DVD (digital video disc or digital versatile disc) is a digital optical disc data storage format. It was invented and developed in 1995 and first released on November 1, 1996, in Japan. The medium can store any kind of digital data and has been widely used to store video programs (watched using DVD players), software and other computer files. DVDs offer significantly higher storage capacity than compact discs (CD) while having the same dimensions. A standard single-layer DVD can store up to 4.7 GB of data, a dual-layer DVD up to 8.5 GB. Dual-layer, double-sided DVDs can store up to a maximum of 17.08 GB. Prerecorded DVDs are mass-produced using molding machines that physically stamp data onto the DVD. Such discs are a form of DVD-ROM because data can only be read and not written or erased. Blank recordable DVD discs (DVD-R and DVD+R) can be recorded once using a DVD recorder and then function as a DVD-ROM. Rewritable DVDs (DVD-RW, DVD+RW, and DVD-RAM) can be recorded and erased many times. DVDs are used in DVD-Video consumer digital video format and less commonly in DVD-Audio consumer digital audio format, as well as for authoring DVD discs written in a special AVCHD format to hold high definition material (often in conjunction with AVCHD format camcorders). DVDs containing other types of information may be referred to as DVD data discs. == Etymology == The Oxford English Dictionary comments that, "In 1995, rival manufacturers of the product initially named digital video disc agreed that, in order to emphasize the flexibility of the format for multimedia applications, the preferred abbreviation DVD would be understood to denote digital versatile disc." The OED also states that in 1995, "The companies said the official name of the format will simply be DVD. Toshiba had been using the name 'digital video disc', but that was switched to 'digital versatile disc' after computer companies complained that it left out their applications." "Digital versatile disc" is the explanation provided in a DVD Forum Primer from 2000 and in the DVD Forum's mission statement, which the purpose is to promote broad acceptance of DVD products on technology, across entertainment, and other industries. Because DVDs became highly popular for the distribution of movies in the 2000s, the term DVD became popularly used in English as a noun to describe specifically a full-length movie released on the format; for example the phrase "to watch a DVD" describes watching a movie on DVD. == History == === Development and launch === Released in 1987, CD Video used analog video encoding on optical discs matching the established standard 120 mm (4.7 in) size of audio CDs. Video CD (VCD) became one of the first formats for distributing digitally encoded films in this format, in 1993. In the same year, two new optical disc storage formats were being developed. One was the Multimedia Compact Disc (MMCD), backed by Philips and Sony (developers of the CD and CD-i), and the other was the Super Density (SD) disc, supported by Toshiba, Time Warner, Matsushita Electric, Hitachi, Mitsubishi Electric, Pioneer, Thomson, and JVC. By the time of the press launches for both formats in January 1995, the MMCD nomenclature had been dropped, and Philips and Sony were referring to their format as Digital Video Disc (DVD). On May 3, 1995, an ad hoc industry technical group formed from five computer companies (IBM, Apple, Compaq, Hewlett-Packard, and Microsoft) issued a press release stating that they would only accept a single format. The group voted to boycott both formats unless the two camps agreed on a single, converged standard. They recruited Lou Gerstner, president of IBM, to pressure the executives of the warring factions. In one significant compromise, the MMCD and SD groups agreed to adopt proposal SD 9, which specified that both layers of the dual-layered disc be read from the same side—instead of proposal SD 10, which would have created a two-sided disc that users would have to turn over. Philips/Sony strongly insisted on the source code, EFMPlus, that Kees Schouhamer Immink had designed for the MMCD, because it makes it possible to apply the existing CD servo technology. Its drawback was a loss from 5 to 4.7 Gigabytes of capacity. As a result, the DVD specification provided a storage capacity of 4.7 GB (4.38 GiB) for a single-layered, single-sided disc and 8.5 GB (7.92 GiB) for a dual-layered, single-sided disc. The DVD specification ended up similar to Toshiba and Matsushita's Super Density Disc, except for the dual-layer option. MMCD was single-sided and optionally dual-layer, whereas SD was two half-thickness, single-layer discs which were pressed separately and then glued together to form a double-sided disc. Philips and Sony decided that it was in their best interests to end the format war, and on September 15, 1995 agreed to unify with companies backing the Super Density Disc to release a single format, with technologies from both. After other compromises between MMCD and SD, the group of computer companies won the day, and a single format was agreed upon. The computer companies also collaborated with the Optical Storage Technology Association (OSTA) on the use of their implementation of the ISO-13346 file system (known as Universal Disk Format) for use on the new DVDs. The format's details were finalized on December 8, 1995. In November 1995, Samsung announced it would start mass-producing DVDs by September 1996. The format launched on November 1, 1996, in Japan, mostly with music video releases. The first major releases from Warner Home Video arrived on December 20, 1996, with four titles being available. The format's release in the U.S. was delayed multiple times, from August 1996, to October 1996, November 1996, before finally settling on early 1997. Players began to be produced domestically that winter, with March 24, 1997, as the U.S. launch date of the format proper in seven test markets. Approximately 32 titles were available on launch day, mainly from the Warner Bros., MGM, and New Line libraries, with the notable inclusion of the 1996 film Twister. However, the launch was planned for the following day (March 25), leading to a distribution change with retailers and studios to prevent similar violations of breaking the street date. The nationwide rollout for the format happened on August 22, 1997. DTS announced in late 1997 that they would be coming onto the format. The sound system company revealed details in a November 1997 online interview, and clarified it would release discs in early 1998. However, this date would be pushed back several times before finally releasing their first titles at the 1999 Consumer Electronics Show. In 2001, blank DVD recordable discs cost the equivalent of $27.34 US dollars in 2022. === Adoption === Movie and home entertainment distributors adopted the DVD format to replace the ubiquitous VHS tape as the primary consumer video distribution format. Immediately following the formal adoption of a unified standard for DVD, two of the four leading video game console companies (Sega and The 3DO Company) said they already had plans to design a gaming console with DVDs as the source medium. Sony stated at the time that they had no plans to use DVD in their gaming systems, despite being one of the developers of the DVD format and eventually the first company to actually release a DVD-based console. Game consoles such as the PlayStation 2, Xbox, and Xbox 360 use DVDs as their source medium for games and other software. Contemporary games for Windows were also distributed on DVD. Early DVDs were mastered using DLT tape, but using DVD-R DL or +R DL eventually became common. TV DVD combos, combining a standard definition CRT TV or an HD flat panel TV with a DVD mechanism under the CRT or on the back of the flat panel, and VCR/DVD combos were also available for purchase. For consumers, DVD soon overtook VHS as the favored choice for home movie releases. In 2001, DVD players outsold VCRs for the first time in the United States. At that time, one in four American households owned a DVD player. By 2007, about 80% of Americans owned a DVD player, a figure that had surpassed VCRs; it was also higher than personal computers or cable television. == Specifications == The DVD specifications created and updated by the DVD Forum are published as so-called DVD Books (e.g. DVD-ROM Book, DVD-Audio Book, DVD-Video Book, DVD-R Book, DVD-RW Book, DVD-RAM Book, DVD-AR (Audio Recording) Book, DVD-VR (Video Recording) Book, etc.). DVD discs are made up of two discs; normally one is blank, and the other contains data. Each disc is 0.6 mm thick, and they are glued together to form a DVD disc. The gluing process must be done carefully to make the disc as flat as possible to avoid both birefringence and "disc tilt", which is when the disc is not perfectly flat, preventing it from being read. Some specifications for mechanical, physical and optical characteristics of DV
Quantum artificial life
Quantum artificial life is the application of quantum algorithms with the ability to simulate biological behavior. Quantum computers offer many potential improvements to processes performed on classical computers, including machine learning and artificial intelligence. Artificial intelligence applications are often inspired by the idea of mimicking human brains through closely related biomimicry. This has been implemented to a certain extent on classical computers (using neural networks), but quantum computers offer many advantages in the simulation of artificial life. Artificial life and artificial intelligence are extremely similar, with minor differences; the goal of studying artificial life is to understand living beings better, while the goal of artificial intelligence is to create intelligent beings. In 2016, Alvarez-Rodriguez et al. developed a proposal for a quantum artificial life algorithm with the ability to simulate life and Darwinian evolution. In 2018, the same research team led by Alvarez-Rodriguez performed the proposed algorithm on the IBM ibmqx4 quantum computer, and received optimistic results. The results accurately simulated a system with the ability to undergo self-replication at the quantum scale. == Artificial life on quantum computers == The growing advancement of quantum computers has led researchers to develop quantum algorithms for simulating life processes. Researchers have designed a quantum algorithm that can accurately simulate Darwinian Evolution. Since the complete simulation of artificial life on quantum computers has only been actualized by one group, this section shall focus on the implementation by Alvarez-Rodriguez, Sanz, Lomata, and Solano on an IBM quantum computer. Individuals were realized as two qubits, one representing the genotype of the individual and the other representing the phenotype. The genotype is copied to transmit genetic information through generations, and the phenotype is dependent on the genetic information as well as the individual's interactions with their environment. In order to set up the system, the state of the genotype is instantiated by some rotation of an ancillary state ( | 0 ⟩ ⟨ 0 | {\displaystyle |0\rangle \langle 0|} ). The environment is a two-dimensional spatial grid occupied by individuals and ancillary states. The environment is divided into cells that are able to possess one or more individuals. Individuals move throughout the grid and occupy cells randomly; when two or more individuals occupy the same cell they interact with each other. === Self replication === The ability to self-replicate is critical for simulating life. Self-replication occurs when the genotype of an individual interacts with an ancillary state, creating a genotype for a new individual; this genotype interacts with a different ancillary state in order to create the phenotype. During this interaction, one would like to copy some information about the initial state into the ancillary state, but by the no cloning theorem, it is impossible to copy an arbitrary unknown quantum state. However, physicists have derived different methods for quantum cloning which does not require the exact copying of an unknown state. The method that has been implemented by Alvarez-Rodriguez et al. is one that involves the cloning of the expectation value of some observable. For a unitary U {\displaystyle U} which copies the expectation value of some set of observables X {\displaystyle {\mathsf {X}}} of state ρ {\displaystyle \rho } into a blank state ρ e {\displaystyle \rho _{e}} , the cloning machine is defined by any ( U , ρ e , X ) {\displaystyle (U,\rho _{e},{\mathsf {X}})} that fulfill the following: ∀ ρ ∀ X ∈ X {\displaystyle \forall \rho \forall X\in {\mathsf {X}}} X ¯ = X 1 ¯ = X 2 ¯ {\displaystyle {\bar {X}}={\bar {X_{1}}}={\bar {X_{2}}}} Where X ¯ {\displaystyle {\bar {X}}} is the mean value of the observable in ρ {\displaystyle \rho } before cloning, X 1 ¯ {\displaystyle {\bar {X_{1}}}} is the mean value of the observable in ρ {\displaystyle \rho } after cloning, and X 2 ¯ {\displaystyle {\bar {X_{2}}}} is the mean value of the observable in ρ e {\displaystyle \rho _{e}} after cloning. Note that the cloning machine has no dependence on ρ {\displaystyle \rho } because we want to be able to clone the expectation of the observables for any initial state. It is important to note that cloning the mean value of the observable transmits more information than is allowed classically. The calculation of the mean value is defined naturally as: X ¯ = T r [ ρ X ] {\displaystyle {\bar {X}}=Tr[\rho X]} , X 1 ¯ = T r [ R X ⊗ I ] {\displaystyle {\bar {X_{1}}}=Tr[RX\otimes I]} , X 2 ¯ = T r [ R I ⊗ X ] {\displaystyle {\bar {X_{2}}}=Tr[RI\otimes X]} where R = U ρ ⊗ ρ e U † {\displaystyle R=U\rho \otimes \rho _{e}U^{\dagger }} The simplest cloning machine clones the expectation value of σ z {\displaystyle \sigma _{z}} in arbitrary state ρ = | ψ ⟩ ⟨ ψ | {\displaystyle \rho =|\psi \rangle \langle \psi |} to ρ e = | 0 ⟩ ⟨ 0 | {\displaystyle \rho _{e}=|0\rangle \langle 0|} using U = C N O T {\displaystyle U=CNOT} . This is the cloning machine implemented for self-replication by Alvarez-Rodriguez et al. The self-replication process clearly only requires interactions between two qubits, and therefore this cloning machine is the only one necessary for self replication. === Interactions === Interactions occur between individuals when the two take up the same space on the environmental grid. The presence of interactions between individuals provides an advantage for shorter-lifespan individuals. When two individuals interact, exchanges of information between the two phenotypes may or may not occur based on their existing values. When both individual's control qubits (genotypes) are alike, no information will be exchanged. When the control qubits differ, the target qubits (phenotype) will be exchanged between the two individuals. This procedure produces a constantly changing predator-prey dynamic in the simulation. Therefore, long-living qubits, with a larger genetic makeup in the simulation, are at a disadvantage. Since information is only exchanged when interacting with an individual of different genetic makeup, the short-lived population has the advantage. === Mutation === Mutations exist in the artificial world with limited probability, equivalent to their occurrence in the real world. There are two ways in which the individual can mutate: through random single qubit rotations and by errors in the self-replication process. There are two different operators that act on the individual and cause mutations. The M operation causes a spontaneous mutation within the individual by rotating a single qubit by parameter θ. The parameter θ is random for each mutation, which creates biodiversity within the artificial environment. The M operation is a unitary matrix which can be described as: M = ( cos ( θ ) s i n ( θ ) s i n ( θ ) − c o s ( θ ) ) {\displaystyle M={\begin{pmatrix}\cos(\theta )&sin(\theta )\\sin(\theta )&-cos(\theta )\end{pmatrix}}} The other possible way for mutations to occur is due to errors in the replication process. Due to the no-cloning theorem, it is impossible to produce perfect copies of systems that are originally in unknown quantum states. However, quantum cloning machines make it possible to create imperfect copies of quantum states, in other words, the process introduces some degree of error. The error that exists in current quantum cloning machines is the root cause for the second kind of mutations in the artificial life experiment. The imperfect cloning operation can be seen as: U M ( θ ) = I 4 + 1 2 ( 0 0 0 1 ) ⊗ ( − 1 1 1 − 1 ) ( c o s θ + i s i n θ + 1 ) {\displaystyle U_{M}(\theta )=\mathrm {I} _{4}+{\frac {1}{2}}{\begin{pmatrix}0&0\\0&1\end{pmatrix}}\otimes {\begin{pmatrix}-1&1\\1&-1\end{pmatrix}}(cos\theta +isin\theta +1)} The two kinds of mutations affect the individual differently. While the spontaneous M operation does not affect the phenotype of the individual, the self-replicating error mutation, UM, alters both the genotype of the individual, and its associated lifetime. The presence of mutations in the quantum artificial life experiment is critical for providing randomness and biodiversity. The inclusion of mutations helps to increase the accuracy of the quantum algorithm. === Death === At the instant the individual is created (when the genotype is copied into the phenotype), the phenotype interacts with the environment. As time evolves, the interaction of the individual with the environment simulates aging which eventually leads to the death of the individual. The death of an individual occurs when the expectation value of σ z {\displaystyle \sigma _{z}} is within some ϵ {\displaystyle \epsilon } of 1 in the phenotype, or, equivalently, when ρ p = | 0 ⟩ ⟨ 0 | {\displaystyle \rho _{p}=|0\rangle \langle 0|} The Lindbladian describes the interaction of the individual with the environment: ρ
Physical schema
A physical data model (or database design) is a representation of a data design as implemented, or intended to be implemented, in a database management system. In the lifecycle of a project it typically derives from a logical data model, though it may be reverse-engineered from a given database implementation. A complete physical data model will include all the database artifacts required to create relationships between tables or to achieve performance goals, such as indexes, constraint definitions, linking tables, partitioned tables or clusters. Analysts can usually use a physical data model to calculate storage estimates; it may include specific storage allocation details for a given database system. As of 2012 seven main databases dominate the commercial marketplace: Informix, Oracle, Postgres, SQL Server, Sybase, IBM Db2 and MySQL. Other RDBMS systems tend either to be legacy databases or used within academia such as universities or further education colleges. Physical data models for each implementation would differ significantly, not least due to underlying operating-system requirements that may sit underneath them. For example: SQL Server runs only on Microsoft Windows operating-systems (Starting with SQL Server 2017, SQL Server runs on Linux. It's the same SQL Server database engine, with many similar features and services regardless of your operating system), while Oracle and MySQL can run on Solaris, Linux and other UNIX-based operating-systems as well as on Windows. This means that the disk requirements, security requirements and many other aspects of a physical data model will be influenced by the RDBMS that a database administrator (or an organization) chooses to use. == Physical schema == Physical schema is a term used in data management to describe how data is to be represented and stored (files, indices, etc.) in secondary storage using a particular database management system (DBMS) (e.g., Oracle RDBMS, Sybase SQL Server, etc.). In the ANSI/SPARC Architecture three schema approach, the internal schema is the view of data that involved data management technology. This is as opposed to an external schema that reflects an individual's view of the data, or the conceptual schema that is the integration of a set of external schemas. The logical schema was the way data were represented to conform to the constraints of a particular approach to database management. At that time the choices were hierarchical and network. Describing the logical schema, however, still did not describe how physically data would be stored on disk drives. That is the domain of the physical schema. Now logical schemas describe data in terms of relational tables and columns, object-oriented classes, and XML tags. A single set of tables, for example, can be implemented in numerous ways, up to and including an architecture where table rows are maintained on computers in different countries.
Sparse identification of non-linear dynamics
Sparse identification of nonlinear dynamics (SINDy) is a data-driven algorithm for obtaining dynamical systems from data. Given a series of snapshots of a dynamical system and its corresponding time derivatives, SINDy performs a sparsity-promoting regression (such as LASSO and sparse Bayesian inference) on a library of nonlinear candidate functions of the snapshots against the derivatives to find the governing equations. This procedure relies on the assumption that most physical systems only have a few dominant terms which dictate the dynamics, given an appropriately selected coordinate system and quality training data. It has been applied to identify the dynamics of fluids, based on proper orthogonal decomposition, as well as other complex dynamical systems, such as biological networks. == Mathematical Overview == First, consider a dynamical system of the form x ˙ = d d t x ( t ) = f ( x ( t ) ) , {\displaystyle {\dot {\textbf {x}}}={\frac {d}{dt}}{\textbf {x}}(t)={\textbf {f}}({\textbf {x}}(t)),} where x ( t ) ∈ R n {\displaystyle {\textbf {x}}(t)\in \mathbb {R} ^{n}} is a state vector (snapshot) of the system at time t {\displaystyle t} and the function f ( x ( t ) ) {\displaystyle {\textbf {f}}({\textbf {x}}(t))} defines the equations of motion and constraints of the system. The time derivative may be either prescribed or numerically approximated from the snapshots. With x {\displaystyle {\textbf {x}}} and x ˙ {\displaystyle {\dot {\textbf {x}}}} sampled at m {\displaystyle m} equidistant points in time ( t 1 , t 2 , ⋯ , t m {\displaystyle t_{1},t_{2},\cdots ,t_{m}} ), these can be arranged into matrices of the form X = [ x T ( t 1 ) x T ( t 2 ) ⋮ x T ( t m ) ] = [ x 1 ( t 1 ) x 2 ( t 1 ) ⋯ x n ( t 1 ) x 1 ( t 2 ) x 2 ( t 2 ) ⋯ x n ( t 2 ) ⋮ ⋮ ⋱ ⋮ x 1 ( t m ) x 2 ( t m ) ⋯ x n ( t m ) ] , {\displaystyle {\bf {{X}={\begin{bmatrix}\mathbf {x} ^{\mathsf {T}}(t_{1})\\\mathbf {x} ^{\mathsf {T}}(t_{2})\\\vdots \\\mathbf {x} ^{\mathsf {T}}(t_{m})\end{bmatrix}}={\begin{bmatrix}x_{1}(t_{1})&x_{2}(t_{1})&\cdots &x_{n}(t_{1})\\x_{1}(t_{2})&x_{2}(t_{2})&\cdots &x_{n}(t_{2})\\\vdots &\vdots &\ddots &\vdots \\x_{1}(t_{m})&x_{2}(t_{m})&\cdots &x_{n}(t_{m})\end{bmatrix}},}}} and similarly for X ˙ {\displaystyle {\dot {\mathbf {X} }}} . Next, a library Θ ( X ) {\displaystyle \mathbf {\Theta } (\mathbf {X} )} of nonlinear candidate functions of the columns of X {\displaystyle {\textbf {X}}} is constructed, which may be constant, polynomial, or more exotic functions (like trigonometric and rational terms, and so on): Θ ( X ) = [ | | | | | | 1 X X 2 X 3 ⋯ sin ( X ) cos ( X ) ⋯ | | | | | | ] {\displaystyle \ \ \ {\bf {{\Theta }({\bf {{X})={\begin{bmatrix}\vline &\vline &\vline &\vline &&\vline &\vline &\\1&{\bf {X}}&{\bf {{X}^{2}}}&{\bf {{X}^{3}}}&\cdots &\sin({\bf {{X})}}&\cos({\bf {{X})}}&\cdots \\\vline &\vline &\vline &\vline &&\vline &\vline &\end{bmatrix}}}}}}} The number of possible model structures from this library is combinatorially high. f ( x ( t ) ) {\displaystyle {\textbf {f}}({\textbf {x}}(t))} is then substituted by Θ ( X ) {\displaystyle {\bf {{\Theta }({\textbf {X}})}}} and a vector of coefficients Ξ = [ ξ 1 ξ 2 ⋯ ξ n ] {\displaystyle {\bf {{\Xi }=\left[{\bf {{\xi }_{1}{\bf {{\xi }_{2}\cdots {\bf {{\xi }_{n}}}}}}}\right]}}} determining the active terms in f ( x ( t ) ) {\displaystyle {\textbf {f}}({\textbf {x}}(t))} : X ˙ = Θ ( X ) Ξ {\displaystyle {\dot {\bf {X}}}={\bf {{\Theta }({\bf {{X}){\bf {\Xi }}}}}}} Because only a few terms are expected to be active at each point in time, an assumption is made that f ( x ( t ) ) {\displaystyle {\textbf {f}}({\textbf {x}}(t))} admits a sparse representation in Θ ( X ) {\displaystyle {\bf {{\Theta }({\textbf {X}})}}} . This then becomes an optimization problem in finding a sparse Ξ {\displaystyle {\bf {\Xi }}} which optimally embeds X ˙ {\displaystyle {\dot {\textbf {X}}}} . In other words, a parsimonious model is obtained by performing least squares regression on the system (4) with sparsity-promoting ( L 1 {\displaystyle L_{1}} ) regularization ξ k = arg min ξ k ′ | | X ˙ k − Θ ( X ) ξ k ′ | | 2 + λ | | ξ k ′ | | 1 , {\displaystyle {\bf {{\xi }_{k}={\underset {\bf {{\xi }'_{k}}}{\arg \min }}\left|\left|{\dot {\bf {X}}}_{k}-{\bf {{\Theta }({\bf {{X}){\bf {{\xi }'_{k}}}}}}}\right|\right|_{2}+\lambda \left|\left|{\bf {{\xi }'_{k}}}\right|\right|_{1},}}} where λ {\displaystyle \lambda } is a regularization parameter. Finally, the sparse set of ξ k {\displaystyle {\bf {{\xi }_{k}}}} can be used to reconstruct the dynamical system: x ˙ k = Θ ( x ) ξ k {\displaystyle {\dot {x}}_{k}={\bf {{\Theta }({\bf {{x}){\bf {{\xi }_{k}}}}}}}}
Quality of Data
Quality of Data (QoD) is a designation coined by L. Veiga, that specifies and describes the required Quality of Service of a distributed storage system from the Consistency point of view of its data. It can be used to support big data management frameworks, Workflow management, and HPC systems (mainly for data replication and consistency). It takes into account data semantics, namely the Time interval of data freshness, the Sequence of tolerable number of outstanding versions of the data read before ore refresh, and the Value divergence allowed before displaying it. Initially it was based on a model from an existing research work regarding vector-field Consistency, awarded the best-paper prize in the ACM/IFIP/Usenix Middleware Conference 2007 and later enhanced for increased scalability and fault-tolerance. This consistency model has been successfully applied and proven in big data key/value store Apache HBase, initially designed as a middleware module seating between clusters from separate data centres. The HBase-QoD coupling minimises bandwidth usage and optimises resources allocation during replication achieving the desired consistency level at a more fine-grained level. QoD is defined by the three-dimensions of vector k=(θ,σ,ν), but with a broader view of the issue, applicable also to large-scale data management techniques in regards to their timely delivery. == Other descriptions == Quality of Data should not be confused with other definitions for data quality such as completeness, validity, and accuracy.
Pattern theory
Pattern theory, formulated by Ulf Grenander, is a mathematical formalism to describe knowledge of the world as patterns. It differs from other approaches to artificial intelligence in that it does not begin by prescribing algorithms and machinery to recognize and classify patterns; rather, it prescribes a vocabulary to articulate and recast the pattern concepts in precise language. Broad in its mathematical coverage, Pattern Theory spans algebra and statistics, as well as local topological and global entropic properties. In addition to the new algebraic vocabulary, its statistical approach is novel in its aim to: Identify the hidden variables of a data set using real world data rather than artificial stimuli, which was previously commonplace. Formulate prior distributions for hidden variables and models for the observed variables that form the vertices of a Gibbs-like graph. Study the randomness and variability of these graphs. Create the basic classes of stochastic models applied by listing the deformations of the patterns. Synthesize (sample) from the models, not just analyze signals with them. The Brown University Pattern Theory Group was formed in 1972 by Ulf Grenander. Many mathematicians are currently working in this group, noteworthy among them being the Fields Medalist David Mumford. Mumford regards Grenander as his "guru" in Pattern Theory.
Certifying algorithm
In theoretical computer science, a certifying algorithm is an algorithm that outputs, together with a solution to the problem it solves, a proof that the solution is correct. A certifying algorithm is said to be efficient if the combined runtime of the algorithm and a proof checker is slower by at most a constant factor than the best known non-certifying algorithm for the same problem. The proof produced by a certifying algorithm should be in some sense simpler than the algorithm itself, for otherwise any algorithm could be considered certifying (with its output verified by running the same algorithm again). Sometimes this is formalized by requiring that a verification of the proof take less time than the original algorithm, while for other problems (in particular those for which the solution can be found in linear time) simplicity of the output proof is considered in a less formal sense. For instance, the validity of the output proof may be more apparent to human users than the correctness of the algorithm, or a checker for the proof may be more amenable to formal verification. Implementations of certifying algorithms that also include a checker for the proof generated by the algorithm may be considered to be more reliable than non-certifying algorithms. For, whenever the algorithm is run, one of three things happens: it produces a correct output (the desired case), it detects a bug in the algorithm or its implication (undesired, but generally preferable to continuing without detecting the bug), or both the algorithm and the checker are faulty in a way that masks the bug and prevents it from being detected (undesired, but unlikely as it depends on the existence of two independent bugs). == Examples == Many examples of problems with checkable algorithms come from graph theory. For instance, a classical algorithm for testing whether a graph is bipartite would simply output a Boolean value: true if the graph is bipartite, false otherwise. In contrast, a certifying algorithm might output a 2-coloring of the graph in the case that it is bipartite, or a cycle of odd length if it is not. Any graph is bipartite if and only if it can be 2-colored, and non-bipartite if and only if it contains an odd cycle. Both checking whether a 2-coloring is valid and checking whether a given odd-length sequence of vertices is a cycle may be performed more simply than testing bipartiteness. Analogously, it is possible to test whether a given directed graph is acyclic by a certifying algorithm that outputs either a topological order or a directed cycle. It is possible to test whether an undirected graph is a chordal graph by a certifying algorithm that outputs either an elimination ordering (an ordering of all vertices such that, for every vertex, the neighbors that are later in the ordering form a clique) or a chordless cycle. And it is possible to test whether a graph is planar by a certifying algorithm that outputs either a planar embedding or a Kuratowski subgraph. The extended Euclidean algorithm for the greatest common divisor of two integers x and y is certifying: it outputs three integers g (the divisor), a, and b, such that ax + by = g. This equation can only be true of multiples of the greatest common divisor, so testing that g is the greatest common divisor may be performed by checking that g divides both x and y and that this equation is correct.