation (PDE) is an equation which computes a function between various partial derivatives of a multivariable function.

The function is often thought of as an "unknown" to be solved for, similar to how $x$ is thought of as an unknown number to be solved for in an algebraic equation like $x^{2} − 3x + 2 = 0$. However, it is usually impossible to write down explicit formulae for solutions of partial differential equations. There is correspondingly a vast amount of modern mathematical and scientific research on methods to numerically approximate solutions of certain partial differential equations using computers. Partial differential equations also occupy a large sector of pure mathematical research, in which the usual questions are, broadly speaking, on the identification of general qualitative features of solutions of various partial differential equations, such as existence, uniqueness, regularity and stability. Among the many open questions are the existence and smoothness of solutions to the Navier–Stokes equations, named as one of the Millennium Prize Problems in 2000.

Partial differential equations are ubiquitous in mathematically oriented scientific fields, such as physics and engineering. For instance, they are foundational in the modern scientific understanding of sound, heat, diffusion, electrostatics, electrodynamics, thermodynamics, fluid dynamics, elasticity, general relativity, and quantum mechanics (Schrödinger equation, Pauli equation etc.). They also arise from many purely mathematical considerations, such as differential geometry and the calculus of variations; among other notable applications, they are the fundamental tool in the proof of the Poincaré conjecture from geometric topology.

Partly due to this variety of sources, there is a wide spectrum of different types of partial differential equations, and methods have been developed for dealing with many of the individual equations which arise. As such, it is usually acknowledged that there is no "general theory" of partial differential equations, with specialist knowledge being somewhat divided between several essentially distinct subfields.

Ordinary differential equations can be viewed as a subclass of partial differential equations, corresponding to functions of a single variable. Stochastic partial differential equations and nonlocal equations are, as of 2020, particularly widely studied extensions of the "PDE" notion. More classical topics, on which there is still much active research, include elliptic and parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations.

Introduction
A function $u(x, y, z)$ of three variables is "harmonic" or "a solution of the Laplace equation" if it satisfies the condition $$\frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}+\frac{\partial^2u}{\partial z^2}=0.$$ Such functions were widely studied in the 19th century due to their relevance for classical mechanics, for example the equilibrium temperature distribution of a homogeneous solid is a harmonic function. If explicitly given a function, it is usually a matter of straightforward computation to check whether or not it is harmonic. For instance $$u(x,y,z) = \frac{1}{\sqrt{x^2 - 2x + y^2 + z^2 + 1}}$$ and $$u(x,y,z) = 2x^2 - y^2 - z^2$$ are both harmonic while $$u(x,y,z)=\sin(xy)+z$$ is not. It may be surprising that the two given examples of harmonic functions are of such a strikingly different form from one another. This is a reflection of the fact that they are not, in any immediate way, both special cases of a "general solution formula" of the Laplace equation. This is in striking contrast to the case of ordinary differential equations (ODEs) roughly similar to the Laplace equation, with the aim of many introductory textbooks being to find algorithms leading to general solution formulas. For the Laplace equation, as for a large number of partial differential equations, such solution formulas fail to exist.

The nature of this failure can be seen more concretely in the case of the following PDE: for a function $v(x, y)$ of two variables, consider the equation $$\frac{\partial^2v}{\partial x\partial y}=0.$$ It can be directly checked that any function $v$ of the form $v(x, y) = f(x) + g(y)$, for any single-variable functions $f$ and $g$ whatsoever, will satisfy this condition. This is far beyond the choices available in ODE solution formulas, which typically allow the free choice of some numbers. In the study of PDEs, one generally has the free choice of functions.

The nature of this choice varies from PDE to PDE. To understand it for any given equation, existence and uniqueness theorems are usually important organizational principles. In many introductory textbooks, the role of existence and uniqueness theorems for ODE can be somewhat opaque; the existence half is usually unnecessary, since one can directly check any proposed solution formula, while the uniqueness half is often only present in the background in order to ensure that a proposed solution formula is as general as possible. By contrast, for PDE, existence and uniqueness theorems are often the only means by which one can navigate through the plethora of different solutions at hand. For this reason, they are also fundamental when carrying out a purely numerical simulation, as one must have an understanding of what data is to be prescribed by the user and what is to be left to the computer to calculate.

To discuss such existence and uniqueness theorems, it is necessary to be precise about the domain of the "unknown function". Otherwise, speaking only in terms such as "a function of two variables", it is impossible to meaningfully formulate the results. That is, the domain of the unknown function must be regarded as part of the structure of the PDE itself.

The following provides two classic examples of such existence and uniqueness theorems. Even though the two PDE in question are so similar, there is a striking difference in behavior: for the first PDE, one has the free prescription of a single function, while for the second PDE, one has the free prescription of two functions. Even more phenomena are possible. For instance, the following PDE, arising naturally in the field of differential geometry, illustrates an example where there is a simple and completely explicit solution formula, but with the free choice of only three numbers and not even one function. \frac{\partial}{\partial y} \frac{\frac{\partial u}{\partial y}}{\sqrt{1 + \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2}}=0,$$ then there are numbers $B$, $U$, and $u$ with $R$. In contrast to the earlier examples, this PDE is nonlinear, owing to the square roots and the squares. A linear PDE is one such that, if it is homogeneous, the sum of any two solutions is also a solution, and any constant multiple of any solution is also a solution.
 * Let $B$ denote the unit-radius disk around the origin in the plane. For any continuous function $U$ on the unit circle, there is exactly one function $f$ on $g$ such that $$\frac{\partial^2u}{\partial x^2} + \frac{\partial^2u}{\partial y^2} = 0$$ and whose restriction to the unit circle is given by $u$.
 * For any functions $x$ and $u$ on the real line $R × (−1, 1)$, there is exactly one function $a$ on $u(x, 0) = f(x)$ such that $$\frac{\partial^2u}{\partial x^2} - \frac{\partial^2u}{\partial y^2} = 0$$ and with $∂u⁄∂y(x, 0) = g(x)$ and $R^{2}$ for all values of $b$.
 * If $c$ is a function on $u(x, y) = ax + by + c$ with $$\frac{\partial}{\partial x} \frac{\frac{\partial u}{\partial x}}{\sqrt{1 + \left(\frac{\partial u}{\partial x}\right)^2 + \left(\frac{\partial u}{\partial y}\right)^2}} +

Well-posedness
Well-posedness refers to a common schematic package of information about a PDE. To say that a PDE is well-posed, one must have: This is, by the necessity of being applicable to several different PDE, somewhat vague. The requirement of "continuity", in particular, is ambiguous, since there are usually many inequivalent means by which it can be rigorously defined. It is, however, somewhat unusual to study a PDE without specifying a way in which it is well-posed.
 * an existence and uniqueness theorem, asserting that by the prescription of some freely chosen functions, one can single out one specific solution of the PDE
 * by continuously changing the free choices, one continuously changes the corresponding solution

The energy method
The energy method is a mathematical procedure that can be used to verify well-posedness of initial-boundary-value-problems (IBVP). In the following example the energy method is used to decide where and which boundary conditions should be imposed such that the resulting IBVP is well-posed. Consider the one-dimensional hyperbolic PDE given by $$\frac{\partial u}{\partial t} + \alpha \frac{\partial u}{\partial x} = 0, \quad x \in [a,b], t > 0,$$

where $$\alpha \neq 0$$ is a constant and $$u(x,t)$$ is an unknown function with initial condition $$u(x,0) = f(x)$$. Multiplying with $$u$$ and integrating over the domain gives $$\int_a^b u \frac{\partial u}{\partial t} \mathrm dx + \alpha \int _a ^b u \frac{\partial u}{\partial x} \mathrm dx = 0.$$

Using that $$\int _a ^b u \frac{\partial u}{\partial t} \mathrm dx = \frac{1}{2} \frac{\partial}{\partial t} \| u \| ^2 \quad \text{and} \quad \int _a ^b u \frac{\partial u}{\partial x} \mathrm dx = \frac{1}{2} u(b,t)^2 - \frac{1}{2} u(a,t)^2,$$ where integration by parts has been used for the first relationship, we get $$\frac{\partial}{\partial t} \| u \| ^2 + \alpha u(b,t)^2 - \alpha u(a,t)^2 = 0.$$

Here $$\| \cdot \|$$ denotes the standard $$L^2$$ norm. For well-posedness we require that the energy of the solution is non-increasing, i.e. that $\frac{\partial}{\partial t} \| u \| ^2 \leq 0$, which is achieved by specifying $$u$$ at $$x = a$$ if $$\alpha > 0$$ and at $$x = b$$ if $$\alpha < 0$$. This corresponds to only imposing boundary conditions at the inflow. Well-posedness allows for growth in terms of data (initial and boundary) and thus it is sufficient to show that $\frac{\partial}{\partial t} \| u \| ^2 \leq 0$ holds when all data are set to zero.

Existence of local solutions
The Cauchy–Kowalevski theorem for Cauchy initial value problems essentially states that if the terms in a partial differential equation are all made up of analytic functions and a certain transversality condition is satisfied (the hyperplane or more generally hypersurface where the initial data are posed must be non-characteristic with respect to the partial differential operator), then on certain regions, there necessarily exist solutions which are as well analytic functions. This is a fundamental result in the study of analytic partial differential equations. Surprisingly, the theorem does not hold in the setting of smooth functions; an example discovered by Hans Lewy in 1957 consists of a linear partial differential equation whose coefficients are smooth (i.e., have derivatives of all orders) but not analytic for which no solution exists. So the Cauchy-Kowalevski theorem is necessarily limited in its scope to analytic functions.

Notation
When writing PDEs, it is common to denote partial derivatives using subscripts. For example: $$u_x = \frac{\partial u}{\partial x},\quad u_{xx} = \frac{\partial^2 u}{\partial x^2},\quad u_{xy} = \frac{\partial^2 u}{\partial y\, \partial x} = \frac{\partial}{\partial y } \left(\frac{\partial u}{\partial x}\right). $$ In the general situation that $u$ is a function of $n$ variables, then $u_{i}$ denotes the first partial derivative relative to the $i$-th input, $u_{ij}$ denotes the second partial derivative relative to the $i$-th and $j$-th inputs, and so on.

The Greek letter $Δ$ denotes the Laplace operator; if $u$ is a function of $n$ variables, then $$\Delta u = u_{11} + u_{22} + \cdots + u_{nn}.$$ In the physics literature, the Laplace operator is often denoted by $∇^{2}$; in the mathematics literature, $∇^{2}u$ may also denote the Hessian matrix of $u$.

Linear equations
A PDE is called linear if it is linear in the unknown and its derivatives. For example, for a function $u$ of $x$ and $y$, a second order linear PDE is of the form $$ a_1(x,y)u_{xx} + a_2(x,y)u_{xy} + a_3(x,y)u_{yx} + a_4(x,y)u_{yy} + a_5(x,y)u_x + a_6(x,y)u_y + a_7(x,y)u = f(x,y) $$ where $a_{i}$ and $f$ are functions of the independent variables $x$ and $y$ only. (Often the mixed-partial derivatives $u_{xy}$ and $u_{yx}$ will be equated, but this is not required for the discussion of linearity.) If the $a_{i}$ are constants (independent of $x$ and $y$) then the PDE is called linear with constant coefficients. If $f$ is zero everywhere then the linear PDE is homogeneous, otherwise it is inhomogeneous. (This is separate from asymptotic homogenization, which studies the effects of high-frequency oscillations in the coefficients upon solutions to PDEs.)

Nonlinear equations
Three main types of nonlinear PDEs are semi-linear PDEs, quasilinear PDEs, and fully nonlinear PDEs.

Nearest to linear PDEs are semi-linear PDEs, where only the highest order derivatives appear as linear terms, with coefficients that are functions of the independent variables. The lower order derivatives and the unknown function may appear arbitrarily. For example, a general second order semi-linear PDE in two variables is $$ a_1(x,y)u_{xx} + a_2(x,y)u_{xy} + a_3(x,y)u_{yx} + a_4(x,y)u_{yy} + f(u_x, u_y, u, x, y) = 0 $$

In a quasilinear PDE the highest order derivatives likewise appear only as linear terms, but with coefficients possibly functions of the unknown and lower-order derivatives: $$ a_1(u_x, u_y, u, x, y)u_{xx} + a_2(u_x, u_y, u, x, y)u_{xy} + a_3(u_x, u_y, u, x, y)u_{yx} + a_4(u_x, u_y, u, x, y)u_{yy} + f(u_x, u_y, u, x, y) = 0 $$ Many of the fundamental PDEs in physics are quasilinear, such as the Einstein equations of general relativity and the Navier–Stokes equations describing fluid motion.

A PDE without any linearity properties is called fully nonlinear, and possesses nonlinearities on one or more of the highest-order derivatives. An example is the Monge–Ampère equation, which arises in differential geometry.

Linear equations of second order
The elliptic/parabolic/hyperbolic classification provides a guide to appropriate initial- and boundary conditions and to the smoothness of the solutions. Assuming $u_{xy} = u_{yx}$, the general linear second-order PDE in two independent variables has the form $$Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots \mbox{(lower order terms)} = 0,$$ where the coefficients $A$, $B$, $C$... may depend upon $x$ and $y$. If $A^{2} + B^{2} + C^{2} > 0$ over a region of the $xy$-plane, the PDE is second-order in that region. This form is analogous to the equation for a conic section: $$Ax^2 + 2Bxy + Cy^2 + \cdots = 0.$$

More precisely, replacing $∂_{x}$ by $X$, and likewise for other variables (formally this is done by a Fourier transform), converts a constant-coefficient PDE into a polynomial of the same degree, with the terms of the highest degree (a homogeneous polynomial, here a quadratic form) being most significant for the classification.

Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant $B^{2} − 4AC$, the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by $B^{2} − AC$ due to the convention of the $xy$ term being $2B$ rather than $B$; formally, the discriminant (of the associated quadratic form) is $(2B)^{2} − 4AC = 4(B^{2} − AC)$, with the factor of 4 dropped for simplicity.

 * 1) $B^{2} − AC < 0$ (elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where $x < 0$. By change of variables, the equation can always be expressed in the form: $$u_{xx} + u_{yy} + \cdots = 0,  $$where x and y correspond to changed variables. Thus justifies Laplace equation as an example of this type.
 * 2) $B^{2} − AC = 0$ (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where $x = 0$. By change of variables, the equation can always be expressed in the form: $$u_{xx} + \cdots = 0,$$where x correspond to changed variables. Thus justifies heat equation, which are of form $u_t - u_{xx} + \cdots = 0      $, as an example of this type.
 * 3) $B^{2} − AC > 0$ (hyperbolic partial differential equation): hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where $x > 0$. By change of variables, the equation can always be expressed in the form: $$u_{xx} - u_{yy} + \cdots = 0,$$where x and y correspond to changed variables. Thus justifies wave equation as an example of this type.

If there are $n$ independent variables $x_{1}, x_{2 }, …, x_{n}$, a general linear partial differential equation of second order has the form $$L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\partial^2 u}{\partial x_i \partial x_j} \quad+ \text{lower-order terms} = 0.$$

The classification depends upon the signature of the eigenvalues of the coefficient matrix $a_{i,j}$.

 * 1) Elliptic: the eigenvalues are all positive or all negative.
 * 2) Parabolic: the eigenvalues are all positive or all negative, except one that is zero.
 * 3) Hyperbolic: there is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
 * 4) Ultrahyperbolic: there is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues.

The theory of elliptic, parabolic, and hyperbolic equations have been studied for centuries, largely centered around or based upon the standard examples of the Laplace equation, the heat equation, and the wave equation.

However, the classification only depends on linearity of the second-order terms and is therefore applicable to semi- and quasilinear PDEs as well. The basic types also extend to hybrids such as the Euler–Tricomi equation; varying from elliptic to hyperbolic for different regions of the domain, as well as higher-order PDEs, but such knowledge is more specialized.

Systems of first-order equations and characteristic surfaces
The classification of partial differential equations can be extended to systems of first-order equations, where the unknown $u$ is now a vector with $m$ components, and the coefficient matrices $A_{ν}$ are $m$ by $m$ matrices for $ν = 1, 2, …, n$. The partial differential equation takes the form $$Lu = \sum_{\nu=1}^{n} A_\nu \frac{\partial u}{\partial x_\nu} + B=0,$$ where the coefficient matrices $A_{ν}$ and the vector $B$ may depend upon $x$ and $u$. If a hypersurface $S$ is given in the implicit form $$\varphi(x_1, x_2, \ldots, x_n)=0,$$ where $φ$ has a non-zero gradient, then $S$ is a characteristic surface for the operator $L$ at a given point if the characteristic form vanishes: $$Q\left(\frac{\partial\varphi}{\partial x_1}, \ldots, \frac{\partial\varphi}{\partial x_n}\right) = \det\left[\sum_{\nu=1}^n A_\nu \frac{\partial \varphi}{\partial x_\nu}\right] = 0.$$

The geometric interpretation of this condition is as follows: if data for $u$ are prescribed on the surface $S$, then it may be possible to determine the normal derivative of $u$ on $S$ from the differential equation. If the data on $S$ and the differential equation determine the normal derivative of $u$ on $S$, then $S$ is non-characteristic. If the data on $S$ and the differential equation do not determine the normal derivative of $u$ on $S$, then the surface is characteristic, and the differential equation restricts the data on $S$: the differential equation is internal to $S$.

 * 1) A first-order system $Lu = 0$ is elliptic if no surface is characteristic for $L$: the values of $u$ on $S$ and the differential equation always determine the normal derivative of $u$ on $S$.
 * 2) A first-order system is hyperbolic at a point if there is a spacelike surface $S$ with normal $ξ$ at that point. This means that, given any non-trivial vector $η$ orthogonal to $ξ$, and a scalar multiplier $λ$, the equation $Q(λξ + η) = 0$ has $m$ real roots $λ_{1}, λ_{2}, …, λ_{m}$. The system is strictly hyperbolic if these roots are always distinct. The geometrical interpretation of this condition is as follows: the characteristic form $Q(ζ) = 0$ defines a cone (the normal cone) with homogeneous coordinates ζ. In the hyperbolic case, this cone has $m$ sheets, and the axis $ζ = λξ$ runs inside these sheets: it does not intersect any of them. But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.

Separation of variables
Linear PDEs can be reduced to systems of ordinary differential equations by the important technique of separation of variables. This technique rests on a feature of solutions to differential equations: if one can find any solution that solves the equation and satisfies the boundary conditions, then it is the solution (this also applies to ODEs). We assume as an ansatz that the dependence of a solution on the parameters space and time can be written as a product of terms that each depend on a single parameter, and then see if this can be made to solve the problem.

In the method of separation of variables, one reduces a PDE to a PDE in fewer variables, which is an ordinary differential equation if in one variable – these are in turn easier to solve.

This is possible for simple PDEs, which are called separable partial differential equations, and the domain is generally a rectangle (a product of intervals). Separable PDEs correspond to diagonal matrices – thinking of "the value for fixed $x$" as a coordinate, each coordinate can be understood separately.

This generalizes to the method of characteristics, and is also used in integral transforms.

Method of characteristics
In special cases, one can find characteristic curves on which the equation reduces to an ODE – changing coordinates in the domain to straighten these curves allows separation of variables, and is called the method of characteristics.

More generally, one may find characteristic surfaces. For a second order partial differential equation solution, see the Charpit method.

Integral transform
An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.

An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.

If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains. The solution for a point source for the heat equation given above is an example of the use of a Fourier integral.

Change of variables
Often a PDE can be reduced to a simpler form with a known solution by a suitable change of variables. For example, the Black–Scholes equation $$ \frac{\partial V}{\partial t} + \tfrac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + rS \frac{\partial V}{\partial S} - rV = 0 $$ is reducible to the heat equation $$ \frac{\partial u}{\partial \tau} = \frac{\partial^2 u}{\partial x^2}$$ by the change of variables $$\begin{align} V(S,t) &= v(x,\tau),\\[5px] x &= \ln\left(S \right),\\[5px] \tau &= \tfrac{1}{2} \sigma^2 (T - t),\\[5px] v(x,\tau) &= e^{-\alpha x-\beta\tau} u(x,\tau). \end{align}$$

Fundamental solution
Inhomogeneous equations can often be solved (for constant coefficient PDEs, always be solved) by finding the fundamental solution (the solution for a point source), then taking the convolution with the boundary conditions to get the solution.

This is analogous in signal processing to understanding a filter by its impulse response.

Superposition principle
The superposition principle applies to any linear system, including linear systems of PDEs. A common visualization of this concept is the interaction of two waves in phase being combined to result in a greater amplitude, for example $sin x + sin x = 2 sin x$. The same principle can be observed in PDEs where the solutions may be real or complex and additive. If $u_{1}$ and $u_{2}$ are solutions of linear PDE in some function space $R$, then $u = c_{1}u_{1} + c_{2}u_{2}$ with any constants $c_{1}$ and $c_{2}$ are also a solution of that PDE in the same function space.

Methods for non-linear equations
There are no generally applicable methods to solve nonlinear PDEs. Still, existence and uniqueness results (such as the Cauchy–Kowalevski theorem) are often possible, as are proofs of important qualitative and quantitative properties of solutions (getting these results is a major part of analysis). Computational solution to the nonlinear PDEs, the split-step method, exist for specific equations like nonlinear Schrödinger equation.

Nevertheless, some techniques can be used for several types of equations. The $h$-principle is the most powerful method to solve underdetermined equations. The Riquier–Janet theory is an effective method for obtaining information about many analytic overdetermined systems.

The method of characteristics can be used in some very special cases to solve nonlinear partial differential equations.

In some cases, a PDE can be solved via perturbation analysis in which the solution is considered to be a correction to an equation with a known solution. Alternatives are numerical analysis techniques from simple finite difference schemes to the more mature multigrid and finite element methods. Many interesting problems in science and engineering are solved in this way using computers, sometimes high performance supercomputers.

Lie group method
From 1870 Sophus Lie's work put the theory of differential equations on a more satisfactory foundation. He showed that the integration theories of the older mathematicians can, by the introduction of what are now called Lie groups, be referred, to a common source; and that ordinary differential equations which admit the same infinitesimal transformations present comparable difficulties of integration. He also emphasized the subject of transformations of contact.

A general approach to solving PDEs uses the symmetry property of differential equations, the continuous infinitesimal transformations of solutions to solutions (Lie theory). Continuous group theory, Lie algebras and differential geometry are used to understand the structure of linear and nonlinear partial differential equations for generating integrable equations, to find its Lax pairs, recursion operators, Bäcklund transform and finally finding exact analytic solutions to the PDE.

Symmetry methods have been recognized to study differential equations arising in mathematics, physics, engineering, and many other disciplines.

Semi-analytical methods
The Adomian decomposition method, the Lyapunov artificial small parameter method, and his homotopy perturbation method are all special cases of the more general homotopy analysis method. These are series expansion methods, and except for the Lyapunov method, are independent of small physical parameters as compared to the well known perturbation theory, thus giving these methods greater flexibility and solution generality.

Numerical solutions
The three most widely used numerical methods to solve PDEs are the finite element method (FEM), finite volume methods (FVM) and finite difference methods (FDM), as well other kind of methods called meshfree methods, which were made to solve problems where the aforementioned methods are limited. The FEM has a prominent position among these methods and especially its exceptionally efficient higher-order version hp-FEM. Other hybrid versions of FEM and Meshfree methods include the generalized finite element method (GFEM), extended finite element method (XFEM), spectral finite element method (SFEM), meshfree finite element method, discontinuous Galerkin finite element method (DGFEM), element-free Galerkin method (EFGM), interpolating element-free Galerkin method (IEFGM), etc.

Finite element method
The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler's method, Runge–Kutta, etc.

Finite difference method
Finite-difference methods are numerical methods for approximating the solutions to differential equations using finite difference equations to approximate derivatives.

Finite volume method
Similar to the finite difference method or finite element method, values are calculated at discrete places on a meshed geometry. "Finite volume" refers to the small volume surrounding each node point on a mesh. In the finite volume method, surface integrals in a partial differential equation that contain a divergence term are converted to volume integrals, using the divergence theorem. These terms are then evaluated as fluxes at the surfaces of each finite volume. Because the flux entering a given volume is identical to that leaving the adjacent volume, these methods conserve mass by design.

Fuzzy differential equation method
Fuzzy differential equation is based on fuzzy logic.

Fuzzy differential inclusion method
Fuzzy differential inclusion method uses fuzzy logic with Laplace transform to solve PDE.

Applications
Partial differential equations are widely used in many fields, such as Astronomy, Cosmology, Quantum mechanics, Heat transfer, Electromagnetism, Fluid dynamics, Elasticity (physics), Elasticity tensor, Tensor operator, Analytic geometry, Artificial intelligence, Deep learning, Language model and Mathematical finance.

Examples

 * Heat equation
 * Wave equation
 * Continuity equation
 * Klein–Gordon equation
 * Laplace's equation
 * Poisson's equation
 * Maxwell's equations
 * Helmholtz equation
 * Lagrange equation
 * Jacobi equation
 * Schrödinger equation
 * Reaction–diffusion system used in AI and deep learning.
 * Scalable Lossless Coding for format conversion in audio and video files to MPEG4.

title: Partial differential equation


text:
Partial derivative

In mathematics, a partial derivative of a function of several variables is its derivative with respect to one of those variables, with the others held constant (as opposed to the total derivative, in which all variables are allowed to vary). Partial derivatives are used in vector calculus and differential geometry.

The partial derivative of a function $$f(x, y, \dots)$$ with respect to the variable $$x$$ is variously denoted by

It can be thought of as the rate of change of the function in the $$x$$-direction.

Sometimes, for $z=f(x, y, \ldots)$, the partial derivative of $$z$$ with respect to $$x$$ is denoted as $$\tfrac{\partial z}{\partial x}.$$ Since a partial derivative generally has the same arguments as the original function, its functional dependence is sometimes explicitly signified by the notation, such as in:

$$f'_x(x, y, \ldots), \frac{\partial f}{\partial x} (x, y, \ldots).$$

The symbol used to denote partial derivatives is ∂. One of the first known uses of this symbol in mathematics is by Marquis de Condorcet from 1770, who used it for partial differences. The modern partial derivative notation was created by Adrien-Marie Legendre (1786), although he later abandoned it; Carl Gustav Jacob Jacobi reintroduced the symbol in 1841.

Definition
Like ordinary derivatives, the partial derivative is defined as a limit. Let $U$ be an open subset of $$\R^n$$ and $$f:U\to\R$$ a function. The partial derivative of $f$ at the point $$\mathbf{a}=(a_1, \ldots, a_n) \in U$$ with respect to the $i$-th variable $x_{i}$ is defined as

$$\begin{align} \frac{\partial }{\partial x_i }f(\mathbf{a}) & = \lim_{h \to 0} \frac{f(a_1, \ldots, a_{i-1}, a_i+h, a_{i+1}\, \ldots ,a_{n})\ - f(a_1, \ldots, a_i, \dots ,a_n)}{h} \\ & = \lim_{h \to 0} \frac{f(\mathbf{a}+h\mathbf{e_i}) - f(\mathbf{a})}{h}\,. \end{align}$$

Where $$\mathbf{e_i}$$ is the unit vector of $i$-th variable $x_{i}$. Even if all partial derivatives $$\partial f / \partial x_i(a)$$ exist at a given point $a$, the function need not be continuous there. However, if all partial derivatives exist in a neighborhood of $a$ and are continuous there, then $f$ is totally differentiable in that neighborhood and the total derivative is continuous. In this case, it is said that $f$ is a $C^{1}$ function. This can be used to generalize for vector valued functions, $f:U \to \R^m$, by carefully using a componentwise argument.

The partial derivative $\frac{\partial f}{\partial x}$ can be seen as another function defined on $U$ and can again be partially differentiated. If the direction of derivative is repeated, it is called a mixed partial derivative. If all mixed second order partial derivatives are continuous at a point (or on a set), $f$ is termed a $C^{2}$ function at that point (or on that set); in this case, the partial derivatives can be exchanged by Clairaut's theorem:

$$\frac{\partial^2f}{\partial x_i \partial x_j} = \frac{\partial^2f} {\partial x_j \partial x_i}.$$

Notation
For the following examples, let $f$ be a function in $x$, $y$, and $z$.

First-order partial derivatives:

$$\frac{ \partial f}{ \partial x} = f'_x = \partial_x f.$$

Second-order partial derivatives:

$$\frac{ \partial^2 f}{ \partial x^2} = f''_{xx} = \partial_{xx} f = \partial_x^2 f.$$

Second-order mixed derivatives:

$$\frac{\partial^2 f}{\partial y \,\partial x} = \frac{\partial}{\partial y} \left( \frac{\partial f}{\partial x} \right) = (f'_{x})'_{y} = f''_{xy} = \partial_{yx} f = \partial_y \partial_x f .$$

Higher-order partial and mixed derivatives:

$$\frac{\partial^{i+j+k} f}{\partial x^i \partial y^j \partial z^k } = f^{(i, j, k)} = \partial_x^i \partial_y^j \partial_z^k f.$$

When dealing with functions of multiple variables, some of these variables may be related to each other, thus it may be necessary to specify explicitly which variables are being held constant to avoid ambiguity. In fields such as statistical mechanics, the partial derivative of $f$ with respect to $x$, holding $y$ and $z$ constant, is often expressed as

$$\left( \frac{\partial f}{\partial x} \right)_{y,z} .$$

Conventionally, for clarity and simplicity of notation, the partial derivative function and the value of the function at a specific point are conflated by including the function arguments when the partial derivative symbol (Leibniz notation) is used. Thus, an expression like

$$\frac{\partial f(x,y,z)}{\partial x}$$

is used for the function, while

$$\frac{\partial f(u,v,w)}{\partial u}$$

might be used for the value of the function at the point $(x,y,z)=(u,v,w)$. However, this convention breaks down when we want to evaluate the partial derivative at a point like $(x,y,z)=(17, u+v, v^2)$. In such a case, evaluation of the function must be expressed in an unwieldy manner as

$$\frac{\partial f(x,y,z)}{\partial x}(17, u+v, v^2)$$

or

$$\left. \frac{\partial f(x,y,z)}{\partial x}\right |_{(x,y,z)=(17, u+v, v^2)}$$

in order to use the Leibniz notation. Thus, in these cases, it may be preferable to use the Euler differential operator notation with $$D_i$$ as the partial derivative symbol with respect to the $i$-th variable. For instance, one would write $$D_1 f(17, u+v, v^2)$$ for the example described above, while the expression $$D_1 f$$ represents the partial derivative function with respect to the first variable.

For higher order partial derivatives, the partial derivative (function) of $$D_i f$$ with respect to the $j$-th variable is denoted $D_j(D_i f)=D_{i,j} f$. That is, $D_j\circ D_i =D_{i,j}$, so that the variables are listed in the order in which the derivatives are taken, and thus, in reverse order of how the composition of operators is usually notated. Of course, Clairaut's theorem implies that $$D_{i,j}=D_{j,i}$$ as long as comparatively mild regularity conditions on $f$ are satisfied.

Gradient
An important example of a function of several variables is the case of a scalar-valued function $$f(x_1, \ldots, x_n)$$ on a domain in Euclidean space $$\R^n$$ (e.g., on $$\R^2$$ or $\R^3$). In this case $f$ has a partial derivative $$\partial f/\partial x_j$$ with respect to each variable $x_{j}$. At the point $a$, these partial derivatives define the vector

$$\nabla f(a) = \left(\frac{\partial f}{\partial x_1}(a), \ldots, \frac{\partial f}{\partial x_n}(a)\right).$$

This vector is called the gradient of $f$ at $a$. If $f$ is differentiable at every point in some domain, then the gradient is a vector-valued function $∇f$ which takes the point $a$ to the vector $∇f(a)$. Consequently, the gradient produces a vector field.

A common abuse of notation is to define the del operator ($∇$) as follows in three-dimensional Euclidean space $$\R^3$$ with unit vectors $\hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}}$:|undefined

$$\nabla = \left[{\frac{\partial}{\partial x}} \right] \hat{\mathbf{i}} + \left[{\frac{\partial}{\partial y}} \right] \hat{\mathbf{j}} + \left[{\frac{\partial}{\partial z}}\right] \hat{\mathbf{k}}$$

Or, more generally, for $n$-dimensional Euclidean space $$\R^n$$ with coordinates $$x_1, \ldots, x_n$$ and unit vectors $\hat{\mathbf{e}}_1, \ldots, \hat{\mathbf{e}}_n$:|undefined

$$\nabla = \sum_{j=1}^n \left[\frac{\partial}{\partial x_j} \right] \hat{\mathbf{e}}_j = \left[\frac{\partial}{\partial x_1} \right] \hat{\mathbf{e}}_1 + \left[\frac{\partial}{\partial x_2} \right] \hat{\mathbf{e}}_2 + \dots + \left[\frac{\partial}{\partial x_n} \right] \hat{\mathbf{e}}_n$$

Example
Suppose that $f$ is a function of more than one variable. For instance,

$$z = f(x,y) = x^2 + xy + y^2 .$$

The graph of this function defines a surface in Euclidean space. To every point on this surface, there are an infinite number of tangent lines. Partial differentiation is the act of choosing one of these lines and finding its slope. Usually, the lines of most interest are those that are parallel to the $y$-plane, and those that are parallel to the $xz$-plane (which result from holding either $xz$ or $xz$ constant, respectively).

To find the slope of the line tangent to the function at $z = x^{2} + xy + y^{2}$ and parallel to the $yz$-plane, we treat $y$ as a constant. The graph and this plane are shown on the right. Below, we see how the function looks on the plane $y = 1$. By finding the derivative of the equation while assuming that $x$ is a constant, we find that the slope of $xz$ at the point $P(1, 1)$ is:

$$\frac{\partial z}{\partial x} = 2x+y.$$

So at $y = 1$, by substitution, the slope is $(x, y)$. Therefore,

$$\frac{\partial z}{\partial x} = 3$$

at the point $(1, 1)$. That is, the partial derivative of $y$ with respect to $y$ at $3$ is $(1, 1)$, as shown in the graph.

The function $f$ can be reinterpreted as a family of functions of one variable indexed by the other variables:

$$f(x,y) = f_y(x) = x^2 + xy + y^2.$$

In other words, every value of $z$ defines a function, denoted $(1, 1)$, which is a function of one variable $x$. That is,

$$f_y(x) = x^2 + xy + y^2.$$

In this section the subscript notation $3$ denotes a function contingent on a fixed value of $f$, and not a partial derivative.

Once a value of $y$ is chosen, say $x$, then $f_{y}$ determines a function $f_{y}$ which traces a curve $f(x,y)$ on the $y$-plane:

$$f_a(x) = x^2 + ax + a^2.$$

In this expression, $y$ is a, not a , so $f_{a}$ is a function of only one real variable, that being $a$. Consequently, the definition of the derivative for a function of one variable applies:

$$f_a'(x) = 2x + a.$$

The above procedure can be performed for any choice of $xz$. Assembling the derivatives together into a function gives a function which describes the variation of $a$ in the $x$ direction:

$$\frac{\partial f}{\partial x}(x,y) = 2x + y.$$

This is the partial derivative of $a$ with respect to $f$. Here '$x$' is a rounded 'd' called the partial derivative symbol; to distinguish it from the letter 'd', '$f$' is sometimes pronounced "partial".

Higher order partial derivatives
Second and higher order partial derivatives are defined analogously to the higher order derivatives of univariate functions. For the function $$f(x, y, ...)$$ the "own" second partial derivative with respect to $x$ is simply the partial derivative of the partial derivative (both with respect to $∂$):

$$\frac{\partial ^2 f}{\partial x^2} \equiv \partial \frac \equiv \frac \equiv f_{xx}.$$

The cross partial derivative with respect to $∂$ and $x$ is obtained by taking the partial derivative of $x$ with respect to $x$, and then taking the partial derivative of the result with respect to $y$, to obtain

$$\frac{\partial ^2 f}{\partial y\, \partial x} \equiv \partial \frac{\partial f / \partial x}{\partial y} \equiv \frac{\partial f_x}{\partial y} \equiv f_{xy}.$$

Schwarz's theorem states that if the second derivatives are continuous, the expression for the cross partial derivative is unaffected by which variable the partial derivative is taken with respect to first and which is taken second. That is,

$$\frac {\partial ^2 f}{\partial x\, \partial y} = \frac{\partial ^2 f}{\partial y\, \partial x}$$

or equivalently $$f_{yx} = f_{xy}.$$

Own and cross partial derivatives appear in the Hessian matrix which is used in the second order conditions in optimization problems. The higher order partial derivatives can be obtained by successive differentiation

Antiderivative analogue
There is a concept for partial derivatives that is analogous to antiderivatives for regular derivatives. Given a partial derivative, it allows for the partial recovery of the original function.

Consider the example of

$$\frac{\partial z}{\partial x} = 2x+y.$$

The so-called partial integral can be taken with respect to $f$ (treating $x$ as constant, in a similar manner to partial differentiation):

$$z = \int \frac{\partial z}{\partial x} \,dx = x^2 + xy + g(y).$$

Here, the constant of integration is no longer a constant, but instead a function of all the variables of the original function except $y$. The reason for this is that all the other variables are treated as constant when taking the partial derivative, so any function which does not involve $x$ will disappear when taking the partial derivative, and we have to account for this when we take the antiderivative. The most general way to represent this is to have the constant represent an unknown function of all the other variables.

Thus the set of functions $x^2 + xy + g(y)$, where $y$ is any one-argument function, represents the entire set of functions in variables $x^{2} + ax + a^{2}$ that could have produced the $x$-partial derivative $2x + y$.

If all the partial derivatives of a function are known (for example, with the gradient), then the antiderivatives can be matched via the above process to reconstruct the original function up to a constant. Unlike in the single-variable case, however, not every set of functions can be the set of all (first) partial derivatives of a single function. In other words, not every vector field is conservative.

Geometry
The volume $x$ of a cone depends on the cone's height $g$ and its radius $x$ according to the formula

$$V(r, h) = \frac{\pi r^2 h}{3}.$$

The partial derivative of $V$ with respect to $h$ is

$$\frac{ \partial V}{\partial r} = \frac{ 2 \pi r h}{3},$$

which represents the rate with which a cone's volume changes if its radius is varied and its height is kept constant. The partial derivative with respect to $r$ equals $\frac{1}{3}\pi r^2$, which represents the rate with which the volume changes if its height is varied and its radius is kept constant.

By contrast, the total derivative of $V$ with respect to $r$ and $h$ are respectively

$$\begin{align} \frac{dV}{dr} &= \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r} + \overbrace{\frac{\pi r^2}{3}}^\frac{ \partial V}{\partial h}\frac{dh}{dr}\,, \\ \frac{dV}{dh} &= \overbrace{\frac{\pi r^2}{3}}^\frac{\partial V}{\partial h} + \overbrace{\frac{2 \pi r h}{3}}^\frac{ \partial V}{\partial r}\frac{dr}{dh}\,. \end{align}$$

The difference between the total and partial derivative is the elimination of indirect dependencies between variables in partial derivatives.

If (for some arbitrary reason) the cone's proportions have to stay the same, and the height and radius are in a fixed ratio $V$,

$$k = \frac{h}{r} = \frac{dh}{dr}.$$

This gives the total derivative with respect to $r$,

$$\frac{dV}{dr} = \frac{2 \pi r h}{3} + \frac{\pi r^2}{3}k\,,$$

which simplifies to

$$\frac{dV}{dr} = k \pi r^2,$$

Similarly, the total derivative with respect to $h$ is

$$\frac{dV}{dh} = \pi r^2.$$

The total derivative with respect to $k$ and $r$ of the volume intended as scalar function of these two variables is given by the gradient vector

$$\nabla V = \left(\frac{\partial V}{\partial r},\frac{\partial V}{\partial h}\right) = \left(\frac{2}{3}\pi rh, \frac{1}{3}\pi r^2\right).$$

Optimization
Partial derivatives appear in any calculus-based optimization problem with more than one choice variable. For example, in economics a firm may wish to maximize profit $f_{a}$ with respect to the choice of the quantities $h$ and $r$ of two different types of output. The first order conditions for this optimization are $x, y$. Since both partial derivatives $π(x, y)$ and $π_{x} = 0 = π_{y}$ will generally themselves be functions of both arguments $h$ and $x$, these two first order conditions form a system of two equations in two unknowns.

Thermodynamics, quantum mechanics and mathematical physics
Partial derivatives appear in thermodynamic equations like Gibbs-Duhem equation, in quantum mechanics as Schrödinger wave equation as well in other equations from mathematical physics. Here the variables being held constant in partial derivatives can be ratio of simple variables like mole fractions $π_{x}$ in the following example involving the Gibbs energies in a ternary mixture system:

$$\bar{G_2}= G + (1-x_2) \left(\frac\right)_{\frac{x_1}{x_3}} $$

Express mole fractions of a component as functions of other components' mole fraction and binary mole ratios:

$\begin{align} x_1 &= \frac{1-x_2}{1+\frac{x_3}{x_1}} \\ x_3 &= \frac{1-x_2}{1+\frac{x_1}{x_3}} \end{align}$

Differential quotients can be formed at constant ratios like those above:

$$\begin{align} \left(\frac{\partial x_1}{\partial x_2}\right)_{\frac{x_1}{x_3}} &= - \frac{x_1}{1-x_2} \\ \left(\frac{\partial x_3}{\partial x_2}\right)_{\frac{x_1}{x_3}} &= - \frac{x_3}{1-x_2} \end{align}$$

Ratios X, Y, Z of mole fractions can be written for ternary and multicomponent systems:

$$\begin{align} X &= \frac{x_3}{x_1 + x_3} \\ Y &= \frac{x_3}{x_2 + x_3} \\ Z &= \frac{x_2}{x_1 + x_2} \end{align}$$

which can be used for solving partial differential equations like:

$$\left(\frac{\partial \mu_2}{\partial n_1}\right)_{n_2, n_3} = \left(\frac{\partial \mu_1}{\partial n_2}\right)_{n_1, n_3}$$

This equality can be rearranged to have differential quotient of mole fractions on one side.

Image resizing
Partial derivatives are key to target-aware image resizing algorithms. Widely known as seam carving, these algorithms require each pixel in an image to be assigned a numerical 'energy' to describe their dissimilarity against orthogonal adjacent pixels. The algorithm then progressively removes rows or columns with the lowest energy. The formula established to determine a pixel's energy (magnitude of gradient at a pixel) depends heavily on the constructs of partial derivatives.

Economics
Partial derivatives play a prominent role in economics, in which most functions describing economic behaviour posit that the behaviour depends on more than one variable. For example, a societal consumption function may describe the amount spent on consumer goods as depending on both income and wealth; the marginal propensity to consume is then the partial derivative of the consumption function with respect to income.

title: Partial derivative


text:
John Goldsmith (linguist)

John Anton Goldsmith (born 1951) is the Edward Carson Waller Distinguished Service Professor at the University of Chicago, with appointments in linguistics and computer science.

Biography
Goldsmith obtained his B.A. at Swarthmore College in 1972, and completed his PhD in Linguistics at MIT in 1976, under the linguist Morris Halle. He was on the faculty of the Department of Linguistics at Indiana University before joining the University of Chicago in 1984. He has taught at the LSA Linguistic Institutes and has held visiting appointments at many universities, such as McGill, Harvard, and UCSD. In 2007, Goldsmith was elected as a Fellow of the American Academy of Arts and Sciences.

Research
Goldsmith's research ranges from phonology to computational linguistics. His PhD thesis introduced autosegmental phonology; the idea that phonological phenomena is a collection of parallel tiers with individual segments, each representing certain features of speech. His more recent research focuses on unsupervised learning of linguistic structure (as demonstrated by the Linguistica project, a body of software that tries to automatically analyze the morphology of a language), as well as extending computational linguistics algorithms to bioinformatics.

Books

 * John A Goldsmith, Bernard Laks, Battle in the Mind Fields, University of Chicago Press, 2018

title: John Goldsmith (linguist)


text:
Autosegmental phonology

Autosegmental phonology is a framework of phonological analysis proposed by John Goldsmith in his PhD thesis in 1976 at the Massachusetts Institute of Technology (MIT).

As a theory of phonological representation, autosegmental phonology developed a formal account of ideas that had been sketched in earlier work by several linguists, notably Bernard Bloch (1948), Charles Hockett (1955) and J. R. Firth (1948). According to such a view, phonological representations consist of more than one linear sequence of segments; each linear sequence constitutes a separate tier. The co-registration of elements (or autosegments) on one tier with those on another is represented by association lines. There is a close relationship between analysis of segments into distinctive features and an autosegmental analysis; each feature in a language appears on exactly one tier.

The working hypothesis of autosegmental analysis is that a large part of phonological generalizations can be interpreted as a restructuring or reorganization of the autosegments in a representation. Clear examples of the usefulness of autosegmental analysis came in early work from the detailed study of African tone languages, as well as the study of vowel and nasal harmony systems. A few years later, John McCarthy proposed an important development by showing that the derivation of words from consonantal roots in Arabic could be analyzed autosegmentally.

In the first decade of the development of the theory, G. N. Clements developed a number of influential aspects of the theory involving harmonic processes, especially vowel harmony and nasal harmony, and John McCarthy generalized the theory to deal with the conjugational system of classical Arabic, on the basis of an autosegmental account of vowel and consonant slots on a central timing tier (see also nonconcatenative morphology).

Structure of autosegmental rules
The autosegmental formalism departs from the depiction of segments as matrices of features in order to show segments as connected groups of individual features. Segments are depicted through vertical listings of features connected by lines. These sets can also underspecify in order to indicate a class rather than a single segment. Environments can be shown by placing other connected sets of features around that which is the focus of the rule. Feature changes are shown by striking through the lines that connect a feature that is lost to the rest of the segment and drawing dotted lines to features that are gained.

Distinctive features
Rather than classify segments using the categories given in the International Phonetic Alphabet, the autosegmental formalism makes use of distinctive features, which provide greater granularity and make identification of natural classes easier. A segment is identified by a +/− dichotomy of a series of binary features, some of which are subfeatures of unary features (place of articulation, in particular, is identified by unary features indicating the active articulator, and binary subfeatures that distinguish further). For example, [p], the voiceless bilabial stop, is indicated [−sonorant, −continuant, −voice, labial], and the set of voiced coronal stops can be indicated [−sonorant, −continuant, +voice, coronal].

Feature dependencies
For unary features to be fully specified, it is necessary to include binary subfeatures that correspond to them. In the autosegmental formalism, this is depicted by placing the binary subfeature at a horizontal offset from the unary feature and connecting them with a line. The next top-level feature in the segment would then be connected to the unary feature as well as opposed to the tone.

Functional groupings
There are situations in which the rule applies not to a particular value of a feature, but to whatever value the feature has. In these situations, it is necessary to include the presence of the feature, but not to specify its value. This can be done by including a placeholder feature composed of ellipses, with an indication of the type of feature. For example, a generic place feature can be indicated [...]P.

Tiers
The autosegmental formalism deals with several separate linear sequences; because of this, a phonological representation is depicted on several distinct tiers. Each of these tiers shows a different language feature.

Segmental tier
The autosegmental tier (also "skeletal tier") contains the features that define the segments articulated in the phonological representation. The descriptions given in the previous section deal with the segmental tier. In the segmental tier, features are assigned to segments.

Timing tier
The timing tier contains timing units that define the lengths of segments in the phonological representation. These timing units are commonly depicted as X's, and are assigned to segments.

Stress tier
The stress tier contains the features that show the distribution of stress in the phonological representation. The features in the stress tier are [+/– stress] and [+/– main], and they are assigned to the stress-bearing units of the language (syllables or moras).

Tone tier
The tone tier contains the features that show the distribution of tones in the phonological representation. The features in the tone tier are [+/– high pitch] and [+/– low pitch], and they are assigned to the tone-bearing units of the language (syllables or moras).

Well-Formedness Condition
As a theory of the dynamic of phonological representations, autosegmental phonology includes a Well-formedness Condition on association lines (each element on one tier that "may" be associated to an element on another tier "must" be associated to such an element, and association lines do not cross) plus an instruction as to what to do in case of a violation of the Well-formedness Condition: add or delete the minimum number of association lines in order to maximally satisfy it. Many of the most interesting predictions of the autosegmental model derive from the automatic effects of the Well-formedness Condition and their independence of language-particular rules.

Place assimilation in nasals
The autosegmental formalism can be especially useful in describing assimilation rules. Using it for such rules makes the relationship between the result of the rule and the environment obvious. It also makes it possible to concisely describe rules that apply to different environments in different ways.

The phenomenon whereby /ɪn/ goes to [ɪn] in such words as and, [ɪŋ] in such words as and , and [ɪm] in such words as and can be represented in the autosegmental formalism. The rule is that a coronal nasal will assimilate to the place of the following consonant. The nasal is depicted by [+nasal] connected to a [coronal]P, and the consonant is depicted to the nasal's right as [...]P. No more specification is necessary because place is the only feature of the following segment that factors into the rule. The assimilation is shown by striking through the line to [coronal]P on the left and drawing a dotted line to the [...]P on the right.

title: Autosegmental phonology