Graphical Kernel System

The Graphical Kernel System (GKS) is a 2D computer graphics system using vector graphics, introduced in 1977. It was suitable for making line and bar charts and similar tasks. A key concept was cross-system portability, based on an underlying coordinate system that could be represented on almost any hardware. GKS is best known as the basis for the graphics in the GEM GUI system used on the Atari ST and as part of Ventura Publisher. A draft international standard was circulated for review in September 1983. Final ratification of the standard was achieved in 1985, making it the first ISO graphics standard. A 3D system modelled on GKS was introduced as PHIGS, which saw some use in the 1980s and early 1990s. == Overview == GKS provides a set of drawing features for two-dimensional vector graphics suitable for charting and similar duties. The calls are designed to be portable across different programming languages, graphics devices and hardware, so that applications written to use GKS will be readily portable to many platforms and devices. GKS was fairly common on computer workstations in the 1980s and early 1990s. GKS formed the basis of Digital Research's GSX which evolved into VDI, one of the core components of GEM. GEM was the native GUI on the Atari ST and was occasionally seen on PCs, particularly in conjunction with Ventura Publisher. GKS was little used commercially outside these markets, but remains in use in some scientific visualization packages. It is also the underlying API defining the Computer Graphics Metafile. One popular application based on an implementation of GKS is the GR Framework, a C library for high-performance scientific visualization that has become a common plotting backend among Julia users. A main developer and promoter of the GKS was José Luis Encarnação, formerly director of the Fraunhofer Institute for Computer Graphics (IGD) in Darmstadt, Germany. GKS has been standardized in the following documents: ANSI standard ANSI X3.124 of 1985. ISO 7942:1985 standard, revised as ISO 7942:1985/Amd 1:1991 and ISO/IEC 7942-1:1994, as well as ISO/IEC 7942-2:1997, ISO/IEC 7942-3:1999 and ISO/IEC 7942-4:1998 The language bindings are ISO standard ISO 8651. GKS-3D (Graphical Kernel System for Three Dimensions) functional definition is ISO standard ISO 8805, and the corresponding C bindings are ISO/IEC 8806. The functionality of GKS is wrapped up as a data model standard in the STEP standard, section ISO 10303-46.

Ampere Computing

Ampere Computing LLC is an American fabless semiconductor company that designs ARM-based central processing units (CPUs) with high core counts for use in cloud computing and data center environments. Founded in 2017 by former Intel president Renée James, the company is headquartered in Santa Clara, California, and operates as an independent subsidiary of SoftBank Group since November 2025. == History == Ampere Computing was founded in fall 2017 by Renée James, ex-President of Intel, with funding from The Carlyle Group. James acquired a team from MACOM Technology Solutions (formerly AppliedMicro) in addition to several industry hires to start the company. Ampere Computing is an ARM architecture licensee and develops its own server microprocessors. Ampere fabricates its products at TSMC. In April 2019, Ampere announced its second major investment round, including investment from Arm Holdings and Oracle Corporation. In June 2019, Nvidia announced a partnership with Ampere to bring support for Compute Unified Device Architecture (CUDA). In November 2019, Nvidia announced a reference design platform for graphics processing unit (GPU)-accelerated ARM-based servers including Ampere. In the first half of 2020, Ampere announced Ampere Altra, an 80-core processor, and Ampere Altra Max, a 128-core processor, without the use of simultaneous multithreading. In March 2020, the company announced a partnership with Oracle. In September 2020, Oracle said it would launch bare-metal and virtual machine instances in early 2021 based on Ampere Altra. In November 2020, Ampere was named one of the top 10 hottest semiconductor startups by CRN. In May 2021, the company announced a partnership with Microsoft. In April 2022, Ampere said that it had filed a confidential prospectus with the U.S. Securities and Exchange Commission, signaling its intent to go public. In June 2022, HPE announced their Gen11 ProLiant system would use Ampere Altra and Ampere Altra Max Cloud Native Processors. In July 2022, Google announced T2A instances using Ampere Altra in the Google cloud and in August 2022 Microsoft announced their instances of Ampere running in Azure. On March 19, 2025, investment holding company SoftBank Group announced it will acquire Ampere Computing for $6.5 billion. The deal finalized in November 2025, with Ampere remaining as an independent subsidiary with its headquarters in Santa Clara, California. == Products == Ampere develops ARM-based computer processors and CPU cores under their Altra brands. These are used in databases, media encoding, web services, network acceleration, mobile gaming, AI inference processing, and other applications and programs that need to scale. On February 5, 2018, Ampere announced the eMAG 8180 featuring 32x Skylark cores fabricated on TSMC's 16FF+ process. It supports a turbo of up to 3.3 GHz with a TDP of 125 W, 8ch 64-bit DDR4, up to 1 TB DDR4 per socket, and 42x PCIe 3.0 Lanes. The Skylark cores were based on AppliedMicro's X-Gene 3. Packet offers servers with the eMAG 8180 and 128 GB DRAM, 480 GB SSD, and 2x 10 Gbit/s networking. On September 19, 2018, Ampere announced the availability of a version featuring 16x Skylark cores. === 2020 === On March 3, 2020, Ampere announced the Ampere Altra featuring 80 cores fabricated on TSMC's N7 process for hyperscale computing. It was the first server-grade processor to include 80 cores and the Q80-30 conserves power by running at 161 W in use. The cores are semi-custom Arm Neoverse N1 cores with Ampere modifications. It supports a frequency of up to 3.3 GHz with TDP of 250 W, 8ch 72-bit DDR4, up to 4 TB DDR4-3200 per socket, 128x PCIe 4.0 Lanes, 1 MB L2 per core and 32 MB SLC. Ampere also announced their roadmap with Ampere Altra Max (2021) in development and AmpereOne (2022) defined. === 2021 === The 128-core Altra Max was released in 2021 and targeted hyperscale cloud providers. It uses the same server socket and platforms as Ampere Altra, and both products have one thread per core. The Altra Max CPUs provide 128 Arm v8.2+ cores per chip and run up to 3.0 GHz. They also support eight channels of DDR4-3200 memory and 128 lanes of PCIe Gen4. Also in 2021, Oracle launched its Oracle Cloud Infrastructure (OCI) using Ampere Altra processors. === 2022 === In February 2022, Ampere and Rigetti Computing announced a strategic partnership to create hybrid quantum-classical computers. The companies will combine Ampere's Altra Max CPUs with Rigetti's Quantum Processing Units (QPU) in cloud-based High-Performance Computing (HPC) environments. In April, Microsoft previewed its Azure Virtual Machines running on the Ampere Altra. The VMs run scale-out workloads, web servers, application servers, open source databases, cloud native .NET applications, Java applications, gaming servers, media servers, and other processes. In May, Ampere announced the sampling of AmpereOne CPUs, 5 nanometer chips based on its in-house Ampere-developed core. AmpereOne will add support for DDR5 main memory and PCIe Gen5 peripherals. On June 28, 2022, HPE became first tier-one server provider to offer compute with optimized cloud-native silicon for service providers and enterprises embracing cloud-native development with new line of HPE ProLiant RL Gen11 servers, using Ampere® Altra® and Ampere® Altra® Max processors, delivering high performance and power efficiency. === 2023 === During April 2023, Ampere released the Altra developer's kit, an IoT Prototype Kit based on Ampere Altra, aimed at cloud developers, available in 32-core, 64-core, and 80-core formats. === 2024 === In May 2024, Ampere updated its AmpereOne roadmap to 256 cores and announced a joint effort with Qualcomm on CPUs and accelerators. == Customers == Ampere's customers include Microsoft Azure, Tencent Cloud, Oracle, ByteDance, Hewlett Packard Enterprise (HPE), Cloudflare, Equinix, Kingsoft Cloud, Meituan, Scaleway, UCloud, Foxconn Industrial Internet, Gigabyte, Inspur, Cruise, Hetzner, Project Ronin, Wiwynn and Google Cloud Platform Cruise uses an Ampere Altra variant for its autonomous driving unit. The CPU was selected because of its throughput and low power consumption. In 2021, Oracle, Microsoft, Tencent, and ByteDance committed to using Ampere's customized chips, first announced in May. In April 2022, Microsoft previewed Ampere Altra processors in its new Azure D-and E- series virtual machines. The Dpsv5 series is built for Linux enterprise application types, and the Epsv5 series is for memory-intensive Linux workloads. They provide up to 64 vCPUs, include VM sizes with 2GiB, 4GiB, and 8GiB per vCPU memory configurations, up to 40 Gbit/s networking, and high-performance local SSD storage. In 2022, Microsoft's Ampere Altra-based Azure servers became the first cloud solution provider server to be Arm SystemReady SR certified. The Azure VMs, powered by Altra processors, were also the first to be SystemReady Virtual Environment standard certified. SystemReady defines a set of firmware and hardware standards as a baseline for system development for software developers, original equipment vendors, and chipmakers.

Harrison White

Harrison Colyar White (March 21, 1930 – May 18, 2024) was an American sociologist who was the Giddings Professor of Sociology at Columbia University. White played an influential role in the “Harvard Revolution” in social networks and the New York School of relational sociology. He is credited with the development of a number of mathematical models of social structure including vacancy chains and blockmodels. He has been a leader of a revolution in sociology that is still in process, using models of social structure that are based on patterns of relations instead of the attributes and attitudes of individuals. Among social network researchers, White is widely respected. For instance, at the 1997 International Network of Social Network Analysis conference, the organizer held a special “White Tie” event, dedicated to White. Social network researcher Emmanuel Lazega refers to him as both “Copernicus and Galileo” because he invented both the vision and the tools. The most comprehensive documentation of his theories can be found in the book Identity and Control, first published in 1992. A major rewrite of the book appeared in June 2008. In 2011, White received the W.E.B. DuBois Career of Distinguished Scholarship Award from the American Sociological Association, which honors "scholars who have shown outstanding commitment to the profession of sociology and whose cumulative work has contributed in important ways to the advancement of the discipline." Before his retirement to live in Tucson, Arizona, White was interested in sociolinguistics and business strategy as well as sociology. == Life and career == === Early years === White was born on March 21, 1930, in Washington, D.C. He had three siblings and his father was a doctor in the US Navy. Although moving around to different Naval bases throughout his adolescence, he considered himself Southern, and Nashville, TN to be his home. At the age of 15, he entered the Massachusetts Institute of Technology (MIT), receiving his undergraduate degree at 20 years of age; five years later, in 1955, he received a doctorate in theoretical physics, also from MIT with John C. Slater as his advisor. His dissertation was titled A quantum-mechanical calculation of inter-atomic force constants in copper. This was published in the Physical Review as "Atomic Force Constants of Copper from Feynman's Theorem" (1958). While at MIT he also took a course with the political scientist Karl Deutsch, who White credits with encouraging him to move toward the social sciences. === Princeton University === After receiving his PhD in theoretical physics, he received a Fellowship from the Ford Foundation to begin his second doctorate in sociology at Princeton University. His dissertation advisor was Marion J. Levy. White also worked with Wilbert Moore, Fred Stephan, and Frank W. Notestein while at Princeton. His cohort was very small, with only four or five other graduate students including David Matza, and Stanley Udy. At the same time, he took up a position as an operations analyst at the Operations Research Office, Johns Hopkins University from 1955 to 1956. During this period, he worked with Lee S. Christie on Queuing with Preemptive Priorities or with Breakdown, which was published in 1958. Christie previously worked alongside mathematical psychologist R. Duncan Luce in the Small Group Laboratory at MIT while White was completing his first PhD in physics also at MIT. While continuing his studies at Princeton, White also spent a year as a fellow at the Center for Advanced Study in the Behavioral Sciences, Stanford University, California where he met Harold Guetzkow. Guetzkow was a faculty member at the Carnegie Institute of Technology, known for his application of simulations to social behavior and long-time collaborator with many other pioneers in organization studies, including Herbert A. Simon, James March, and Richard Cyert. Upon meeting Simon through his mutual acquaintance with Guetzkow, White received an invitation to move from California to Pittsburgh to work as an assistant professor of Industrial Administration and Sociology at the Graduate School of Industrial Administration, Carnegie Institute of Technology (later Carnegie-Mellon University), where he stayed for a couple of years, between 1957 and 1959. In an interview, he claimed to have fought with the dean, Leyland Bock, to have the word "sociology" included in his title. It was also during his time at the Stanford Center for Advanced Study that White met his first wife, Cynthia A. Johnson, who was a graduate of Radcliffe College, where she had majored in art history. The couple's joint work on the French Impressionists, Canvases and Careers (1965) and “Institutional Changes in the French Painting World” (1964), originally grew out of a seminar on art in 1957 at the Center for Advanced Study led by Robert Wilson. White originally hoped to use sociometry to map the social structure of French art to predict shifts, but he had an epiphany that it was not social structure but institutional structure which explained the shift. It was also during these years that White, still a graduate student in sociology, wrote and published his first social scientific work, "Sleep: A Sociological Interpretation" in Acta Sociologica in 1960, together with Vilhelm Aubert, a Norwegian sociologist. This work was a phenomenological examination of sleep which attempted to "demonstrate that sleep was more than a straightforward biological activity... [but rather also] a social event". For his dissertation, White carried out empirical research on a research and development department in a manufacturing firm, consisting of interviews and a 110-item questionnaire with managers. He specifically used sociometric questions, which he used to model the "social structure" of relationships between various departments and teams in the organization. In May 1960 he submitted as his doctoral dissertation, titled Research and Development as a Pattern in Industrial Management: A Case Study in Institutionalisation and Uncertainty, earning a PhD in sociology from Princeton University. His first publication based on his dissertation was ''Management conflict and sociometric structure'' in the American Journal of Sociology. === University of Chicago === In 1959 James Coleman left the University of Chicago to found a new department of social relations at Johns Hopkins University, this left a vacancy open for a mathematical sociologist like White. He moved to Chicago to start working as an associate professor at the Department of Sociology. At that time, highly influential sociologists, such as Peter Blau, Mayer Zald, Elihu Katz, Everett Hughes, Erving Goffman were there. As Princeton only required one year in residence, and White took the opportunity to take positions at Johns Hopkins, Stanford, and Carnegie while still working on his dissertation, it was at Chicago that White credits as being his "real socialization in a way, into sociology." It was here that White advised his first two graduate students Joel H. Levine and Morris Friedell, both who went on to make contributions to social network analysis in sociology. While at the Center for Advanced Study, White began learning anthropology and became fascinated with kinship. During his stay at the University of Chicago White was able to finish An Anatomy of Kinship, published in 1963 within the Prentice-Hall series in Mathematical Analysis of Social Behavior, with James Coleman and James March as chief editors. The book received significant attention from many mathematical sociologists of the time, and contributed greatly to establish White as a model builder. === The Harvard Revolution === In 1963, White left Chicago to be an associate professor of sociology at the Harvard Department of Social Relations—the same department founded by Talcott Parsons and still heavily influenced by the structural-functionalist paradigm of Parsons. As White previously only taught graduate courses at Carnegie and Chicago, his first undergraduate course was An Introduction to Social Relations (see Influence) at Harvard, which became infamous among network analysts. As he "thought existing textbooks were grotesquely unscientific," the syllabus of the class was noted for including few readings by sociologists, and comparatively more readings by anthropologists, social psychologists, and historians. White was also a vocal critic of what he called the "attributes and attitudes" approach of Parsonsian sociology, and came to be the leader of what has been variously known as the “Harvard Revolution," the "Harvard breakthrough," or the "Harvard renaissance" in social networks. He worked closely with small group researchers George C. Homans and Robert F. Bales, which was largely compatible with his prior work in organizational research and his efforts to formalize network analysis. Overlapping White's early years, Charles Tilly, a graduate of the Harvard Department of Social

Sufficient dimension reduction

In statistics, sufficient dimension reduction (SDR) is a paradigm for analyzing data that combines the ideas of dimension reduction with the concept of sufficiency. Dimension reduction has long been a primary goal of regression analysis. Given a response variable y and a p-dimensional predictor vector x {\displaystyle {\textbf {x}}} , regression analysis aims to study the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , the conditional distribution of y {\displaystyle y} given x {\displaystyle {\textbf {x}}} . A dimension reduction is a function R ( x ) {\displaystyle R({\textbf {x}})} that maps x {\displaystyle {\textbf {x}}} to a subset of R k {\displaystyle \mathbb {R} ^{k}} , k < p, thereby reducing the dimension of x {\displaystyle {\textbf {x}}} . For example, R ( x ) {\displaystyle R({\textbf {x}})} may be one or more linear combinations of x {\displaystyle {\textbf {x}}} . A dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} is said to be sufficient if the distribution of y ∣ R ( x ) {\displaystyle y\mid R({\textbf {x}})} is the same as that of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, no information about the regression is lost in reducing the dimension of x {\displaystyle {\textbf {x}}} if the reduction is sufficient. == Graphical motivation == In a regression setting, it is often useful to summarize the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} graphically. For instance, one may consider a scatterplot of y {\displaystyle y} versus one or more of the predictors or a linear combination of the predictors. A scatterplot that contains all available regression information is called a sufficient summary plot. When x {\displaystyle {\textbf {x}}} is high-dimensional, particularly when p ≥ 3 {\displaystyle p\geq 3} , it becomes increasingly challenging to construct and visually interpret sufficiency summary plots without reducing the data. Even three-dimensional scatter plots must be viewed via a computer program, and the third dimension can only be visualized by rotating the coordinate axes. However, if there exists a sufficient dimension reduction R ( x ) {\displaystyle R({\textbf {x}})} with small enough dimension, a sufficient summary plot of y {\displaystyle y} versus R ( x ) {\displaystyle R({\textbf {x}})} may be constructed and visually interpreted with relative ease. Hence sufficient dimension reduction allows for graphical intuition about the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} , which might not have otherwise been available for high-dimensional data. Most graphical methodology focuses primarily on dimension reduction involving linear combinations of x {\displaystyle {\textbf {x}}} . The rest of this article deals only with such reductions. == Dimension reduction subspace == Suppose R ( x ) = A T x {\displaystyle R({\textbf {x}})=A^{T}{\textbf {x}}} is a sufficient dimension reduction, where A {\displaystyle A} is a p × k {\displaystyle p\times k} matrix with rank k ≤ p {\displaystyle k\leq p} . Then the regression information for y ∣ x {\displaystyle y\mid {\textbf {x}}} can be inferred by studying the distribution of y ∣ A T x {\displaystyle y\mid A^{T}{\textbf {x}}} , and the plot of y {\displaystyle y} versus A T x {\displaystyle A^{T}{\textbf {x}}} is a sufficient summary plot. Without loss of generality, only the space spanned by the columns of A {\displaystyle A} need be considered. Let η {\displaystyle \eta } be a basis for the column space of A {\displaystyle A} , and let the space spanned by η {\displaystyle \eta } be denoted by S ( η ) {\displaystyle {\mathcal {S}}(\eta )} . It follows from the definition of a sufficient dimension reduction that F y ∣ x = F y ∣ η T x , {\displaystyle F_{y\mid x}=F_{y\mid \eta ^{T}x},} where F {\displaystyle F} denotes the appropriate distribution function. Another way to express this property is y ⊥ ⊥ x ∣ η T x , {\displaystyle y\perp \!\!\!\perp {\textbf {x}}\mid \eta ^{T}{\textbf {x}},} or y {\displaystyle y} is conditionally independent of x {\displaystyle {\textbf {x}}} , given η T x {\displaystyle \eta ^{T}{\textbf {x}}} . Then the subspace S ( η ) {\displaystyle {\mathcal {S}}(\eta )} is defined to be a dimension reduction subspace (DRS). === Structural dimensionality === For a regression y ∣ x {\displaystyle y\mid {\textbf {x}}} , the structural dimension, d {\displaystyle d} , is the smallest number of distinct linear combinations of x {\displaystyle {\textbf {x}}} necessary to preserve the conditional distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} . In other words, the smallest dimension reduction that is still sufficient maps x {\displaystyle {\textbf {x}}} to a subset of R d {\displaystyle \mathbb {R} ^{d}} . The corresponding DRS will be d-dimensional. === Minimum dimension reduction subspace === A subspace S {\displaystyle {\mathcal {S}}} is said to be a minimum DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} if it is a DRS and its dimension is less than or equal to that of all other DRSs for y ∣ x {\displaystyle y\mid {\textbf {x}}} . A minimum DRS S {\displaystyle {\mathcal {S}}} is not necessarily unique, but its dimension is equal to the structural dimension d {\displaystyle d} of y ∣ x {\displaystyle y\mid {\textbf {x}}} , by definition. If S {\displaystyle {\mathcal {S}}} has basis η {\displaystyle \eta } and is a minimum DRS, then a plot of y versus η T x {\displaystyle \eta ^{T}{\textbf {x}}} is a minimal sufficient summary plot, and it is (d + 1)-dimensional. == Central subspace == If a subspace S {\displaystyle {\mathcal {S}}} is a DRS for y ∣ x {\displaystyle y\mid {\textbf {x}}} , and if S ⊂ S drs {\displaystyle {\mathcal {S}}\subset {\mathcal {S}}_{\text{drs}}} for all other DRSs S drs {\displaystyle {\mathcal {S}}_{\text{drs}}} , then it is a central dimension reduction subspace, or simply a central subspace, and it is denoted by S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . In other words, a central subspace for y ∣ x {\displaystyle y\mid {\textbf {x}}} exists if and only if the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} of all dimension reduction subspaces is also a dimension reduction subspace, and that intersection is the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} . The central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does not necessarily exist because the intersection ⋂ S drs {\textstyle \bigcap {\mathcal {S}}_{\text{drs}}} is not necessarily a DRS. However, if S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} does exist, then it is also the unique minimum dimension reduction subspace. === Existence of the central subspace === While the existence of the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} is not guaranteed in every regression situation, there are some rather broad conditions under which its existence follows directly. For example, consider the following proposition from Cook (1998): Let S 1 {\displaystyle {\mathcal {S}}_{1}} and S 2 {\displaystyle {\mathcal {S}}_{2}} be dimension reduction subspaces for y ∣ x {\displaystyle y\mid {\textbf {x}}} . If x {\displaystyle {\textbf {x}}} has density f ( a ) > 0 {\displaystyle f(a)>0} for all a ∈ Ω x {\displaystyle a\in \Omega _{x}} and f ( a ) = 0 {\displaystyle f(a)=0} everywhere else, where Ω x {\displaystyle \Omega _{x}} is convex, then the intersection S 1 ∩ S 2 {\displaystyle {\mathcal {S}}_{1}\cap {\mathcal {S}}_{2}} is also a dimension reduction subspace. It follows from this proposition that the central subspace S y ∣ x {\displaystyle {\mathcal {S}}_{y\mid x}} exists for such x {\displaystyle {\textbf {x}}} . == Methods for dimension reduction == There are many existing methods for dimension reduction, both graphical and numeric. For example, sliced inverse regression (SIR) and sliced average variance estimation (SAVE) were introduced in the 1990s and continue to be widely used. Although SIR was originally designed to estimate an effective dimension reducing subspace, it is now understood that it estimates only the central subspace, which is generally different. More recent methods for dimension reduction include likelihood-based sufficient dimension reduction, estimating the central subspace based on the inverse third moment (or kth moment), estimating the central solution space, graphical regression, envelope model, and the principal support vector machine. For more details on these and other methods, consult the statistical literature. Principal components analysis (PCA) and similar methods for dimension reduction are not based on the sufficiency principle. === Example: linear regression === Consider the regression model y = α + β T x + ε , where ε ⊥ ⊥ x . {\displaystyle y=\alpha +\beta ^{T}{\textbf {x}}+\varepsilon ,{\text{ where }}\varepsilon \perp \!\!\!\perp {\textbf {x}}.} Note that the distribution of y ∣ x {\displaystyle y\mid {\textbf {x}}} is the same as the distribution of y ∣ β T x {\displ

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

Problem solving

Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to get from point A to B) to complex issues in business and technical fields. The former is an example of simple problem solving (SPS) addressing one issue, whereas the latter is complex problem solving (CPS) with multiple interrelated obstacles. Another classification of problem-solving tasks is into well-defined problems with specific obstacles and goals, and ill-defined problems in which the current situation is troublesome but it is not clear what kind of resolution to aim for. Similarly, one may distinguish formal or fact-based problems requiring psychometric intelligence, versus socio-emotional problems which depend on the changeable emotions of individuals or groups, such as tactful behavior, fashion, or gift choices. Solutions require sufficient resources and knowledge to attain the goal. Professionals such as lawyers, doctors, programmers, and consultants are largely problem solvers for issues that require technical skills and knowledge beyond general competence. Many businesses have found profitable markets by recognizing a problem and creating a solution: the more widespread and inconvenient the problem, the greater the opportunity to develop a scalable solution. There are many specialized problem-solving techniques and methods in fields such as science, engineering, business, medicine, mathematics, computer science, philosophy, and social organization. The mental techniques to identify, analyze, and solve problems are studied in psychology and cognitive sciences. Also widely researched are the mental obstacles that prevent people from finding solutions; problem-solving impediments include confirmation bias, mental set, and functional fixedness. == Definition == The term problem solving has a slightly different meaning depending on the discipline. For instance, it is a mental process in psychology and a computerized process in computer science. There are two different types of problems: ill-defined and well-defined; different approaches are used for each. Well-defined problems have specific end goals and clearly expected solutions, while ill-defined problems do not. Well-defined problems allow for more initial planning than ill-defined problems. Solving problems sometimes involves dealing with pragmatics (the way that context contributes to meaning) and semantics (the interpretation of the problem). The ability to understand what the end goal of the problem is, and what rules could be applied, represents the key to solving the problem. Sometimes a problem requires abstract thinking or coming up with a creative solution. Problem solving has two major domains: mathematical problem solving and personal problem solving. Each concerns some difficulty or barrier that is encountered. === Psychology === Problem solving in psychology refers to the process of finding solutions to problems encountered in life. Solutions to these problems are usually situation- or context-specific. The process starts with problem finding and problem shaping, in which the problem is discovered and simplified. The next step is to generate possible solutions and evaluate them. Finally a solution is selected to be implemented and verified. Problems have an end goal to be reached; how you get there depends upon problem orientation (problem-solving coping style and skills) and systematic analysis. Mental health professionals study the human problem-solving processes using methods such as introspection, behaviorism, simulation, computer modeling, and experiment. Social psychologists look into the person-environment relationship aspect of the problem and independent and interdependent problem-solving methods. Problem solving has been defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills. Empirical research shows many different strategies and factors influence everyday problem solving. Rehabilitation psychologists studying people with frontal lobe injuries have found that deficits in emotional control and reasoning can be re-mediated with effective rehabilitation and could improve the capacity of injured persons to resolve everyday problems. Interpersonal everyday problem solving is dependent upon personal motivational and contextual components. One such component is the emotional valence of "real-world" problems, which can either impede or aid problem-solving performance. Researchers have focused on the role of emotions in problem solving, demonstrating that poor emotional control can disrupt focus on the target task, impede problem resolution, and lead to negative outcomes such as fatigue, depression, and inertia. In conceptualization,human problem solving consists of two related processes: problem orientation, and the motivational/attitudinal/affective approach to problematic situations and problem-solving skills. People's strategies cohere with their goals and stem from the process of comparing oneself with others. === Cognitive sciences === Among the first experimental psychologists to study problem solving were the Gestaltists in Germany, such as Karl Duncker in The Psychology of Productive Thinking (1935). Perhaps best known is the work of Allen Newell and Herbert A. Simon. Experiments in the 1960s and early 1970s asked participants to solve relatively simple, well-defined, but not previously seen laboratory tasks. These simple problems, such as the Tower of Hanoi, admitted optimal solutions that could be found quickly, allowing researchers to observe the full problem-solving process. Researchers assumed that these model problems would elicit the characteristic cognitive processes by which more complex "real world" problems are solved. An outstanding problem-solving technique found by this research is the principle of decomposition. === Computer science === Much of computer science and artificial intelligence involves designing automated systems to solve a specified type of problem: to accept input data and calculate a correct or adequate response, reasonably quickly. Algorithms are recipes or instructions that direct such systems, written into computer programs. Steps for designing such systems include problem determination, heuristics, root cause analysis, de-duplication, analysis, diagnosis, and repair. Analytic techniques include linear and nonlinear programming, queuing systems, and simulation. A large, perennial obstacle is to find and fix errors in computer programs: debugging. === Logic === Formal logic concerns issues like validity, truth, inference, argumentation, and proof. In a problem-solving context, it can be used to formally represent a problem as a theorem to be proved, and to represent the knowledge needed to solve the problem as the premises to be used in a proof that the problem has a solution. The use of computers to prove mathematical theorems using formal logic emerged as the field of automated theorem proving in the 1950s. It included the use of heuristic methods designed to simulate human problem solving, as in the Logic Theory Machine, developed by Allen Newell, Herbert A. Simon and J. C. Shaw, as well as algorithmic methods such as the resolution principle developed by John Alan Robinson. In addition to its use for finding proofs of mathematical theorems, automated theorem-proving has also been used for program verification in computer science. In 1958, John McCarthy proposed the advice taker, to represent information in formal logic and to derive answers to questions using automated theorem-proving. An important step in this direction was made by Cordell Green in 1969, who used a resolution theorem prover for question-answering and for such other applications in artificial intelligence as robot planning. The resolution theorem-prover used by Cordell Green bore little resemblance to human problem solving methods. In response to criticism of that approach from researchers at MIT, Robert Kowalski developed logic programming and SLD resolution, which solves problems by problem decomposition. He has advocated logic for both computer and human problem solving and computational logic to improve human thinking. === Engineering === When products or processes fail, problem solving techniques can be used to develop corrective actions that can be taken to prevent further failures. Such techniques can also be applied to a product or process prior to an actual failure event—to predict, analyze, and mitigate a potential problem in advance. Techniques such as failure mode and effects analysis can proactively reduce the likelihood of problems. In either the reactive or the proactive case, it is necessary to build a causal explanation through a process of diagnosis. In deriving an explanation of effects in terms of causes, abduction generates new ideas or hypothes

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