A codebook is a type of document used for gathering and storing cryptography codes. Originally, codebooks were often literally books, but today "codebook" is a byword for the complete record of a series of codes, regardless of physical format. == Cryptography == In cryptography, a codebook is a document used for implementing a code. A codebook contains a lookup table for coding and decoding; each word or phrase has one or more strings which replace it. To decipher messages written in code, corresponding copies of the codebook must be available at either end. The distribution and physical security of codebooks presents a special difficulty in the use of codes compared to the secret information used in ciphers, the key, which is typically much shorter. The United States National Security Agency documents sometimes use codebook to refer to block ciphers; compare their use of combiner-type algorithm to refer to stream ciphers. Codebooks come in two forms, one-part or two-part: In one-part codes, the plaintext words and phrases and the corresponding code words are in the same alphabetical order. They are organized similar to a standard dictionary. Such codes are half the size of two-part codes but are more vulnerable since an attacker who recovers some code word meanings can often infer the meaning of nearby code words. One-part codes may be used simply to shorten messages for transmission or have their security enhanced with superencryption methods, such as adding a secret number to numeric code words. In two-part codes, one part is for converting plaintext to ciphertext, the other for the opposite purpose. They are usually organized similarly to a language translation dictionary, with plaintext words (in the first part) and ciphertext words (in the second part) presented like dictionary headwords. The earliest known use of a codebook system was by Gabriele de Lavinde in 1379 working for the Antipope Clement VII. Two-part codebooks go back as least as far as Antoine Rossignol in the 1800s. From the 15th century until the middle of the 19th century, nomenclators (named after nomenclator) were the most used cryptographic method. Codebooks with superencryption were the most used cryptographic method of World War I. The JN-25 code used in World War II used a codebook of 30,000 code groups superencrypted with 30,000 random additives. The book used in a book cipher or the book used in a running key cipher can be any book shared by sender and receiver and is different from a cryptographic codebook. == Social sciences == In social sciences, a codebook is a document containing a list of the codes used in a set of data to refer to variables and their values, for example locations, occupations, or clinical diagnoses. == Data compression == Codebooks were also used in 19th- and 20th-century commercial codes for the non-cryptographic purpose of data compression. Codebooks are used in relation to precoding and beamforming in mobile networks such as 5G and LTE. The usage is standardized by 3GPP, for example in the document TS 38.331, NR; Radio Resource Control (RRC); Protocol specification.
WebGPU Shading Language
WebGPU Shading Language (WGSL, internet media type: text/wgsl) is a high-level shading language and the normative shader language for the WebGPU API on the web. WGSL's syntax is influenced by Rust and is designed with strong static validation, explicit resource binding, and portability in mind for secure execution in browsers. In web contexts, WebGPU implementations accept WGSL source and perform compilation to platform-specific intermediate forms (for example, to SPIR‑V, DXIL, or MSL via the user agent), but such backends are not exposed to web content. == History and background == Graphics on the web historically used WebGL, with shaders written in GLSL ES. As applications demanded more modern GPU features and finer control over compute and graphics pipelines, the W3C's GPU for the Web Community Group and Working Group created WebGPU and its companion shading language, WGSL, to provide a secure, portable model suitable for the web platform. WGSL was developed to be human-readable, avoid undefined behavior common in legacy shading languages, and align closely with WebGPU's resource and validation model. == Design goals == WGSL's design emphasizes: Safety and determinism suitable for web security constraints (extensive static validation and well-defined semantics). Portability across diverse GPU backends via an abstract resource model shared with WebGPU. Readability and explicitness (no preprocessor, minimal implicit conversions, explicit address spaces and bindings). Alignment with modern GPU features (compute, storage buffers, textures, atomics) while retaining a familiar C/Rust-like syntax. == Language overview == === Types and values === Core scalar types include bool, i32, u32, and f32. Vectors (e.g., vec2, vec3, vec4) and matrices (up to 4×4) are available for floating-point element types. Optional f16 (half precision) may be enabled via a WebGPU feature; availability is implementation-dependent. Atomic types (atomic
Active networking
Active networking is a communication pattern that allows packets flowing through a telecommunications network to dynamically modify the operation of the network. Active network architecture is composed of execution environments (similar to a unix shell that can execute active packets), a node operating system capable of supporting one or more execution environments. It also consists of active hardware, capable of routing or switching as well as executing code within active packets. This differs from the traditional network architecture which seeks robustness and stability by attempting to remove complexity and the ability to change its fundamental operation from underlying network components. Network processors are one means of implementing active networking concepts. Active networks have also been implemented as overlay networks. == What does it offer? == Active networking allows the possibility of highly tailored and rapid "real-time" changes to the underlying network operation. This enables such ideas as sending code along with packets of information allowing the data to change its form (code) to match the channel characteristics. The smallest program that can generate a sequence of data can be found in the definition of Kolmogorov complexity. The use of real-time genetic algorithms within the network to compose network services is also enabled by active networking. == How it relates to other networking paradigms == Active networking relates to other networking paradigms primarily based upon how computing and communication are partitioned in the architecture. === Active networking and software-defined networking === Active networking is an approach to network architecture with in-network programmability. The name derives from a comparison with network approaches advocating minimization of in-network processing, based on design advice such as the "end-to-end argument". Two major approaches were conceived: programmable network elements ("switches") and capsules, a programmability approach that places computation within packets traveling through the network. Treating packets as programs later became known as "active packets". Software-defined networking decouples the system that makes decisions about where traffic is sent (the control plane) from the underlying systems that forward traffic to the selected destination (the data plane). The concept of a programmable control plane originated at the University of Cambridge in the Systems Research Group, where (using virtual circuit identifiers available in Asynchronous Transfer Mode switches) multiple virtual control planes were made available on a single physical switch. Control Plane Technologies (CPT) was founded to commercialize this concept. == Fundamental challenges == Active network research addresses the nature of how best to incorporate extremely dynamic capability within networks. In order to do this, active network research must address the problem of optimally allocating computation versus communication within communication networks. A similar problem related to the compression of code as a measure of complexity is addressed via algorithmic information theory. One of the challenges of active networking has been the inability of information theory to mathematically model the active network paradigm and enable active network engineering. This is due to the active nature of the network in which communication packets contain code that dynamically change the operation of the network. Fundamental advances in information theory are required in order to understand such networks. == Nanoscale active networks == As the limit in reduction of transistor size is reached with current technology, active networking concepts are being explored as a more efficient means accomplishing computation and communication. More on this can be found in nanoscale networking.
Digital redlining
Digital redlining is the practice of creating and perpetuating inequities between already marginalized groups specifically through the use of digital technologies, digital content, and the internet. The concept of digital redlining is an extension of the practice of redlining in housing discrimination, a historical legal practice in the United States and Canada dating back to the 1930s where red lines were drawn on maps to indicate poor and primarily black neighborhoods that were deemed unsuitable for loans or further development, which created great economic disparities between neighborhoods. The term was popularized by Dr. Chris Gilliard, a privacy scholar, who defines digital redlining as "the creation and maintenance of tech practices, policies, pedagogies, and investment decisions that enforce class boundaries and discriminate against specific groups". Though digital redlining is related to the digital divide and techniques such as weblining and personalization, it is distinct from these concepts as part of larger complex systemic issues. It can refer to practices that create inequities of access to technology services in geographical areas, such as when internet service providers decide to not service specific geographic areas because they are perceived to be not as profitable and thus reduce access to crucial services and civic participation. It can also be used to refer to inequities caused by the policies and practices of digital technologies. For instance, with these methods inequities are accomplished through divisions that are created via algorithms which are hidden from the technology user; the use of big data and analytics allow for a much more nuanced form of discrimination that can target specific vulnerable populations. These algorithmic means are enabled through the use of unregulated data technologies that apply a score to individuals that statistically categorize personality traits or tendencies which are similar to a credit score but are proprietary to the technology companies and not under outside oversight. == Digital redlining and geography == While the roots of redlining lie in excluding populations based on geography, digital redlining occurs in both geographical and non-geographical contexts. An example of both contexts can be found in the charges brought against Facebook on March 28 of 2019, by the United States Department of Housing and Urban Development (HUD). HUD charged Facebook with violating the Fair Housing Act of 1968 by "encouraging, enabling, and causing housing discrimination through the company's advertising platform." HUD stated that Facebook allowed advertisers to “exclude people who live in a specified area from seeing an ad by drawing a red line around that area.” The discrimination called out by HUD included those that were racist, homophobic, ableist, and classist. Besides this example of geographically based digital redlining, HUD also charged that Facebook used profile information and designations to exclude classes of people. The charges stated: "Facebook enabled advertisers to exclude people whom Facebook classified as parents; non-American-born; non-Christian; interested in accessibility; interested in Hispanic culture; or a wide variety of other interests that closely align with the Fair Housing Act’s protected classes" Several media outlets pointed out HUDs own history of housing discrimination through redlining, the establishment of the Fair Housing Act to combat redlining, and how the digital platform was recreating this discriminatory practice. === Digital redlining within a geographical context === Although digital redlining refers to a complex and varied set of practices, it has been most commonly applied to practices with a geographical dimension. Common examples include when an internet service providers decide to not service specific geographic areas because those areas are seen to be not as profitable, resulting in discrimination against low-income communities, with resulting impacts on access to crucial services and civic participation. AT&T has faced specific scrutiny for this form of digital redlining, it has been reported that AT&T has been classist in its offerings of broadband internet service in areas that are more impoverished. Geographically based digital redlining can also apply to digital content or the distribution of goods sold online. Geographically based games such as Pokémon Go have been shown to offer more virtual stops and rewards in geographic areas that are less ethnically and racially diverse. In 2016, Amazon was rebuked for not offering their Prime same-day delivery service to many communities that were largely African American and had incomes that were beneath the national average. Even services such as email can be impacted, with many email administrators creating filters for flagging particular email messages as spam based on the geographical origin of the message. === Digital redlining based on personal identity === Although often aligned with discrimination that falls into a geographically based context digital redlining also refers to when vulnerable populations are targeted for or excluded from specific content or access to the internet in a way that harms them based on some aspect of their identity. Trade schools and community colleges, which typically have a more working class student body, have been found to block public internet content from their students where elite research institutions do not. The use of big data and analytics allow for a much more nuanced form of discrimination that can target specific vulnerable populations. For example, Facebook has been criticized for providing tools that allow advertisers to target ads by ethnic affinity and gender, effectively blocking minorities from seeing specific ads for housing and employment. In October 2019, a major class action lawsuit was filed against Facebook alleging gender and age discrimination in financial advertising. A broad array of consumers can be particularly vulnerable to digital redlining when it is used outside of a geographical context. Besides targeting vulnerable populations based on traditional and legally recognized classifications such as race, gender, age, etc., it has been shown that personal data mined and then resold by brokers can be used to target those who have been identified as suffering from Alzheimer's or dementia, or simply identified as impulse buyers or gullible. == Term distinctions == === Distinctions between weblining and digital redlining === Earlier distinctions have been made between weblining—the process of charging customers different prices based on profile information --- and internet or digital redlining, with digital redlining being focused not on pricing but access. As early as 2002 the Gale Encyclopedia of E-Commerce puts forth the distinction more in use today: weblining is the pervasive and generally accepted (or at least tolerated) practice of personalizing access to products and services in ways invisible to the user; digital redlining is when such personalized, data-driven schemes perpetuate traditional advantages of privileged demographics. As weblining has become more ubiquitous, the term has fallen out of use in favor of the more general term personalization. === Distinctions between the digital divide and digital redlining === Scholars have often drawn connections between the digital divide and digital redlining. In practice, the digital divide is seen as one of a number of impacts of digital redlining, and digital redlining is one of a number of ways in which the divide is maintained or extended. == Criticisms == A 2001 report looked to find if the reason for a gap in access to broadband internet by low-income and minority populations was due to a lack of availability or due to other factors. The report found that there was "little evidence of digital redlining based on income or black or Hispanic concentrations" but that there was mixed evidence of redlining based on areas in which Native American or Asian populations were larger.
User-generated content
User-generated content (UGC), alternatively known as user-created content (UCC), is content generated by users of the Internet such as images, videos, audio, text, testimonials, software, and user interactions. Online content aggregation platforms such as social media, discussion forums and wikis by their interactive and social nature, no longer produce multimedia content but provide tools to produce, collaborate, and share a variety of content, which can affect the attitudes and behaviors of the audience in various aspects. This transforms the role of consumers from passive spectators to active participants. User-generated content is used for a wide range of applications, including problem processing, news, entertainment, customer engagement, advertising, gossip, research and more. It is an example of the democratization of content production and the flattening of traditional media hierarchies. The BBC adopted a user-generated content platform for its websites in 2005, and Time magazine named "You" as the Person of the Year in 2006, referring to the rise in the production of UGC on Web 2.0 platforms. CNN also developed a similar user-generated content platform, known as iReport. There are other examples of news channels implementing similar protocols, especially in the immediate aftermath of a catastrophe or terrorist attack. Social media users can provide key eyewitness content and information that may otherwise have been inaccessible. Since 2020, there has been an increasing number of businesses who are utilizing User Generated Content (UGC) to promote their products and services. Several factors significantly influence how UGC is received, including the quality of the content, the credibility of the creator, and viewer engagement. These elements can impact users' perceptions and trust towards the brand, as well as influence the buying intentions of potential customers. UGC has proven to be an effective method for brands to connect with consumers, drawing their attention through the sharing of experiences and information on social media platforms. Due to new media and technology affordances, such as low cost and low barriers to entry, the Internet is an easy platform to create and dispense user-generated content, allowing the dissemination of information at a rapid pace in the wake of an event. == Definition == The advent of user-generated content marked a shift among media organizations from creating online content to providing facilities for amateurs to publish their own content. User-generated content has also been characterized as citizen media as opposed to the "packaged goods media" of the past century. Citizen Media is audience-generated feedback and news coverage. People give their reviews and share stories in the form of user-generated and user-uploaded audio and user-generated video. The former is a two-way process in contrast to the one-way distribution of the latter. Conversational or two-way media is a key characteristic of so-called Web 2.0, which encourages the publishing of one's own content and commenting on other people's content. The role of the passive audience, therefore, has shifted since the birth of new media, and an ever-growing number of participatory users are taking advantage of these interactive opportunities, especially on the Internet, to create independent content. Grassroots experimentation then generated an innovation in sounds, artists, techniques, and associations with audiences, which then are being used in mainstream media. The active, participatory, and creative audience is prevailing today with relatively accessible media, tools, and applications, and its culture is in turn affecting mass media corporations and global audiences. The Organisation for Economic Co-operation and Development (OECD) has defined three core variables for UGC: Accessible Content: User-generated content (UGC) is publicly produced through platforms located on the Internet and is available to any individual browsing such a publicly accessible website or a public social media account. There are other contexts where users must remain in a community or closed group to access and publish on such platforms (for example, wikis). This is a way of differentiating that although the content is accessible to the audience, there are certain restrictions for the users who generates the content. Creative effort: Creative effort was put into creating the work or adapting existing works to construct a new one; i.e. users must add their own value to the work. UGC often also has a collaborative element to it, as is the case with websites that users can edit collaboratively. For example, merely copying a portion of a television show and posting it to an online video website (an activity frequently seen on the UGC sites) would not be considered UGC. However, uploading photographs, expressing one's thoughts in a blog post or creating a new music video could be considered UGC. Yet the minimum amount of creative effort is hard to define and depends on the context. Creation outside of professional routines and practices: User-generated content is generally created outside of professional routines and practices. It often does not have an institutional or a commercial market context. In extreme cases, UGC may be produced by non-professionals without the expectation of profit or remuneration. Motivating factors include connecting with peers, achieving a certain level of fame, notoriety, or prestige, and the desire to express oneself. == Media pluralism == According to Cisco, in 2016 an average of 96,000 petabytes was transferred monthly over the Internet, more than twice as many as in 2012. In 2016, the number of active websites surpassed 1 billion, up from approximately 700 million in 2012. Reaching 1.66 billion daily active users in Q4 2019, Facebook has emerged as the most popular social media platform globally. Other social media platforms are also dominant at the regional level such as: Twitter in Japan, Naver in the Republic of Korea, Instagram (owned by Facebook) and LinkedIn (owned by Microsoft) in Africa, VKontakte (VK) and Odnoklassniki (eng. Classmates) in Russia and other countries in Central and Eastern Europe, WeChat and QQ in China. However, a concentration phenomenon is occurring globally giving dominance to a few online platforms that become popular for some unique features they provide, most commonly for the added privacy they offer users through disappearing messages or end-to-end encryption (e.g. Jami, Signal, Snapchat, Telegram, Viber, and WhatsApp), but they have tended to occupy niches and to facilitate the exchanges of information that remain rather invisible to larger audiences. Production of freely accessible information has been increasing since 2012. In January 2017, Wikipedia had more than 43 million articles, almost twice as many as in January 2012. This corresponded to a progressive diversification of content and an increase in contributions in languages other than English. In 2017, less than 12 percent of Wikipedia content was in English, down from 18 percent in 2012. Graham, Straumann, and Hogan say that the increase in the availability and diversity of content has not radically changed the structures and processes for the production of knowledge. For example, while content on Africa has dramatically increased, a significant portion of this content has continued to be produced by contributors operating from North America and Europe, rather than from Africa itself. == History == The massive, multi-volume Oxford English Dictionary was exclusively composed of user-generated content. In 1857, Richard Chenevix Trench of the London Philological Society sought public contributions throughout the English-speaking world for the creation of the first edition of the OED. As Simon Winchester recounts: So what we're going to do, if I have your agreement that we're going to produce such a dictionary, is that we're going to send out invitations, were going to send these invitations to every library, every school, every university, every book shop that we can identify throughout the English-speaking world... everywhere where English is spoken or read with any degree of enthusiasm, people will be invited to contribute words. And the point is, the way they do it, the way they will be asked and instructed to do it, is to read voraciously and whenever they see a word, whether it's a preposition or a sesquipedalian monster, they are to... if it interests them and if where they read it, they see it in a sentence that illustrates the way that that word is used, offers the meaning of the day to that word, then they are to write it on a slip of paper... the top left-hand side you write the word, the chosen word, the catchword, which in this case is 'twilight'. Then the quotation, the quotation illustrates the meaning of the word. And underneath it, the citation, where it came from, whether it was printed or whether it was in manuscri
Eigenface
An eigenface ( EYE-gən-) is the name given to a set of eigenvectors when used in the computer vision problem of human face recognition. The approach of using eigenfaces for recognition was developed by Sirovich and Kirby and used by Matthew Turk and Alex Pentland in face classification. The eigenvectors are derived from the covariance matrix of the probability distribution over the high-dimensional vector space of face images. The eigenfaces themselves form a basis set of all images used to construct the covariance matrix. This produces dimension reduction by allowing the smaller set of basis images to represent the original training images. Classification can be achieved by comparing how faces are represented by the basis set. == History == The eigenface approach began with a search for a low-dimensional representation of face images. Sirovich and Kirby showed that principal component analysis could be used on a collection of face images to form a set of basis features. These basis images, known as eigenpictures, could be linearly combined to reconstruct images in the original training set. If the training set consists of M images, principal component analysis could form a basis set of N images, where N < M. The reconstruction error is reduced by increasing the number of eigenpictures; however, the number needed is always chosen less than M. For example, if you need to generate a number of N eigenfaces for a training set of M face images, you can say that each face image can be made up of "proportions" of all the K "features" or eigenfaces: Face image1 = (23% of E1) + (2% of E2) + (51% of E3) + ... + (1% En). In 1991 M. Turk and A. Pentland expanded these results and presented the eigenface method of face recognition. In addition to designing a system for automated face recognition using eigenfaces, they showed a way of calculating the eigenvectors of a covariance matrix such that computers of the time could perform eigen-decomposition on a large number of face images. Face images usually occupy a high-dimensional space and conventional principal component analysis was intractable on such data sets. Turk and Pentland's paper demonstrated ways to extract the eigenvectors based on matrices sized by the number of images rather than the number of pixels. Once established, the eigenface method was expanded to include methods of preprocessing to improve accuracy. Multiple manifold approaches were also used to build sets of eigenfaces for different subjects and different features, such as the eyes. == Generation == A set of eigenfaces can be generated by performing a mathematical process called principal component analysis (PCA) on a large set of images depicting different human faces. Informally, eigenfaces can be considered a set of "standardized face ingredients", derived from statistical analysis of many pictures of faces. Any human face can be considered to be a combination of these standard faces. For example, one's face might be composed of the average face plus 10% from eigenface 1, 55% from eigenface 2, and even −3% from eigenface 3. Remarkably, it does not take many eigenfaces combined together to achieve a fair approximation of most faces. Also, because a person's face is not recorded by a digital photograph, but instead as just a list of values (one value for each eigenface in the database used), much less space is taken for each person's face. The eigenfaces that are created will appear as light and dark areas that are arranged in a specific pattern. This pattern is how different features of a face are singled out to be evaluated and scored. There will be a pattern to evaluate symmetry, whether there is any style of facial hair, where the hairline is, or an evaluation of the size of the nose or mouth. Other eigenfaces have patterns that are less simple to identify, and the image of the eigenface may look very little like a face. The technique used in creating eigenfaces and using them for recognition is also used outside of face recognition: handwriting recognition, lip reading, voice recognition, sign language/hand gestures interpretation and medical imaging analysis. Therefore, some do not use the term eigenface, but prefer to use 'eigenimage'. === Practical implementation === To create a set of eigenfaces, one must: Prepare a training set of face images. The pictures constituting the training set should have been taken under the same lighting conditions, and must be normalized to have the eyes and mouths aligned across all images. They must also be all resampled to a common pixel resolution (r × c). Each image is treated as one vector, simply by concatenating the rows of pixels in the original image, resulting in a single column with r × c elements. For this implementation, it is assumed that all images of the training set are stored in a single matrix T, where each column of the matrix is an image. Subtract the mean. The average image a has to be calculated and then subtracted from each original image in T. Calculate the eigenvectors and eigenvalues of the covariance matrix S. Each eigenvector has the same dimensionality (number of components) as the original images, and thus can itself be seen as an image. The eigenvectors of this covariance matrix are therefore called eigenfaces. They are the directions in which the images differ from the mean image. Usually this will be a computationally expensive step (if at all possible), but the practical applicability of eigenfaces stems from the possibility to compute the eigenvectors of S efficiently, without ever computing S explicitly, as detailed below. Choose the principal components. Sort the eigenvalues in descending order and arrange eigenvectors accordingly. The number of principal components k is determined arbitrarily by setting a threshold ε on the total variance. Total variance v = ( λ 1 + λ 2 + . . . + λ n ) {\displaystyle v=(\lambda _{1}+\lambda _{2}+...+\lambda _{n})} , n = number of components, and λ {\displaystyle \lambda } represents component eigenvalue. k is the smallest number that satisfies ( λ 1 + λ 2 + . . . + λ k ) v > ϵ {\displaystyle {\frac {(\lambda _{1}+\lambda _{2}+...+\lambda _{k})}{v}}>\epsilon } These eigenfaces can now be used to represent both existing and new faces: we can project a new (mean-subtracted) image on the eigenfaces and thereby record how that new face differs from the mean face. The eigenvalues associated with each eigenface represent how much the images in the training set vary from the mean image in that direction. Information is lost by projecting the image on a subset of the eigenvectors, but losses are minimized by keeping those eigenfaces with the largest eigenvalues. For instance, working with a 100 × 100 image will produce 10,000 eigenvectors. In practical applications, most faces can typically be identified using a projection on between 100 and 150 eigenfaces, so that most of the 10,000 eigenvectors can be discarded. === Matlab example code === Here is an example of calculating eigenfaces with Extended Yale Face Database B. To evade computational and storage bottleneck, the face images are sampled down by a factor 4×4=16. Note that although the covariance matrix S generates many eigenfaces, only a fraction of those are needed to represent the majority of the faces. For example, to represent 95% of the total variation of all face images, only the first 43 eigenfaces are needed. To calculate this result, implement the following code: === Computing the eigenvectors === Performing PCA directly on the covariance matrix of the images is often computationally infeasible. If small images are used, say 100 × 100 pixels, each image is a point in a 10,000-dimensional space and the covariance matrix S is a matrix of 10,000 × 10,000 = 108 elements. However the rank of the covariance matrix is limited by the number of training examples: if there are N training examples, there will be at most N − 1 eigenvectors with non-zero eigenvalues. If the number of training examples is smaller than the dimensionality of the images, the principal components can be computed more easily as follows. Let T be the matrix of preprocessed training examples, where each column contains one mean-subtracted image. The covariance matrix can then be computed as S = TTT and the eigenvector decomposition of S is given by S v i = T T T v i = λ i v i {\displaystyle \mathbf {Sv} _{i}=\mathbf {T} \mathbf {T} ^{T}\mathbf {v} _{i}=\lambda _{i}\mathbf {v} _{i}} However TTT is a large matrix, and if instead we take the eigenvalue decomposition of T T T u i = λ i u i {\displaystyle \mathbf {T} ^{T}\mathbf {T} \mathbf {u} _{i}=\lambda _{i}\mathbf {u} _{i}} then we notice that by pre-multiplying both sides of the equation with T, we obtain T T T T u i = λ i T u i {\displaystyle \mathbf {T} \mathbf {T} ^{T}\mathbf {T} \mathbf {u} _{i}=\lambda _{i}\mathbf {T} \mathbf {u} _{i}} Meaning that, if ui is an eigenvector of TTT, then vi = Tui is an eigenvector of S. If we have
Bulletin (service)
Bulletin was an online newsletter platform launched by Facebook on July 6, 2021, that allows notable writers to make announcements directly to their subscribers. Its competitors included Substack, of which Bulletin was called a "near-clone." Writers participating in the platform's launch included Malcolm Gladwell, Mitch Albom, Tan France, Jessica Yellin, Jane Wells, Erin Andrews and Dorie Greenspan. Facebook CEO Mark Zuckerberg stated that Bulletin represented the first time that the company had "built a project that is directly for journalists and individual writers." In October 2022 Meta announced the shutdown of Bulletin. The platform went into read only mode in January 2023 and became unavailable in April 2023. == History == Facebook announced Bulletin as its online newsletter platform on June 29, 2021. and launched by the company on July 6, 2021. Facebook CEO Mark Zuckerberg touted the service by saying that Bulletin represented the first time that the company had "built a project that is directly for journalists and individual writers." Writers participating in the platform's launch included Malcolm Gladwell, Mitch Albom, Tan France, Jessica Yellin, Jane Wells, Erin Andrews and Dorie Greenspan. == Reception == Unlike competitor such as Substack, Facebook indicated upon service's launch that it would not take a cut of subscription fees of writers using that platform. According to Washington Post technology writer Will Oremus, the move was criticized by those who viewed it as a form of predatory pricing intended by Facebook to force those competitors out of business. Sandeep Vaheesan, legal director of the think tank Open Markets, called for the government to reexamine predatory pricing as a violation of antitrust law, saying, "We want companies to compete by making better products, investing in new equipment and tech — not purely relying on their financial advantages to capture market share."