DiscoVision is the name of several things related to the video LaserDisc format. It was the original name of the "Reflective Optical Videodisc System" format later known as "LaserVision" or LaserDisc. == Description == MCA DiscoVision, Inc. was a division of entertainment giant MCA (Music Corporation of America), established in 1969 to develop and sell an optical videodisc system. MCA released discs pressed in Carson and Costa Mesa, California on the DiscoVision label from the format's Atlanta, Georgia launch in 1978 to 1982 and the release of the film The Four Seasons. DiscoVision titles included films from Universal Pictures, Paramount Pictures, Warner Bros. Pictures, and Disney content. Agreements were made with Columbia Pictures and United Artists, though no discs were released on the DiscoVision label from either studio. Most of these companies later established their own labels for the format, the first being Paramount with a dozen movies released on the Paramount Home Video label in the summer of 1981. The successor to MCA DiscoVision, DiscoVision Associates (DVA), was the result of a partnership between IBM and MCA. It was hoped that the merger would provide the basis for improvement of the quality of DiscoVision pressings, but no appreciable improvement ever took hold. In 1981, responsibility for the laser videodisc was sold to Pioneer Electronic Corporation, after MCA Discovision had previously started a partnership in 1977 with Pioneer, Universal Pioneer, to produce the Pioneer PR-7820 player (the first industrial model of DiscoVision player from 1978), as well as establishing disc pressing plants in Japan. As part of the partnership, Pioneer, in association with MCA, had a disc replication facility in Kofu, Japan that produced discs. Some of the last DiscoVision label discs were manufactured by Pioneer in Japan. In the same year, MCA discontinued their DiscoVision branding, due to the sale of the technology to Pioneer (who then rebranded the format as LaserDisc) and in turn rebranded their laserdisc releases, now fabricated by Pioneer, under the MCA Videodisc banner; this was changed to the "MCA Home Video" name for both its VHS and videodisc releases. Some of DiscoVision's technical staff went on to form MCA Video Games, in an effort to produce video game cartridges. DiscoVision Associates later evolved into a patent holding company which manages and licenses intellectual property related to LaserDisc, Compact Disc, and optical disc technologies, as well as other non-disc related fields. In 1989, Pioneer acquired DiscoVision Associates where it continues to license its technologies independently. As the portfolio of patent expired, the presence of DiscoVision became less visible. However, it established the success of a patent holding company, which other companies are stimulated to generate royalty income from their own patent portfolio.
DAVI
The Dutch Automated Vehicle Initiative (DAVI) is a research and demonstration initiative developing automated vehicles for use on public roads. The project is unique in that, besides simply making driverless cars, it also focuses on having automated vehicles share information among each other. The aim is to have the cars help to avoid traffic congestion by reducing the safety distance between the cars (from 2 seconds to 0.5 seconds) and avoiding sudden traffic slow-downs due to maneuvers undertaken by drivers.
Artificial intuition
Artificial intuition is a theoretical capacity of an artificial software to function similarly to human consciousness, specifically in the capacity of human consciousness known as intuition. == Comparison of human and the theoretically artificial == Intuition is the function of the mind, the experience of which, is described as knowledge based on "a hunch", resulting (as the word itself does) from "contemplation" or "insight". Psychologist Jean Piaget showed that intuitive functioning within the normally developing human child at the Intuitive Thought Substage of the preoperational stage occurred at from four to seven years of age. In Carl Jung's concept of synchronicity, the concept of "intuitive intelligence" is described as something like a capacity that transcends ordinary-level functioning to a point where information is understood with a greater depth than is available in more simple rationally-thinking entities. Artificial intuition is theoretically (or otherwise) a sophisticated function of an artifice that is able to interpret data with depth and locate hidden factors functioning in Gestalt psychology, and that intuition in the artificial mind would, in the context described here, be a bottom-up process upon a macroscopic scale identifying something like the archetypal (see τύπος). To create artificial intuition supposes the possibility of the re-creation of a higher functioning of the human mind, with capabilities such as what might be found in semantic memory and learning. The transferral of the functioning of a biological system to synthetic functioning is based upon modeling of functioning from knowledge of cognition and the brain, for instance as applications of models of artificial neural networks from the research done within the discipline of computational neuroscience. == Application software contributing to its development == The notion of a process of a data-interpretative synthesis has already been found in a computational-linguistic software application that has been created for use in an internal security context. The software integrates computed data based specifically on objectives incorporating a paradigm described as "religious intuitive" (hermeneutic), functional to a degree that represents advances upon the performance of generic lexical data mining.
Ontology (information science)
In information science, an ontology encompasses a representation, formal naming, and definitions of the categories, properties, and relations between the concepts, data, or entities that pertain to one, many, or all domains of discourse. More simply, an ontology is a way of showing the properties of a subject area and how they are related, by defining a set of terms and relational expressions that represent the entities in that subject area. The field which studies ontologies so conceived is sometimes referred to as applied ontology. Every academic discipline or field, in creating its terminology, thereby lays the groundwork for an ontology. Each uses ontological assumptions to frame explicit theories, research and applications. Improved ontologies may improve problem solving within that domain, interoperability of data systems, and discoverability of data. Translating research papers within every field is a problem made easier when experts from different countries maintain a controlled vocabulary of jargon between each of their languages. For instance, the definition and ontology of economics is a primary concern in Marxist economics, but also in other subfields of economics. An example of economics relying on information science occurs in cases where a simulation or model is intended to enable economic decisions, such as determining what capital assets are at risk and by how much (see risk management). What ontologies in both information science and philosophy have in common is the attempt to represent entities, including both objects and events, with all their interdependent properties and relations, according to a system of categories. In both fields, there is considerable work on problems of ontology engineering (e.g., Quine and Kripke in philosophy, Sowa and Guarino in information science), and debates concerning to what extent normative ontology is possible (e.g., foundationalism and coherentism in philosophy, BFO and Cyc in artificial intelligence). Applied ontology is considered by some as a successor to prior work in philosophy. However many current efforts are more concerned with establishing controlled vocabularies of narrow domains than with philosophical first principles, or with questions such as the mode of existence of fixed essences or whether enduring objects (e.g., perdurantism and endurantism) may be ontologically more primary than processes. Artificial intelligence has retained considerable attention regarding applied ontology in subfields like natural language processing within machine translation and knowledge representation, but ontology editors are being used often in a range of fields, including biomedical informatics and industry. Such efforts often use ontology editing tools such as Protégé. == Ontology in philosophy == Ontology is a branch of philosophy and intersects areas such as metaphysics, epistemology, and philosophy of language, as it considers how knowledge, language, and perception relate to the nature of reality. Metaphysics deals with questions like "what exists?" and "what is the nature of reality?". One of five traditional branches of philosophy, metaphysics is concerned with exploring existence through properties, entities and relations such as those between particulars and universals, intrinsic and extrinsic properties, or essence and existence. Metaphysics has been an ongoing topic of discussion since recorded history. == Etymology == The compound word ontology combines onto-, from the Greek ὄν, on (gen. ὄντος, ontos), i.e. "being; that which is", which is the present participle of the verb εἰμί, eimí, i.e. "to be, I am", and -λογία, -logia, i.e. "logical discourse", see classical compounds for this type of word formation. While the etymology is Greek, the oldest extant record of the word itself, the Neo-Latin form ontologia, appeared in 1606 in the work Ogdoas Scholastica by Jacob Lorhard (Lorhardus) and in 1613 in the Lexicon philosophicum by Rudolf Göckel (Goclenius). The first occurrence in English of ontology as recorded by the OED (Oxford English Dictionary, online edition, 2008) came in Archeologia Philosophica Nova or New Principles of Philosophy (1663) by Gideon Harvey. == Formal ontology == Since the mid-1970s, researchers in the field of artificial intelligence (AI) have recognized that knowledge engineering is the key to building large and powerful AI systems. AI researchers argued that they could create new ontologies as computational models that enable certain kinds of automated reasoning, which was only marginally successful. In the 1980s, the AI community began to use the term ontology to refer to both a theory of a modeled world and a component of knowledge-based systems. In particular, David Powers introduced the word ontology to AI to refer to real world or robotic grounding, publishing in 1990 literature reviews emphasizing grounded ontology in association with the call for papers for a AAAI Summer Symposium Machine Learning of Natural Language and Ontology, with an expanded version published in SIGART Bulletin and included as a preface to the proceedings. Some researchers, drawing inspiration from philosophical ontologies, viewed computational ontology as a kind of applied philosophy. In 1993, the widely cited web page and paper "Toward Principles for the Design of Ontologies Used for Knowledge Sharing" by Tom Gruber used ontology as a technical term in computer science closely related to earlier idea of semantic networks and taxonomies. Gruber introduced the term as a specification of a conceptualization: An ontology is a description (like a formal specification of a program) of the concepts and relationships that can formally exist for an agent or a community of agents. This definition is consistent with the usage of ontology as set of concept definitions, but more general. And it is a different sense of the word than its use in philosophy. Attempting to distance ontologies from taxonomies and similar efforts in knowledge modeling that rely on classes and inheritance, Gruber stated (1993): Ontologies are often equated with taxonomic hierarchies of classes, class definitions, and the subsumption relation, but ontologies need not be limited to these forms. Ontologies are also not limited to conservative definitions, that is, definitions in the traditional logic sense that only introduce terminology and do not add any knowledge about the world (Enderton, 1972). To specify a conceptualization, one needs to state axioms that do constrain the possible interpretations for the defined terms. Recent experimental ontology frameworks have also explored resonance-based AI-human co-evolution structures, such as IAMF (Illumination AI Matrix Framework), OntoMotoOS (a meta-operating system concept for ethical and ontological AI–human co-evolution), and PSRT (Phase-Structural Reality Theory across multi-scale ontological layers). Though not yet widely adopted in academic discourse, such models propose phased approaches to ethical harmonization and structural emergence. As refinement of Gruber's definition Feilmayr and Wöß (2016) stated: "An ontology is a formal, explicit specification of a shared conceptualization that is characterized by high semantic expressiveness required for increased complexity." == Formal ontology components == Contemporary ontologies share many structural similarities, regardless of the language in which they are expressed. Most ontologies describe individuals (instances), classes (concepts), attributes and relations. === Types === ==== Domain ontology ==== A domain ontology (or domain-specific ontology) represents concepts which belong to a realm of the world, such as biology or politics. Each domain ontology typically models domain-specific definitions of terms. For example, the word card has many different meanings. An ontology about the domain of poker would model the "playing card" meaning of the word, while an ontology about the domain of computer hardware would model the "punched card" and "video card" meanings. Since domain ontologies are written by different people, they represent concepts in very specific and unique ways, and are often incompatible within the same project. As systems that rely on domain ontologies expand, they often need to merge domain ontologies by hand-tuning each entity or using a combination of software merging and hand-tuning. This presents a challenge to the ontology designer. Different ontologies in the same domain arise due to different languages, different intended usage of the ontologies, and different perceptions of the domain (based on cultural background, education, ideology, etc.). At present, merging ontologies that are not developed from a common upper ontology is a largely manual process and therefore time-consuming and expensive. Domain ontologies that use the same upper ontology to provide a set of basic elements with which to specify the meanings of the domain ontology entities can be merged with less effo
Computer and information science
Computer and information science (CIS; also known as information and computer science) is a field that emphasizes both computing and informatics, upholding the strong association between the fields of information sciences and computer sciences and treating computers as a tool rather than a field. Information science is one with a long history, unlike the relatively very young field of computer science, and is primarily concerned with gathering, storing, disseminating, sharing and protecting any and all forms of information. It is a broad field, covering a myriad of different areas but is often referenced alongside computer science because of the incredibly useful nature of computers and computer programs in helping those studying and doing research in the field – particularly in helping to analyse data and in spotting patterns too broad for a human to intuitively perceive. While information science is sometimes confused with information theory, the two have vastly different subject matter. Information theory focuses on one particular mathematical concept of information while information science is focused on all aspects of the processes and techniques of information. Computer science, in contrast, is less focused on information and its different states, but more, in a very broad sense, on the use of computers – both in theory and practice – to design and implement algorithms in order to aid the processing of information during the different states described above. It has strong foundations in the field of mathematics, as the very first recognised practitioners of the field were renowned mathematicians such as Alan Turing. Information science and computing began to converge in the 1950s and 1960s, as information scientists started to realize the many ways computers would improve information storage and retrieval. == Terminology == Due to the distinction between computers and computing, some of the research groups refer to computing or datalogy. The French refer to computer science as the term informatique. The term information and communications technology (ICT), refers to how humans communicate with using machines and computers, making a distinction from information and computer science, which is how computers use and gain information. Informatics is also distinct from computer science, which encompasses the study of logic and low-level computing issues. == Education == Universities may confer degrees with a major in computer and information science, not to be confused with a more specific Bachelor of Computer Science or respective graduate computer science degrees. The QS World University Rankings is one of the most widely recognised and distinguished university comparisons. They ranked the top 10 universities for computer science and information systems in 2015. They are: Massachusetts Institute of Technology (MIT) Stanford University University of Oxford Carnegie Mellon University Harvard University University of California, Berkeley (UCB) University of Cambridge The Hong Kong University of Science and Technology Swiss Federal Institute of Technology (ETH Zurich) Princeton University A Computer Information Science degree gives students both network and computing knowledge which is needed to design, develop, and assist information systems which helps to solve business problems and to support business problems and to support business operations and decision making at a managerial level also. == Areas of information and computer science == Due to the nature of this field, many topics are also shared with computer science and information systems. The discipline of Information and Computer Science spans a vast range of areas from basic computer science theory (algorithms and computational logic) to in depth analysis of data manipulation and use within technology. === Programming theory === The process of taking a given algorithm and encoding it into a language that can be understood and executed by a computer. There are many different types of programming languages and various different types of computers, however, they all have the same goal: to turn algorithms into machine code. Popular programming languages used within the academic study of CIS include, but are not limited to: Java, Python, C#, C++, Perl, Ruby, Pascal, Swift, Visual Basic. === Information and information systems === The academic study of software and hardware systems that process large quantities and data, support large scale data management and how data can be used. This is where the field is unique from the standard study of computer science. The area of information systems focuses on the networks of hardware and software that are required to process, manipulate and distribute such data. === Computer systems and organisations === The process of analysing computer architecture and various logic circuits. This involves looking at low level computer processes at bit level computation. This is an in-depth look into the hardware processing of a computational system, involving looking at the basic structure of a computer and designing such systems. This can also involve evaluating complex circuit diagrams, and being able to construct these to solve a main problem. The main purpose behind this area of study is to achieve an understanding of how computers function on a basic level, often through tracing machine operations. === Machines, languages, and computation === This is the study into fundamental computer algorithms, which are the basis to computer programs. Without algorithms, no computer programs would exist. This also involves the process of looking into various mathematical functions behind computational algorithms, basic theory and functional (low level) programming. In an academic setting, this area would introduce the fundamental mathematical theorems and functions behind theoretical computer science which are the building blocks for other areas in the field. Complex topics such as; proofs, algebraic functions and sets will be introduced during studies of CIS. == Developments == Information and computer science is a field that is rapidly developing with job prospects for students being extremely promising with 75.7% of graduates gaining employment. Also the IT industry employs one in twenty of the workforce with it predicted to increase nearly five times faster than the average of the UK and between 2012 and 2017 more than half a million people will be needed within the industry and the fact that nine out of ten tech firms are suffering from candidate shortages which is having a negative impact on their business as it delays the creation and development of new products, and it's predicted in the US that in the next decade there will be more than one million jobs in the technology sector than computer science graduates to fill them. Because of this programming is now being taught at an earlier age with an aim to interest students from a young age into computer and information science hopefully leading more children to study this at a higher level. For example, children in England will now be exposed to computer programming at the age of 5 due to an updated national curriculum. == Employment == Due to the wide variety of jobs that now involve computer and information science related tasks, it is difficult to provide a comprehensive list of possible jobs in this area, but some of the key areas are artificial intelligence, software engineering and computer networking and communication. Work in this area also tends to require sufficient understanding of mathematics and science. Moreover, jobs that having a CIS degree can lead to, include: systems analyst, network administrator, system architect, information systems developer, web programmer, or software developer. The earning potential for CIS graduates is quite promising. A 2013 survey from the National Association of Colleges and Employers (NACE) found that the average starting salary for graduates who earned a degree in a computer related field was $59,977, up 4.3% from the prior year. This is higher than other popular degrees such as business ($54,234), education ($40,480) and math and sciences ($42,724). Furthermore, Payscale ranked 129 college degrees based on their graduates earning potential with engineering, math, science, and technology fields dominating the ranking. With eight computer related degrees appearing among the top 30. With the lowest starting salary for these jobs being $49,900. A Rasmussen College article describes various jobs CIS graduates may obtain with software applications developers at the top making a median income of $98,260. According to the National Careers Service an Information Scientist can expect to earn £24,000+ per year as a starting salary.
Language resource
In linguistics and language technology, a language resource is a "[composition] of linguistic material used in the construction, improvement and/or evaluation of language processing applications, (...) in language and language-mediated research studies and applications." According to Bird & Simons (2003), this includes data, i.e. "any information that documents or describes a language, such as a published monograph, a computer data file, or even a shoebox full of handwritten index cards. The information could range in content from unanalyzed sound recordings to fully transcribed and annotated texts to a complete descriptive grammar", tools, i.e., "computational resources that facilitate creating, viewing, querying, or otherwise using language data", and advice, i.e., "any information about what data sources are reliable, what tools are appropriate in a given situation, what practices to follow when creating new data". The latter aspect is usually referred to as "best practices" or "(community) standards". In a narrower sense, language resource is specifically applied to resources that are available in digital form, and then, "encompassing (a) data sets (textual, multimodal/multimedia and lexical data, grammars, language models, etc.) in machine readable form, and (b) tools/technologies/services used for their processing and management". == Typology == As of May 2020, no widely used standard typology of language resources has been established (current proposals include the LREMap, METASHARE, and, for data, the LLOD classification). Important classes of language resources include data lexical resources, e.g., machine-readable dictionaries, linguistic corpora, i.e., digital collections of natural language data, linguistic data bases such as the Cross-Linguistic Linked Data collection, tools linguistic annotations and tools for creating such annotations in a manual or semiautomated fashion (e.g., tools for annotating interlinear glossed text such as Toolbox and FLEx, or other language documentation tools), applications for search and retrieval over such data (corpus management systems), for automated annotation (part-of-speech tagging, syntactic parsing, semantic parsing, etc.), metadata and vocabularies vocabularies, repositories of linguistic terminology and language metadata, e.g., MetaShare (for language resource metadata), the ISO 12620 data category registry (for linguistic features, data structures and annotations within a language resource), or the Glottolog database (identifiers for language varieties and bibliographical database). == Language resource publication, dissemination and creation == A major concern of the language resource community has been to develop infrastructures and platforms to present, discuss and disseminate language resources. Selected contributions in this regard include: a series of International Conferences on Language Resources and Evaluation (LREC), the European Language Resources Association (ELRA, EU-based), and the Linguistic Data Consortium (LDC, US-based), which represent commercial hosting and dissemination platforms for language resources, the Open Languages Archives Community (OLAC), which provides and aggregates language resource metadata, the Language Resources and Evaluation Journal (LREJ), the European Language Grid is a European platform for language technologies (eg services), data and resources. As for the development of standards and best practices for language resources, these are subject of several community groups and standardization efforts, including ISO Technical Committee 37: Terminology and other language and content resources (ISO/TC 37), developing standards for all aspects of language resources, W3C Community Group Best Practices for Multilingual Linked Open Data (BPMLOD), working on best practice recommendations for publishing language resources as Linked Data or in RDF, W3C Community Group Linked Data for Language Technology (LD4LT), working on linguistic annotations on the web and language resource metadata, W3C Community Group Ontology-Lexica (OntoLex), working on lexical resources, the Open Linguistics working group of the Open Knowledge Foundation, working on conventions for publishing and linking open language resources, developing the Linguistic Linked Open Data cloud, the Text Encoding Initiative (TEI), working on XML-based specifications for language resources and digitally edited text.
Zassenhaus algorithm
In mathematics, the Zassenhaus algorithm is a method to calculate a basis for the intersection and sum of two subspaces of a vector space. It is named after Hans Zassenhaus, but no publication of this algorithm by him is known. It is used in computer algebra systems. == Algorithm == === Input === Let V be a vector space and U, W two finite-dimensional subspaces of V with the following spanning sets: U = ⟨ u 1 , … , u n ⟩ {\displaystyle U=\langle u_{1},\ldots ,u_{n}\rangle } and W = ⟨ w 1 , … , w k ⟩ . {\displaystyle W=\langle w_{1},\ldots ,w_{k}\rangle .} Finally, let B 1 , … , B m {\displaystyle B_{1},\ldots ,B_{m}} be linearly independent vectors so that u i {\displaystyle u_{i}} and w i {\displaystyle w_{i}} can be written as u i = ∑ j = 1 m a i , j B j {\displaystyle u_{i}=\sum _{j=1}^{m}a_{i,j}B_{j}} and w i = ∑ j = 1 m b i , j B j . {\displaystyle w_{i}=\sum _{j=1}^{m}b_{i,j}B_{j}.} === Output === The algorithm computes the base of the sum U + W {\displaystyle U+W} and a base of the intersection U ∩ W {\displaystyle U\cap W} . === Algorithm === The algorithm creates the following block matrix of size ( ( n + k ) × ( 2 m ) ) {\displaystyle ((n+k)\times (2m))} : ( a 1 , 1 a 1 , 2 ⋯ a 1 , m a 1 , 1 a 1 , 2 ⋯ a 1 , m ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ a n , 1 a n , 2 ⋯ a n , m a n , 1 a n , 2 ⋯ a n , m b 1 , 1 b 1 , 2 ⋯ b 1 , m 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ b k , 1 b k , 2 ⋯ b k , m 0 0 ⋯ 0 ) {\displaystyle {\begin{pmatrix}a_{1,1}&a_{1,2}&\cdots &a_{1,m}&a_{1,1}&a_{1,2}&\cdots &a_{1,m}\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\a_{n,1}&a_{n,2}&\cdots &a_{n,m}&a_{n,1}&a_{n,2}&\cdots &a_{n,m}\\b_{1,1}&b_{1,2}&\cdots &b_{1,m}&0&0&\cdots &0\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\b_{k,1}&b_{k,2}&\cdots &b_{k,m}&0&0&\cdots &0\end{pmatrix}}} Using elementary row operations, this matrix is transformed to the row echelon form. Then, it has the following shape: ( c 1 , 1 c 1 , 2 ⋯ c 1 , m ∙ ∙ ⋯ ∙ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ c q , 1 c q , 2 ⋯ c q , m ∙ ∙ ⋯ ∙ 0 0 ⋯ 0 d 1 , 1 d 1 , 2 ⋯ d 1 , m ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 d ℓ , 1 d ℓ , 2 ⋯ d ℓ , m 0 0 ⋯ 0 0 0 ⋯ 0 ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 0 0 ⋯ 0 0 0 ⋯ 0 ) {\displaystyle {\begin{pmatrix}c_{1,1}&c_{1,2}&\cdots &c_{1,m}&\bullet &\bullet &\cdots &\bullet \\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\c_{q,1}&c_{q,2}&\cdots &c_{q,m}&\bullet &\bullet &\cdots &\bullet \\0&0&\cdots &0&d_{1,1}&d_{1,2}&\cdots &d_{1,m}\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\0&0&\cdots &0&d_{\ell ,1}&d_{\ell ,2}&\cdots &d_{\ell ,m}\\0&0&\cdots &0&0&0&\cdots &0\\\vdots &\vdots &&\vdots &\vdots &\vdots &&\vdots \\0&0&\cdots &0&0&0&\cdots &0\end{pmatrix}}} Here, ∙ {\displaystyle \bullet } stands for arbitrary numbers, and the vectors ( c p , 1 , c p , 2 , … , c p , m ) {\displaystyle (c_{p,1},c_{p,2},\ldots ,c_{p,m})} for every p ∈ { 1 , … , q } {\displaystyle p\in \{1,\ldots ,q\}} and ( d p , 1 , … , d p , m ) {\displaystyle (d_{p,1},\ldots ,d_{p,m})} for every p ∈ { 1 , … , ℓ } {\displaystyle p\in \{1,\ldots ,\ell \}} are nonzero. Then ( y 1 , … , y q ) {\displaystyle (y_{1},\ldots ,y_{q})} with y i := ∑ j = 1 m c i , j B j {\displaystyle y_{i}:=\sum _{j=1}^{m}c_{i,j}B_{j}} is a basis of U + W {\displaystyle U+W} and ( z 1 , … , z ℓ ) {\displaystyle (z_{1},\ldots ,z_{\ell })} with z i := ∑ j = 1 m d i , j B j {\displaystyle z_{i}:=\sum _{j=1}^{m}d_{i,j}B_{j}} is a basis of U ∩ W {\displaystyle U\cap W} . === Proof of correctness === First, we define π 1 : V × V → V , ( a , b ) ↦ a {\displaystyle \pi _{1}:V\times V\to V,(a,b)\mapsto a} to be the projection to the first component. Let H := { ( u , u ) ∣ u ∈ U } + { ( w , 0 ) ∣ w ∈ W } ⊆ V × V . {\displaystyle H:=\{(u,u)\mid u\in U\}+\{(w,0)\mid w\in W\}\subseteq V\times V.} Then π 1 ( H ) = U + W {\displaystyle \pi _{1}(H)=U+W} and H ∩ ( 0 × V ) = 0 × ( U ∩ W ) {\displaystyle H\cap (0\times V)=0\times (U\cap W)} . Also, H ∩ ( 0 × V ) {\displaystyle H\cap (0\times V)} is the kernel of π 1 | H {\displaystyle {\pi _{1}|}_{H}} , the projection restricted to H. Therefore, dim ( H ) = dim ( U + W ) + dim ( U ∩ W ) {\displaystyle \dim(H)=\dim(U+W)+\dim(U\cap W)} . The Zassenhaus algorithm calculates a basis of H. In the first m columns of this matrix, there is a basis y i {\displaystyle y_{i}} of U + W {\displaystyle U+W} . The rows of the form ( 0 , z i ) {\displaystyle (0,z_{i})} (with z i ≠ 0 {\displaystyle z_{i}\neq 0} ) are obviously in H ∩ ( 0 × V ) {\displaystyle H\cap (0\times V)} . Because the matrix is in row echelon form, they are also linearly independent. All rows which are different from zero ( ( y i , ∙ ) {\displaystyle (y_{i},\bullet )} and ( 0 , z i ) {\displaystyle (0,z_{i})} ) are a basis of H, so there are dim ( U ∩ W ) {\displaystyle \dim(U\cap W)} such z i {\displaystyle z_{i}} s. Therefore, the z i {\displaystyle z_{i}} s form a basis of U ∩ W {\displaystyle U\cap W} . == Example == Consider the two subspaces U = ⟨ ( 1 − 1 0 1 ) , ( 0 0 1 − 1 ) ⟩ {\displaystyle U=\left\langle \left({\begin{array}{r}1\\-1\\0\\1\end{array}}\right),\left({\begin{array}{r}0\\0\\1\\-1\end{array}}\right)\right\rangle } and W = ⟨ ( 5 0 − 3 3 ) , ( 0 5 − 3 − 2 ) ⟩ {\displaystyle W=\left\langle \left({\begin{array}{r}5\\0\\-3\\3\end{array}}\right),\left({\begin{array}{r}0\\5\\-3\\-2\end{array}}\right)\right\rangle } of the vector space R 4 {\displaystyle \mathbb {R} ^{4}} . Using the standard basis, we create the following matrix of dimension ( 2 + 2 ) × ( 2 ⋅ 4 ) {\displaystyle (2+2)\times (2\cdot 4)} : ( 1 − 1 0 1 1 − 1 0 1 0 0 1 − 1 0 0 1 − 1 5 0 − 3 3 0 0 0 0 0 5 − 3 − 2 0 0 0 0 ) . {\displaystyle \left({\begin{array}{rrrrrrrr}1&-1&0&1&&1&-1&0&1\\0&0&1&-1&&0&0&1&-1\\\\5&0&-3&3&&0&0&0&0\\0&5&-3&-2&&0&0&0&0\end{array}}\right).} Using elementary row operations, we transform this matrix into the following matrix: ( 1 0 0 0 ∙ ∙ ∙ ∙ 0 1 0 − 1 ∙ ∙ ∙ ∙ 0 0 1 − 1 ∙ ∙ ∙ ∙ 0 0 0 0 1 − 1 0 1 ) {\displaystyle \left({\begin{array}{rrrrrrrrr}1&0&0&0&&\bullet &\bullet &\bullet &\bullet \\0&1&0&-1&&\bullet &\bullet &\bullet &\bullet \\0&0&1&-1&&\bullet &\bullet &\bullet &\bullet \\\\0&0&0&0&&1&-1&0&1\end{array}}\right)} (Some entries have been replaced by " ∙ {\displaystyle \bullet } " because they are irrelevant to the result.) Therefore ( ( 1 0 0 0 ) , ( 0 1 0 − 1 ) , ( 0 0 1 − 1 ) ) {\displaystyle \left(\left({\begin{array}{r}1\\0\\0\\0\end{array}}\right),\left({\begin{array}{r}0\\1\\0\\-1\end{array}}\right),\left({\begin{array}{r}0\\0\\1\\-1\end{array}}\right)\right)} is a basis of U + W {\displaystyle U+W} , and ( ( 1 − 1 0 1 ) ) {\displaystyle \left(\left({\begin{array}{r}1\\-1\\0\\1\end{array}}\right)\right)} is a basis of U ∩ W {\displaystyle U\cap W} .