On a Red Station, Drifting is a 2012 science fiction novella by Aliette de Bodard. Set in her Xuya Universe, it focuses on two women aboard a space station with a failing artificial intelligence. It received critical acclaim, becoming a finalist for the 2012 Nebula Award for Best Novella, the 2013 Hugo Award for Best Novella, and the 2013 Locus Award for Best Novella. == Plot == Lê Thi Linh is a magistrate of the Dai Viet Empire who is forced to flee her planet after criticizing the Emperor’s wartime policies. At the same time, rebel groups seize control of her planet and kill most of her subordinates. Linh seeks refuge with her distant relatives on Prosper Station. Prosper is controlled by an artificial intelligence called the Honoured Ancestress. Lê Thi Quyen, Linh’s cousin by marriage, manages the day-to-day operations of Prosper while her husband is away at war. Quyen and Linh immediately fall into conflict. Quyen’s brother-in-law Huu Hieu sells his mem-implants, which are copies of their ancestors’ consciousnesses. Meanwhile, the Honoured Ancestress experiences increasingly severe technical problems. Hieu and Linh become close. Hieu plans use the money from the sale of the implants to leave Prosper and marry his lover on a different station. Linh is upset knowing that she will never be able to leave. A visiting cousin, Lady Oahn, provides schematics for the repair of the Honoured Ancestress. In an effort to hurt Quyen, Linh writes an unflattering poem at a banquet honoring Oanh. In doing so, she reveals that Hieu is trying to leave Prosper. Hieu attempts suicide out of shame, but Linh rescues him. Quyen is able to repair the Honoured Ancestress, restoring her functionality at the expense of erasing many of her memories. The Emperor’s Embroidered Guard arrives at Prosper Station in search of Linh. Linh finds the missing mem-implants and returns them to Quyen. Quyen and Linh briefly reconcile before Linh is arrested and removed from Prosper Station. == Major themes == A review in Kirkus wrote that the novel's "familiar setting" was a "departure point" for the novel to explore its themes. The novel explores family ties; almost everyone on Prosper Station is related in some fashion. Additionally, the use of ancestors' mem-implants further explores the concept of family ties, with some descendants being considered more "worthy" than others due to their higher number of implants. The novel also explores questions of worth, as those who fail at ability tests are often forced to become the "lesser partners" in marriages and are discriminated against due to their perceived lack of achievement. The author notes that it is interesting that gender plays no role in the question of worth, and that the majority of the men in the story are actually the "lesser partner" in their marriage. == Style == The novel is divided into three sections. Liz Bourke wrote that each section builds thematically "towards an emotional crescendo". == Reception == Writing for Locus, Liz Bourke praised the novel's exploration of interpersonal conflict between Linh and Quyen, writing that "essentially subverts the popularly-understood derogatory overtones of 'domestic conflict'". Bourke also praised the story's tension, calling it "so well-strung the prose practically vibrates under its influence". A review for Kirkus stated that the novel is a "beautifully realized story and the characters, plot, theme and writing are expertly crafted." === Awards ===
Computational heuristic intelligence
Computational heuristic intelligence (CHI) refers to specialized programming techniques in computational intelligence (also called artificial intelligence, or AI). These techniques have the express goal of avoiding complexity issues, also called NP-hard problems, by using human-like techniques. They are best summarized as the use of exemplar-based methods (heuristics), rather than rule-based methods (algorithms). Hence the term is distinct from the more conventional computational algorithmic intelligence, or symbolic AI. An example of a CHI technique is the encoding specificity principle of Tulving and Thompson. In general, CHI principles are problem solving techniques used by people, rather than programmed into machines. It is by drawing attention to this key distinction that the use of this term is justified in a field already replete with confusing neologisms. Note that the legal systems of all modern human societies employ both heuristics (generalisations of cases) from individual trial records as well as legislated statutes (rules) as regulatory guides. Another recent approach to the avoidance of complexity issues is to employ feedback control rather than feedforward modeling as a problem-solving paradigm. This approach has been called computational cybernetics, because (a) the term 'computational' is associated with conventional computer programming techniques which represent a strategic, compiled, or feedforward model of the problem, and (b) the term 'cybernetic' is associated with conventional system operation techniques which represent a tactical, interpreted, or feedback model of the problem. Of course, real programs and real problems both contain both feedforward and feedback components. A real example which illustrates this point is that of human cognition, which clearly involves both perceptual (bottom-up, feedback, sensor-oriented) and conceptual (top-down, feedforward, motor-oriented) information flows and hierarchies. The AI engineer must choose between mathematical and cybernetic problem solution and machine design paradigms. This is not a coding (program language) issue, but relates to understanding the relationship between the declarative and procedural programming paradigms. The vast majority of STEM professionals never get the opportunity to design or implement pure cybernetic solutions. When pushed, most responders will dismiss the importance of any difference by saying that all code can be reduced to a mathematical model anyway. Unfortunately, not only is this belief false, it fails most spectacularly in many AI scenarios. Mathematical models are not time agnostic, but by their very nature are pre-computed, i.e. feedforward. Dyer [2012] and Feldman [2004] have independently investigated the simplest of all somatic governance paradigms, namely control of a simple jointed limb by a single flexor muscle. They found that it is impossible to determine forces from limb positions- therefore, the problem cannot have a pre-computed (feedforward) mathematical solution. Instead, a top-down command bias signal changes the threshold feedback level in the sensorimotor loop, e.g. the loop formed by the afferent and efferent nerves, thus changing the so-called ‘equilibrium point’ of the flexor muscle/ elbow joint system. An overview of the arrangement reveals that global postures and limb position are commanded in feedforward terms, using global displacements (common coding), with the forces needed being computed locally by feedback loops. This method of sensorimotor unit governance, which is based upon what Anatol Feldman calls the ‘equilibrium Point’ theory, is formally equivalent to a servomechanism such as a car's ‘cruise control’.
Diffbot
Diffbot is a developer of machine learning and computer vision algorithms and public APIs for extracting data from web pages / web scraping to create a knowledge base. == Overview == The company has gained interest from its application of computer vision technology to web pages, wherein it visually parses a web page for important elements and returns them in a structured format. In 2015 Diffbot announced it was working on its version of an automated "knowledge graph" by crawling the web and using its automatic web page extraction to build a large database of structured web data. In 2019 Diffbot released their Knowledge Graph which has since grown to include over two billion entities (corporations, people, articles, products, discussions, and more), and ten trillion "facts." == Features == The company's products allow software developers to analyze web home pages and article pages, and extract the "important information" while ignoring elements deemed not core to the primary content. In August 2012 the company released its Page Classifier API, which automatically categorizes web pages into specific "page types". As part of this, Diffbot analyzed 750,000 web pages shared on the social media service Twitter and revealed that photos, followed by articles and videos, are the predominant web media shared on the social network. In September 2020 the company released a Natural Language Processing API for automatically building Knowledge Graphs from text. The company raised $2 million in funding in May 2012 from investors including Andy Bechtolsheim and Sky Dayton. Diffbot's customers include Adobe, AOL, Cisco, DuckDuckGo, eBay, Instapaper, Microsoft, Onswipe and Springpad.
Jensen Huang
Jen-Hsun "Jensen" Huang (Chinese: 黃仁勳; Wade–Giles: Huáng Jén-hsūn; Tâi-lô: N̂g Jîn-hun; born February 17, 1963) is a Taiwanese and American business executive and electrical engineer who is the founder, president, and CEO of Nvidia, the world's most valuable company. As of 2026, Forbes estimates his net worth at over US$200 billion, making him the seventh-wealthiest individual in the world. The son of Taiwanese immigrants, Huang spent his childhood in Taiwan and Thailand before moving to the United States, where he was a student in Kentucky and Oregon. After earning a master's degree from Stanford University, Huang launched Nvidia in 1993 from a Denny's restaurant in San Jose, California, at age 30 and has remained its president and CEO ever since. He led the company out of near-bankruptcy during the 1990s and oversaw its expansion into GPU production, high-performance computing, and artificial intelligence (AI). Under Huang, Nvidia experienced rapid growth during the AI boom, becoming the first company to reach a market capitalization of over $5 trillion in October 2025. In 2021 and 2024, Time magazine included Huang in their list of the most influential people. In 2025, he was named as one of the "Architects of AI" for Time's Person of the Year. == Early life and education == Huang was born in Taipei, Taiwan, on February 17, 1963, and moved to the southern city of Tainan as a child. He is the younger of two sons of Huang Hsing-tai, a chemical engineer at an oil refinery, and Lo Tsai-hsiu, a schoolteacher. They were a middle-class Taiwanese family that relocated often, and were native speakers of Taiwanese Hokkien. Each day, Jensen's mother randomly selected 10 words from the dictionary to teach her sons English. When he was five years old, Huang's family moved to Thailand to support his father's refinery career and remained there for approximately four years. He attended Ruamrudee International School while in Bangkok. In the late 1960s, Hsing-tai traveled from Taiwan to New York City to train under an air conditioning company and, after returning home, resolved to send his sons to the United States. At age nine, Jensen, despite not yet being able to speak English fluently, was sent by his parents to live in the United States. He and his older brother moved in 1973 to live with an uncle in Tacoma, Washington, escaping widespread social unrest in Thailand. Both Huang's aunt and uncle were recent immigrants to Washington state; they accidentally enrolled him and his brother in the Oneida Baptist Institute, a religious reform academy in Kentucky for troubled youth, mistakenly believing it to be a prestigious boarding school. In order to afford the academy's tuition, Jensen's parents sold nearly all their possessions. When he was 10 years old, Huang lived with his older brother in the Oneida boys' dormitory. Each student was expected to work every day, and his brother was assigned to perform manual labor on a nearby tobacco farm. Because he was too young to attend classes at the reform academy, Huang was educated at a separate public school—the Oneida Elementary school in Oneida, Kentucky—arriving as "an undersized Asian immigrant with long hair and heavily accented English" and was frequently bullied and beaten. In Oneida, Huang cleaned toilets every day, learned to play table-tennis, joined the swimming team, and appeared in Sports Illustrated at age 14. He taught his illiterate roommate, a "17-year-old covered in tattoos and knife scars," how to read in exchange for being taught how to bench press. In 2002, Huang said he remembered his life in Kentucky "more vividly than just about any other". Two years after Huang arrived in Oneida, his parents moved to the United States and settled in Beaverton, Oregon, after which the brothers withdrew from school in Kentucky to live back with them. As a teenager, Huang attended Aloha High School in Aloha, Oregon, where he excelled academically. He skipped two grades, graduated at age 16, and became a nationally ranked table-tennis player in addition to being a member of its mathematics, computer, and science clubs. In 1977, the school purchased an Apple II computer. Huang used the machine to play Super Star Trek, a text-based game, and to program in BASIC, creating his own version of Snake. Beginning at age 15, Huang got his first job working the graveyard shift at a local Denny's restaurant as a dishwasher, busboy, and waiter from 1978 to 1983. After high school, he chose to enroll at Oregon State University due to its low in-state tuition. He studied electrical engineering and graduated in 1984 with a bachelor's degree with highest honors. Huang later recalled, "I was the youngest kid in school, in class" and the only student who "looked like a child". Years later, while working as a microchip designer in Silicon Valley, he concurrently pursued graduate night classes at Stanford University, where he earned a master's degree in electrical engineering in 1992. == AMD and LSI Logic == After graduating from college, Huang was a microchip designer in Silicon Valley. He was recruited for positions at Texas Instruments, Advanced Micro Devices (AMD), and LSI Logic, ultimately choosing the California-based AMD due to already being familiar with the company. Huang designed AMD microprocessors while simultaneously attending Stanford and raising his two children. However, when he heard of new chip design processes at LSI Logic, Huang left AMD to assume a role as a technical officer at the LSI Corporation, working under a startup company, Sun Microsystems, where he met engineers Chris Malachowsky and Curtis Priem. LSI was in contract with Sun Microsystems and had introduced Huang to Malachowsky and Priem, who were working on a new graphics accelerator card. While the three produced the card's manufacturing process, the relationship between Malachowsky and Priem became strained as the two disputed the chip's design, leading to infighting; according to Malachowsky, they "broke every tool that LSI Logic had in their standard portfolio". In 1989, Huang, Malachowsky, and Priem finalized the accelerator, which they called the "GX graphics engine". GX was a widespread financial success; the sales of the graphics engine contributed to Sun Microsystem's revenue increasing from $262 million in 1987 to $656 million in 1990, and Huang was promoted to be the director of LSI's CoreWare, a division that manufactured chips for hardware vendors. == Nvidia == === Founding (1993) === When business began to slow for Sun Microsystems after 1990, Huang, along with Priem and Malachowsky, each resigned their jobs to pursue a venture together in making graphics chips for PC games. They initially named their new company "NVision" until Huang suggested that the company be named "Nvidia" based on the Latin word invidia, as Priem wanted competitors to turn "green with envy". They eventually dropped the "i" to honor the NV1 chip that they were then developing. The three met frequently in 1992 at a Denny's roadside diner in East San Jose to formulate a business plan. Huang chose for them to meet at Denny's due to his prior work experience at the restaurant chain and because it was "quieter than home and had cheap coffee". The three founded the company during one meeting at a breakfast booth at the diner. To formally incorporate the company, Huang found a lawyer, James Gaither of Cooley Godward, who demanded the $200 in cash in Huang's pockets to capitalize the company. After that meeting, Huang went back to Priem and Malachowsky to ask each of them for $200 for their respective shares of the company, which meant that Nvidia's initial capital was $600. On April 5, 1993, Huang personally signed Nvidia's original articles of incorporation into effect. Although he left LSI, Huang remained in good standing with the company and was able to secure funding for Nvidia from LSI's CEO, Wilfred Corrigan, who introduced Huang to venture capitalist Don Valentine. An account cited how Huang's presentation pitch went badly. Valentine, the leader of Sequoia Capital, chose to invest in Nvidia through Corrigan's support, as did Sutter Hill Ventures. The funding enabled Nvidia to begin development efforts toward its first chip and to begin paying wages for its employees. By the first day of operation, Huang was made Nvidia's president and CEO. Even though Huang, at age 30, was younger than Priem and Malachowsky, both Priem and Malachowsky believed that he was prepared to be CEO. According to Priem, "we basically deferred to Jensen on day one" and told Huang, "you're in charge of running the company—all the stuff Chris and I don't know how to do". === President and CEO (1993–present) === As of 2024, Huang has been Nvidia's chief executive for over three decades, a tenure described by The Wall Street Journal as "almost unheard of in fast-moving Silicon Valley". He owns 3.6% of Nvidia's stock, which went public in 1999. He earned US$24.6 million as CEO i
Ratio Club
The Ratio Club was a small British informal dining club from 1949 to 1958 of young psychiatrists, psychologists, physiologists, mathematicians and engineers who met to discuss issues in cybernetics. == History == The idea of the club arose from a symposium on animal behaviour held in July 1949 by the Society of Experimental Biology in Cambridge. The club was founded by the neurologist John Bates, with other notable members such as W. Ross Ashby. The name Ratio was suggested by Albert Uttley, it being the Latin root meaning "computation or the faculty of mind which calculates, plans and reasons". He pointed out that it is also the root of rationarium, meaning a statistical account, and ratiocinatius, meaning argumentative. The use was probably inspired by an earlier suggestion by Donald Mackay of the 'MR club', from Machina ratiocinatrix, a term used by Norbert Wiener in the introduction to his then recently published book Cybernetics, or Control and Communication in the Animal and the Machine. Wiener used the term in reference to calculus ratiocinator, a calculating machine constructed by Leibniz. The initial membership was W. Ross Ashby, Horace Barlow, John Bates, George Dawson, Thomas Gold, W. E. Hick, Victor Little, Donald MacKay, Turner McLardy, P. A. Merton, John Pringle, Harold Shipton, Donald Sholl, Eliot Slater, Albert Uttley, W. Grey Walter and John Hugh Westcott. Alan Turing joined after the first meeting with I. J. Good, Philip Woodward and William Rushton added soon after. Giles Brindley attended several meetings as a guest. Warren McCulloch made presentations to the club twice, the first time at its inaugural meeting (a talk which the members found disappointing), and became a correspondent with and supporter of a number of its members. Others who attended at least one Ratio Club event as guests included Walter Pitts, Claude Shannon, J.Z. Young, C.H. Waddington, Peter Elias, J. C. R. Licklider, Oliver Selfridge, Benoît Mandelbrot, Colin Cherry and Anthony Oettinger. One one occasion I.J. Good brought along the then director of the USA's National Security Agency (presumably either Ralph Canine or John Samford given the dates). Several members admired the work of psychologist and philosopher Kenneth Craik and considered him an important influence; according to Husbands and Holland "there is no doubt Craik would have been a leading member of the club" had he not died young in 1945. The club has been considered the most influential cybernetics group in the UK, and many of its members went on to become prominent scientists.
Tensor operator
In pure and applied mathematics, quantum mechanics and computer graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which apply the notion of the spherical basis and spherical harmonics. The spherical basis closely relates to the description of angular momentum in quantum mechanics and spherical harmonic functions. The coordinate-free generalization of a tensor operator is known as a representation operator. == The general notion of scalar, vector, and tensor operators == In quantum mechanics, physical observables that are scalars, vectors, and tensors, must be represented by scalar, vector, and tensor operators, respectively. Whether something is a scalar, vector, or tensor depends on how it is viewed by two observers whose coordinate frames are related to each other by a rotation. Alternatively, one may ask how, for a single observer, a physical quantity transforms if the state of the system is rotated. Consider, for example, a system consisting of a molecule of mass M {\displaystyle M} , traveling with a definite center of mass momentum, p z ^ {\displaystyle p{\mathbf {\hat {z}} }} , in the z {\displaystyle z} direction. If we rotate the system by 90 ∘ {\displaystyle 90^{\circ }} about the y {\displaystyle y} axis, the momentum will change to p x ^ {\displaystyle p{\mathbf {\hat {x}} }} , which is in the x {\displaystyle x} direction. The center-of-mass kinetic energy of the molecule will, however, be unchanged at p 2 / 2 M {\displaystyle p^{2}/2M} . The kinetic energy is a scalar and the momentum is a vector, and these two quantities must be represented by a scalar and a vector operator, respectively. By the latter in particular, we mean an operator whose expected values in the initial and the rotated states are p z ^ {\displaystyle p{\mathbf {\hat {z}} }} and p x ^ {\displaystyle p{\mathbf {\hat {x}} }} . The kinetic energy on the other hand must be represented by a scalar operator, whose expected value must be the same in the initial and the rotated states. In the same way, tensor quantities must be represented by tensor operators. An example of a tensor quantity (of rank two) is the electrical quadrupole moment of the above molecule. Likewise, the octupole and hexadecapole moments would be tensors of rank three and four, respectively. Other examples of scalar operators are the total energy operator (more commonly called the Hamiltonian), the potential energy, and the dipole-dipole interaction energy of two atoms. Examples of vector operators are the momentum, the position, the orbital angular momentum, L {\displaystyle {\mathbf {L} }} , and the spin angular momentum, S {\displaystyle {\mathbf {S} }} . (Fine print: Angular momentum is a vector as far as rotations are concerned, but unlike position or momentum it does not change sign under space inversion, and when one wishes to provide this information, it is said to be a pseudovector.) Scalar, vector and tensor operators can also be formed by products of operators. For example, the scalar product L ⋅ S {\displaystyle {\mathbf {L} }\cdot {\mathbf {S} }} of the two vector operators, L {\displaystyle {\mathbf {L} }} and S {\displaystyle {\mathbf {S} }} , is a scalar operator, which figures prominently in discussions of the spin–orbit interaction. Similarly, the quadrupole moment tensor of our example molecule has the nine components Q i j = ∑ α q α ( 3 r α , i r α , j − r α 2 δ i j ) . {\displaystyle Q_{ij}=\sum _{\alpha }q_{\alpha }\left(3r_{\alpha ,i}r_{\alpha ,j}-r_{\alpha }^{2}\delta _{ij}\right).} Here, the indices i {\displaystyle i} and j {\displaystyle j} can independently take on the values 1, 2, and 3 (or x {\displaystyle x} , y {\displaystyle y} , and z {\displaystyle z} ) corresponding to the three Cartesian axes, the index α {\displaystyle \alpha } runs over all particles (electrons and nuclei) in the molecule, q α {\displaystyle q_{\alpha }} is the charge on particle α {\displaystyle \alpha } , and r α , i {\displaystyle r_{\alpha ,i}} is the i {\displaystyle i} -th component of the position of this particle. Each term in the sum is a tensor operator. In particular, the nine products r α , i r α , j {\displaystyle r_{\alpha ,i}r_{\alpha ,j}} together form a second rank tensor, formed by taking the outer product of the vector operator r α {\displaystyle {\mathbf {r} }_{\alpha }} with itself. == Rotations of quantum states == === Quantum rotation operator === The rotation operator about the unit vector n (defining the axis of rotation) through angle θ is U [ R ( θ , n ^ ) ] = exp ( − i θ ℏ n ^ ⋅ J ) {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right)} where J = (Jx, Jy, Jz) are the rotation generators (also the angular momentum matrices): J x = ℏ 2 ( 0 1 0 1 0 1 0 1 0 ) J y = ℏ 2 ( 0 i 0 − i 0 i 0 − i 0 ) J z = ℏ ( − 1 0 0 0 0 0 0 0 1 ) {\displaystyle J_{x}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&1&0\\1&0&1\\0&1&0\end{pmatrix}}\,\quad J_{y}={\frac {\hbar }{\sqrt {2}}}{\begin{pmatrix}0&i&0\\-i&0&i\\0&-i&0\end{pmatrix}}\,\quad J_{z}=\hbar {\begin{pmatrix}-1&0&0\\0&0&0\\0&0&1\end{pmatrix}}} and let R ^ = R ^ ( θ , n ^ ) {\displaystyle {\widehat {R}}={\widehat {R}}(\theta ,{\hat {\mathbf {n} }})} be a rotation matrix. According to the Rodrigues' rotation formula, the rotation operator then amounts to U [ R ( θ , n ^ ) ] = 1 1 − i sin θ ℏ n ^ ⋅ J − 1 − cos θ ℏ 2 ( n ^ ⋅ J ) 2 . {\displaystyle U[R(\theta ,{\hat {\mathbf {n} }})]=1\!\!1-{\frac {i\sin \theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} -{\frac {1-\cos \theta }{\hbar ^{2}}}({\hat {\mathbf {n} }}\cdot \mathbf {J} )^{2}.} An operator Ω ^ {\displaystyle {\widehat {\Omega }}} is invariant under a unitary transformation U if Ω ^ = U † Ω ^ U ; {\displaystyle {\widehat {\Omega }}={U}^{\dagger }{\widehat {\Omega }}U;} in this case for the rotation U ^ ( R ) {\displaystyle {\widehat {U}}(R)} , Ω ^ = U ( R ) † Ω ^ U ( R ) = exp ( i θ ℏ n ^ ⋅ J ) Ω ^ exp ( − i θ ℏ n ^ ⋅ J ) . {\displaystyle {\widehat {\Omega }}={U(R)}^{\dagger }{\widehat {\Omega }}U(R)=\exp \left({\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right){\widehat {\Omega }}\exp \left(-{\frac {i\theta }{\hbar }}{\hat {\mathbf {n} }}\cdot \mathbf {J} \right).} === Angular momentum eigenkets === The orthonormal basis set for total angular momentum is | j , m ⟩ {\displaystyle |j,m\rangle } , where j is the total angular momentum quantum number and m is the magnetic angular momentum quantum number, which takes values −j, −j + 1, ..., j − 1, j. A general state within the j subspace | ψ ⟩ = ∑ m c j m | j , m ⟩ {\displaystyle |\psi \rangle =\sum _{m}c_{jm}|j,m\rangle } rotates to a new state by: | ψ ¯ ⟩ = U ( R ) | ψ ⟩ = ∑ m c j m U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =U(R)|\psi \rangle =\sum _{m}c_{jm}U(R)|j,m\rangle } Using the completeness condition: I = ∑ m ′ | j , m ′ ⟩ ⟨ j , m ′ | {\displaystyle I=\sum _{m'}|j,m'\rangle \langle j,m'|} we have | ψ ¯ ⟩ = I U ( R ) | ψ ⟩ = ∑ m m ′ c j m | j , m ′ ⟩ ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle |{\bar {\psi }}\rangle =IU(R)|\psi \rangle =\sum _{mm'}c_{jm}|j,m'\rangle \langle j,m'|U(R)|j,m\rangle } Introducing the Wigner D matrix elements: D ( R ) m ′ m ( j ) = ⟨ j , m ′ | U ( R ) | j , m ⟩ {\displaystyle {D(R)}_{m'm}^{(j)}=\langle j,m'|U(R)|j,m\rangle } gives the matrix multiplication: | ψ ¯ ⟩ = ∑ m m ′ c j m D m ′ m ( j ) | j , m ′ ⟩ ⇒ | ψ ¯ ⟩ = D ( j ) | ψ ⟩ {\displaystyle |{\bar {\psi }}\rangle =\sum _{mm'}c_{jm}D_{m'm}^{(j)}|j,m'\rangle \quad \Rightarrow \quad |{\bar {\psi }}\rangle =D^{(j)}|\psi \rangle } For one basis ket: | j , m ¯ ⟩ = ∑ m ′ D ( R ) m ′ m ( j ) | j , m ′ ⟩ {\displaystyle |{\overline {j,m}}\rangle =\sum _{m'}{D(R)}_{m'm}^{(j)}|j,m'\rangle } For the case of orbital angular momentum, the eigenstates | ℓ , m ⟩ {\displaystyle |\ell ,m\rangle } of the orbital angular momentum operator L and solutions of Laplace's equation on a 3d sphere are spherical harmonics: Y ℓ m ( θ , ϕ ) = ⟨ θ , ϕ | ℓ , m ⟩ = ( 2 ℓ + 1 ) 4 π ( ℓ − m ) ! ( ℓ + m ) ! P ℓ m ( cos θ ) e i m ϕ {\displaystyle Y_{\ell }^{m}(\theta ,\phi )=\langle \theta ,\phi |\ell ,m\rangle ={\sqrt {{(2\ell +1) \over 4\pi }{(\ell -m)! \over (\ell +m)!}}}\,P_{\ell }^{m}(\cos {\theta })\,e^{im\phi }} where Pℓm is an associated Legendre polynomial, ℓ is the orbital angular momentum quantum number, and m is the orbital magnetic quantum number which takes the values −ℓ, −ℓ + 1, ... ℓ − 1, ℓ The formalism of spherical harmonics have wide applications in applied mathematics, and are closely related to the formalism of spherical tensors, as shown below. Spherical harmonics are functions of the polar and azimuthal angles, ϕ and θ respectively, which can be conveniently collected into a unit vector n(θ, ϕ) pointing in the direction of those angles, in the Cartesian basis it is: n ^ ( θ , ϕ ) = cos ϕ sin θ e x + s
Predictive Model Markup Language
The Predictive Model Markup Language (PMML) is an XML-based predictive model interchange format conceived by Robert Lee Grossman, then the director of the National Center for Data Mining at the University of Illinois at Chicago. PMML provides a way for analytic applications to describe and exchange predictive models produced by data mining and machine learning algorithms. It supports common models such as logistic regression and other feedforward neural networks. Version 0.9 was published in 1998. Subsequent versions have been developed by the Data Mining Group. Since PMML is an XML-based standard, the specification comes in the form of an XML schema. PMML itself is a mature standard with over 30 organizations having announced products supporting PMML. == PMML components == A PMML file can be described by the following components: Header: contains general information about the PMML document, such as copyright information for the model, its description, and information about the application used to generate the model such as name and version. It also contains an attribute for a timestamp which can be used to specify the date of model creation. Data Dictionary: contains definitions for all the possible fields used by the model. It is here that a field is defined as continuous, categorical, or ordinal (attribute optype). Depending on this definition, the appropriate value ranges are then defined as well as the data type (such as, string or double). Data Transformations: transformations allow for the mapping of user data into a more desirable form to be used by the mining model. PMML defines several kinds of simple data transformations. Normalization: map values to numbers, the input can be continuous or discrete. Discretization: map continuous values to discrete values. Value mapping: map discrete values to discrete values. Functions (custom and built-in): derive a value by applying a function to one or more parameters. Aggregation: used to summarize or collect groups of values. Model: contains the definition of the data mining model. E.g., A multi-layered feedforward neural network is represented in PMML by a "NeuralNetwork" element which contains attributes such as: Model Name (attribute modelName) Function Name (attribute functionName) Algorithm Name (attribute algorithmName) Activation Function (attribute activationFunction) Number of Layers (attribute numberOfLayers) This information is then followed by three kinds of neural layers which specify the architecture of the neural network model being represented in the PMML document. These attributes are NeuralInputs, NeuralLayer, and NeuralOutputs. Besides neural networks, PMML allows for the representation of many other types of models including support vector machines, association rules, Naive Bayes classifier, clustering models, text models, decision trees, and different regression models. Mining Schema: a list of all fields used in the model. This can be a subset of the fields as defined in the data dictionary. It contains specific information about each field, such as: Name (attribute name): must refer to a field in the data dictionary Usage type (attribute usageType): defines the way a field is to be used in the model. Typical values are: active, predicted, and supplementary. Predicted fields are those whose values are predicted by the model. Outlier Treatment (attribute outliers): defines the outlier treatment to be use. In PMML, outliers can be treated as missing values, as extreme values (based on the definition of high and low values for a particular field), or as is. Missing Value Replacement Policy (attribute missingValueReplacement): if this attribute is specified then a missing value is automatically replaced by the given values. Missing Value Treatment (attribute missingValueTreatment): indicates how the missing value replacement was derived (e.g. as value, mean or median). Targets: allows for post-processing of the predicted value in the format of scaling if the output of the model is continuous. Targets can also be used for classification tasks. In this case, the attribute priorProbability specifies a default probability for the corresponding target category. It is used if the prediction logic itself did not produce a result. This can happen, e.g., if an input value is missing and there is no other method for treating missing values. Output: this element can be used to name all the desired output fields expected from the model. These are features of the predicted field and so are typically the predicted value itself, the probability, cluster affinity (for clustering models), standard error, etc. The latest release of PMML, PMML 4.1, extended Output to allow for generic post-processing of model outputs. In PMML 4.1, all the built-in and custom functions that were originally available only for pre-processing became available for post-processing too. == PMML 4.0, 4.1, 4.2 and 4.3 == PMML 4.0 was released on June 16, 2009. Examples of new features included: Improved Pre-Processing Capabilities: Additions to built-in functions include a range of Boolean operations and an If-Then-Else function. Time Series Models: New exponential Smoothing models; also place holders for ARIMA, Seasonal Trend Decomposition, and Spectral density estimation, which are to be supported in the near future. Model Explanation: Saving of evaluation and model performance measures to the PMML file itself. Multiple Models: Capabilities for model composition, ensembles, and segmentation (e.g., combining of regression and decision trees). Extensions of Existing Elements: Addition of multi-class classification for Support Vector Machines, improved representation for Association Rules, and the addition of Cox Regression Models. PMML 4.1 was released on December 31, 2011. New features included: New model elements for representing Scorecards, k-Nearest Neighbors (KNN) and Baseline Models. Simplification of multiple models. In PMML 4.1, the same element is used to represent model segmentation, ensemble, and chaining. Overall definition of field scope and field names. A new attribute that identifies for each model element if the model is ready or not for production deployment. Enhanced post-processing capabilities (via the Output element). PMML 4.2 was released on February 28, 2014. New features include: Transformations: New elements for implementing text mining New built-in functions for implementing regular expressions: matches, concat, and replace Simplified outputs for post-processing Enhancements to Scorecard and Naive Bayes model elements PMML 4.3 was released on August 23, 2016. New features include: New Model Types: Gaussian Process Bayesian Network New built-in functions Usage clarifications Documentation improvements Version 4.4 was released in November 2019. == Release history == == Data Mining Group == The Data Mining Group is a consortium managed by the Center for Computational Science Research, Inc., a nonprofit founded in 2008. The Data Mining Group also developed a standard called Portable Format for Analytics, or PFA, which is complementary to PMML.