Data remanence is the residual representation of digital data that remains even after attempts have been made to remove or erase the data. This residue may result from data being left intact by a nominal file deletion operation, by reformatting of storage media that does not remove data previously written to the media, or through physical properties of the storage media that allow previously written data to be recovered. Data remanence may make inadvertent disclosure of sensitive information possible should the storage media be released into an uncontrolled environment (e.g., thrown in refuse containers or lost). Various techniques have been developed to counter data remanence. These techniques are classified as clearing, purging/sanitizing, or destruction. Specific methods include overwriting, degaussing, encryption, and media destruction. Effective application of countermeasures can be complicated by several factors, including media that are inaccessible, media that cannot effectively be erased, advanced storage systems that maintain histories of data throughout the data's life cycle, and persistence of data in memory that is typically considered volatile. Several standards exist for the secure removal of data and the elimination of data remanence. == Causes == Many operating systems, file managers, and other software provide a facility where a file is not immediately deleted when the user requests that action. Instead, the file is moved to a holding area (i.e. the "trash"), making it easy for the user to undo a mistake. Similarly, many software products automatically create backup copies of files that are being edited, to allow the user to restore the original version, or to recover from a possible crash (autosave feature). Even when an explicit deleted file retention facility is not provided or when the user does not use it, operating systems do not actually remove the contents of a file when it is deleted unless they are aware that explicit erasure commands are required, like on a solid-state drive. (In such cases, the operating system will issue the Serial ATA TRIM command or the SCSI UNMAP command to let the drive know to no longer maintain the deleted data.) Instead, they simply remove the file's entry from the file system directory because this requires less work and is therefore faster, and the contents of the file—the actual data—remain on the storage medium. The data will remain there until the operating system reuses the space for new data. In some systems, enough filesystem metadata are also left behind to enable easy undeletion by commonly available utility software. Even when undelete has become impossible, the data, until it has been overwritten, can be read by software that reads disk sectors directly. Computer forensics often employs such software. Likewise, reformatting, repartitioning, or reimaging a system is unlikely to write to every area of the disk, though all will cause the disk to appear empty or, in the case of reimaging, empty except for the files present in the image, to most software. Finally, even when the storage media is overwritten, physical properties of the media may permit recovery of the previous contents. In most cases however, this recovery is not possible by just reading from the storage device in the usual way, but requires using laboratory techniques such as disassembling the device and directly accessing/reading from its components. § Complications below gives further explanations for causes of data remanence. == Countermeasures == There are three levels commonly recognized for eliminating remnant data: === Clearing === Clearing is the removal of sensitive data from storage devices in such a way that there is assurance that the data may not be reconstructed using normal system functions or software file/data recovery utilities. The data may still be recoverable, but not without special laboratory techniques. Clearing is typically an administrative protection against accidental disclosure within an organization. For example, before a hard drive is re-used within an organization, its contents may be cleared to prevent their accidental disclosure to the next user. === Purging === Purging or sanitizing is the physical rewrite of sensitive data from a system or storage device done with the specific intent of rendering the data unrecoverable at a later time. Purging, proportional to the sensitivity of the data, is generally done before releasing media beyond control, such as before discarding old media, or moving media to a computer with different security requirements. === Destruction === The storage media is made unusable for conventional equipment. Effectiveness of destroying the media varies by medium and method. Depending on recording density of the media, and/or the destruction technique, this may leave data recoverable by laboratory methods. Conversely, destruction using appropriate techniques is the most secure method of preventing retrieval. == Specific methods == === Overwriting === A common method used to counter data remanence is to overwrite the storage media with new data. This is often called wiping or shredding a disk or file, by analogy to common methods of destroying print media, although the mechanism bears no similarity to these. Because such a method can often be implemented in software alone, and may be able to selectively target only part of the media, it is a popular, low-cost option for some applications. Overwriting is generally an acceptable method of clearing, as long as the media is writable and not damaged. The simplest overwrite technique writes the same data everywhere—often just a pattern of all zeros. At a minimum, this will prevent the data from being retrieved simply by reading from the media again using standard system functions. The UEFI in modern machines may offer an ATA class disk erase function as well. The ATA-6 standard governs secure erases specifications. Bitlocker is whole disk encryption and illegible without the key. Writing a fresh GPT allows a new file system to be established. Blocks will set empty but LBA read is illegible. New data will be unaffected and work fine. In an attempt to counter more advanced data recovery techniques, specific overwrite patterns and multiple passes have often been prescribed. These may be generic patterns intended to eradicate any trace signatures; an example is the seven-pass pattern 0xF6, 0x00, 0xFF,
Sanad (government app)
Sanad (Arabic: سند) is the official digital identity and e-government services application of the Hashemite Kingdom of Jordan. Developed and managed by the Ministry of Digital Economy and Entrepreneurship, the app provides a unified platform for accessing a range of public services and personal records digitally. == Overview == Launched in February 2020, Sanad is part of Jordan's broader digital transformation strategy aimed at improving public service delivery and enhancing administrative efficiency. The app allows users to authenticate their identity digitally and access over 550 services from more than 50 government and private sector entities. == Features == Sanad provides a wide array of services, including: Viewing and managing official digital documents Applying for government services (e.g., jordanian passport issuance or renewal, health insurance) Accessing personal records (e.g., pension, property ownership) Digitally signing documents Paying utility bills and traffic fines Receiving and tracking official notifications The app is available on iOS, Android, and HarmonyOS platforms and supports both Arabic and English languages. == Digital Identity == A core feature of Sanad is the digital identity system, which enables secure login and authentication for all integrated services. Users must activate their digital identity at designated Sanad stations across Jordan to access the full suite of services. == Adoption and Impact == As of 2025, more than 1.6 million Jordanians have activated their digital identities through Sanad. The app has played a significant role in streamlining government interactions and reducing the need for in-person visits, especially during the COVID-19 pandemic. == Recent Developments == In 2025, the Ministry launched an updated version of the app with enhanced user experience and new services, including the e-passport issuance feature.
ArchiMate
ArchiMate ( AR-ki-mayt) is an open and independent enterprise architecture modeling language to support the description, analysis and visualization of architecture within and across business domains in an unambiguous way. ArchiMate is a technical standard from The Open Group and is based on concepts from the now superseded IEEE 1471 standard. It is supported by various tool vendors and consulting firms. ArchiMate is also a registered trademark of The Open Group. The Open Group has a certification program for ArchiMate users, software tools and courses. ArchiMate distinguishes itself from other languages such as Unified Modeling Language (UML) and Business Process Modeling and Notation (BPMN) by its enterprise modelling scope. Also, UML and BPMN are meant for a specific use and they are quite heavy – containing about 150 (UML) and 250 (BPMN) modeling concepts whereas ArchiMate works with just about 50 (in version 2.0). The goal of ArchiMate is to be ”as small as possible”, not to cover every edge scenario imaginable. To be easy to learn and apply, ArchiMate was intentionally restricted “to the concepts that suffice for modeling the proverbial 80% of practical cases". == Overview == ArchiMate offers a common language for describing the construction and operation of business processes, organizational structures, information flows, IT systems, and technical infrastructure. This insight helps the different stakeholders to design, assess, and communicate the consequences of decisions and changes within and between these business domains. The main concepts and relationships of the ArchiMate language can be seen as a framework, the so-called Archimate Framework: It divides the enterprise architecture into a business, application and technology layer. In each layer, three aspects are considered: active elements, an internal structure and elements that define use or communicate information. One of the objectives of the ArchiMate language is to define the relationships between concepts in different architecture domains. The concepts of this language therefore hold the middle between the detailed concepts, which are used for modeling individual domains (for example, the Unified Modeling Language (UML) for modeling software products), and Business Process Model and Notation (BPMN), which is used for business process modeling. == History == ArchiMate is partly based on the now superseded IEEE 1471 standard. It was developed in the Netherlands by a project team from the Telematica Instituut in cooperation with several Dutch partners from government, industry and academia. Among the partners were Ordina NV, Radboud Universiteit Nijmegen, the Leiden Institute for Advanced Computer Science (LIACS) and the Centrum Wiskunde & Informatica (CWI). Later, tests were performed in organizations such as ABN AMRO, the Dutch Tax and Customs Administration and the ABP. The development process lasted from July 2002 to December 2004, and took about 35 person years and approximately 4 million euros. The development was funded by the Dutch government (Dutch Tax and Customs Administration), and business partners, including ABN AMRO and the ABP Pension Fund. In 2008 the ownership and stewardship of ArchiMate was transferred to The Open Group. It is now managed by the ArchiMate Forum within The Open Group. In February 2009 The Open Group published the ArchiMate 1.0 standard as a formal technical standard. In January 2012 the ArchiMate 2.0 standard, and in 2013 the ArchiMate 2.1 standard was released. In June 2016, the Open Group released version 3.0 of the ArchiMate Specification. An update to Archimate 3.0.1 came out in August 2017. Archimate 3.1 was published 5 November 2019. The latest version of the ArchiMate Specification is version 3.2 released October 2022. Version 3.0 adds enhanced support for capability-oriented strategic modelling, new entities representing physical resources (for modelling the ingredients, equipment and transport resources used in the physical world) and a generic metamodel showing the entity types and the relationships between them. == ArchiMate framework == === Core framework === The main concepts and elements of the ArchiMate language are being presented as ArchiMate core framework. It consists of three layers and three aspects. This creates a matrix of combinations. Every layer has its passive structure, behavior and active structure aspects. ==== Layers ==== ArchiMate has a layered and service-oriented look on architectural models. The higher layers make use of services that are provided by the lower layers. Although, at an abstract level, the concepts that are used within each layer are similar, we define more concrete concepts that are specific for a certain layer. In this context, we distinguish three main layers: The business layer is about business processes, services, functions and events of business units. This layer "offers products and services to external customers, which are realized in the organization by business processes performed by business actors and roles". The application layer is about software applications that "support the components in the business with application services". The technology layer deals "with the hardware and communication infrastructure to support the application layer. This layer offers infrastructural services needed to run applications, realized by computer and communication hardware and system software". Each of these main layers can be further divided in sub-layers. For example, in the business layer, the primary business processes realising the products of a company may make use of a layer of secondary (supporting) business processes; in the application layer, the end-user applications may make use of generic services offered by supporting applications. On top of the business layer, a separate environment layer may be added, modelling the external customers that make use of the services of the organisation (although these may also be considered part of the business layer). In line with service orientation, the most important relation between layers is formed by use relations, which show how the higher layers make use of the services of lower layers. However, a second type of link is formed by realisation relations: elements in lower layers may realise comparable elements in higher layers; e.g., a ‘data object’ (application layer) may realise a ‘business object’ (business layer); or an ‘artifact’ (technology layer) may realise either a ‘data object’ or an ‘application component’ (application layer). ==== Aspects ==== Passive structure is the set of entities on which actions are conducted. In the business layer the example would be information objects, in the application layer data objects and in the technology layer, they could include physical objects. Behavior refers to the processes and functions performed by the actors. "Structural elements are assigned to behavioral elements, to show who or what displays the behavior". Active structure is the set of entities that display some behavior, e.g. business actors, devices, or application components. === Full framework === The Full ArchiMate framework is enriched by the physical layer, which was added to allow modeling of “physical equipment, materials, and distribution networks” and was not present in the previous version. The implementation and migration layer adds elements that allow architects to model a state of transition, to mark parts of the architecture that are temporary for the purpose, as the name says, of implementation and migration. Strategy layer adds three elements: resource, capability and course of action. These elements help to incorporate strategic dimension to the ArchiMate language by allowing it to depict the usage of resources and capabilities in order to achieve some strategic goals. Finally, there is a motivation aspect that allows different stakeholders to describe the motivation of specific actors or domains, which can be quite important when looking at one thing from several different angles. It adds several elements like stakeholder, value, driver, goal, meaning etc. == ArchiMate language == The ArchiMate language is formed as a top-level and is hierarchical. On the top, there is a model. A model is a collection of concepts. A concept can be either an element or a relationship. An element can be either of behavior type, structure, motivation or a so-called composite element (which means that it does not fit just one aspect of the framework, but two or more). The functionality of all concepts without a dependency on a specific layer is described by the generic metamodel. This layer-independent description of concepts is useful when trying to understand the mechanics of the Archimate language. === Concepts === ==== Elements ==== The generic elements are distributed into the same categories as the layers: Active structure elements Behavior elements Passive structure elements Motivation elements Active structure e
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.
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.
Princh
Princh is a Danish software company, which is headquartered in Aarhus, Denmark. Founded in 2015, Princh develops cloud printing and electronic payment products. The company is headquartered in the city of Aarhus. While utilizing a smartphone or web app, users can locate a nearby printer to their current location, get directions to access said printer, and/or authorize a print and pay for the print job in question. The product is available as a native mobile apps for Android and iOS, as well as on web and desktop products for businesses and libraries. The app connects a network of printer owners and users around the world. Princh supports an array of printable files. == History == The company was founded in 2015. The company is currently based in the southern part of Aarhus. The Princh printing service was officially launched on June 23, 2015. Currently, Princh is available as a service in a multitude of locations such as print shops, libraries, hotels, or universities. Princh is a popular printing and payment product among libraries and can among other places be found in Denmark, Sweden, Norway, Germany, United Kingdom, United States, and Canada. == How it works == With the Princh app, users will be able to locate their nearest printer. Once the user is at the printer, the user chooses the document to be printed out and shares it with the Princh app. The user then selects the desired nearby printer entering the printer ID number or scanning the QR-code located on top of the printer, pays electronically and the print job is processed by the printer. Printer owners get access to a personal control panel where they can set printing prices and monitor all Princh activity for their business. == Notes and references ==
Whitehead's algorithm
Whitehead's algorithm is a mathematical algorithm in group theory for solving the automorphic equivalence problem in the finite rank free group Fn. The algorithm is based on a classic 1936 paper of J. H. C. Whitehead. It is still unknown (except for the case n = 2) if Whitehead's algorithm has polynomial time complexity. == Statement of the problem == Let F n = F ( x 1 , … , x n ) {\displaystyle F_{n}=F(x_{1},\dots ,x_{n})} be a free group of rank n ≥ 2 {\displaystyle n\geq 2} with a free basis X = { x 1 , … , x n } {\displaystyle X=\{x_{1},\dots ,x_{n}\}} . The automorphism problem, or the automorphic equivalence problem for F n {\displaystyle F_{n}} asks, given two freely reduced words w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} whether there exists an automorphism φ ∈ Aut ( F n ) {\displaystyle \varphi \in \operatorname {Aut} (F_{n})} such that φ ( w ) = w ′ {\displaystyle \varphi (w)=w'} . Thus the automorphism problem asks, for w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} whether Aut ( F n ) w = Aut ( F n ) w ′ {\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'} . For w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} one has Aut ( F n ) w = Aut ( F n ) w ′ {\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'} if and only if Out ( F n ) [ w ] = Out ( F n ) [ w ′ ] {\displaystyle \operatorname {Out} (F_{n})[w]=\operatorname {Out} (F_{n})[w']} , where [ w ] , [ w ′ ] {\displaystyle [w],[w']} are conjugacy classes in F n {\displaystyle F_{n}} of w , w ′ {\displaystyle w,w'} accordingly. Therefore, the automorphism problem for F n {\displaystyle F_{n}} is often formulated in terms of Out ( F n ) {\displaystyle \operatorname {Out} (F_{n})} -equivalence of conjugacy classes of elements of F n {\displaystyle F_{n}} . For an element w ∈ F n {\displaystyle w\in F_{n}} , | w | X {\displaystyle |w|_{X}} denotes the freely reduced length of w {\displaystyle w} with respect to X {\displaystyle X} , and ‖ w ‖ X {\displaystyle \|w\|_{X}} denotes the cyclically reduced length of w {\displaystyle w} with respect to X {\displaystyle X} . For the automorphism problem, the length of an input w {\displaystyle w} is measured as | w | X {\displaystyle |w|_{X}} or as ‖ w ‖ X {\displaystyle \|w\|_{X}} , depending on whether one views w {\displaystyle w} as an element of F n {\displaystyle F_{n}} or as defining the corresponding conjugacy class [ w ] {\displaystyle [w]} in F n {\displaystyle F_{n}} . == History == The automorphism problem for F n {\displaystyle F_{n}} was algorithmically solved by J. H. C. Whitehead in a classic 1936 paper, and his solution came to be known as Whitehead's algorithm. Whitehead used a topological approach in his paper. Namely, consider the 3-manifold M n = # i = 1 n S 2 × S 1 {\displaystyle M_{n}=\#_{i=1}^{n}\mathbb {S} ^{2}\times \mathbb {S} ^{1}} , the connected sum of n {\displaystyle n} copies of S 2 × S 1 {\displaystyle \mathbb {S} ^{2}\times \mathbb {S} ^{1}} . Then π 1 ( M n ) ≅ F n {\displaystyle \pi _{1}(M_{n})\cong F_{n}} , and, moreover, up to a quotient by a finite normal subgroup isomorphic to Z 2 n {\displaystyle \mathbb {Z} _{2}^{n}} , the mapping class group of M n {\displaystyle M_{n}} is equal to Out ( F n ) {\displaystyle \operatorname {Out} (F_{n})} ; see. Different free bases of F n {\displaystyle F_{n}} can be represented by isotopy classes of "sphere systems" in M n {\displaystyle M_{n}} , and the cyclically reduced form of an element w ∈ F n {\displaystyle w\in F_{n}} , as well as the Whitehead graph of [ w ] {\displaystyle [w]} , can be "read-off" from how a loop in general position representing [ w ] {\displaystyle [w]} intersects the spheres in the system. Whitehead moves can be represented by certain kinds of topological "swapping" moves modifying the sphere system. Subsequently, Rapaport, and later, based on her work, Higgins and Lyndon, gave a purely combinatorial and algebraic re-interpretation of Whitehead's work and of Whitehead's algorithm. The exposition of Whitehead's algorithm in the book of Lyndon and Schupp is based on this combinatorial approach. Culler and Vogtmann, in their 1986 paper that introduced the Outer space, gave a hybrid approach to Whitehead's algorithm, presented in combinatorial terms but closely following Whitehead's original ideas. == Whitehead's algorithm == Our exposition regarding Whitehead's algorithm mostly follows Ch.I.4 in the book of Lyndon and Schupp, as well as. === Overview === The automorphism group Aut ( F n ) {\displaystyle \operatorname {Aut} (F_{n})} has a particularly useful finite generating set W {\displaystyle {\mathcal {W}}} of Whitehead automorphisms or Whitehead moves. Given w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} the first part of Whitehead's algorithm consists of iteratively applying Whitehead moves to w , w ′ {\displaystyle w,w'} to take each of them to an "automorphically minimal" form, where the cyclically reduced length strictly decreases at each step. Once we find automorphically these minimal forms u , u ′ {\displaystyle u,u'} of w , w ′ {\displaystyle w,w'} , we check if ‖ u ‖ X = ‖ u ′ ‖ X {\displaystyle \|u\|_{X}=\|u'\|_{X}} . If ‖ u ‖ X ≠ ‖ u ′ ‖ X {\displaystyle \|u\|_{X}\neq \|u'\|_{X}} then w , w ′ {\displaystyle w,w'} are not automorphically equivalent in F n {\displaystyle F_{n}} . If ‖ u ‖ X = ‖ u ′ ‖ X {\displaystyle \|u\|_{X}=\|u'\|_{X}} , we check if there exists a finite chain of Whitehead moves taking u {\displaystyle u} to u ′ {\displaystyle u'} so that the cyclically reduced length remains constant throughout this chain. The elements w , w ′ {\displaystyle w,w'} are not automorphically equivalent in F n {\displaystyle F_{n}} if and only if such a chain exists. Whitehead's algorithm also solves the search automorphism problem for F n {\displaystyle F_{n}} . Namely, given w , w ′ ∈ F n {\displaystyle w,w'\in F_{n}} , if Whitehead's algorithm concludes that Aut ( F n ) w = Aut ( F n ) w ′ {\displaystyle \operatorname {Aut} (F_{n})w=\operatorname {Aut} (F_{n})w'} , the algorithm also outputs an automorphism φ ∈ Aut ( F n ) {\displaystyle \varphi \in \operatorname {Aut} (F_{n})} such that φ ( w ) = w ′ {\displaystyle \varphi (w)=w'} . Such an element φ ∈ Aut ( F n ) {\displaystyle \varphi \in \operatorname {Aut} (F_{n})} is produced as the composition of a chain of Whitehead moves arising from the above procedure and taking w {\displaystyle w} to w ′ {\displaystyle w'} . === Whitehead automorphisms === A Whitehead automorphism, or Whitehead move, of F n {\displaystyle F_{n}} is an automorphism τ ∈ Aut ( F n ) {\displaystyle \tau \in \operatorname {Aut} (F_{n})} of F n {\displaystyle F_{n}} of one of the following two types: There is a permutation σ ∈ S n {\displaystyle \sigma \in S_{n}} of { 1 , 2 , … , n } {\displaystyle \{1,2,\dots ,n\}} such that for i = 1 , … , n {\displaystyle i=1,\dots ,n} τ ( x i ) = x σ ( i ) ± 1 {\displaystyle \tau (x_{i})=x_{\sigma (i)}^{\pm 1}} Such τ {\displaystyle \tau } is called a Whitehead automorphism of the first kind. There is an element a ∈ X ± 1 {\displaystyle a\in X^{\pm 1}} , called the multiplier, such that for every x ∈ X ± 1 {\displaystyle x\in X^{\pm 1}} τ ( x ) ∈ { x , x a , a − 1 x , a − 1 x a } . {\displaystyle \tau (x)\in \{x,xa,a^{-1}x,a^{-1}xa\}.} Such τ {\displaystyle \tau } is called a Whitehead automorphism of the second kind. Since τ {\displaystyle \tau } is an automorphism of F n {\displaystyle F_{n}} , it follows that τ ( a ) = a {\displaystyle \tau (a)=a} in this case. Often, for a Whitehead automorphism τ ∈ Aut ( F n ) {\displaystyle \tau \in \operatorname {Aut} (F_{n})} , the corresponding outer automorphism in Out ( F n ) {\displaystyle \operatorname {Out} (F_{n})} is also called a Whitehead automorphism or a Whitehead move. ==== Examples ==== Let F 4 = F ( x 1 , x 2 , x 3 , x 4 ) {\displaystyle F_{4}=F(x_{1},x_{2},x_{3},x_{4})} . Let τ : F 4 → F 4 {\displaystyle \tau :F_{4}\to F_{4}} be a homomorphism such that τ ( x 1 ) = x 2 x 1 , τ ( x 2 ) = x 2 , τ ( x 3 ) = x 2 x 3 x 2 − 1 , τ ( x 4 ) = x 4 {\displaystyle \tau (x_{1})=x_{2}x_{1},\quad \tau (x_{2})=x_{2},\quad \tau (x_{3})=x_{2}x_{3}x_{2}^{-1},\quad \tau (x_{4})=x_{4}} Then τ {\displaystyle \tau } is actually an automorphism of F 4 {\displaystyle F_{4}} , and, moreover, τ {\displaystyle \tau } is a Whitehead automorphism of the second kind, with the multiplier a = x 2 − 1 {\displaystyle a=x_{2}^{-1}} . Let τ ′ : F 4 → F 4 {\displaystyle \tau ':F_{4}\to F_{4}} be a homomorphism such that τ ′ ( x 1 ) = x 1 , τ ′ ( x 2 ) = x 1 − 1 x 2 x 1 , τ ′ ( x 3 ) = x 1 − 1 x 3 x 1 , τ ′ ( x 4 ) = x 1 − 1 x 4 x 1 {\displaystyle \tau '(x_{1})=x_{1},\quad \tau '(x_{2})=x_{1}^{-1}x_{2}x_{1},\quad \tau '(x_{3})=x_{1}^{-1}x_{3}x_{1},\quad \tau '(x_{4})=x_{1}^{-1}x_{4}x_{1}} Then τ ′ {\displaystyle \tau '} is actually an inner automorphism of F 4 {\displaystyle F_{4}} given by conjugation by x 1 {\displaystyle x_{1}} , and, moreover, τ ′ {\displaystyle \