A class of sets is said to shatter another set if it is possible to "pick out" any element of that set using intersection. The concept of shattered sets plays an important role in Vapnik–Chervonenkis theory, also known as VC-theory. Shattering and VC-theory are used in the study of empirical processes as well as in statistical computational learning theory. == Definition == Suppose A is a set and C is a class of sets. The class C shatters the set A if for each subset a of A, there is some element c of C such that a = c ∩ A . {\displaystyle a=c\cap A.} Equivalently, C shatters A when their intersection is equal to A's power set: P(A) = { c ∩ A | c ∈ C }. We employ the letter C to refer to a "class" or "collection" of sets, as in a Vapnik–Chervonenkis class (VC-class). The set A is often assumed to be finite because, in empirical processes, we are interested in the shattering of finite sets of data points. == Example == We will show that the class of all discs in the plane (two-dimensional space) does not shatter every set of four points on the unit circle, yet the class of all convex sets in the plane does shatter every finite set of points on the unit circle. Let A be a set of four points on the unit circle and let C be the class of all discs. To test where C shatters A, we attempt to draw a disc around every subset of points in A. First, we draw a disc around the subsets of each isolated point. Next, we try to draw a disc around every subset of point pairs. This turns out to be doable for adjacent points, but impossible for points on opposite sides of the circle. Any attempt to include those points on the opposite side will necessarily include other points not in that pair. Hence, any pair of opposite points cannot be isolated out of A using intersections with class C and so C does not shatter A. As visualized below: Because there is some subset which can not be isolated by any disc in C, we conclude then that A is not shattered by C. And, with a bit of thought, we can prove that no set of four points is shattered by this C. However, if we redefine C to be the class of all elliptical discs, we find that we can still isolate all the subsets from above, as well as the points that were formerly problematic. Thus, this specific set of 4 points is shattered by the class of elliptical discs. Visualized below: With a bit of thought, we could generalize that any set of finite points on a unit circle could be shattered by the class of all convex sets (visualize connecting the dots). == Shatter coefficient == To quantify the richness of a collection C of sets, we use the concept of shattering coefficients (also known as the growth function). For a collection C of sets s ⊂ Ω {\displaystyle s\subset \Omega } , Ω {\displaystyle \Omega } being any space, often a sample space, we define the nth shattering coefficient of C as S C ( n ) = max ∀ x 1 , x 2 , … , x n ∈ Ω card { { x 1 , x 2 , … , x n } ∩ s , s ∈ C } {\displaystyle S_{C}(n)=\max _{\forall x_{1},x_{2},\dots ,x_{n}\in \Omega }\operatorname {card} \{\,\{\,x_{1},x_{2},\dots ,x_{n}\}\cap s,s\in C\}} where card {\displaystyle \operatorname {card} } denotes the cardinality of the set and x 1 , x 2 , … , x n ∈ Ω {\displaystyle x_{1},x_{2},\dots ,x_{n}\in \Omega } is any set of n points,. S C ( n ) {\displaystyle S_{C}(n)} is the largest number of subsets of any set A of n points that can be formed by intersecting A with the sets in collection C. For example, if set A contains 3 points, its power set, P ( A ) {\displaystyle P(A)} , contains 2 3 = 8 {\displaystyle 2^{3}=8} elements. If C shatters A, its shattering coefficient(3) would be 8 and S C ( 2 ) {\displaystyle S_{C}(2)} would be 2 2 = 4 {\displaystyle 2^{2}=4} . However, if one of those sets in P ( A ) {\displaystyle P(A)} cannot be obtained through intersections in c, then S C ( 3 ) {\displaystyle S_{C}(3)} would only be 7. If none of those sets can be obtained, S C ( 3 ) {\displaystyle S_{C}(3)} would be 0. Additionally, if S C ( 2 ) = 3 {\displaystyle S_{C}(2)=3} , for example, then there is an element in the set of all 2-point sets from A that cannot be obtained from intersections with C. It follows from this that S C ( 3 ) {\displaystyle S_{C}(3)} would also be less than 8 (i.e. C would not shatter A) because we have already located a "missing" set in the smaller power set of 2-point sets. This example illustrates some properties of S C ( n ) {\displaystyle S_{C}(n)} : S C ( n ) ≤ 2 n {\displaystyle S_{C}(n)\leq 2^{n}} for all n because { s ∩ A | s ∈ C } ⊆ P ( A ) {\displaystyle \{s\cap A|s\in C\}\subseteq P(A)} for any A ⊆ Ω {\displaystyle A\subseteq \Omega } . If S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} , that means there is a set of cardinality n, which can be shattered by C. If S C ( N ) < 2 N {\displaystyle S_{C}(N)<2^{N}} for some N > 1 {\displaystyle N>1} then S C ( n ) < 2 n {\displaystyle S_{C}(n)<2^{n}} for all n ≥ N {\displaystyle n\geq N} . The third property means that if C cannot shatter any set of cardinality N then it can not shatter sets of larger cardinalities. == Vapnik–Chervonenkis class == If A cannot be shattered by C, there will be a smallest value of n that makes the shatter coefficient(n) less than 2 n {\displaystyle 2^{n}} because as n gets larger, there are more sets that could be missed. Alternatively, there is also a largest value of n for which the S C ( n ) {\displaystyle S_{C}(n)} is still 2 n {\displaystyle 2^{n}} , because as n gets smaller, there are fewer sets that could be omitted. The extreme of this is S C ( 0 ) {\displaystyle S_{C}(0)} (the shattering coefficient of the empty set), which must always be 2 0 = 1 {\displaystyle 2^{0}=1} . These statements lends themselves to defining the VC dimension of a class C as: V C ( C ) = min n { n : S C ( n ) < 2 n } {\displaystyle VC(C)={\underset {n}{\min }}\{n:S_{C}(n)<2^{n}\}\,} or, alternatively, as V C 0 ( C ) = max n { n : S C ( n ) = 2 n } . {\displaystyle VC_{0}(C)={\underset {n}{\max }}\{n:S_{C}(n)=2^{n}\}.\,} Note that V C ( C ) = V C 0 ( C ) + 1. {\displaystyle VC(C)=VC_{0}(C)+1.} . The VC dimension is usually defined as V C 0 {\displaystyle VC_{0}} , the largest cardinality of points chosen that will still shatter A (i.e. n such that S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} ). Altneratively, if for any n there is a set of cardinality n which can be shattered by C, then S C ( n ) = 2 n {\displaystyle S_{C}(n)=2^{n}} for all n and the VC dimension of this class C is infinite. A class with finite VC dimension is called a Vapnik–Chervonenkis class or VC class. A class C is uniformly Glivenko–Cantelli if and only if it is a VC class.
Fyre (software)
Fyre, formerly de Jong Explorer, is a cross-platform tool for producing artwork based on histograms of iterated chaotic functions. It implements the Peter de Jong map in a fixed function pipeline through either a GTK GUI frontend, or a command line facility for easier rendering of high-resolution, high quality images. The program was renamed from de Jong Explorer to Fyre simply because 'It wasn't taken yet' and so that in the future, it could support more functions than just the standard Peter de Jong map. Fyre features a sidebar on the left to which the user can input the required variables and on the right is displayed the result of the equation. == Extra features == Additional image manipulation tools such as Gaussian blurs and Gamma controls are included in the program. The advantage to using them directly within Fyre is that the image accuracy and quality do not decline. Fyre features animation capabilities so that a user can link together several maps and create uncompressed AVIs from them. However, the uncompressed animation files are very large and so should be compressed with a separate tool, such as mencoder. == Peter de Jong Map == For most values of a,b,c and d the point (x,y) moves chaotically. The resulting image is a map of the probability that the point lies within the area represented by each pixel. Therefore, the longer that the user lets Fyre render for, the larger the probability map becomes and the more accurate the resulting image.
Social media use by the Islamic State
The Islamic State is widely known for its posting of disturbing content, such as beheading videos, on the internet. This propaganda is disseminated through websites and many social media platforms such as Twitter, Facebook, Telegram, and YouTube. By utilizing social media, the organization has garnered a strong following and successfully recruited tens of thousands of followers from around the world. In response to its successful use of social media, many websites and social media platforms have banned accounts and removed content promoting the Islamic State from their platforms. == Background == The Islamic State is a Jihadist militant group and a former unrecognised proto-state. The group sophisticatedly utilizes social media as a tool for spreading its message and for international recruitment. == Target audience == IS targets a variety of different groups both in the Middle East and Western Countries. There are a wide variety of motives for why fighters may be prompted to join IS. Researchers from Quantum cite nine attributes characteristic of a fighter looking to join IS: status seeking, identity seeking, revenge, redemption, thrill, ideology, justice, and death. The standard IS recruit, both from the Middle East and Western countries, is relatively young. The average age of IS fighters is around 26 years old, with 86% of recruits being male. Middle Eastern recruits come from economically disadvantaged backgrounds in Northern Iraq. Recent destruction in the Iraq War and Syrian Civil War has created hatred of Western Powers in the region. By 2025, researchers identified a significant shift toward targeting minors and adolescents, a phenomenon dubbed the "Alt-Jihad." This younger demographic is targeted not through theological arguments, but through a "victimhood-revenge" narrative that blends extremist ideology with pop-culture aesthetics in gaming environments like Roblox and Minecraft. In 2024 alone, 42 minors were arrested in Europe for involvement in IS-related plotting or propaganda. Western recruits are often second or third-generation immigrants. Computer scientists Zeeshan ul-hassan Usmani also found that the majority of the Western recruits do not feel "at home" in their home country. As a result, these fighters often have desires to go abroad and escape conditions in their home country. In addition to recruitment, IS's social media presence is also meant to intimidate and spread terror around the world. IS's posting of beheadings and other execution videos primarily target the Western world. == Content and messages == IS produces propaganda videos that range from video executions to full-length documentaries. The videos have a high production quality and incorporate montages, slow motion scenes, and are often accompanied by a short dialogue. IS has a dedicated team of over 100 media insurgents dedicated to recording these videos. While the group previously relied on glossy magazines like Dabiq, post-territorial strategies have shifted focus to the weekly newsletter Al-Naba. Unlike previous publications designed for recruitment, Al-Naba serves as a "central pillar" of the group's media strategy, focusing on bureaucratic reporting and military statistics to project a narrative of endurance and maintain internal cohesion among dispersed fighters. The IS executions typically consist of beheadings or mass shootings in retaliation to western intervention in IS territory. The particular videos that IS often post include executions of "enemies of the Caliphate," which often consist of westerners or Jordanian nationals. Most infamously, an executioner nicknamed Jihadi John was seen in many of these videos prior to his death in 2015. Jihadi John is notorious for executing many US, UK, and Japanese citizens such as Steven Sotloff, David Haines, and Alan Henning. In many of the videos and materials produced by IS, there is the theme of inclusion and brotherhood. Additionally, the videos also focus on three main messages: Convey narrative of global war and ultimate victory Radicalize populations globally Encourage international lone state actor and small cell attacks in support of IS These messages can be seen throughout all content produced by the Islamic State such as war documentaries, execution videos, and Rumiyah (magazine). == Social media usage == From 2013 to 2014, the organization primarily used mainstream platforms such as Twitter, Facebook, and YouTube. In 2014, these large social media platforms removed IS content. Since then, IS has chosen to utilize social media platforms that either protect their content or allow for content to quickly be reposted. These platforms of choice are Telegram, Justpaste.it, and Surespot, until the latter's shutdown in 2022. By 2025, the group had further diversified into decentralized platforms like Rocket.Chat and TamTam to evade moderation. IS also implements marketing initiatives like “Jihadist Follow Friday,” which encourages users to follow new IS-related accounts each Friday. This specific hashtag mirrors commonly used hashtags such as #motivation monday or #throwbackthursday. To augment their online presence and popularity, the organization encourages their followers to use a plethora of Arabic hashtags, which translate to #theFridayofSupportingISIS, and #CalamityWillBefalltheUS. This allows them to gain followers each week while promoting their community and message on a weekly basis. === Twitter === During 2014, there were an estimated 46,000 to 90,000 Twitter accounts that advocated for IS or were run by supporters of the group. In 2015, Twitter reported that it banned 125,000 IS sympathetic accounts. In 2016, it published an update of 325,000 deleted accounts. Though many accounts have been suspended, IS supporters often create new accounts. Twitter defines those who recreate accounts as “resurgents” and explains that these are often difficult accounts to remove completely, since they tend to pop back up in alternate forms. It is estimated that approximately 20% of all IS affiliated Twitter accounts can be traced back to fake accounts created by the same user. Many of these accounts are traced back to the “Baqiya family,” which is an online network of thousands of IS followers. Many of these accounts are active during important IS military victories. During the IS march on Mosul, there were about 42,000 tweets on Twitter supporting the invasion. === Telegram === During 2014, IS became very active on Telegram after many major social media platforms banned IS content and sympathetic accounts. Telegram is an encrypted messaging application. The platform by nature is created as an end-to-end user encryption platform. Further, it also has special features such as the self-destruct timer which erase all evidence and messages. The app has a user data protection policy because violating this policy could potentially damage the app’s brand of customer privacy. Government agencies have been unable to break Telegram's encryption technology. On Telegram, IS often uses the hashtag #KhilafahNews to attract their users. Telegram is used by IS to plan social media campaigns on alternate platforms. The organization also uses Telegram as an anchor platform to connect with their user base when their other accounts are banned on Twitter and Facebook. On 28 February 2016 a video was uploaded threatening to expose the najaasah and shoot the hesitates. Produced by Ibn-Altayb and distributed by Al-Hayat, the video shows footage of Bruxelles attacks and the victims. In July 2017, Telegram came under scrutiny from the media and news media outlets. It has been documented that IS gunmen have used this app to maintain contact with IS leaders in Raqqa days before terror attacks in Turkey, Berlin, and St. Petersburg. Despite concerns from Western media, there has been little to no action taken against IS accounts on Telegram. In April 2019 a video was uploaded in which they urged lone wolves to attempt to attack during the Holy Week in Sevilla and Málaga. In Sevilla, a jihadist who intended to perform a lone wolf attack was arrested. === TikTok === In October 2019, it was reported that IS recruitment content was discovered on TikTok. Approximately two dozen accounts were subsequently shut down in response. By 2025, TikTok had evolved into a "low-threshold" gateway for extremist recruitment, characterized by researchers as part of a "Virtual Caliphate Complex." Nearly 93 unofficial IS support groups, known as "feeder groups," were found to be repackaging official IS content into short-form videos with pink hearts, catchy music, and internet memes to evade detection and appeal to the "TikTok generation." This content often promotes a "victimhood-revenge" narrative rather than complex theology, specifically designed to radicalize minors. === Justpaste.it === Justpaste.it, an anonymous photo and text sharing website, has also been utilized heavily. With the option to lock images, the website allows anonymous
Hydration (web development)
In web development, hydration or rehydration is a technique in which client-side JavaScript converts a web page that is static from the perspective of the web browser, delivered either through static rendering or server-side rendering, into a dynamic web page by attaching event handlers to the HTML elements in the DOM. Because the HTML is pre-rendered on a server, this allows for a fast "first contentful paint" (when useful data is first displayed to the user), but there is a period of time afterward where the page appears to be fully loaded and interactive, but is not until the client-side JavaScript is executed and event handlers have been attached. Frameworks that use hydration include Next.js and Nuxt. React v16.0 introduced a "hydrate" function, which hydrates an element, in its API. == Variations == === Streaming server-side rendering === Streaming server-side rendering allows one to send HTML in chunks that the browser can progressively render as it is received. This can provide a fast first paint and first contentful paint as HTML markup arrives to users faster. === Progressive rehydration === In progressive rehydration, individual pieces of a server-rendered application are “booted up” over time, rather than the current common approach of initializing the entire application at once. This can help reduce the amount of JavaScript required to make pages interactive, since client-side upgrading of low priority parts of the page can be deferred to prevent blocking the main thread. It can also help avoid one of the most common server-side rendering rehydration pitfalls, where a server-rendered DOM tree gets destroyed and then immediately rebuilt – most often because the initial synchronous client-side render required data that wasn't quite ready, perhaps awaiting Promise resolution. === Partial rehydration === Partial rehydration has proven difficult to implement. This approach is an extension of the idea of progressive rehydration, where the individual pieces (components/views/trees) to be progressively rehydrated are analyzed and those with little interactivity or no reactivity are identified. For each of these mostly-static parts, the corresponding JavaScript code is then transformed into inert references and decorative functionality, reducing their client-side footprint to near-zero. The partial hydration approach comes with its own issues and compromises. It poses some interesting challenges for caching, and client-side navigation means it cannot be assumed that server-rendered HTML for inert parts of the application will be available without a full page load. One framework that supports partial rehydration is Elder.js, which is based on Svelte. === Trisomorphic rendering === Trisomorphic rendering is a technique which uses streaming server-side rendering for initial/non-JavaScript navigations, and then uses service workers to take on rendering of HTML for navigations after it has been installed. This can keep cached components and templates up to date and enables SPA-style navigations for rendering new views in the same session. This approach works best when one can share the same templating and routing code between the server, client page, and service worker.
Digital media in education
Digital media in education refers to the use of digital technologies to support and enhance teaching and learning processes. This includes the application of multiple digital software applications, devices, and online platforms as tools for learning. Learners interact with these technologies to access, analyze, evaluate, and create media content and communication in various forms. The integration of digital media in education has dramatically increased over time, significantly transforming traditional educational practices. When viewed through a global and inclusive lens, digital education should be guided by principles of equity, inclusion, and public infrastructure to ensure meaningful participation of all learners. == History == === 20th century === Technological advances in the 20th century, particularly the invention of the Internet, laid the foundation for incorporating technology into education. In the early 1900s, the overhead projector and instructional radio broadcasts were among the first technologies used for educational purposes. The introduction of computers in classrooms occurred in 1950, when a flight simulation program was developed to train pilots at the Massachusetts Institute of Technology. However, access to computers remained extremely limited for several decades. In 1964, John Kemeny and Thomas Kurtz developed the BASIC programming language, which simplified computer interaction and introduced time-sharing, enabling multiple users to work on the same system simultaneously. This innovation made computing increasingly accessible for educational settings. By the 1980s, schools began to show more interest in computers as companies released mass-market devices to the public. Networking further enabled the interconnection of computers into unified communication systems, which proved more efficient and cost-effective than previous stand-alone machines. This development prompted wider adoption of computing in educational institutions. The invention of the World Wide Web in 1992 further simplified internet navigation and sparked further interest in educational settings. Initially, computers were integrated into school curricula for tasks such as word processing, spreadsheet creation, and data organization. By the late 1990s, the Internet became a research tool, functioning as a vast library. By 1999, 99% of public school teachers in the United States reported having access to at least one computer in their schools, and 84% had a computer available in their classrooms. The emergence of World Wide Web also contributed to the development of learning management systems (LMS), which allowed educators to create online teaching environments for content storage, student activities, discussions, and assignments. Advances in digital compression and high-speed Internet made video creation and distribution more affordable, fostering the use of the systems designed for recording lectures. These tools were often incorporated into learning management platforms, supporting the expansion of fully online courses. === 21st century === By 2002, the Massachusetts Institute of Technology began offering recorded lectures to the public, marking a significant milestone in the movement toward accessible online education. The launch of YouTube in 2005 further transformed educational content distribution. Educators increasingly uploaded lectures and instructional videos on platforms with initiatives like Khan Academy, which was active in 2006, contributing to You Tube's role as a prominent educational resource. In 2007, Apple launched iTunesU, another platform for sharing educational resources and videos. Meanwhile, learning management systems gained popularity, with Blackboard and Canvas becoming two of the most widely used platforms with Canvas's release in 2008. That same year also marked the introduction of the first Massive Open Online Course (MOOC), which provided open access to webinars and expert-led instructions for global learners. As technology evolved, traditional projectors were gradually replaced by interactive whiteboards, which enabled educators to integrate digital tools more effectively in their classrooms. By 2009, 97% of classrooms in the United States had at least one computer, and 93% had Internet access. The COVID-19 pandemic, which forced schools across the world to close, significantly impacted education with schools shifting to distance education. Students attended classes remotely using devices such as laptops, phones, and tablets, supported by digital platforms that facilitated at-home learning environments. However, adapting assessment methods to the new learning environment posed certain challenges. A study conducted by Eddie M. Mulenga and José M. Marbán on Zambian students during the pandemic revealed difficulties in adapting to digital learning, particularly in subjects like mathematics. Similar issues were reported among students in Romania, where the transition to virtual learning presented significant obstacles in engagement and adaptability. === Post-pandemic developments === In the period following the onset of COVID-19, education systems worldwide rapidly adopted digital solutions to maintain continuity of learning and teaching. By the end of March 2020, all 46 OECD and partners countries closed some or all of their schools nationwide. By June 2020, the length of school closures in these countries ranged from 7 to over 18 weeks. These disruptions in formal education prompted governments and educators to quickly adopt digital learning. This global shift to online education highlighted considerable inequalities in digital access, although many systems struggled with inequitable access, especially in regions lacking devices, stable internet connections, or conducive home learning environments. Stimultaneously, commercial educational technology (ed-tech) companies introduced rapid digital solutions to the disruption caused by the pandemic. This led to what has been described as a "seller's market," where the urgency of implementation may cause the prioritization of availability and scale over pedagogical and equity considerations. In the post-pandemic era, digital media in education continues to evolve. It increasingly intersects with artificial intelligence (AI) technologies such as adaptive learning platforms, AI-enabled content generation, and personalized learning environments. These tools enhance global engagement and access but also raise concerns about infrastructure, inclusivity, ethical implementation as well as critical pedagogies. Scholars recommend that educators and policymakers adopt inclusive practices, prioritize equitable infrastructure, and develop critical digital literacy. Facer and Selwyn also emphasize the need for public digital infrastructure and sustainable and justice-oriented policies that empower all learners. Overall, these perspectives reflect a growing consensus that digital media in education should be implemented critically to promote inclusive, multimodal, and future-oriented learning environments.
Sherwood Applied Business Security Architecture
SABSA (Sherwood Applied Business Security Architecture) is a model and methodology for developing a risk-driven enterprise information security architecture and service management, to support critical business processes. It was developed independently from the Zachman Framework, but has a similar structure. The primary characteristic of the SABSA model is that everything must be derived from an analysis of the business requirements for security, especially those in which security has an enabling function through which new business opportunities can be developed and exploited. The process analyzes the business requirements at the outset, and creates a chain of traceability through the strategy and concept, design, implementation, and ongoing ‘manage and measure’ phases of the lifecycle to ensure that the business mandate is preserved. Framework tools created from practical experience further support the whole methodology. The model is layered, with the top layer being the business requirements definition stage. At each lower layer a new level of abstraction and detail is developed, going through the definition of the conceptual architecture, logical services architecture, physical infrastructure architecture and finally at the lowest layer, the selection of technologies and products (component architecture). The SABSA model itself is generic and can be the starting point for any organization, but by going through the process of analysis and decision-making implied by its structure, it becomes specific to the enterprise, and is finally highly customized to a unique business model. It becomes in reality the enterprise security architecture, and it is central to the success of a strategic program of information security management within the organization. SABSA is a particular example of a methodology that can be used both for IT (information technology) and OT (operational technology) environments. == SABSA matrix == Note: The above is the original SABSA Matrix, which is still valid today, but it has been expanded by a comprehensive service management matrix and updated in some detail and terminology areas. In the words of David Lynas, SABSA author, "The SABSA Matrix and the SABSA Service Management Matrix have not been updated since the late 90s. We have redesigned them to deliver the improvements your feedback has requested over the years. We have not fundamentally changed the structure or principles of the matrices (very few elements have changed position) but have focused on terminology update and consistency." The new versions can be downloaded (along with the 2009 revision of the SABSA White Paper and other important documents like the SABSA Certification Roadmap) at the SABSA Members' Web Site.
Kimchi (software)
Kimchi is a web management tool to manage Kernel-based Virtual Machine (KVM) infrastructure. Developed with HTML5, Kimchi is developed to intuitively manage KVM guests, create storage pools, manage network interfaces (bridges, VLANs, NAT), and perform other related tasks. The name is an extended acronym for KVM infrastructure management. It is an Apache-licensed project hosted on GitHub, and incubated by oVirt.org.