Messaging Layer Security (MLS) is a security layer for end-to-end encrypted messages. It is maintained by the MLS working group of the Internet Engineering Task Force (IETF), and is designed to provide an efficient and practical security mechanism for groups as large as 50,000 and for those who access chat systems from multiple devices. == Security properties == Security properties of MLS include message confidentiality, message integrity and authentication, membership authentication, asynchronicity, forward secrecy, post-compromise security, and scalability. == History == The idea was born in 2016 and first discussed in an unofficial meeting during IETF 96 in Berlin with attendees from Wire, Mozilla and Cisco. Initial ideas were based on pairwise encryption for secure 1:1 and group communication. In 2017, an academic paper introducing Asynchronous Ratcheting Trees was published by the University of Oxford and Facebook setting the focus on more efficient encryption schemes. The first BoF took place in February 2018 at IETF 101 in London. The founding members are Mozilla, Facebook, Wire, Google, Twitter, University of Oxford, and INRIA. On March 29, 2023, the IETF approved publication of Messaging Layer Security (MLS) as a new standard. It was officially published on July 19, 2023. At that time, Google announced it intended to add MLS to the end to end encryption used by Google Messages over Rich Communication Services (RCS). In March 2025, the GSMA announced the Universal Profile 3.0 standard of RCS would support MLS and Apple announced it would support this RCS standard on Apple Messages. Both Google Messages and Apple Messages began the rollout of MLS E2EE over RCS in May 2026. Matrix is one of the protocols declaring migration to MLS. In 2026, Discord rolled out end-to-end encryption on voice and video calls, using MLS for scalable group key exchanges. Research on adding post-quantum cryptography (PQC) to MLS is ongoing. The IETF has prepared an Internet-Draft using PQC algorithms in MLS. == Implementations ==
Acoustic model
An acoustic model is used in automatic speech recognition to represent the relationship between an audio signal and the phonemes or other linguistic units that make up speech. The model is learned from a set of audio recordings and their corresponding transcripts. It is created by taking audio recordings of speech, and their text transcriptions, and using software to create statistical representations of the sounds that make up each word. == Background == Modern speech recognition systems use both an acoustic model and a language model to represent the statistical properties of speech. The acoustic model models the relationship between the audio signal and the phonetic units in the language. The language model is responsible for modeling the word sequences in the language. These two models are combined to get the top-ranked word sequences corresponding to a given audio segment. Most modern speech recognition systems operate on the audio in small chunks known as frames with an approximate duration of 10ms per frame. The raw audio signal from each frame can be transformed by applying the mel-frequency cepstrum. The coefficients from this transformation are commonly known as mel-frequency cepstral coefficients (MFCCs) and are used as an input to the acoustic model along with other features. Recently, the use of convolutional neural networks has led to major improvements in acoustic modeling. == Speech audio characteristics == Audio can be encoded at different sampling rates (i.e. samples per second – the most common being: 8, 16, 32, 44.1, 48, and 96 kHz), and different bits per sample (the most common being: 8-bits, 16-bits, 24-bits or 32-bits). Speech recognition engines work best if the acoustic model they use was trained with speech audio which was recorded at the same sampling rate/bits per sample as the speech being recognized. == Telephony-based speech recognition == The limiting factor for telephony based speech recognition is the bandwidth at which speech can be transmitted. For example, a standard land-line telephone only has a bandwidth of 64 kbit/s at a sampling rate of 8 kHz and 8-bits per sample (8000 samples per second 8-bits per sample = 64000 bit/s). Therefore, for telephony based speech recognition, acoustic models should be trained with 8 kHz/8-bit speech audio files. In the case of voice over IP, the codec determines the sampling rate/bits per sample of speech transmission. Codecs with a higher sampling rate/bits per sample for speech transmission (which improve the sound quality) necessitate acoustic models trained with audio data that matches that sampling rate/bits per sample. == Desktop-based speech recognition == For speech recognition on a standard desktop PC, the limiting factor is the sound card. Most sound cards today can record at sampling rates of between 16–48 kHz of audio, with bit rates of 8- to 16-bits per sample, and playback at up to 96 kHz. As a general rule, a speech recognition engine works better with acoustic models trained with speech audio data recorded at higher sampling rates/bits per sample. But using audio with too high a sampling rate/bits per sample can slow the recognition engine down. A compromise is needed. Thus for desktop speech recognition, the current standard is acoustic models trained with speech audio data recorded at sampling rates of 16 kHz/16 bits per sample.
IBM Retail Store Systems
This article describes IBM point of sale equipment from 1973 with the introduction of the IBM 3650 till 1986 with the introduction of the IBM 4680. IBM continued to announced new retail products until the sale of the IBM Retail Store Solutions business to Toshiba TEC, announced on 17 April 17 2012. == Background == IBM began selling retail point of sale systems starting in 1973 with the IBM 3650 Retail Store System aimed at department and chain stores and the IBM 3660 Supermarket System designed for supermarkets. The IBM 3650 was announced alongside other IBM vertical industry systems such as the IBM 3600 Finance Communication System, and the IBM 3790 communications system, the combination of which IBM described as a "revolution in terminal based systems". All of these systems relied on a significant number of developments across IBM: New chips: Large Scale Integration allowed advanced Field Effect Transistor logic chips that packed far more transistors onto a new metalized one-inch square ceramic substrate Gas panels: Developed as an alternative to cathode ray tubes, the neon argon gas panel provided clear and flicker-free images. Modem communications: Synchronous Data Link Control provided lower-cost communications over telephone lines New disks: The "Gulliver" disk file that supplied a hard drive smaller than three cubic feet and also the "Igar" diskette drive Smaller printers: A disk printer system called "spica" that used a rotating disk print element with engraved print elements that are struck by a single hammer as the disk rotates Belt printers: A new system, known as "Lynx," using a removable belt that was significantly cheaper, quieter and simpler than earlier chain printers Keyboards: New keyboard technology called "Calico" that could build a wide variety of keyboards using common manufacturing facilities Power supplies: Transistorised Switching Regulators or TsRs: compact power supplies that are one third to one-fourth the size of previous generations === Store Loop (SLOOP) architecture === The 36xx retail terminals are connected to the store controller via a loop also called a Store Loop, similar to that used by the IBM 3600 Finance System. If a terminal detects an error, it runs a self-diagnosis routine, displays an error code to the operator, and uses bypass circuitry to remove itself from the loop and allow the loop to continue operating. If the loop fails, the most downstream terminal transmits an error code to the controller. Intermittent errors are written to disk on the store controller. === Supplies Manufacturing === While IBM's Data Processing Division created the retail store systems, it's Information Record Division (IRD) also saw signifiant opportunity in manufacturing supplies for retail systems. As an example in their Dayton NJ plant they used a high-speed Webtron press to create up to 1 million magnet merchandise tags per shift. == IBM 3650 Retail Store System == The 3650 System is a family of products designed to computerise a retail store, both at the point of sale and for back office store management functions. It includes a method to generate encoded tickets for merchandise, rather than use the Universal Product Code (UPC). The key devices for the system were as follows: === Shop Floor === ==== 3653 Point of Sale Terminal ==== Designed for the store floor, it is a loop attached device with: a wire matrix printer with 3 stations: cash receipt, sales-check and transaction journal. a keyboard with 10 numeric keys and 19 function keys an 8 digit display and description lights. in addition to the 8 digits it also displays the following characters: "$", "." and "-" operator guidance panel with 20 backlit captions status indicators a cash drawer a check verification station. Options include a wand magnet label reader with a 4 foot flexible cord, and locks for the journal tape and the till cover. The terminal effectively loads its software remotely from the 3651 over the loop, which IBM calls an IML (initial microcode load). It can also be IMLed locally using a tape cassette recorder. IBM later offered a choice of OEM Wand Attachments that could be ordered by RPQ that could use OCR or scan UPCs, instead of a wand magnet label reader. Only one wand could be attached to a specific 3653. There are two models: Model 1, which is not programmable. Was announced 10 August 1973. Model P1, which is customer programmable. Has 36 KB of storage expandable to 60 KB. Was announced 13 October 1978. === Back office equipment === ==== 3651 Store Controller ==== Controls data flow inside either a single store or multiple stores and sends retail transactions to a mainframe using a modem. For point of sale it performed functions such as: Automatic price lookup from a master price file Automatic distribution of net sales by up to 54 departments Automatic application of applicable discounts and sales taxes Automatic control of food stamp maximums Check authorization facilities For back office it also helped report preparation such as: store summary individual cashier performance store office reconciliation sales by up to 54 departments Current inquiries for department sales; cashier performance & cash position; store cash position. Inquiries and changes to the master price records and operator authorization control records. Setting the time and date for the internal clock. Running the customer checkouts in training mode. Printing of messages received from the host mainframe Entry of messages to send to the host mainframe Reporting of customer stock returns Updating the system with data received from the mainframe Preparing shelf Labels Basic features include: Each loop attaches up to 63 or 64 terminals depending on traffic volumes and desired response times Has an error and operator panel. There were many models including: A25 Has a 5 MB internal disk. Has 60K of memory expandable to 76KB. Supports one store loop. Attaches to 3275, 3653 and 3663. Announced 19 May 1978, withdrawn 19 February 1981 B25 Same as a A25 with a 9.2 MB internal disk. Announced 19 May 1978 C25 Announced 15 May 1981, withdrawn 15 December 1987 A50 Has a 5 MB internal disk. Announced 5 May 1975. Announced 10 August 1973, withdrawn 15 December 1987 B50 Same as B50 with a 9.2 MB internal disk. Announced 5 May 1975, withdrawn 15 December 1987 A60 Has a 5 MB internal disk. Has an integrated 3669. Attaches up to 24 3663 terminals. Announced 11 October 1973, withdrawn 15 December 1987 B60 Same as A60 with a 9.3 MB internal disk. Announced 17 November 1975, withdrawn 15 December 1987 A75 Has 5 MB internal disk. Has 60K of memory expandable to 124KB. Supports one to three store loops. Attaches to 3275, 3653, 3657, 3784 and 3663 terminals. Announced 19 May 1978 B75 Same as A75 with 9.3 MB internal disk. Announced 19 May 1978, withdrawn 15 December 1987 C75 Same as A75 with 18.6 MB internal disk. Announced 19 May 1978, withdrawn 15 December 1987 D75 Same as A75 with 27.9 MB internal disk. Announced 19 May 1978, withdrawn 15 December 1987 There were also two additional models that could be used instead of the 3651: 7480 Model 1: Has a 18.6 MB internal disk 7480 Model 2: Has a 27.9 MB internal disk ==== 3872 Modem ==== Used to attach to a 3659 for remote loops. Each 3872 can attach three 3659s. ==== 3659 Remote Communication Unit ==== Connected to an IBM 3872 and provides a remote loop for up to 64 point of sale terminals. Announced 10 August 1973, withdrawn 15 December 1987 (Model 2, announced 17 March 1976, withdrawn 20 December 1982) Intended to be used in a back office location like the store manager's office or the data entry office ==== 3275-3 Display Station ==== It is a loop attached display terminal with printer attachment hardware ==== 3784 Line Printer ==== A belt printer for higher-volume end-of-day reporting. The maximum print speed is 155 Ipm using a 48 character set. ==== 3657 Ticket Unit ==== Used to print tickets and encoded labels to attach to store merchandise. It is a loop attached device. It prints the following: 1" by 1" adhesive backed labels with up to 11 characters at 500 tickets per minute. IBM sold these in rolls of 9000 1" x 2" tickets with up to 42 encoded characters and two lines of print of up to 21 characters at 250 tickets per minute. IBM sold these in rolls of 2800 1" x 3" tickets with up to 79 encoded characters and two lines of print of up to 32 characters at 167 tickets per minute. IBM sold these in rolls of 1900 It can also batch read the tickets for validation, separating good tickets from bad ones into two cartridges. Announced 10 August 1973, withdrawn 15 December 1987 ==== 7481 Data Storage Unit ==== This optional unit is used to record transaction data and initialize terminals if the store controller is not available. It uses a built in tape drive to store this data. === Early deployments === The first customer installation of a 3650 was at a Dillard's department store in Little Rock, Arkansas, in late 1974. They placed arou
Digital scrapbooking
Digital scrapbooking is the term for the creation of a new 2D artwork by re-combining various graphic elements. It is a form of scrapbooking that is done using a personal computer, digital or scanned photos and computer graphics software. It is a relatively new form of the traditional print scrapbooking. Recent advances in technology now enable the craft to be pursued on tablets and smart devices utilising imaging apps as well as hobby specific apps, some of which have been created specifically by brands for use with their own image products. Digital scrapbooking kits are available to purchase and download at many websites that specialize in the craft. Kits contain graphics and word-art and are usually themed and color-coordinated. They usually consist of a mix of background images and "cut out" [extracted] images containing alpha channels. Once a kit has been downloaded to the computer or device, it can then be used over and over again to make new scrapbook pages (scrapbook layouts) within the software program that one chooses to use, often in combination with the users's own family photographs, scanned keepsakes and other unique personal elements scanned on a flatbed scanner. Scanning is usually done at 300dpi, to make the resulting images suitable for print. == Licensing and Copyright == Kits are sometimes licensed differently from other forms of traditional royalty-free stock images that may be purchased per-item or in sets at online stock photography sites. Some kit packs will be wholly royalty-free, but some kit makers may restrict usage to non-commercial work only. Some may specifically forbid the use of their work in projects for commercial gain, for example greetings cards and gift tags that may be made with their kits. Licensing often varies from kit to kit, even from the same maker. Some kits include derivative works of public domain material. In contrast to stock, creators of digital scrapbooking kits often require a credit or byline to indicate that their image elements have been used in a new creation. == Uses == Some artistic individuals combine digital scrapbooking with traditional scrapbooking to create what's known as hybrid scrapbooking projects. Hybrid scrapbooking involves creating layouts on the computer using digital supplies that will then be printed and combined with traditional supplies such as buttons, ribbons and other elements. Conversely, a hybrid scrapbook project may also be created using traditional paper supplies and augmented with digital elements that have been printed and cut out specifically for use on the project. Journaling may be done within the software programs to accompany images and to create digital storybooks, or scrapbooks, which are then published in photo books via various popular print-on-demand services, printed and added to traditional scrapbooks, burned to CDs or posted on the Web. Digital Scrapbooking may also be done online by uploading photos to a specialist scrapbooking website and utilising their custom built platforms and decorative image elements to complete the projects for print to finished products, for example photo books and holiday greeting cards. == Market Size == The traditional scrapbooking market appeared to decline somewhat in the USA since 2010, probably due to the 2008 financial crisis, and the digital scrapbooking market (being potentially a much cheaper form of scrapbooking) may have increased accordingly. Both markets currently appear to have recovered lost ground and expanded since the beginning of the COVID-19 pandemic as many people sought to productively fill their time during lockdowns, quarantines and self-isolation / stay at home directions. == Digital scrapbooking software == The main software programs that are typically used are Adobe Photoshop, Adobe Photoshop Elements, paint.net (freeware), Filter Forge, Corel Paintshop Pro, and GIMP. Additionally Adobe offer the Photoshop iOS product using the same code base as the desktop version to drive the app version. == Digital scrapbooking supplies == Digital scrapbooking supplies are downloaded from the Internet and then stored on a computer or external hardrive, DVD or CD media, SD cards, or in the cloud, to be used as needed. Both paid and free digital scrapbooking supplies available from numerous designers on their blogs or in e-commerce stores either as solo designers or as part of a wide cohort of designers working cooperatively in large full service e-commerce websites. Usually designed at 300ppi image resolution, digital scrapbooking product offerings and supplies often include: Full coordinated kits containing digital background “papers”, decorative alphabets, and diverse embellishments generally containing a mixture of .JPG and .PNG files; "Quick pages", flattened files containing a completed page layout with transparent photo windows in .PNG file format; Digital templates, fully layered layouts i.e. pages that have had the composition pre-designed ready for use in an imaging program or app, fully customizable for color schemes, kit choices, photographs and other embellishments, generally supplied in either .PSD or .TIF file format; Hybrid “quick pages”, i.e. layouts that are both fully designed and fully layered for customization, generally supplied in either .PSD or .TIF file format; Adobe Photoshop actions, brushes, custom shapes, paths and styles, saved in their respective native Photoshop file formats; and Corel PaintShop Pro equivalent tools.
Brand networking
Brand networking is the engagement of a social networking service around a brand by providing consumers with a platform of relevant content, elements of participation, and a currency, score, or ranking. Brand networking creates communities that serve as interactive destinations to encourage brand participation online and off. This evolved level of user participation with the brand facilitates strong relationships with consumers, leverages sales, and generates fan equity. The concept builds on the marketing literature on brand communities, which describes specialized, non-geographically bound groups of consumers organized around shared interest in a brand, and on subsequent research on social-media-based brand communities that examines how such groups operate when embedded in general-purpose networking platforms. == History == The development and growth of social networking in the early 2000s gave birth to brand networking. Brands saw the immediate potential to reach and interact with consumers through online platforms like Facebook and MySpace. At first, the ability to reach consumers through these platforms was inadequate; brands had the option to join as members or simply advertise on these sites. The potential existed to not only display advertisements to consumers, but to encourage them to interact with the brand. This is when brands made the shift to create their own networking platforms. Less evolved attempts to connect brands with consumers via networking are typically built as online platforms meant only to complement a product/service and are limited in functionality. Typically these sites offer consumers the opportunity to interact through discussion boards and group pages. The Guiding Light Community was built to complement the popular CBS television soap opera. The site offers members reward points for contributing content to discussion boards and blogs (which is all geared toward the show). == Structure == Brand networking is more than the utilization of a social networking platform; it is connecting consumers together and constructing relationships directly with the brand. Three key elements, in unity, create effective brand networking: relevant content, elements of participation, and a competitive currency. Websites in conjunction with other media types (television, radio, print) present content around a vertical industry, sector of interest, or cultural and social issues for a brand. This can be in areas such as health, marketing, or business, or any content relevant to the brand message. Such content is not only provided by the brand but also in the form of consumer-generated media. Research on brand-related user-generated content across major platforms suggests that the form and tone of consumer contributions vary by platform, with promotional content more common on some networks and response-oriented content on others. A brand provides participation with consumers online and offline. This is accomplished through the combination of typical social networking features online, such as personalised pages, friend lists, groups, and messaging, alongside elements of involvement offline. This is not simply connecting an online platform with mobile devices, but providing separate mobile features jointly with a secondary media type to drive online usage and build relationships with the brand on the go. By participating in mobile campaigns, users are interacting with the brand outside of traditional brick and mortar or e-commerce destinations. Empirical work on consumer brand engagement in social media frames such participation along cognitive, affective, and behavioural dimensions. The final element of brand networking involves incentivising participation with the other two elements. The addition of a currency or point system acts as an anchor to the brand and network and creates a competitive dynamic between consumers. These points are distributed for activity carried out outside of the networking site. By incentivising usage offline, the brand image is reinforced for the consumer and strengthens the relationship. Consumers are turned into promoters for both the brand and the users' benefit. The use of points, badges, leaderboards, and similar mechanics is described in the marketing literature as gamification, and has been linked to higher participation rates in mobile and loyalty programmes. == Fan equity == Fan equity is the idea that by locking in consumers to a brand, they are turned into fans of the brand. As fans, they promote, interact, and consume on a daily basis and become assets. Apple Inc. is one example of a company often cited as possessing fan equity. Customers of Apple are extremely brand loyal and are assets to the company. Creating a fan-generated brand is a difficult but effective method of business. Through the use of brand networking, a company is able to build a consumer or fan base that provides a strong relationship between business and consumers. The trust is formed and fans do a lot of work for the brand by word of mouth. Peer-to-peer channels are the strongest means of communication for a brand, but also one in which the brand can only influence and not control. Subsequent research links community engagement with brand trust, identifying community engagement as a mediator between social-media brand community participation and trust. This method of business is argued to be a relationship handled by the brand generally for its own gain. Many fans do not realise the work they are doing for companies by using their product or service. Facebook is a fan-based brand that has become a global phenomenon through customer use, with social media features such as sharing and commenting. With the growth of social media, marketing and advertising through social media has continued to expand. Brands can display and promote their products or services at a fast rate, with consumers sharing and contributing to the brand on a global scale. This can also be seen as online word of mouth exposure that can produce positive or negative feedback for brands. Once consumers become fans they are typically loyal, which can create positive word of mouth for a brand. Fans become a valuable asset, boosting the status and reputation of a brand. Different perceptions of brands can be linked to a person's origin or religion, which creates a difficulty when trying to enter a market or gain market share. Businesses need to be aware of the types of products or services they introduce to a specific market, ensuring they are culturally sensitive. Fan pages are created on social media to maintain the relationship between brands and consumers. By engaging and interacting with consumers, brands obtain fans and produce positive imaging. Some fans become attached to brands and are often encouraged to remain as fans through the use of celebrities endorsing the brand. Research on parasocial interaction in social-media environments suggests that one-sided emotional bonds that consumers form with endorsers and brand personae help convert ordinary followers into engaged fans.
Kolmogorov–Arnold Networks
Kolmogorov–Arnold Networks (KANs) are a type of artificial neural network architecture inspired by the Kolmogorov–Arnold representation theorem, also known as the superposition theorem. Unlike traditional multilayer perceptrons (MLPs), which rely on fixed activation functions and linear weights, KANs replace each weight with a learnable univariate function, often represented using splines. == History == KANs (Kolmogorov–Arnold Networks) were proposed by Liu et al. (2024) as a generalization of the Kolmogorov–Arnold representation theorem (KART), aiming to outperform MLPs in small-scale AI and scientific tasks. Before KANs, numerous studies explored KART's connections to neural networks or used it as a basis for designing new network architectures. In the 1980s and 1990s, early research applied KART to neural network design. Kůrková et al. (1992), Hecht-Nielsen (1987), and Nees (1994) established theoretical foundations for multilayer networks based on KART. Igelnik et al. (2003) introduced the Kolmogorov Spline Network using cubic splines to model complex functions. Sprecher (1996, 1997) introduced numerical methods for building network layers, while Nakamura et al. (1993) created activation functions with guaranteed approximation accuracy. These works linked KART's theoretical potential with practical neural network implementation. KART has also been used in other computational and theoretical fields. Coppejans (2004) developed nonparametric regression estimators using B-splines, Bryant (2008) applied it to high-dimensional image tasks, Liu (2015) investigated theoretical applications in optimal transport and image encryption, and more recently, Polar and Poluektov (2021) used Urysohn operators for efficient KART construction, while Fakhoury et al. (2022) introduced ExSpliNet, integrating KART with probabilistic trees and multivariate B-splines for improved function approximation. == Architecture == KANs are based on the Kolmogorov–Arnold representation theorem, which was linked to the 13th Hilbert problem. Given x = ( x 1 , x 2 , … , x n ) {\displaystyle x=(x_{1},x_{2},\dots ,x_{n})} consisting of n variables, a multivariate continuous function f ( x ) {\displaystyle f(x)} can be represented as: f ( x ) = f ( x 1 , … , x n ) = ∑ q = 1 2 n + 1 Φ q ( ∑ p = 1 n φ q , p ( x p ) ) {\displaystyle f(x)=f(x_{1},\dots ,x_{n})=\sum _{q=1}^{2n+1}\Phi _{q}\left(\sum _{p=1}^{n}\varphi _{q,p}(x_{p})\right)} (1) This formulation contains two nested summations: an outer and an inner sum. The outer sum ∑ q = 1 2 n + 1 {\displaystyle \sum _{q=1}^{2n+1}} aggregates 2 n + 1 {\displaystyle 2n+1} terms, each involving a function Φ q : R → R {\displaystyle \Phi _{q}:\mathbb {R} \to \mathbb {R} } . The inner sum ∑ p = 1 n {\displaystyle \sum _{p=1}^{n}} computes n terms for each q, where each term φ q , p : [ 0 , 1 ] → R {\displaystyle \varphi _{q,p}:[0,1]\to \mathbb {R} } is a continuous function of the single variable x p {\displaystyle x_{p}} . The inner continuous functions φ q , p {\displaystyle \varphi _{q,p}} are universal, independent of f {\displaystyle f} , while the outer functions Φ q {\displaystyle \Phi _{q}} depend on the specific function f {\displaystyle f} being represented. The representation (1) holds for all multivariate functions f {\displaystyle f} as proved in . If f {\displaystyle f} is continuous, then the outer functions Φ q {\displaystyle \Phi _{q}} are continuous; if f {\displaystyle f} is discontinuous, then the corresponding Φ q {\displaystyle \Phi _{q}} are generally discontinuous, while the inner functions φ q , p {\displaystyle \varphi _{q,p}} remain the same universal functions. Liu et al. proposed the name KAN. A general KAN network consisting of L layers takes x to generate the output as: K A N ( x ) = ( Φ L − 1 ∘ Φ L − 2 ∘ ⋯ ∘ Φ 1 ∘ Φ 0 ) x {\displaystyle \mathrm {KAN} (x)=(\Phi ^{L-1}\circ \Phi ^{L-2}\circ \cdots \circ \Phi ^{1}\circ \Phi ^{0})x} (3) Here, Φ l {\displaystyle \Phi ^{l}} is the function matrix of the l-th KAN layer or a set of pre-activations. Let i denote the neuron of the l-th layer and j the neuron of the (l+1)-th layer. The activation function φ j , i l {\displaystyle \varphi _{j,i}^{l}} connects (l, i) to (l+1, j): φ j , i l , l = 0 , … , L − 1 , i = 1 , … , n l , j = 1 , … , n l + 1 {\displaystyle \varphi _{j,i}^{l},\quad l=0,\dots ,L-1,\;i=1,\dots ,n_{l},\;j=1,\dots ,n_{l+1}} (4) where nl is the number of nodes of the l-th layer. Thus, the function matrix Φ l {\displaystyle \Phi ^{l}} can be represented as an n l + 1 × n l {\displaystyle n_{l+1}\times n_{l}} matrix of activations: x l + 1 = ( φ 1 , 1 l ( ⋅ ) φ 1 , 2 l ( ⋅ ) ⋯ φ 1 , n l l ( ⋅ ) φ 2 , 1 l ( ⋅ ) φ 2 , 2 l ( ⋅ ) ⋯ φ 2 , n l l ( ⋅ ) ⋮ ⋮ ⋱ ⋮ φ n l + 1 , 1 l ( ⋅ ) φ n l + 1 , 2 l ( ⋅ ) ⋯ φ n l + 1 , n l l ( ⋅ ) ) x l {\displaystyle x^{l+1}={\begin{pmatrix}\varphi _{1,1}^{l}(\cdot )&\varphi _{1,2}^{l}(\cdot )&\cdots &\varphi _{1,n_{l}}^{l}(\cdot )\\\varphi _{2,1}^{l}(\cdot )&\varphi _{2,2}^{l}(\cdot )&\cdots &\varphi _{2,n_{l}}^{l}(\cdot )\\\vdots &\vdots &\ddots &\vdots \\\varphi _{n_{l+1},1}^{l}(\cdot )&\varphi _{n_{l+1},2}^{l}(\cdot )&\cdots &\varphi _{n_{l+1},n_{l}}^{l}(\cdot )\end{pmatrix}}x^{l}} == Implementations == To make the KAN layers optimizable, the inner function is formed by the combination of spline and basic functions as the formula: φ ( x ) = w b b ( x ) + w s spline ( x ) {\displaystyle \varphi (x)=w_{b}\,b(x)+w_{s}\,{\text{spline}}(x)} where b ( x ) {\displaystyle b(x)} is the basic function, usually defined as s i l u ( x ) = x / ( 1 + e x ) {\displaystyle silu(x)=x/(1+e^{x})} and w b {\displaystyle w_{b}} is the base weight matrix. Also, w s {\displaystyle w_{s}} is the spline weight matrix and spline ( x ) {\displaystyle {\text{spline}}(x)} is the spline function. The spline function can be a sum of B-splines. spline ( x ) = ∑ i c i B i ( x ) {\displaystyle {\text{spline}}(x)=\sum _{i}c_{i}B_{i}(x)} Many studies suggested to use other polynomial and curve functions instead of B-spline to create new KAN variants. == Functions used == The choice of functional basis strongly influences the performance of KANs. Common function families include: B-splines: Provide locality, smoothness, and interpretability; they are the most widely used in current implementations. RBFs (include Gaussian RBFs): Capture localized features in data and are effective in approximating functions with non-linear or clustered structures. Chebyshev polynomials: Offer efficient approximation with minimized error in the maximum norm, making them useful for stable function representation. Rational function: Useful for approximating functions with singularities or sharp variations, as they can model asymptotic behavior better than polynomials. Fourier series: Capture periodic patterns effectively and are particularly useful in domains such as physics-informed machine learning. Wavelet functions (DoG, Mexican hat, Morlet, and Shannon): Used for feature extraction as they can capture both high-frequency and low-frequency data components. Piecewise linear functions: Provide efficient approximation for multivariate functions in KANs. == Usage == In some modern neural architectures like convolutional neural networks (CNNs), recurrent neural networks (RNNs), and Transformers, KANs are typically used as drop-in substitutes for MLP layers. Despite KANs' general-purpose design, researchers have created and used them for a number of tasks: Scientific machine learning (SciML): Function fitting, partial differential equations (PDEs) and physical/mathematical laws. Continual learning: KANs better preserve previously learned information during incremental updates, avoiding catastrophic forgetting due to the locality of spline adjustments. Graph neural networks: Extensions such as Kolmogorov–Arnold Graph Neural Networks (KA-GNNs) integrate KAN modules into message-passing architectures, showing improvements in molecular property prediction tasks. Sensor data processing: Kolmogorov–Arnold Networks (KANs) have recently been applied to sensor data processing due to their ability to model complex nonlinear relationships with relatively few parameters and improved interpretability compared to conventional multilayer perceptrons. Applications include industrial soft sensors, biomedical signal analysis, remote sensing, and environmental monitoring systems. == Drawbacks == KANs can be computationally intensive and require a large number of parameters due to their use of polynomial functions to capture data.
Power cycling
Power cycling is the act of turning a piece of equipment, usually a computer, off and then on again. Reasons for power cycling include having an electronic device reinitialize its set of configuration parameters or recover from an unresponsive state of its mission critical functionality, such as in a crash or hang situation. Power cycling can also be used to reset network activity inside a modem. It can also be among the first steps for troubleshooting an issue. == Overview == Power cycling can be done manually, usually using the power switch on the device, or remotely, through some type of external device connected to the power input. In the data center environment, remote control power cycling can usually be done through a power distribution unit, over the network. In the home environment, this can be done through home automation powerline communications. Most Internet service providers publish a "how-to" on their website showing their customers the correct procedure to power cycle their devices. Power cycling is a common diagnostic procedure usually performed first when a computer system freezes. However, frequently power cycling a computer can cause thermal stress. Reset has an equal effect on the software but may be less problematic for the hardware as power is not interrupted. == Historical uses == On all Apollo missions to the moon, the landing radar was required to acquire the surface before a landing could be attempted. But on Apollo 14, the landing radar was unable to lock on. Mission control told the astronauts to cycle the power. They did, the radar locked on just in time, and the landing was completed. During the Rosetta mission to comet 67P/Churyumov–Gerasimenko, the Philae lander did not return the expected telemetry on awakening after arrival at the comet. The problem was diagnosed as "somehow a glitch in the electronics", engineers cycled the power, and the lander awoke correctly. During the launch of the billion dollar AEHF-6 satellite on 26 March 2020 by an Atlas V rocket from Cape Canaveral Space Force Station in Florida, a hold was called at T-46 seconds due to hydraulic system not responding as expected. The launch crew turned it off and back on, and the launch proceeded normally. In 2023 the Interstellar Boundary Explorer spacecraft stopped responding to commands after an anomaly. When gentler techniques failed, NASA resorted to rebooting the spacecraft with the remote equivalent of a power cycle.