Information security is the practice of protecting information by mitigating information risks. It is part of information risk management. It typically involves preventing or reducing the probability of unauthorized or inappropriate access to data or the unlawful use, disclosure, disruption, deletion, corruption, modification, inspection, recording, or devaluation of information. It also involves actions intended to reduce the adverse impacts of such incidents. Protected information may take any form, e.g., electronic or physical, tangible (e.g., paperwork), or intangible (e.g., knowledge). Information security's primary focus is the balanced protection of data confidentiality, integrity, and availability (known as the CIA triad, unrelated to the US government organization) while maintaining a focus on efficient policy implementation, all without hampering organization productivity. This is largely achieved through a structured risk management process. To standardize this discipline, academics and professionals collaborate to offer guidance, policies, and industry standards on passwords, antivirus software, firewalls, encryption software, legal liability, security awareness and training, and so forth. This standardization may be further driven by a wide variety of laws and regulations that affect how data is accessed, processed, stored, transferred, and destroyed. While paper-based business operations are still prevalent, requiring their own set of information security practices, enterprise digital initiatives are increasingly being emphasized, with information assurance now typically being dealt with by information technology (IT) security specialists. These specialists apply information security to technology (most often some form of computer system). IT security specialists are almost always found in any major enterprise/establishment due to the nature and value of the data within larger businesses. They are responsible for keeping all of the technology within the company secure from malicious attacks that often attempt to acquire critical private information or gain control of the internal systems. There are many specialist roles in Information Security including securing networks and allied infrastructure, securing applications and databases, security testing, information systems auditing, business continuity planning, electronic record discovery, and digital forensics. == Standards == Information security standards are guidelines generally outlined in published materials that aim to protect a user's or an organization's cyber environment from threats. This environment includes the users themselves, hardware such as devices and networks, software such as applications or services, and any information in storage or transit. These standards comprise security concepts, technologies, and guidelines to deal with an adverse event. They may also include assessment criteria and certification for organizations implementing a minimum level of security. These standards are developed by various international and national bodies to prevent or mitigate cyber-attacks, ensure consistency among developers, and establish a minimum standard in industries susceptible to an attack. The ISO/IEC 27000 family, published by the International Organization for Standardization (ISO) and the International Electrotechnical Commission (IEC), provides information about the guidelines and requirements for an Information Security Management System (ISMS). The Common Criteria (ISO/IEC 15408) provides guidelines on evaluating and certifying the security of a system. The IEC 62443 establishes security standards for automation and control systems. Similarly, the ISO/SAE 21434, ETSI EN 303 645, and EN 18031 provide standards for road vehicles, the Internet of Things, and radio-based systems respectively. The NIST Cybersecurity Framework (NIST CSF) is a set of guidelines developed by the U.S. National Institute of Standards and Technology to help organizations with risk management. NIST also publishes various Federal Information Processing Standards (FIPS) and Special Publications. The United Kingdom has introduced Cyber Essentials, which is a certification scheme to protect organizations against common security threats. The Australian Cyber Security Centre publishes the Essential Eight mitigation strategies. The Payment Card Industry Data Security Standard (PCI DSS) regulates handling of cardholder data in order to reduce credit card fraud. UL has published standards related to specific industries such as UL 2900-2-3 for security and life safety signaling systems and UL-2900-2-1 for healthcare and wellness systems. == Threats == Information security threats come in many different forms. Some of the most common threats today are software attacks, theft of intellectual property, theft of identity, theft of equipment or information, sabotage, and information extortion. Viruses, worms, phishing attacks, and Trojan horses are a few common examples of software attacks. The theft of intellectual property has also been an extensive issue for many businesses. Identity theft is the attempt to act as someone else usually to obtain that person's personal information or to take advantage of their access to vital information through social engineering. Sabotage usually consists of the destruction of an organization's website in an attempt to cause loss of confidence on the part of its customers. Information extortion consists of theft of a company's property or information as an attempt to receive a payment in exchange for returning the information or property back to its owner, as with ransomware. One of the most functional precautions against these attacks is to conduct periodical user awareness. Governments, military, corporations, financial institutions, hospitals, non-profit organizations, and private businesses amass a great deal of confidential information about their employees, customers, products, research, and financial status. Should confidential information about a business's customers or finances or new product line fall into the hands of a competitor or hacker, a business and its customers could suffer widespread, irreparable financial loss, as well as damage to the company's reputation. From a business perspective, information security must be balanced against cost; the Gordon-Loeb Model provides a mathematical economic approach for addressing this concern. For the individual, information security has a significant effect on privacy, which is viewed very differently in various cultures. == History == Since the early days of communication, diplomats and military commanders understood that it was necessary to provide some mechanism to protect the confidentiality of correspondence and to have some means of detecting tampering. Julius Caesar is credited with the invention of the Caesar cipher c. 50 B.C., which was created in order to prevent his secret messages from being read should a message fall into the wrong hands. However, for the most part protection was achieved through the application of procedural handling controls. Sensitive information was marked up to indicate that it should be protected and transported by trusted persons, guarded and stored in a secure environment or strong box. As postal services expanded, governments created official organizations to intercept, decipher, read, and reseal letters (e.g., the U.K.'s Secret Office, founded in 1653). In the mid-nineteenth century more complex classification systems were developed to allow governments to manage their information according to the degree of sensitivity. For example, the British Government codified this, to some extent, with the publication of the Official Secrets Act in 1889. Section 1 of the law concerned espionage and unlawful disclosures of information, while Section 2 dealt with breaches of official trust. A public interest defense was soon added to defend disclosures in the interest of the state. A similar law was passed in India in 1889, The Indian Official Secrets Act, which was associated with the British colonial era and used to crack down on newspapers that opposed the Raj's policies. A newer version was passed in 1923 that extended to all matters of confidential or secret information for governance. By the time of the First World War, multi-tier classification systems were used to communicate information to and from various fronts, which encouraged greater use of code making and breaking sections in diplomatic and military headquarters. Encoding became more sophisticated between the wars as machines were employed to scramble and unscramble information. The establishment of computer security inaugurated the history of information security. The need for such appeared during World War II. The volume of information shared by the Allied countries during the Second World War necessitated formal alignment of classification systems and procedural controls. An arcane range of markings evol
Distributed Common Ground System
The Distributed Common Ground System (DCGS) is a system which produces military intelligence for multiple branches of the American military. == DCGS Programs == DCGS-N - DCGS for the United States Navy DCGS-A - DCGS for the United States Army AF DCGS - DCGS for the United States Air Force DCGS-MC - DCGS for the United States Marine Corps DCGS-SOF - DCGS for the United States Special Operations Forces IS&A Support Center - DCGS-A Help Desk for the United States Army - https://dcgsahelp.max.gov/ - Max.gov sunset 15 December 2023 == Description == While in U.S. Air Force use, the system produces intelligence collected by the U-2 Dragonlady, RQ-4 Global Hawk, MQ-9 Reaper and MQ-1 Predator. The previous system of similar use was the Deployable Ground Station (DGS), which was first deployed in July 1994. Subsequent version of DGS were developed from 1995 through 2009. Although officially designated a "weapons system", it consists of computer hardware and software connected together in a computer network, devoted to processing and dissemination of information such as images. The 480th Intelligence, Surveillance and Reconnaissance Wing of the Air Combat Command operates and maintains the USAF system. A plan envisioned in 1998 was to develop interoperable systems for the Army and Navy, in addition to the Air Force. By 2006, version 10.6 was deployed by the Air Force, and a version known as DCGS-A was developed for the Army. After a 2010 report by General Michael T. Flynn, the program was intended to use cloud computing and be as easy to use as an iPad, which soldiers over a few years were commonly using. By April 2011, project manager Colonel Charles Wells announced version 3 of the Army system (code named "Griffin") was being deployed in the US war in Afghanistan. In January 2012, the United States Army Communications-Electronics Research, Development and Engineering Center hosted a meeting based on the DCGS-A early experience. It brought together technology providers in the hope of developing more integrated systems using cloud computing with open architectures, compared to previously specialized custom-built systems. A major contractor was Lockheed Martin, with computers supplied by Silicon Graphics International out of its Chippewa Falls, Wisconsin office. Software known as the Analyst's Notebook, originally developed by i2 Limited, was included in DCGS-A. IBM acquired i2 in 2011. Some US Army personnel reported using a Palantir Technologies product to improve their ability to predict locations of improvised explosive devices. An April 2012 report recommending further study after initial success. Palantir software was rated easy to use, but did not have the flexibility and wide number of data sources of DCGS-A. In July 2012, Congressman Duncan D. Hunter (from California, the state where Palantir is based) complained of US DoD obstacles to its wider use. Although a limited test in August 2011 by the Test and Evaluation Command had recommended deployment, operation problems of DCGS-A included the baseline system was "not operationally effective" with reboots on average about every 8 hours. A set of improvements was identified in November 2012. The press reported some of the shortcomings uncovered by General Genaro Dellarocco in the tests. The ambitious goal of integrating 473 data sources for 75 million reports proved to be challenging, after spending an estimated $2.3 billion on the Army system alone. In May 2013 Politico reported that Palantir lobbyists and some anonymous returning veterans continued to advocate the use of its software, despite its interoperability limits. In particular, members of special forces and US Marines were not required to use the official Army system. Similar stories appeared in other publications, with Army representatives (such as Major General Mary A. Legere) citing the limitations of various systems. Congressman Hunter was a member of the House Armed Services Committee which required a review of the program, after two other members of congress sent an open letter to Secretary of Defense Leon Panetta. The Senate Defense Appropriations Subcommittee included testimony from Army Chief of Staff General Ray Odierno. The 130th Engineer Brigade (United States) has found the system to be "unstable, slow, not friendly and a major hindrance to operations". The equivalent system for the United States Navy was planned for initial deployment by 2015, and within a shipboard network called Consolidated Afloat Networks and Enterprise Services (CANES) by 2016. Some early testing was announced in 2009 aboard the aircraft carrier USS Harry Truman. A portion of the software, a distributed data framework for the DCGS integration backbone (DIB) version 4, was submitted to an open-source software repository of the Codice Foundation on GitHub. The framework was new for DIB version 4, replacing the legacy DIB portal with an Ozone Widget Framework interface. It was written in the Java programming language. == DCGS-A == Distributed Common Ground System-Army (DCGS-A) is the United States Army's primary system to post data, process information, and disseminate Intelligence, Surveillance and Reconnaissance (ISR) information about the threat, weather, and terrain to echelons. DCGS-A provides commanders the ability to task battle-space sensors and receive intelligence information from multiple sources. === Promotion === An August 17, 2011, UPI article quoted i2 Chief Executive Officer Robert Griffin who commented on DCGS-A's best-of-breed approach to development. The article detailed the Army contracting with i2 for Analyst's Notebook software. "With its open architecture, Analyst's Notebook supports the Army's strategy to employ and integrate best-of-breed solutions from across the industry to meet the dynamic needs users face in the field on a daily basis." A February 1, 2012, article in the Army web page quoted Mark Kitz, DCGS-A technical director. DCGS-A "uses the latest in cloud technology to rapidly gather, collaborate and share intelligence data from multiple sources to deliver a common operating picture. DCGS-A is able to rapidly adapt to changing operational environments by leveraging an iterative development model and open architecture allowing for collaboration with multiple government, industry and academic partners." A July 2012 article in SIGNAL Magazine, monthly publication of the Armed Forces Communications and Electronics Association, promoted DCGS-A as taking advantage of technological environments with which young soldiers are familiar. The article quoted the DCGS-A program manager, Col. Charles Wells on the systems benefits. The article also included Lockheed Martin's DCGS-A program manager. The Milwaukee Journal Sentinel published an article May 4, 2012, about Wisconsin-located companies helping DCGS-A with cloud computing technology. The article promoted the speed when cloud computing processes intelligence and cost savings by analyzing data in the field. === The U.S. Army's 2011 Posture Statement === The U.S. Army released its 2011 Army Posture Statement March 2. It included a statement on DCGS-A: “The Distributed Common Ground System-Army (DCGS-A) is the Army's premier intelligence, surveillance, and reconnaissance (ISR) enterprise for the tasking of sensors, analysis and processing of data, exploitation of data, and dissemination of intelligence (TPED) across all echelons. It is the Army component of the larger Defense Intelligence Information Enterprise (DI2E) and interoperable with other Service DCGS programs. Under the DI2E framework, USD (I) hopes to provide COCOM Joint Intelligence Operations Centers (JIOCs) capabilities interoperable with DCGS-A through a Cloud/widget approach. DCGS-A connects tactical, operational, and theater-level commanders to hundreds of intelligence and intelligence-related data sources at all classification levels and allows them to focus efforts of the entire ISR community on their information requirements. === Comparisons === Some Ground Commanders who describe DCGS-A as "unwieldy and unreliable, hard to learn and difficult to use," supporting alternative software from Palantir Technologies. Palantir software supports small unit situational awareness, but is not sufficiently funded to support the broader role that DCGS-A fulfills. == Operators == 480th Intelligence, Surveillance and Reconnaissance Wing 9th Intelligence Squadron 13th Intelligence Squadron 548th Intelligence, Surveillance and Reconnaissance Group 548 Operational Support Squadron 48th Intelligence Squadron 101st Intelligence Squadron 113th Air Support Operations Squadron 127th Command and Control Squadron 161st Intelligence Squadron
Sum of absolute transformed differences
The sum of absolute transformed differences (SATD) is a block matching criterion widely used in fractional motion estimation for video compression. It works by taking a frequency transform, usually a Hadamard transform, of the differences between the pixels in the original block and the corresponding pixels in the block being used for comparison. The transform itself is often of a small block rather than the entire macroblock. For example, in x264, a series of 4×4 blocks are transformed rather than doing the more processor-intensive 16×16 transform. == Comparison to other metrics == SATD is slower than the sum of absolute differences (SAD), both due to its increased complexity and the fact that SAD-specific MMX and SSE2 instructions exist, while there are no such instructions for SATD. However, SATD can still be optimized considerably with SIMD instructions on most modern CPUs. The benefit of SATD is that it more accurately models the number of bits required to transmit the residual error signal. As such, it is often used in video compressors, either as a way to drive and estimate rate explicitly, such as in the Theora encoder (since 1.1 alpha2), as an optional metric used in wide motion searches, such as in the Microsoft VC-1 encoder, or as a metric used in sub-pixel refinement, such as in x264.
NETtalk (artificial neural network)
NETtalk is an artificial neural network that learns to pronounce written English text by supervised learning. It takes English text as input, and produces a matching phonetic transcriptions as output. It is the result of research carried out in the mid-1980s by Terrence Sejnowski and Charles Rosenberg. The intent behind NETtalk was to construct simplified models that might shed light on the complexity of learning human level cognitive tasks, and their implementation as a connectionist model that could also learn to perform a comparable task. The authors trained it by backpropagation. The network was trained on a large amount of English words and their corresponding pronunciations, and is able to generate pronunciations for unseen words with a high level of accuracy. The output of the network was a stream of phonemes, which fed into DECtalk to produce audible speech. It achieved popular success, appearing on the Today show. From the point of view of modeling human cognition, NETtalk does not specifically model the image processing stages and letter recognition of the visual cortex. Rather, it assumes that the letters have been pre-classified and recognized. It is NETtalk's task to learn proper associations between the correct pronunciation with a given sequence of letters based on the context in which the letters appear. A similar architecture was subsequently used for the opposite task, that of converting continuous speech signal to a phoneme sequence. == Training == The training dataset was a 20,008-word subset of the Brown Corpus, with manually annotated phoneme and stress for each letter. The development process was described in a 1993 interview. It took three months -- 250 person-hours -- to create the training dataset, but only a few days to train the network. After it was run successfully on this, the authors tried it on a phonological transcription of an interview with a young Latino boy from a barrio in Los Angeles. This resulted in a network that reproduced his Spanish accent. The original NETtalk was implemented on a Ridge 32, which took 0.275 seconds per learning step (one forward and one backward pass). Training NETtalk became a benchmark to test for the efficiency of backpropagation programs. For example, an implementation on Connection Machine-1 (with 16384 processors) ran at 52x speedup. An implementation on a 10-cell Warp ran at 340x speedup. The following table compiles the benchmark scores as of 1988. Speed is measured in "millions of connections per second" (MCPS). For example, the original NETtalk on Ridge 32 took 0.275 seconds per forward-backward pass, giving 18629 / 10 6 0.275 = 0.068 {\displaystyle {\frac {18629/10^{6}}{0.275}}=0.068} MCPS. Relative times are normalized to the MicroVax. == Architecture == The network had three layers and 18,629 adjustable weights, large by the standards of 1986. There were worries that it would overfit the dataset, but it was trained successfully. The input of the network has 203 units, divided into 7 groups of 29 units each. Each group is a one-hot encoding of one character. There are 29 possible characters: 26 letters, comma, period, and word boundary (whitespace). To produce the pronunciation of a single character, the network takes the character itself, as well as 3 characters before and 3 characters after it. The hidden layer has 80 units. The output has 26 units. 21 units encode for articulatory features (point of articulation, voicing, vowel height, etc.) of phonemes, and 5 units encode for stress and syllable boundaries. Sejnowski studied the learned representation in the network, and found that phonemes that sound similar are clustered together in representation space. The output of the network degrades, but remains understandable, when some hidden neurons are removed.
Conference on Computer Vision and Pattern Recognition
The Conference on Computer Vision and Pattern Recognition is an annual conference on computer vision and pattern recognition. == Affiliations == The conference was first held in 1983 in Washington, DC, organized by Takeo Kanade and Dana H. Ballard. From 1985 to 2010 it was sponsored by the IEEE Computer Society. In 2011 it was also co-sponsored by University of Colorado Colorado Springs. Since 2012 it has been co-sponsored by the IEEE Computer Society and the Computer Vision Foundation, which provides open access to the conference papers. == Scope == The conference considers a wide range of topics related to computer vision and pattern recognition—basically any topic that is extracting structures or answers from images or video or applying mathematical methods to data to extract or recognize patterns. Common topics include object recognition, image segmentation, motion estimation, 3D reconstruction, and deep learning. The conference generally has less than 30% acceptance rates for all papers and less than 5% for oral presentations. It is managed by a rotating group of volunteers who are chosen in a public election at the Pattern Analysis and Machine Intelligence-Technical Community (PAMI-TC) meeting four years before the meeting. The conference uses a multi-tier double-blind peer review process. The program chairs, who cannot submit papers, select area chairs who manage the reviewers for their subset of submissions. == Location and time == The conference is usually held in June in North America. == Awards == === Best Paper Award === These awards are picked by committees delegated by the program chairs of the conference. === Longuet-Higgins Prize === The Longuet-Higgins Prize recognizes papers from ten years ago that have made a significant impact on computer vision research. === PAMI Young Researcher Award === The Pattern Analysis and Machine Intelligence Young Researcher Award is an award given by the Technical Committee on Pattern Analysis and Machine Intelligence of the IEEE Computer Society to a researcher within 7 years of completing their Ph.D. for outstanding early career research contributions. Candidates are nominated by the computer vision community, with winners selected by a committee of senior researchers in the field. This award was originally instituted in 2012 by the journal Image and Vision Computing, also presented at the conference, and the journal continues to sponsor the award. === PAMI Thomas S. Huang Memorial Prize === The Thomas Huang Memorial Prize was established at the 2020 conference and is awarded annually starting from 2021 to honor researchers who are recognized as examples in research, teaching/mentoring, and service to the computer vision community.
Web container
A web container (also known as a servlet container; and compare "webcontainer") is the component of a web server that interacts with Jakarta Servlets. A web container is responsible for managing the lifecycle of servlets, mapping a URL to a particular servlet and ensuring that the URL requester has the correct access-rights. A web container handles requests to servlets, Jakarta Server Pages (JSP) files, and other types of files that include server-side code. The Web container creates servlet instances, loads and unloads servlets, creates and manages request and response objects, and performs other servlet-management tasks. A web container implements the web component contract of the Jakarta EE architecture. This architecture specifies a runtime environment for additional web components, including security, concurrency, lifecycle management, transaction, deployment, and other services. == List of Servlet containers == The following is a list of notable applications which implement the Jakarta Servlet specification from Eclipse Foundation, divided depending on whether they are directly sold or not. === Open source Web containers === Apache Tomcat (formerly Jakarta Tomcat) is an open source web container available under the Apache Software License. Apache Tomcat 6 and above are operable as general application container (prior versions were web containers only) Apache Geronimo is a full Java EE 6 implementation by Apache Software Foundation. Enhydra, from Lutris Technologies. GlassFish from Eclipse Foundation (an application server, but includes a web container). Jetty, from the Eclipse Foundation. Also supports SPDY and WebSocket protocols. Open Liberty, from IBM, is a fully compliant Jakarta EE server Virgo from Eclipse Foundation provides modular, OSGi based web containers implemented using embedded Tomcat and Jetty. Virgo is available under the Eclipse Public License. WildFly (formerly JBoss Application Server) is a full Java EE implementation by Red Hat, division JBoss. === Commercial Web containers === iPlanet Web Server, from Oracle. JBoss Enterprise Application Platform from Red Hat, division JBoss is subscription-based/open-source Jakarta EE-based application server. WebLogic Application Server, from Oracle Corporation (formerly developed by BEA Systems). Orion Application Server, from IronFlare. Resin Pro, from Caucho Technology. IBM WebSphere Application Server. SAP NetWeaver.
Inverted pendulum
An inverted pendulum is a pendulum that has its center of mass above its pivot point. It is unstable and falls over without additional help. It can be suspended stably in this inverted position by using a control system to monitor the angle of the pole and move the pivot point horizontally back under the center of mass when it starts to fall over, keeping it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is used as a benchmark for testing control strategies. It is often implemented with the pivot point mounted on a cart that can move horizontally under control of an electronic servo system as shown in the photo; this is called a cart and pole apparatus. Most applications limit the pendulum to 1 degree of freedom by affixing the pole to an axis of rotation. Whereas a normal pendulum is stable when hanging downward, an inverted pendulum is inherently unstable, and must be actively balanced in order to remain upright; this can be done either by applying a torque at the pivot point, by moving the pivot point horizontally as part of a feedback system, changing the rate of rotation of a mass mounted on the pendulum on an axis parallel to the pivot axis and thereby generating a net torque on the pendulum, or by oscillating the pivot point vertically. A simple demonstration of moving the pivot point in a feedback system is achieved by balancing an upturned broomstick on the end of one's finger. A second type of inverted pendulum is a tiltmeter for tall structures, which consists of a wire anchored to the bottom of the foundation and attached to a float in a pool of oil at the top of the structure that has devices for measuring movement of the neutral position of the float away from its original position. == Overview == A pendulum with its bob hanging directly below the support pivot is at a stable equilibrium point, where it remains motionless because there is no torque on the pendulum. If displaced from this position, it experiences a restoring torque that returns it toward the equilibrium position. A pendulum with its bob in an inverted position, supported on a rigid rod directly above the pivot, 180° from its stable equilibrium position, is at an unstable equilibrium point. At this point again there is no torque on the pendulum, but the slightest displacement away from this position causes a gravitation torque on the pendulum that accelerates it away from equilibrium, causing it to fall over. In order to stabilize a pendulum in this inverted position, a feedback control system can be used, which monitors the pendulum's angle and moves the position of the pivot point sideways when the pendulum starts to fall over, to keep it balanced. The inverted pendulum is a classic problem in dynamics and control theory and is widely used as a benchmark for testing control algorithms (PID controllers, state-space representation, neural networks, fuzzy control, genetic algorithms, etc.). Variations on this problem include multiple links, allowing the motion of the cart to be commanded while maintaining the pendulum, and balancing the cart-pendulum system on a see-saw. The inverted pendulum is related to rocket or missile guidance, where the center of gravity is located behind the center of drag causing aerodynamic instability. The understanding of a similar problem can be shown by simple robotics in the form of a balancing cart. Balancing an upturned broomstick on the end of one's finger is a simple demonstration, and the problem is solved by self-balancing personal transporters such as the Segway PT, the self-balancing hoverboard and the self-balancing unicycle. Another way that an inverted pendulum may be stabilized, without any feedback or control mechanism, is by oscillating the pivot rapidly up and down. This is called Kapitza's pendulum. If the oscillation is sufficiently strong (in terms of its acceleration and amplitude) then the inverted pendulum can recover from perturbations in a strikingly counterintuitive manner. If the driving point moves in simple harmonic motion, the pendulum's motion is described by the Mathieu equation. == Equations of motion == The equations of motion of inverted pendulums are dependent on what constraints are placed on the motion of the pendulum. Inverted pendulums can be created in various configurations resulting in a number of Equations of Motion describing the behavior of the pendulum. === Stationary pivot point === In a configuration where the pivot point of the pendulum is fixed in space, the equation of motion is similar to that for an uninverted pendulum. The equation of motion below assumes no friction or any other resistance to movement, a rigid massless rod, and the restriction to 2-dimensional movement. θ ¨ − g ℓ sin θ = 0 {\displaystyle {\ddot {\theta }}-{g \over \ell }\sin \theta =0} Where θ ¨ {\displaystyle {\ddot {\theta }}} is the angular acceleration of the pendulum, g {\displaystyle g} is the standard gravity on the surface of the Earth, ℓ {\displaystyle \ell } is the length of the pendulum, and θ {\displaystyle \theta } is the angular displacement measured from the equilibrium position. When θ ¨ {\displaystyle {\ddot {\theta }}} added to both sides, it has the same sign as the angular acceleration term: θ ¨ = g ℓ sin θ {\displaystyle {\ddot {\theta }}={g \over \ell }\sin \theta } Thus, the inverted pendulum accelerates away from the vertical unstable equilibrium in the direction initially displaced, and the acceleration is inversely proportional to the length. Tall pendulums fall more slowly than short ones. Derivation using torque and moment of inertia: The pendulum is assumed to consist of a point mass, of mass m {\displaystyle m} , affixed to the end of a massless rigid rod, of length ℓ {\displaystyle \ell } , attached to a pivot point at the end opposite the point mass. The net torque of the system must equal the moment of inertia times the angular acceleration: τ n e t = I θ ¨ {\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=I{\ddot {\theta }}} The torque due to gravity providing the net torque: τ n e t = m g ℓ sin θ {\displaystyle {\boldsymbol {\tau }}_{\mathrm {net} }=mg\ell \sin \theta \,\!} Where θ {\displaystyle \theta \ } is the angle measured from the inverted equilibrium position. The resulting equation: I θ ¨ = m g ℓ sin θ {\displaystyle I{\ddot {\theta }}=mg\ell \sin \theta \,\!} The moment of inertia for a point mass: I = m R 2 {\displaystyle I=mR^{2}} In the case of the inverted pendulum the radius is the length of the rod, ℓ {\displaystyle \ell } . Substituting in I = m ℓ 2 {\displaystyle I=m\ell ^{2}} m ℓ 2 θ ¨ = m g ℓ sin θ {\displaystyle m\ell ^{2}{\ddot {\theta }}=mg\ell \sin \theta \,\!} Mass and ℓ 2 {\displaystyle \ell ^{2}} is divided from each side resulting in: θ ¨ = g ℓ sin θ {\displaystyle {\ddot {\theta }}={g \over \ell }\sin \theta } === Inverted pendulum on a cart === An inverted pendulum on a cart consists of a mass m {\displaystyle m} at the top of a pole of length ℓ {\displaystyle \ell } pivoted on a horizontally moving base as shown in the adjacent image. The cart is restricted to linear motion and is subject to forces resulting in or hindering motion. === Essentials of stabilization === The essentials of stabilizing the inverted pendulum can be summarized qualitatively in three steps. 1. If the tilt angle θ {\displaystyle \theta } is to the right, the cart must accelerate to the right and vice versa. 2. The position of the cart x {\displaystyle x} relative to track center is stabilized by slightly modulating the null angle (the angle error that the control system tries to null) by the position of the cart, that is, null angle = θ + k x {\displaystyle =\theta +kx} where k {\displaystyle k} is small. This makes the pole want to lean slightly toward track center and stabilize at track center where the tilt angle is exactly vertical. Any offset in the tilt sensor or track slope that would otherwise cause instability translates into a stable position offset. A further added offset gives position control. 3. A normal pendulum subject to a moving pivot point such as a load lifted by a crane, has a peaked response at the pendulum radian frequency of ω p = g / ℓ {\displaystyle \omega _{p}={\sqrt {g/\ell }}} . To prevent uncontrolled swinging, the frequency spectrum of the pivot motion should be suppressed near ω p {\displaystyle \omega _{p}} . The inverted pendulum requires the same suppression filter to achieve stability. As a consequence of the null angle modulation strategy, the position feedback is positive, that is, a sudden command to move right produces an initial cart motion to the left followed by a move right to rebalance the pendulum. The interaction of the pendulum instability and the positive position feedback instability to produce a stable system is a feature that makes the mathematical analysis an interesting and challenging problem. === From Lagrange's equations === The equations of motion c