AI Art Kingdom

AI Art Kingdom — independent reviews, comparisons, pricing and step-by-step guides on Aizhi.

Adobe After Effects

Adobe After Effects is a digital effects, motion graphics, and compositing application developed by Adobe Inc.; it is used for animation and in the post-production process of film making, video games and television production. Among other things, After Effects can be used for keying, tracking, compositing, and animation. It also functions as a very basic non-linear editor, audio editor, and media transcoder. In 2019, the program won an Academy Award for scientific and technical achievement. == History == After Effects was originally created by David Herbstman, David Simons, Daniel Wilk, David M. Cotter, and Russell Belfer at the Company of Science and Art in Providence, Rhode Island. The first two versions of the software, 1.0 (January 1993) and 1.1, were released there by the company. CoSA with After Effects was acquired by Aldus Corporation in July 1993, which in turn was acquired by Adobe in 1994. Adobe acquired PageMaker as well. Adobe's first new release of After Effects was version 3.0. == Third-party integrations == After Effects functionality can be extended through a variety of third-party integrations. The most common integrations are: plug-ins, scripts, and extensions. === Plug-ins === Plug-ins are predominantly written in C or C++ and extend the functionality of After Effects, allowing for more advanced features such as particle systems, physics engines, 3D effects, and the ability to bridge the gap between After Effects and another. === Scripts === After Effects Scripts are a series of commands written in both JavaScript and the ExtendScript language. After Effects Scripts, unlike plug-ins, can only access the core functionality of After Effects. Scripts are often developed to automate repetitive tasks, to simplify complex After Effects features, or to perform complex calculations that would otherwise take a long time to complete. Scripts can also use some functionality not directly exposed through the graphical user interface. === Extensions === After Effects Extensions offer the ability to extend After Effects functionality through modern web development technologies like HTML5, and Node.js, without the need for C++. After Effects Extensions make use of Adobe's Common Extensibility Platform or CEP Panels, which means they can be built to interact with other Adobe CC apps.
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Artificial intelligence industry in Taiwan

The artificial intelligence (AI) industry in Taiwan refers to the development, application, and commercialization of artificial intelligence technologies within Taiwan. The industry has grown alongside Taiwan's established strengths in semiconductor manufacturing and information and communications technology (ICT), and is supported by government policy, research institutions, and private sector participation. AI development in Taiwan has focused on integrating hardware capabilities with software applications across sectors such as manufacturing, healthcare, and smart infrastructure. Artificial intelligence has been identified as a strategic area of development in Taiwan since the late 2010s. While Taiwan has historically played a limited role in early theoretical and expert-system phases of AI development, its position in global electronics manufacturing has provided a foundation for participation in the contemporary era of machine learning and data-driven AI systems. Taiwan's AI industry is characterized by a strong hardware base, particularly in semiconductor production and AI server manufacturing, combined with increasing investment in software, data infrastructure, and applied AI services. The sector has been shaped by global demand for computing power, advances in deep learning, and the expansion of AI applications in industrial and commercial contexts. == Government policy and development == The Taiwanese government has promoted AI development through a series of national strategies. In 2017, the Ministry of Science and Technology launched the "AI Grand Strategy for a Small Country" initiative, investing approximately US$517 million between 2017 and 2021 to support research, infrastructure, and talent development. This initiative aimed to build a domestic AI ecosystem by funding research centers, expanding data infrastructure, and supporting industrial adoption. The Executive Yuan also introduced the AI Taiwan Action Plan 1.0 (2018–2021), which focused on integrating AI technologies into existing industries and strengthening research and development capabilities. A subsequent plan, AI Taiwan Action Plan 2.0 (2023–2026), expanded the focus to include ethical governance, regulatory frameworks, and risk management in response to the growth of generative AI technologies. In 2023, the Taiwan AI Center of Excellence (Taiwan AICoE), a government-backed hub, was established by the National Science and Technology Council to accelerate AI development, foster international collaboration, and train talent in Taiwan. It acts as a specialized think tank focusing on creating a "smart technology island" by integrating AI resources and developing trusted, human-centric AI technologies. In 2024, the Taiwan Chip-based Industrial Innovation Program (CbI) was launched by the Executive Yuan as a 10-year, NT$300 billion (US$9.3 billion) initiative to leverage Taiwan's semiconductor dominance, driving innovation in AI, smart mobility, manufacturing, and healthcare. It aims to combine generative AI with IC technology, cultivate talent, and attract global startups to build a "Silicon Island". In parallel, the Taiwanese government has explored legislative frameworks such as a proposed Artificial Intelligence Fundamental Act in December 2025, addressing issues including data protection, safety standards, and intellectual property. == Industrial structure == === Semiconductor and hardware foundation === Taiwan's AI industry is closely linked to its semiconductor sector. In 2020, Taiwan accounted for approximately 77.3% of the global wafer foundry market and 57.7% of packaging and testing, with a 20.1% share in integrated circuit (IC) design. These capabilities provide critical infrastructure for AI systems, which rely on high-performance computing hardware. Taiwanese firms are also involved in the production of AI servers and related components, contributing significantly to global supply chains for data centers and cloud computing. The integration of chip design, manufacturing, and assembly has enabled Taiwan to play a central role in providing the computational resources required for AI development. On 20 November 2025, Google established the "Google Taiwan AI Infrastructure R&D Center", second only to its US headquarters and largest AI hardware infrastructure engineering center outside of the United States. === Software and services === Compared to its hardware capabilities, Taiwan's AI software sector is less developed. The absence of large-scale global AI platform companies has been noted as a structural limitation. As a result, much of Taiwan's AI industry focuses on applied solutions, including customization of existing AI models for specific industries. Therefore, efforts to strengthen software capabilities have included investment in research institutions, startup ecosystems, and collaborations between academia and industry. == Applications == === Smart manufacturing === AI has been widely applied in Taiwan's manufacturing sector, which is a major component of the economy. Applications include process automation, predictive maintenance, quality control, and fault detection. AI-enabled smart manufacturing systems aim to improve efficiency, reduce production costs, and enhance product quality. Taiwan's manufacturing industry has incorporated AI technologies into production lines, particularly in electronics and machinery sectors. === Healthcare === The use of AI in healthcare in Taiwan has expanded in areas such as medical imaging, diagnostics, and drug development. AI systems are used to analyze CT scans, MRI data, and other clinical information to support diagnosis and treatment planning. Taiwan's healthcare sector, which includes medical devices, pharmaceuticals, and medical services, has benefited from the integration of AI technologies, particularly in precision medicine and clinical decision support systems. A notable example of AI healthcare deployment in Taiwan is the collaboration between Siemens Healthineers, Ever Fortune AI, and Asia University Hospital. === Edge computing and IoT === AI applications in Taiwan increasingly involve edge computing, where data processing occurs near the source rather than in centralized cloud systems. This approach reduces latency and bandwidth requirements and is used in smart devices, sensors, and industrial equipment. Edge AI technologies are applied in areas such as smart appliances, industrial automation, and transportation systems. == Education and talent development == Human capital development has been a key focus of Taiwan's AI strategy. The Taiwan AI Academy, established in 2018 with support from Academia Sinica and industry partners, provides training programs for professionals and students aimed at accelerating the adoption of artificial intelligence technologies across industries. The academy offers a range of courses, including executive-level programs, technical training, and specialized tracks in areas such as smart manufacturing, smart healthcare, and edge AI. These programs are designed to provide intensive and practical instruction over relatively short periods. A notable component of the curriculum is project-based learning, in which participants are required to complete proof-of-concept (POC) projects addressing real-world industrial problems. These projects are often developed further for implementation within companies, facilitating technology transfer and commercialization. Between 2018 and 2021, more than 8,000 individuals completed AI training programs across campuses in Taipei, Hsinchu, Taichung, and Tainan. Graduates of the academy have contributed to the introduction of AI systems in sectors such as manufacturing, healthcare, and finance, supporting broader industrial transformation efforts. In addition to the Taiwan AI Academy, universities and research institutions in Taiwan play a significant role in AI education and research. Leading universities have expanded programs in computer science, data science, and machine learning, while research institutes conduct applied and fundamental studies in artificial intelligence. Collaboration between academia, government, and industry is a common feature of Taiwan's AI ecosystem, with joint research projects, internship programs, and technology incubation initiatives supporting talent development. Government-supported initiatives have also sought to attract and retain AI talent, including funding for graduate education, international collaboration programs, and incentives for industry–academic partnerships. These efforts aim to address talent shortages and strengthen Taiwan's capacity in both applied and foundational AI research. == Regulation and governance == Taiwan has developed guidelines and policy frameworks to address the risks associated with AI technologies. In 2023, the Executive Yuan issued guidelines for the use of generative AI in government agencies, focusing on data security and privacy. Ongoing policy discussions hav
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Information flow

In discourse-based grammatical theory, information flow is any tracking of referential information by speakers. Information may be new, i.e., just introduced into the conversation; given, i.e., already active in the speakers' consciousness; or old, i.e., no longer active. The various types of activation, and how these are defined, are model-dependent. Information flow affects grammatical structures such as: Word order (topic, focus, and afterthought constructions). Active, passive, or middle voice. Choice of deixis, such as articles; "medial" deictics such as Spanish ese and Japanese sore are generally determined by the familiarity of a referent rather than by physical distance. Overtness of information, such as whether an argument of a verb is indicated by a lexical noun phrase, a pronoun, or not mentioned at all. Clefting: Splitting a single clause into two clauses, each with its own verb, e.g. ‘The chicken turtles tasted like chicken.’ becomes ‘It was the chicken turtle | that tasted like chicken.’ In this case, clefting is used to shift the focus of the sentence to the subject, the chicken turtle. Front focus: Placing at the start (front) of a sentence information that would normally occur later in the sentence, to give it extra prominence. For example, in pop culture, Yoda's speech often utilizes such syntactic construction, such as when he says 'much to learn you still have' to Luke Skywalker. End focus (or end weight): Given or familiar information followed by new information. This gives prominence to the final part of the sentences and can enable suspense to build, e.g. ‘Through the door came a gigantic wolf’.(Umer Prince)
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Recording format

A recording format is a format for encoding data for storage on a storage medium. The format can be container information such as sectors on a disk, or user/audience information (content) such as analog stereo audio. Multiple levels of encoding may be achieved in one format. For example, a text encoded page may contain HTML and XML encoding, combined in a plain text file format, using either EBCDIC or ASCII character encoding, on a UDF digitally formatted disk. In electronic media, the primary format is the encoding that requires hardware to interpret (decode) data; while secondary encoding is interpreted by secondary signal processing methods, usually computer software. == Recording container formats == A container format is a system for dividing physical storage space or virtual space for data. Data space can be divided evenly by a system of measurement, or divided unevenly with meta data. A grid may divide physical or virtual space with physical or virtual (dividers) borders, evenly or unevenly. Just as a physical container (such as a file cabinet) is divided by physical borders (such as drawers and file folders), data space is divided by virtual borders. Meta data such as a unit of measurement, address, or meta tags act as virtual borders in a container format. A template may be considered an abstract format for containing a solution as well as the content itself. Systems of measurement Metric system Geographic coordinate system Page grid Film formats Audio data format Video tape format Disk format File format Meta data Text formatting Template Data structure == Raw content formats == A raw content format is a system of converting data to displayable information. Raw content formats may either be recorded in secondary signal processing methods such as a software container format (e.g. digital audio, digital video) or recorded in the primary format. A primary raw content format may be directly observable (e.g. image, sound, motion, smell, sensation) or physical data which only requires hardware to display it, such as a phonographic needle and diaphragm or a projector lamp and magnifying glass.
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Level set (data structures)

In computer science, a level set is a data structure designed to represent discretely sampled dynamic level sets of functions. A common use of this form of data structure is in efficient image rendering. The underlying method constructs a signed distance field that extends from the boundary, and can be used to solve the motion of the boundary in this field. == Chronological developments == The powerful level-set method is due to Osher and Sethian 1988. However, the straightforward implementation via a dense d-dimensional array of values, results in both time and storage complexity of O ( n d ) {\displaystyle O(n^{d})} , where n {\displaystyle n} is the cross sectional resolution of the spatial extents of the domain and d {\displaystyle d} is the number of spatial dimensions of the domain. === Narrow band === The narrow band level set method, introduced in 1995 by Adalsteinsson and Sethian, restricted most computations to a thin band of active voxels immediately surrounding the interface, thus reducing the time complexity in three dimensions to O ( n 2 ) {\displaystyle O(n^{2})} for most operations. Periodic updates of the narrowband structure, to rebuild the list of active voxels, were required which entailed an O ( n 3 ) {\displaystyle O(n^{3})} operation in which voxels over the entire volume were accessed. The storage complexity for this narrowband scheme was still O ( n 3 ) . {\displaystyle O(n^{3}).} Differential constructions over the narrow band domain edge require careful interpolation and domain alteration schemes to stabilise the solution. === Sparse field === This O ( n 3 ) {\displaystyle O(n^{3})} time complexity was eliminated in the approximate "sparse field" level set method introduced by Whitaker in 1998. The sparse field level set method employs a set of linked lists to track the active voxels around the interface. This allows incremental extension of the active region as needed without incurring any significant overhead. While consistently O ( n 2 ) {\displaystyle O(n^{2})} efficient in time, O ( n 3 ) {\displaystyle O(n^{3})} storage space is still required by the sparse field level set method. See for implementation details. === Sparse block grid === The sparse block grid method, introduced by Bridson in 2003, divides the entire bounding volume of size n 3 {\displaystyle n^{3}} into small cubic blocks of m 3 {\displaystyle m^{3}} voxels each. A coarse grid of size ( n / m ) 3 {\displaystyle (n/m)^{3}} then stores pointers only to those blocks that intersect the narrow band of the level set. Block allocation and deallocation occur as the surface propagates to accommodate to the deformations. This method has a suboptimal storage complexity of O ( ( n m ) 3 + m 3 n 2 ) {\displaystyle O\left((nm)3+m^{3}n^{2}\right)} , but retains the constant time access inherent to dense grids. === Octree === The octree level set method, introduced by Strain in 1999 and refined by Losasso, Gibou and Fedkiw, and more recently by Min and Gibou uses a tree of nested cubes of which the leaf nodes contain signed distance values. Octree level sets currently require uniform refinement along the interface (i.e. the narrow band) in order to obtain sufficient precision. This representation is efficient in terms of storage, O ( n 2 ) , {\displaystyle O(n^{2}),} and relatively efficient in terms of access queries, O ( log n ) . {\displaystyle O(\log \,n).} An advantage of the level method on octree data structures is that one can solve the partial differential equations associated with typical free boundary problems that use the level set method. The CASL research group has developed this line of work in computational materials, computational fluid dynamics, electrokinetics, image-guided surgery and controls. === Run-length encoded === The run-length encoding (RLE) level set method, introduced in 2004, applies the RLE scheme to compress regions away from the narrow band to just their sign representation while storing with full precision the narrow band. The sequential traversal of the narrow band is optimal and storage efficiency is further improved over the octree level set. The addition of an acceleration lookup table allows for fast O ( log ⁡ r ) {\displaystyle O(\log r)} random access, where r is the number of runs per cross section. Additional efficiency is gained by applying the RLE scheme in a dimensional recursive fashion, a technique introduced by Nielsen & Museth's similar DT-Grid. === Hash Table Local Level Set === The Hash Table Local Level Set method was introduced in 2011 by Eyiyurekli and Breen and extended in 2012 by Brun, Guittet, and Gibou, only computes the level set data in a band around the interface, as in the Narrow Band Level-Set Method, but also only stores the data in that same band. A hash table data structure is used, which provides an O ( 1 ) {\displaystyle O(1)} access to the data. However, Brun et al. conclude that their method, while being easier to implement, performs worse than a quadtree implementation. They find that as it is, [...] a quadtree data structure seems more adapted than the hash table data structure for level-set algorithms. Three main reasons for worse efficiency are listed: to obtain accurate results, a rather large band is required close to the interface, which counterbalances the absence of grid nodes far from the interface; the performances are deteriorated by extrapolation procedures on the outer edges of the local grid and the width of the band restricts the time step and slows down the method. === Point-based === Corbett in 2005 introduced the point-based level set method. Instead of using a uniform sampling of the level set, the continuous level set function is reconstructed from a set of unorganized point samples via moving least squares.
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MaPS S.A.

MaPS S.A. is a software editor founded in 2011 by Thierry Muller. The company is headquartered in Luxembourg. Its platform, called MaPS System, provides Data Management software for Multichannel Marketing. == History and funding == The first version of MaPS System was released under the agency Prem1um S.A. in 2005 in the partnership with Pingroom agency. In combination with MaPS System, Prem1um also provided various consulting services in Marketing, Publishing and Sales. This is where MaPS System takes its names (M stands for Marketing, P for Publishing and S for Sales). In 2011, after being successful, Prem1um S.A. decided to enable the software MaPS System to operate independently under MaPS S.A., as a separate company and editor of the software. The first financial supports were provided by Malta ICI, a Venture Capital firm, and the local partner Chameleon Invest, a seed-capital fund led by Business Angels, who invested €900,000. In a second investment round in 2014 led by Newion Investments, a Venture Capital firm, €1.4 Million were raised, thus amounting to total assets of €2.2 Million. In 2016, the company was taken over by three private investors. In 2018, after two years of continuous growth and European expansion in Belgium, Germany and Switzerland, MaPS S.A acquired Awevo, an e-commerce web agency. == Products == The services included in MaPS System range from the data centralization, Data Governance to an optimized Multichannel Marketing. The software currently includes more than 35 modules for Master Data Management, Product Information Management, Digital Asset Management, Business Process Management including catalogue Publishing features. == Certifications == In 2019, MaPS System and Awevo received "Made in Luxembourg" label, given to the companies whose services are entirely designed in Luxembourg, without any production or development offshoring. MaPS System is a member of ICT Cluster by Luxinnovation.
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Brian Deer Classification System

The Brian Deer Classification System (BDC) is a library classification system used to organize materials in libraries with specialized Indigenous collections. The system was created in the mid-1970s by Canadian librarian A. Brian Deer, a Kahnawake Mohawk. It has been adapted for use in a British Columbia version, and also by a small number of First Nations libraries in Canada. == History and usage == Deer designed his classification system while working in the library of the National Indian Brotherhood (now the Assembly of First Nations) from 1974 to 1976. Instead of using a standard library classification scheme, such as that of the Library of Congress, he created a new system to organize the library's historic indigenous research materials and papers. He later worked at the library of the Union of British Columbia Indian Chiefs, where he developed a system for its holdings. He returned to Kahnawake, working at its Cultural Centre at Kahnawake and the Kahnawake Branch branch of the Mohawk Nation Office. His system was flexible, and he created new forms for their collections. The new systems Deer created were designed specifically for the materials in each collection according to the concerns of local Indigenous people at the time (for example, categories included land claims, treaty rights, resource management, and Elders' stories). Between 1978 and 1980, the system was adapted for use in British Columbia by Gene Joseph and Keltie McCall while they were working at the Union of British Columbia Indian Chiefs, becoming known as BDC-BC. Joseph later adapted it further for use in the Xwi7xwa Library at University of British Columbia, Vancouver. Though the Brian Deer Classification was not created as a universal classification solution for Indigenous resources, the system has provided a foundation for specialized libraries to create their own localized classification schemes. Variations of the Brian Deer Classification System are used in a small number of Canadian libraries. One prominent library using BDC is the X̱wi7x̱wa Library at the University of British Columbia, which uses a British Columbia-focused version of BDC along with First Nations House of Learning subject headings. The Union of British Columbia Indian Chiefs Resource Centre issued a revised BDC-BC in 2014, with the goal of providing users with a more flexible and culturally appropriate approach to organizing their resources. The Aanischaaukamikw Cree Cultural Institute in Oujé-Bougoumou, Quebec, implemented a local adaptation of BDC when they opened in 2012. In 2020 the Carrier Sekani Tribal Council in Prince George, British Columbia, shifted from organizing its library with the Dewey Decimal Classification to using a version of the BDC. They added new subject heading categories for topics of local interest such as the crisis of Missing and murdered Indigenous women. Simon Fraser University Library began developing the Indigenous Curriculum Resource Centre (ICRC) in 2020, with the physical space opening in 2023. The ICRC is Call to Action 21 of SFU's Aboriginal Reconciliation Council's final report, Walk This Path With Us. Through its collection, the ICRC supports those interested in learning about how and why decolonizing pedagogy and teaching practices are important. The physical items in the collection are catalogued using a modified Brian Deer Classification system. In 2022 Kwantlen Polytechnic University’s χʷəχʷéy̓əm Indigenous Collection released a revised BDC-BC System. This BDC contains works exclusively with Indigenous authored materials and expands the cuttering systems of previous BDC, with the result that much of the collection reflects a spatial relationality. The implementation of this BDC was possible due to the tireless work at Xwi7xwa Library, Union of British Columbia Indian Chiefs Resource Centre, and Simon Fraser University Library's Indigenous Curriculum Resource Centre. == Structure == The high-level organizational structure of BDC reflects a First Nations worldview, with an emphasis on relationships between and among people, animals, and the land. Subcategories demonstrate the relationships among First Nations by grouping them geographically as opposed to alphabetically; the latter is a practice frequently used for specific topics in the Library of Congress Classification. The top-level hierarchy of the X̱wi7x̱wa Library adaptation of BDC-BC demonstrates the emphasis on access to subjects prioritized by a First Nation collection: Reference Materials Local History History International Education Economic Development Housing and Community Development Criminal Justice System Constitution (Canada) and First Nations Self Government Rights and Title Natural Resources Community Resources Health World View Fine Arts Languages Literature The system is not designed to provide a comprehensive description of all topics of interest to North American Indigenous peoples; in addition, its use is limited in scope, being intended for small and specialized libraries. While English is used in the classification scheme as a common language among First Nations peoples and non-Indigenous library users, Indigenous spellings and terminology that local library users would expect to find are used to provide access. Short and easily remembered call numbers are used to facilitate use by both library workers and patrons, with the recognition that Indigenous libraries often have a small staff and limited resources to devote to cataloging. Beyond its simplicity, one potential drawback of the system is its shortage of clear guidelines for application, which provides flexibility but can also result in inconsistencies within and between library catalogs. Because few libraries use the BDC and there are limited examples for use as case studies, implementing the system and keeping it up-to-date can prove a challenge for libraries with limited resources. However, X̱wi7x̱wa Library head librarian Ann Doyle describes the system as "an important part of the body of Indigenous scholarship" that should be retained as a reflection of Indigenous worldviews, as well as for ease of access for Indigenous library users.
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Magic Quadrant

Magic Quadrant (MQ) is a series of market research reports published by research and advisory firm Gartner that rely on proprietary qualitative data analysis methods to demonstrate market trends, such as direction, maturity, and participants. Their analyses are conducted for several specific technology industries and are updated every 1–2 years: once an updated report has been published, its predecessor is "retired". == Rating == Gartner rates vendors upon two criteria: completeness of vision and ability to execute. Completeness of vision – Reflects the vendor's innovation, and whether the vendor drives or follows the market. Ability to execute – Summarizes factors such as the vendor's financial viability, market responsiveness, product development, sales channels and customer base. The two component scores lead to a vendor position in one of four quadrants: === Leaders === Vendors in the "Leaders" quadrant have the highest composite scores for their completeness of vision and ability to execute. A vendor in the Leaders quadrant has the market share, credibility, and marketing & sales capabilities needed to drive the acceptance of new technologies. These vendors demonstrate a clear understanding of market needs, they are innovators and thought leaders, and they have well-articulated plans that customers and prospects can use when designing their infrastructures and strategies. In addition, they have a presence in the five major geographical regions, consistent financial performance, and broad platform support. === Challengers === Vendors in the "Challengers" quadrant have high scores mainly for their ability to execute. They both participate in the market and execute well enough to be a serious threat to vendors in the "Leaders" quadrant. They have strong products, as well as sufficiently credible market position and resources to sustain continued growth. Financial viability is not an issue for vendors in the "Challengers" quadrant, but they lack the size and influence of vendors in the "Leaders" quadrant due to their relative lack of vision. === Visionaries === Vendors in the "Visionaries" quadrant have high scores mainly for their completeness of vision. They deliver innovative products that address operationally or financially important end-user problems at a broad scale, but have not yet demonstrated the ability to capture market share or maintain sustainable levels of profitability. Visionary vendors are frequently privately held companies and acquisition targets for larger, established companies. The likelihood of acquisition often reduces the risks associated with installing their systems. === Niche Players === Vendors in the "Niche Players" quadrant have relatively low scores for both their ability to execute and their completeness of vision. They are often narrowly focused on specific market or vertical segments. This quadrant often also includes vendors that are adapting their existing products to enter the market under consideration, or larger vendors having difficulty developing and executing on their vision. == Gartner Critical Capabilities == Gartner Critical Capabilities complement Magic Quadrant analysis to offer deeper insight into the products and services offered by multiple vendors by a comparative analysis that scores competing products or services against a set of critical differentiators identified by Gartner. Gartner has periodically ended Magic Quadrant listings for IT Service Management, Web Content Management, and other industries as those markets have fully matured or other factors rendered the analytic framework inapplicable. == Criticism == The Magic Quadrant, and analysts in general, skew the market: according to research, by applying their methodologies to describe a market, they change that marketplace to fit their tools. Another criticism is that open source vendors are not considered sufficiently by analysts like Gartner, as has been published in an online discussion between a VP from Talend and a German Research VP from Gartner. On May 29, 2009 (2009-05-29), software vendor ZL Technologies filed a federal lawsuit against Gartner that challenged the "legitimacy" of Gartner's Magic Quadrant rating system. Gartner filed a motion to dismiss by claiming First Amendment protection since it contends that its MQ reports contain "pure opinion", which legally means opinions that are not based on fact. The court threw out the ZL case because it lacked a specific complaint. The decision was upheld on appeal.
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Screen space directional occlusion

Screen space directional occlusion (SSDO) is a computer graphics technique enhancing screen space ambient occlusion (SSAO) by taking direction into account to sample the ambient light (both the light coming directly at an object, as well as the light reflected off of the object directly behind it), to better approximate global illumination. SSDO was introduced by Tobias Ritschel, Thorsten Grosch, and Hans-Peter Seidel in their 2009 ACM Symposium on Interactive 3D Graphics and Games paper Approximating dynamic global illumination in image space, which describes it as extending SSAO to directional occlusion with one diffuse indirect bounce of light; later literature notes that SSDO still suffers from common screen-space artifacts such as noise and banding. == Method == The original SSDO paper describes a two-pass screen-space approach, with one pass for direct lighting and a second pass for indirect bounces. Later literature describes SSDO as assuming a general shadowing direction that allows color bleeding and a single light bounce.
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Algorithm engineering

Algorithm engineering focuses on the design, analysis, implementation, optimization, profiling and experimental evaluation of computer algorithms, bridging the gap between algorithmics theory and practical applications of algorithms in software engineering. It is a general methodology for algorithmic research. == Origins == In 1995, a report from an NSF-sponsored workshop "with the purpose of assessing the current goals and directions of the Theory of Computing (TOC) community" identified the slow speed of adoption of theoretical insights by practitioners as an important issue and suggested measures to reduce the uncertainty by practitioners whether a certain theoretical breakthrough will translate into practical gains in their field of work, and tackle the lack of ready-to-use algorithm libraries, which provide stable, bug-free and well-tested implementations for algorithmic problems and expose an easy-to-use interface for library consumers. But also, promising algorithmic approaches have been neglected due to difficulties in mathematical analysis. The term "algorithm engineering" was first used with specificity in 1997, with the first Workshop on Algorithm Engineering (WAE97), organized by Giuseppe F. Italiano. == Difference from algorithm theory == Algorithm engineering does not intend to replace or compete with algorithm theory, but tries to enrich, refine and reinforce its formal approaches with experimental algorithmics (also called empirical algorithmics). This way it can provide new insights into the efficiency and performance of algorithms in cases where the algorithm at hand is less amenable to algorithm theoretic analysis, formal analysis pessimistically suggests bounds which are unlikely to appear on inputs of practical interest, the algorithm relies on the intricacies of modern hardware architectures like data locality, branch prediction, instruction stalls, instruction latencies which the machine model used in Algorithm Theory is unable to capture in the required detail, the crossover between competing algorithms with different constant costs and asymptotic behaviors needs to be determined. == Methodology == Some researchers describe algorithm engineering's methodology as a cycle consisting of algorithm design, analysis, implementation and experimental evaluation, joined by further aspects like machine models or realistic inputs. They argue that equating algorithm engineering with experimental algorithmics is too limited, because viewing design and analysis, implementation and experimentation as separate activities ignores the crucial feedback loop between those elements of algorithm engineering. === Realistic models and real inputs === While specific applications are outside the methodology of algorithm engineering, they play an important role in shaping realistic models of the problem and the underlying machine, and supply real inputs and other design parameters for experiments. === Design === Compared to algorithm theory, which usually focuses on the asymptotic behavior of algorithms, algorithm engineers need to keep further requirements in mind: Simplicity of the algorithm, implementability in programming languages on real hardware, and allowing code reuse. Additionally, constant factors of algorithms have such a considerable impact on real-world inputs that sometimes an algorithm with worse asymptotic behavior performs better in practice due to lower constant factors. === Analysis === Some problems can be solved with heuristics and randomized algorithms in a simpler and more efficient fashion than with deterministic algorithms. Unfortunately, this makes even simple randomized algorithms difficult to analyze because there are subtle dependencies to be taken into account. === Implementation === Huge semantic gaps between theoretical insights, formulated algorithms, programming languages and hardware pose a challenge to efficient implementations of even simple algorithms, because small implementation details can have rippling effects on execution behavior. The only reliable way to compare several implementations of an algorithm is to spend an considerable amount of time on tuning and profiling, running those algorithms on multiple architectures, and looking at the generated machine code. === Experiments === See: Experimental algorithmics === Application engineering === Implementations of algorithms used for experiments differ in significant ways from code usable in applications. While the former prioritizes fast prototyping, performance and instrumentation for measurements during experiments, the latter requires thorough testing, maintainability, simplicity, and tuning for particular classes of inputs. === Algorithm libraries === Stable, well-tested algorithm libraries like LEDA play an important role in technology transfer by speeding up the adoption of new algorithms in applications. Such libraries reduce the required investment and risk for practitioners, because it removes the burden of understanding and implementing the results of academic research. == Conferences == Two main conferences on Algorithm Engineering are organized annually, namely: Symposium on Experimental Algorithms (SEA), established in 1997 (formerly known as WEA). SIAM Meeting on Algorithm Engineering and Experiments (ALENEX), established in 1999. The 1997 Workshop on Algorithm Engineering (WAE'97) was held in Venice (Italy) on September 11–13, 1997. The Third International Workshop on Algorithm Engineering (WAE'99) was held in London, UK in July 1999. The first Workshop on Algorithm Engineering and Experimentation (ALENEX99) was held in Baltimore, Maryland on January 15–16, 1999. It was sponsored by DIMACS, the Center for Discrete Mathematics and Theoretical Computer Science (at Rutgers University), with additional support from SIGACT, the ACM Special Interest Group on Algorithms and Computation Theory, and SIAM, the Society for Industrial and Applied Mathematics.
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Algorism

Algorism is the technique of performing basic arithmetic by writing numbers in place value form and applying a set of memorized rules and facts to the digits. One who practices algorism is known as an algorist. This positional notation system has largely superseded earlier calculation systems that used a different set of symbols for each numerical magnitude, such as Roman numerals, and in some cases required a device such as an abacus. == Etymology == The word algorism comes from the name Al-Khwārizmī (c. 780–850), a Persian mathematician, astronomer, geographer and scholar in the House of Wisdom in Baghdad, whose name means "the native of Khwarezm", which is now in modern-day Uzbekistan. He wrote a treatise in Arabic language in the 9th century, which was translated into Latin in the 12th century under the title Algoritmi de numero Indorum. This title means "Algoritmi on the numbers of the Indians", where "Algoritmi" was the translator's Latinization of Al-Khwarizmi's name. Al-Khwarizmi was the most widely read mathematician in Europe in the late Middle Ages, primarily through his other book, the Algebra. In late medieval Latin, algorismus, the corruption of his name, simply meant the "decimal number system" that is still the meaning of modern English algorism. During the 17th century, the French form for the word – but not its meaning – was changed to algorithm, following the model of the word logarithm, this form alluding to the ancient Greek arithmos = number. English adopted the French very soon afterwards, but it wasn't until the late 19th century that "algorithm" took on the meaning that it has in modern English. In English, it was first used about 1230 and then by Chaucer in 1391. Another early use of the word is from 1240, in a manual titled Carmen de Algorismo composed by Alexandre de Villedieu. It begins thus: Haec algorismus ars praesens dicitur, in qua / Talibus Indorum fruimur bis quinque figuris. which translates as: This present art, in which we use those twice five Indian figures, is called algorismus. The word algorithm also derives from algorism, a generalization of the meaning to any set of rules specifying a computational procedure. Occasionally algorism is also used in this generalized meaning, especially in older texts. == History == Starting with the integer arithmetic developed in India using base 10 notation, Al-Khwārizmī along with other mathematicians in medieval Islam, documented new arithmetic methods and made many other contributions to decimal arithmetic (see the articles linked below). These included the concept of the decimal fractions as an extension of the notation, which in turn led to the notion of the decimal point. This system was popularized in Europe by Leonardo of Pisa, now known as Fibonacci.
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Informedia Digital Library

The Informedia Digital Library is an ongoing research program at Carnegie Mellon University to build search engines and information visualization technology for many types of media. The program has carried out research on spoken document retrieval, video information retrieval, video segmentation, face recognition, and cross-language information retrieval. The Lycos search engine was an early product of the Informedia Digital Library Project. The project is led by Howard Wactlar. Researchers on the project have included: Michael Mauldin, Alex Hauptmann, Michael Christel, Michael Witbrock, Raj Reddy, Takeo Kanade and Scott Stevens.
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Artisse AI

Artisse AI is a Hong Kong-based technology company founded by William Wu. The company developed a mobile photography application using generative artificial intelligence to transform selfies into high-quality, personalized images. The app allows users to visualize themselves in various scenarios, outfits, and hairstyles, and they can adjust lighting and ambiance to match their preferences. The app launched in 2023 across multiple markets, including the United States, United Kingdom, Japan, South Korea, Canada, and Australia. By January 2024, users had generated over 5 million images. That same month, the company secured $6.7 million in seed funding to support product development and marketing. == History == Artisse was originally founded in South Korea in 2022 by William Wu. The early concept was connected to a virtual idol initiative developed in collaboration with a K-pop agency, intended to support Wu's blockchain gaming business. The project later evolved into a standalone AI photography application. The current version of the Artisse app was developed following the company's relocation to Hong Kong in 2022. In January 2024, Artisse secured $6.7 million in seed funding, led by The London Fund. The investment was aimed at supporting product development, marketing, and user acquisition. Artisse uses an AI algorithm to create hyperrealistic images from uploaded photos. The app generates personalized images by combining generative AI technology, a global pool of licensed talent, and finished art services. The app works with individual users and businesses, offering professional-grade photos and advertisement images. According to the British newspaper Evening Standard the company has developed the world's first and most advanced AI photographer. It captures 15-30 photos of the user and generates 2D images, placing them in various outfits and locations worldwide. === Catheron Gaming === Artisse AI originated from Catheon Gaming, a blockchain gaming and entertainment company founded in 2021 by William Wu. Catheon Gaming published more than 30 Web3 titles in its first year, developed a blockchain game distribution platform, and offered advisory services to external developers. In 2022, HSBC and KPMG listed Catheon Gaming among the "Top 10 Emerging Giants" in the Asia–Pacific region, selected from a pool of more than 6,000 startups. In June 2023, Catheon Gaming was rebranded as Artisse Interactive, creating two divisions: Artisse Gaming, which continued blockchain and Web3 game development, and Artisse AI, which focused on generative photography technology. == Technology == Artisse uses a proprietary generative AI model combined with open-source imaging frameworks and diffusion models. Users are prompted to upload between 15 and 30 personal images, allowing the AI to train a personalized model in 30 to 40 minutes. After training, the app generates new images based on either textual or visual prompts, with options to adjust elements such as clothing, hairstyles, lighting, and backgrounds. To enhance realism, the app integrates augmented reality features and image refinement tools. The company has introduced features to address representation issues related to body shape and skin tone, although concerns persist about the ethical implications of altering personal traits. == Products == === Artisse mobile app === Available on iOS and Android platforms in 35 languages. Users initially receive 25 free images, after which the app adopts a subscription pricing model ranging from approximately $6 to $30 per month. By early 2024, the app reported around 4,000 paying subscribers out of more than 200,000 downloads. === Business and enterprise services === Artisse provides B2B solutions for creating marketing imagery and partners with agencies like Iconic Management to enable cost-effective virtual photoshoots. Additional features in development include virtual try-on capabilities and augmented reality integration for fashion retail. == Reception == Media coverage has noted the app's photorealistic image outputs with some sources highlighting its ease of use. However, concerns have been raised regarding image authenticity, algorithmic biases, and the potential impact on professional photography and modeling. Artisse has been widely covered by media outlets including TechCrunch, PetaPixel, Forbes Australia, and The Evening Standard. These publications discussed the app's integration of generative AI technology within the consumer photography space, its growing market influence, and its rapid adoption by users worldwide.
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Maritime Informatics

Maritime Informatics is a thematic topic within the broader discipline of informatics. It can be considered as both a field of study and domain of application. As an application domain, it is the outlet of innovations originating from data science and artificial intelligence; as a field of study, it is positioned between computer science and marine engineering. == Beginnings of maritime informatics == As a result of the increasing levels of digitalisation occurring in the maritime sector starting around 2010 and stimulated by the EU-endorsed MonaLisa project for sea traffic management (STM), a number of academics and shipping industry leaders recognised that the maritime transportation sector would benefit from a specific field of study and application to be known as Maritime Informatics - the use of information systems, data sharing and data analytics in the business and operations of maritime transportation. They considered that it would lead to improvements in efficiency, safety, resilience, and ecological sustainability - all of which are currently lacking for many aspects of sea transport. One of the first public airings of the concept of Maritime Informatics was a presentation delivered on 11 September 2014 in Gothenburg, Sweden. A proposal for an inaugural minitrack on Maritime Informatics was accepted for the 2015 Americas Conference on Information Systems in Puerto Rico where three papers were presented. Since then numerous publications has been brought forward captured at www.maritimeinformatics.org and in late 2020 the first reference book on Maritime Informatics was co-written by 81 expert contributors (47 practitioners and 34 researchers) from 20 countries. Most impactful authors and journals in the domain have been documented in a review paper. Dimitrios Zissis, Luca Cazzanti and Leonardo M. Millefiori are the top three authors; top journals and conferences include Ocean Engineering, Proceedings of the 12th ACM International Conference on Distributed and Event-based Systems, Sensors, the international Conference On Engineering, Technology And Innovation, Expert Systems With Applications, IEEE Access, and Journal of Navigation. == Background == The shipping industry has several particular organisational aspects that are recognised and taken into account in maritime informatics: It is predominantly a self-organising ecosystem Many activities are undertaken as part of episodic tight coupling There is a so-called maritime stack There is increasing pressure to balance capital productivity and energy efficiency There is the potential virtuous interplay between different types of systems == Data sharing == Digital data sharing is key to the all-important, arguably fundamental, data analytics aspects of maritime informatics because it opens the way for better access to relevant and reliable data. As in land-based commerce, digital data sharing is a growing phenomenon in maritime operations - though there is a way to go. It is enabling greater transparency for all those involved in the transportation of goods and passengers, not least being the end-customer. This leads to better and more informed decision-making and planning by all those involved. The push for digitalisation and data sharing is being pursued both by governments and the commercial sector. For example, the Member States of the IMO agreed a mandatory requirement for their governments to introduce electronic information exchange between ships and ports as from 8 April 2019. Meanwhile, commercial operators, particularly in the container lines are putting systems in place for sharing data for mutual benefit in their operations. Data sharing is an important aspect of the Port Collaborative Decision Making (PortCDM) and Port Call Optimization initiatives, both of which seek to improve the coordination, synchronization and efficiency of the port call process by enabling a common and shared situational awareness among all those involved. == Standardisation == The availability and sharing of relevant digital data underpins maritime informatics and is key to more effective and efficient coordination and synchronisation in the predominantly self-organising ecosystem that is maritime transportation. For this to occur, a high priority underpinning maritime informatics is the encouragement of standardised digital data exchange and data sharing, leading, in turn, to improvements in shipping analytics. Improved availability of data will support better historical analysis, now-casting and forecasting. The International Maritime Organization (IMO) FAL Committee is taking the lead in ensuring that the common terms used in the various standards being developed or in use in the maritime sector are compatible and therefore interoperable as far as is practicable, by creating and maintaining The IMO Compendium on Facilitation and Electronic Business. The IMO Compendium consists of an IMO Data Set and IMO Reference Data Model agreed by the main organisations involved in the development of standards for the electronic exchange of information related to the FAL Convention: the World Customs Organization (WCO), the United Nations Economic Commission for Europe (UNECE) and the International Organization for Standardization (ISO). There are several other prominent international governmental and non-governmental organisations actively contributing to the ongoing standardisation and harmonisation process including the UN Electronic Data Interchange for Administration, Commerce and Transport (UN EDIFACT), the Digital Container Shipping Association (DCSA), the International Harbour Masters Association (IHMA) and BIMCO - the world's largest direct-membership organisation for shipowners, charterers, shipbrokers and agents.
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Berlekamp–Rabin algorithm

In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field F p {\displaystyle \mathbb {F} _{p}} with p {\displaystyle p} elements. The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers. == History == The method was proposed by Elwyn Berlekamp in his 1970 work on polynomial factorization over finite fields. His original work lacked a formal correctness proof and was later refined and modified for arbitrary finite fields by Michael Rabin. In 1986 René Peralta proposed a similar algorithm for finding square roots in F p {\displaystyle \mathbb {F} _{p}} . In 2000 Peralta's method was generalized for cubic equations. == Statement of problem == Let p {\displaystyle p} be an odd prime number. Consider the polynomial f ( x ) = a 0 + a 1 x + ⋯ + a n x n {\textstyle f(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}} over the field F p ≃ Z / p Z {\displaystyle \mathbb {F} _{p}\simeq \mathbb {Z} /p\mathbb {Z} } of remainders modulo p {\displaystyle p} . The algorithm should find all λ {\displaystyle \lambda } in F p {\displaystyle \mathbb {F} _{p}} such that f ( λ ) = 0 {\textstyle f(\lambda )=0} in F p {\displaystyle \mathbb {F} _{p}} . == Algorithm == === Randomization === Let f ( x ) = ( x − λ 1 ) ( x − λ 2 ) ⋯ ( x − λ n ) {\textstyle f(x)=(x-\lambda _{1})(x-\lambda _{2})\cdots (x-\lambda _{n})} . Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivial divisors and factorize them recursively. To do this, consider the polynomial f z ( x ) = f ( x − z ) = ( x − λ 1 − z ) ( x − λ 2 − z ) ⋯ ( x − λ n − z ) {\textstyle f_{z}(x)=f(x-z)=(x-\lambda _{1}-z)(x-\lambda _{2}-z)\cdots (x-\lambda _{n}-z)} where z {\displaystyle z} is some element of F p {\displaystyle \mathbb {F} _{p}} . If one can represent this polynomial as the product f z ( x ) = p 0 ( x ) p 1 ( x ) {\displaystyle f_{z}(x)=p_{0}(x)p_{1}(x)} then in terms of the initial polynomial it means that f ( x ) = p 0 ( x + z ) p 1 ( x + z ) {\displaystyle f(x)=p_{0}(x+z)p_{1}(x+z)} , which provides needed factorization of f ( x ) {\displaystyle f(x)} . === Classification of === F p {\displaystyle \mathbb {F} _{p}} elements Due to Euler's criterion, for every monomial ( x − λ ) {\displaystyle (x-\lambda )} exactly one of following properties holds: The monomial is equal to x {\displaystyle x} if λ = 0 {\displaystyle \lambda =0} , The monomial divides g 0 ( x ) = ( x ( p − 1 ) / 2 − 1 ) {\textstyle g_{0}(x)=(x^{(p-1)/2}-1)} if λ {\displaystyle \lambda } is quadratic residue modulo p {\displaystyle p} , The monomial divides g 1 ( x ) = ( x ( p − 1 ) / 2 + 1 ) {\textstyle g_{1}(x)=(x^{(p-1)/2}+1)} if λ {\displaystyle \lambda } is quadratic non-residual modulo p {\displaystyle p} . Thus if f z ( x ) {\displaystyle f_{z}(x)} is not divisible by x {\displaystyle x} , which may be checked separately, then f z ( x ) {\displaystyle f_{z}(x)} is equal to the product of greatest common divisors gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} and gcd ( f z ( x ) ; g 1 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{1}(x))} . === Berlekamp's method === The property above leads to the following algorithm: Explicitly calculate coefficients of f z ( x ) = f ( x − z ) {\displaystyle f_{z}(x)=f(x-z)} , Calculate remainders of x , x 2 , x 2 2 , x 2 3 , x 2 4 , … , x 2 ⌊ log 2 ⁡ p ⌋ {\textstyle x,x^{2},x^{2^{2}},x^{2^{3}},x^{2^{4}},\ldots ,x^{2^{\lfloor \log _{2}p\rfloor }}} modulo f z ( x ) {\displaystyle f_{z}(x)} by squaring the current polynomial and taking remainder modulo f z ( x ) {\displaystyle f_{z}(x)} , Using exponentiation by squaring and polynomials calculated on the previous steps calculate the remainder of x ( p − 1 ) / 2 {\textstyle x^{(p-1)/2}} modulo f z ( x ) {\textstyle f_{z}(x)} , If x ( p − 1 ) / 2 ≢ ± 1 ( mod f z ( x ) ) {\textstyle x^{(p-1)/2}\not \equiv \pm 1{\pmod {f_{z}(x)}}} then gcd {\displaystyle \gcd } mentioned below provide a non-trivial factorization of f z ( x ) {\displaystyle f_{z}(x)} , Otherwise all roots of f z ( x ) {\displaystyle f_{z}(x)} are either residues or non-residues simultaneously and one has to choose another z {\displaystyle z} . If f ( x ) {\displaystyle f(x)} is divisible by some non-linear primitive polynomial g ( x ) {\displaystyle g(x)} over F p {\displaystyle \mathbb {F} _{p}} then when calculating gcd {\displaystyle \gcd } with g 0 ( x ) {\displaystyle g_{0}(x)} and g 1 ( x ) {\displaystyle g_{1}(x)} one will obtain a non-trivial factorization of f z ( x ) / g z ( x ) {\displaystyle f_{z}(x)/g_{z}(x)} , thus algorithm allows to find all roots of arbitrary polynomials over F p {\displaystyle \mathbb {F} _{p}} . === Modular square root === Consider equation x 2 ≡ a ( mod p ) {\textstyle x^{2}\equiv a{\pmod {p}}} having elements β {\displaystyle \beta } and − β {\displaystyle -\beta } as its roots. Solution of this equation is equivalent to factorization of polynomial f ( x ) = x 2 − a = ( x − β ) ( x + β ) {\textstyle f(x)=x^{2}-a=(x-\beta )(x+\beta )} over F p {\displaystyle \mathbb {F} _{p}} . In this particular case problem it is sufficient to calculate only gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} . For this polynomial exactly one of the following properties will hold: GCD is equal to 1 {\displaystyle 1} which means that z + β {\displaystyle z+\beta } and z − β {\displaystyle z-\beta } are both quadratic non-residues, GCD is equal to f z ( x ) {\displaystyle f_{z}(x)} which means that both numbers are quadratic residues, GCD is equal to ( x − t ) {\displaystyle (x-t)} which means that exactly one of these numbers is quadratic residue. In the third case GCD is equal to either ( x − z − β ) {\displaystyle (x-z-\beta )} or ( x − z + β ) {\displaystyle (x-z+\beta )} . It allows to write the solution as β = ( t − z ) ( mod p ) {\textstyle \beta =(t-z){\pmod {p}}} . === Example === Assume we need to solve the equation x 2 ≡ 5 ( mod 11 ) {\textstyle x^{2}\equiv 5{\pmod {11}}} . For this we need to factorize f ( x ) = x 2 − 5 = ( x − β ) ( x + β ) {\displaystyle f(x)=x^{2}-5=(x-\beta )(x+\beta )} . Consider some possible values of z {\displaystyle z} : Let z = 3 {\displaystyle z=3} . Then f z ( x ) = ( x − 3 ) 2 − 5 = x 2 − 6 x + 4 {\displaystyle f_{z}(x)=(x-3)^{2}-5=x^{2}-6x+4} , thus gcd ( x 2 − 6 x + 4 ; x 5 − 1 ) = 1 {\displaystyle \gcd(x^{2}-6x+4;x^{5}-1)=1} . Both numbers 3 ± β {\displaystyle 3\pm \beta } are quadratic non-residues, so we need to take some other z {\displaystyle z} . Let z = 2 {\displaystyle z=2} . Then f z ( x ) = ( x − 2 ) 2 − 5 = x 2 − 4 x − 1 {\displaystyle f_{z}(x)=(x-2)^{2}-5=x^{2}-4x-1} , thus gcd ( x 2 − 4 x − 1 ; x 5 − 1 ) ≡ x − 9 ( mod 11 ) {\textstyle \gcd(x^{2}-4x-1;x^{5}-1)\equiv x-9{\pmod {11}}} . From this follows x − 9 = x − 2 − β {\textstyle x-9=x-2-\beta } , so β ≡ 7 ( mod 11 ) {\displaystyle \beta \equiv 7{\pmod {11}}} and − β ≡ − 7 ≡ 4 ( mod 11 ) {\textstyle -\beta \equiv -7\equiv 4{\pmod {11}}} . A manual check shows that, indeed, 7 2 ≡ 49 ≡ 5 ( mod 11 ) {\textstyle 7^{2}\equiv 49\equiv 5{\pmod {11}}} and 4 2 ≡ 16 ≡ 5 ( mod 11 ) {\textstyle 4^{2}\equiv 16\equiv 5{\pmod {11}}} . == Correctness proof == The algorithm finds factorization of f z ( x ) {\displaystyle f_{z}(x)} in all cases except for ones when all numbers z + λ 1 , z + λ 2 , … , z + λ n {\displaystyle z+\lambda _{1},z+\lambda _{2},\ldots ,z+\lambda _{n}} are quadratic residues or non-residues simultaneously. According to theory of cyclotomy, the probability of such an event for the case when λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are all residues or non-residues simultaneously (that is, when z = 0 {\displaystyle z=0} would fail) may be estimated as 2 − k {\displaystyle 2^{-k}} where k {\displaystyle k} is the number of distinct values in λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} . In this way even for the worst case of k = 1 {\displaystyle k=1} and f ( x ) = ( x − λ ) n {\displaystyle f(x)=(x-\lambda )^{n}} , the probability of error may be estimated as 1 / 2 {\displaystyle 1/2} and for modular square root case error probability is at most 1 / 4 {\displaystyle 1/4} . == Complexity == Let a polynomial have degree n {\displaystyle n} . We derive the algorithm's complexity as follows: Due to the binomial theorem ( x − z ) k = ∑ i = 0 k ( k i ) ( − z ) k − i x i {\textstyle (x-z)^{k}=\sum \limits _{i=0}^{k}{\binom {k}{i}}(-z)^{k-i}x^{i}} , we may transition from f ( x ) {\displaystyle f(x)} to f ( x − z ) {\displaystyle f(x-z)} in O ( n 2 ) {\displaystyle O(n^{2})} time. Polynomial multiplication a
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