In number theory, Berlekamp's root finding algorithm, also called the Berlekamp–Rabin algorithm, is the probabilistic method of finding roots of polynomials over the field F p {\displaystyle \mathbb {F} _{p}} with p {\displaystyle p} elements. The method was discovered by Elwyn Berlekamp in 1970 as an auxiliary to the algorithm for polynomial factorization over finite fields. The algorithm was later modified by Rabin for arbitrary finite fields in 1979. The method was also independently discovered before Berlekamp by other researchers. == History == The method was proposed by Elwyn Berlekamp in his 1970 work on polynomial factorization over finite fields. His original work lacked a formal correctness proof and was later refined and modified for arbitrary finite fields by Michael Rabin. In 1986 René Peralta proposed a similar algorithm for finding square roots in F p {\displaystyle \mathbb {F} _{p}} . In 2000 Peralta's method was generalized for cubic equations. == Statement of problem == Let p {\displaystyle p} be an odd prime number. Consider the polynomial f ( x ) = a 0 + a 1 x + ⋯ + a n x n {\textstyle f(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}} over the field F p ≃ Z / p Z {\displaystyle \mathbb {F} _{p}\simeq \mathbb {Z} /p\mathbb {Z} } of remainders modulo p {\displaystyle p} . The algorithm should find all λ {\displaystyle \lambda } in F p {\displaystyle \mathbb {F} _{p}} such that f ( λ ) = 0 {\textstyle f(\lambda )=0} in F p {\displaystyle \mathbb {F} _{p}} . == Algorithm == === Randomization === Let f ( x ) = ( x − λ 1 ) ( x − λ 2 ) ⋯ ( x − λ n ) {\textstyle f(x)=(x-\lambda _{1})(x-\lambda _{2})\cdots (x-\lambda _{n})} . Finding all roots of this polynomial is equivalent to finding its factorization into linear factors. To find such factorization it is sufficient to split the polynomial into any two non-trivial divisors and factorize them recursively. To do this, consider the polynomial f z ( x ) = f ( x − z ) = ( x − λ 1 − z ) ( x − λ 2 − z ) ⋯ ( x − λ n − z ) {\textstyle f_{z}(x)=f(x-z)=(x-\lambda _{1}-z)(x-\lambda _{2}-z)\cdots (x-\lambda _{n}-z)} where z {\displaystyle z} is some element of F p {\displaystyle \mathbb {F} _{p}} . If one can represent this polynomial as the product f z ( x ) = p 0 ( x ) p 1 ( x ) {\displaystyle f_{z}(x)=p_{0}(x)p_{1}(x)} then in terms of the initial polynomial it means that f ( x ) = p 0 ( x + z ) p 1 ( x + z ) {\displaystyle f(x)=p_{0}(x+z)p_{1}(x+z)} , which provides needed factorization of f ( x ) {\displaystyle f(x)} . === Classification of === F p {\displaystyle \mathbb {F} _{p}} elements Due to Euler's criterion, for every monomial ( x − λ ) {\displaystyle (x-\lambda )} exactly one of following properties holds: The monomial is equal to x {\displaystyle x} if λ = 0 {\displaystyle \lambda =0} , The monomial divides g 0 ( x ) = ( x ( p − 1 ) / 2 − 1 ) {\textstyle g_{0}(x)=(x^{(p-1)/2}-1)} if λ {\displaystyle \lambda } is quadratic residue modulo p {\displaystyle p} , The monomial divides g 1 ( x ) = ( x ( p − 1 ) / 2 + 1 ) {\textstyle g_{1}(x)=(x^{(p-1)/2}+1)} if λ {\displaystyle \lambda } is quadratic non-residual modulo p {\displaystyle p} . Thus if f z ( x ) {\displaystyle f_{z}(x)} is not divisible by x {\displaystyle x} , which may be checked separately, then f z ( x ) {\displaystyle f_{z}(x)} is equal to the product of greatest common divisors gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} and gcd ( f z ( x ) ; g 1 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{1}(x))} . === Berlekamp's method === The property above leads to the following algorithm: Explicitly calculate coefficients of f z ( x ) = f ( x − z ) {\displaystyle f_{z}(x)=f(x-z)} , Calculate remainders of x , x 2 , x 2 2 , x 2 3 , x 2 4 , … , x 2 ⌊ log 2 p ⌋ {\textstyle x,x^{2},x^{2^{2}},x^{2^{3}},x^{2^{4}},\ldots ,x^{2^{\lfloor \log _{2}p\rfloor }}} modulo f z ( x ) {\displaystyle f_{z}(x)} by squaring the current polynomial and taking remainder modulo f z ( x ) {\displaystyle f_{z}(x)} , Using exponentiation by squaring and polynomials calculated on the previous steps calculate the remainder of x ( p − 1 ) / 2 {\textstyle x^{(p-1)/2}} modulo f z ( x ) {\textstyle f_{z}(x)} , If x ( p − 1 ) / 2 ≢ ± 1 ( mod f z ( x ) ) {\textstyle x^{(p-1)/2}\not \equiv \pm 1{\pmod {f_{z}(x)}}} then gcd {\displaystyle \gcd } mentioned below provide a non-trivial factorization of f z ( x ) {\displaystyle f_{z}(x)} , Otherwise all roots of f z ( x ) {\displaystyle f_{z}(x)} are either residues or non-residues simultaneously and one has to choose another z {\displaystyle z} . If f ( x ) {\displaystyle f(x)} is divisible by some non-linear primitive polynomial g ( x ) {\displaystyle g(x)} over F p {\displaystyle \mathbb {F} _{p}} then when calculating gcd {\displaystyle \gcd } with g 0 ( x ) {\displaystyle g_{0}(x)} and g 1 ( x ) {\displaystyle g_{1}(x)} one will obtain a non-trivial factorization of f z ( x ) / g z ( x ) {\displaystyle f_{z}(x)/g_{z}(x)} , thus algorithm allows to find all roots of arbitrary polynomials over F p {\displaystyle \mathbb {F} _{p}} . === Modular square root === Consider equation x 2 ≡ a ( mod p ) {\textstyle x^{2}\equiv a{\pmod {p}}} having elements β {\displaystyle \beta } and − β {\displaystyle -\beta } as its roots. Solution of this equation is equivalent to factorization of polynomial f ( x ) = x 2 − a = ( x − β ) ( x + β ) {\textstyle f(x)=x^{2}-a=(x-\beta )(x+\beta )} over F p {\displaystyle \mathbb {F} _{p}} . In this particular case problem it is sufficient to calculate only gcd ( f z ( x ) ; g 0 ( x ) ) {\displaystyle \gcd(f_{z}(x);g_{0}(x))} . For this polynomial exactly one of the following properties will hold: GCD is equal to 1 {\displaystyle 1} which means that z + β {\displaystyle z+\beta } and z − β {\displaystyle z-\beta } are both quadratic non-residues, GCD is equal to f z ( x ) {\displaystyle f_{z}(x)} which means that both numbers are quadratic residues, GCD is equal to ( x − t ) {\displaystyle (x-t)} which means that exactly one of these numbers is quadratic residue. In the third case GCD is equal to either ( x − z − β ) {\displaystyle (x-z-\beta )} or ( x − z + β ) {\displaystyle (x-z+\beta )} . It allows to write the solution as β = ( t − z ) ( mod p ) {\textstyle \beta =(t-z){\pmod {p}}} . === Example === Assume we need to solve the equation x 2 ≡ 5 ( mod 11 ) {\textstyle x^{2}\equiv 5{\pmod {11}}} . For this we need to factorize f ( x ) = x 2 − 5 = ( x − β ) ( x + β ) {\displaystyle f(x)=x^{2}-5=(x-\beta )(x+\beta )} . Consider some possible values of z {\displaystyle z} : Let z = 3 {\displaystyle z=3} . Then f z ( x ) = ( x − 3 ) 2 − 5 = x 2 − 6 x + 4 {\displaystyle f_{z}(x)=(x-3)^{2}-5=x^{2}-6x+4} , thus gcd ( x 2 − 6 x + 4 ; x 5 − 1 ) = 1 {\displaystyle \gcd(x^{2}-6x+4;x^{5}-1)=1} . Both numbers 3 ± β {\displaystyle 3\pm \beta } are quadratic non-residues, so we need to take some other z {\displaystyle z} . Let z = 2 {\displaystyle z=2} . Then f z ( x ) = ( x − 2 ) 2 − 5 = x 2 − 4 x − 1 {\displaystyle f_{z}(x)=(x-2)^{2}-5=x^{2}-4x-1} , thus gcd ( x 2 − 4 x − 1 ; x 5 − 1 ) ≡ x − 9 ( mod 11 ) {\textstyle \gcd(x^{2}-4x-1;x^{5}-1)\equiv x-9{\pmod {11}}} . From this follows x − 9 = x − 2 − β {\textstyle x-9=x-2-\beta } , so β ≡ 7 ( mod 11 ) {\displaystyle \beta \equiv 7{\pmod {11}}} and − β ≡ − 7 ≡ 4 ( mod 11 ) {\textstyle -\beta \equiv -7\equiv 4{\pmod {11}}} . A manual check shows that, indeed, 7 2 ≡ 49 ≡ 5 ( mod 11 ) {\textstyle 7^{2}\equiv 49\equiv 5{\pmod {11}}} and 4 2 ≡ 16 ≡ 5 ( mod 11 ) {\textstyle 4^{2}\equiv 16\equiv 5{\pmod {11}}} . == Correctness proof == The algorithm finds factorization of f z ( x ) {\displaystyle f_{z}(x)} in all cases except for ones when all numbers z + λ 1 , z + λ 2 , … , z + λ n {\displaystyle z+\lambda _{1},z+\lambda _{2},\ldots ,z+\lambda _{n}} are quadratic residues or non-residues simultaneously. According to theory of cyclotomy, the probability of such an event for the case when λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} are all residues or non-residues simultaneously (that is, when z = 0 {\displaystyle z=0} would fail) may be estimated as 2 − k {\displaystyle 2^{-k}} where k {\displaystyle k} is the number of distinct values in λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} . In this way even for the worst case of k = 1 {\displaystyle k=1} and f ( x ) = ( x − λ ) n {\displaystyle f(x)=(x-\lambda )^{n}} , the probability of error may be estimated as 1 / 2 {\displaystyle 1/2} and for modular square root case error probability is at most 1 / 4 {\displaystyle 1/4} . == Complexity == Let a polynomial have degree n {\displaystyle n} . We derive the algorithm's complexity as follows: Due to the binomial theorem ( x − z ) k = ∑ i = 0 k ( k i ) ( − z ) k − i x i {\textstyle (x-z)^{k}=\sum \limits _{i=0}^{k}{\binom {k}{i}}(-z)^{k-i}x^{i}} , we may transition from f ( x ) {\displaystyle f(x)} to f ( x − z ) {\displaystyle f(x-z)} in O ( n 2 ) {\displaystyle O(n^{2})} time. Polynomial multiplication a
OpenIO
OpenIO offered object storage for a wide range of high-performance applications. OpenIO was founded in 2015 by Laurent Denel (CEO), Jean-François Smigielski (CTO) and five other co-founders; it leveraged open source software, developed since 2006, based on a grid technology that enabled dynamic behaviour and supported heterogenous hardware. In October 2017 OpenIO was completed a $5 million funding rounds. In July 2020 OpenIO had been acquired by OVH and withdrawn from the market to become the core technology of OVHcloud object storage offering. == Software == OpenIO is a software-defined object store that supports S3 and can be deployed on-premises, cloud-hosted or at the edge, on any hardware mix. It has been designed from the beginning for performance and cost-efficiency at any scale, and it has been optimized for Big Data, HPC and AI. OpenIO stores objects within a flat structure within a massively distributed directory with indirections, which allows the data query path to be independent of the number of nodes and the performance not to be affected by the growth of capacity. Servers are organized as a grid of nodes massively distributed, where each node takes part in directory and storage services, which ensures that there is no single point of failure and that new nodes are automatically discovered and immediately available without the need to rebalance data. The software is built on top of a technology that ensures optimal data placement based on real-time metrics and allows the addition or removal of storage devices with automatic performance and load impact optimization. For data protection OpenIO has synchronous and asynchronous replication with multiple copies, and an erasure coding implementation based on Reed-Solomon that can be deployed in one data center or geo-distributed or stretched clusters. The software has a feature that catches all events that occur in the cluster and can pass them up in the stack or to applications running on OpenIO nodes. This enables event-driven computing directly into the storage infrastructure. The open source code is available on Github and it is licensed under AGPL3 for server code and LGPL3 for client code. == Performance == OpenIO claimed in 2019 to have reached 1.372 Tbit/s write speed (171 GB/s) on a cluster of 350 physical machines. The benchmark scenario, conducted under production conditions with standard hardware (commodity servers with 7200 rpm HDDs), consisted in backing up a 38 PB Hadoop datalake via the DistCp command. This level of performance marked, according to analysts, the arrival of a new generation of object storage technologies oriented toward high performance and hyper-scalability.
Digital Cinema Package
A Digital Cinema Package (DCP) is a collection of digital files used to store and convey digital cinema (DC) audio, image, and data streams. The term was popularized by Digital Cinema Initiatives, LLC in its original recommendation for packaging DC contents. However, the industry tends to apply the term to the structure more formally known as the composition. A DCP is a container format for compositions, a hierarchical file structure that represents a title version. The DCP may carry a partial composition (e.g. not a complete set of files), a single complete composition, or multiple and complete compositions. The composition consists of a Composition Playlist (in XML format) that defines the playback sequence of a set of Track Files. Track Files carry the essence (audio, image, subtitles), which is wrapped using Material eXchange Format (MXF). Track Files must contain only one essence type. Two track files at a minimum must be present in every composition (see SMPTE ST429-2 D-Cinema Packaging – DCP Constraints, or Cinepedia): a track file carrying picture essence, and a track file carrying audio essence. The composition, consisting of a Composition Playlist (CPL) and associated track files, are distributed as a Digital Cinema Package (DCP). A composition is a complete representation of a title version, while the DCP need not carry a full composition. However, as already noted, it is commonplace in the industry to discuss the title in terms of a DCP, as that is the deliverable to the cinema. The Picture Track File essence is compressed using JPEG 2000 and the Audio Track File carries a 24-bit linear PCM uncompressed multichannel WAV file. Encryption may optionally be applied to the essence of a track file to protect it from unauthorized use. The encryption used is AES 128-bit in CBC mode. In practice, there are two versions of composition in use. The original version is called Interop DCP. In 2009, a specification was published by SMPTE (SMPTE ST 429-2 Digital Cinema Packaging – DCP Constraints) for what is commonly referred to as SMPTE DCP. SMPTE DCP is similar but not backwards compatible with Interop DCP, resulting in an uphill effort to transition the industry from Interop DCP to SMPTE DCP. SMPTE DCP requires significant constraints to ensure success in the field, as shown by ISDCF. While legacy support for Interop DCP is necessary for commercial products, new productions are encouraged to be distributed in SMPTE DCP. == Technical specifications == The DCP root folder (in the storage medium) contains a number of files, some used to store the image and audio contents, and some other used to organize and manage the whole playlist. === Picture MXF files === Picture contents may be stored in one or more reels corresponding to one or more MXF files. Each reel contains pictures as MPEG-2 or JPEG 2000 essence, depending on the adopted codec. MPEG-2 is no longer compliant with the DCI specification. JPEG 2000 is the only accepted compression format. Supported frame rates are: SMPTE (JPEG 2000) 24, 25, 30, 48, 50, and 60 fps @ 2K 24, 25, and 30 fps @ 4K 24 and 48 fps @ 2K stereoscopic MXF Interop (JPEG 2000) – Deprecated 24 and 48 fps @ 2K (MXF Interop can be encoded at 25 frame/s but support is not guaranteed) 24 fps @ 4K 24 fps @ 2K stereoscopic MXF Interop (MPEG-2) – Deprecated 23.976 and 24 fps @ 1920 × 1080 Maximum frame sizes are 2048 × 1080 for 2K DC, and 4096 × 2160 for 4K DC. Common formats are: SMPTE (JPEG 2000) Flat (1998 × 1080 or 3996 × 2160), = 1.85:1 aspect ratio Scope (2048 × 858 or 4096 × 1716), ~2.39:1 aspect ratio HDTV (1920 × 1080 or 3840 × 2160), 16:9 aspect ratio (~1.78:1) (although not specifically defined in the DCI specification, this resolution is DCI compliant per section 8.4.3.2). Full (2048 × 1080 or 4096 × 2160) (~1.9:1 aspect ratio, official name by DCI is Full Container. Not widely accepted in cinemas.) MXF Interop (MPEG-2) – Deprecated Full Frame (1920 × 1080) 12 bits per component precision (36 bits total per pixel) XYZ' colorspace; the prime mark indicates gamma encoding (gamma=2.6) Maximum bit rate is 250 Mbit/s (1.3 MBytes per frame at 24 frame per second) === Sound MXF files === Sound contents are also stored in reels corresponding to picture reels in number and duration. In case of multilingual features, separate reels are required to convey different languages. Each file contains linear PCM essence. Sampling rate is 48,000 or 96,000 samples per second Sample precision of 24 bits Linear mapping (no companding) Up to 16 independent channels === Asset map file === List of all files included in the DCP, in XML format. === Composition playlist file === Defines the playback order during presentation. The order is saved in XML format in this file; each picture and sound reel is identified by its UUID. In the following example, a reel is composed by picture and sound: === Packing list file or package key list (PKL) === All files in the composition are hashed and their hash is stored here, in XML format. This file is generally used during ingestion in a digital cinema server to verify if data have been corrupted or tampered with in some way. For example, an MXF picture reel is identified by the following
Commercial skipping
Commercial skipping is a feature of some digital video recorders that makes it possible to automatically skip commercials in recorded programs. This feature created controversy, with major television networks and movie studios claiming it violates copyright and should be banned. == History == After the video cassette recorder (VCR) became popular in the 1980s, the television industry began studying the impact of users fast forwarding through commercials. Advertising agencies fought the trend by making them more entertaining. For many years, video recorders manufactured for the Japanese market have been able to skip advertisements automatically, which is done by detecting when foreign language audio overdub tracks provided for many programmes go silent, as advertisements were broadcast with a single language only. The first digital video recorder (DVR) with a built-in commercial skipping feature was ReplayTV with its "4000 Series" and "5000 Series" units. In 2002, the main television networks and movie studios sued ReplayTV, claiming that skipping advertisements during replay violates copyright. Later, five owners of ReplayTV represented by Electronic Frontier Foundation and attorneys Ira Rothken and Richard Wiebe countersued, asking the federal judge to uphold consumers' rights to record TV shows and skip commercials, claiming that features like commercial skipping help parents protect their kids from excessive consumerism. ReplayTV ended up filing for bankruptcy in 2003 after fighting a copyright infringement suit over the ReplayTV's ability to skip commercials. === Commercial skipping software === In addition to the DVR devices which existed in the private market since the late 1990s, towards the mid-2000s, due to the significant advances in home computers, Home theater PCs started gaining popularity in the private market and many users began using their Home theater PCs in their living room for entertainment purposes. Following this, many DVR programs were developed, including popular programs such as Windows Media Center, which contained all of the features of the DVR devices in addition to advanced features such as HDTV and the use of Multiple TV Tuner Cards. Some independent developers began developing independent software capable of skipping the commercial segments when playing recorded videos, and permanently removing the commercial segments from recorded video files. By 2014, many DVR programs such as Windows Media Center, SageTV and MythTV had the capability to skip commercials segments in recorded TV broadcasts after installing third-party add-ons such as DVRMSToolbox, Comskip and ShowAnalyzer, which use various advanced techniques to locate the commercial segments in the video files and save their locations to text files. The text files can also be fed into programs such as MEncoder or DVRMSToolboxGUI which can delete the commercial segments from the recorded video files. A few third-party tools such as MCEBuddy automate detection and removal/marking of commercials. One of the weaknesses of commercial skippers is that, operating automatically, they may misidentify program material as a commercial. Some programs like MCEBuddy provide the ability to fine-tune commercial detection for groups of files (e.g. by channel or country) and provide tools to manually fine-tune commercial segments for individual files. In May 2012, the US Dish Network began offering a DVR with what it calls AutoHop. The device would automatically skip commercials when displaying programming that the viewer had previously recorded with the PrimeTime Anytime feature. It does not skip ads on any live programs. US broadcasters were angered at the news, and FOX embarked on legal action. Most, but not all, of Fox's claims were dismissed; ultimately an agreement was reached whereby AutoHop would only become available for Fox stations seven days after a program is transmitted; terms of the settlement were not disclosed. == The future of TV advertisements == The introduction of digital video recorders and services with skipping and fast-forward capabilities enables viewers to avoid viewing interruptive advertisements in recorded programs, either manually or automatically. While advertising separate to television shows can be skipped, advertising in TV shows themselves ("product placement") cannot be skipped. Streaming services such as Hulu show shorter advertisements with a countdown timer and tailored to the viewers interests, asking interactive questions like "Is this ad relevant to you?".
WebGL
WebGL (short for Web Graphics Library) is a JavaScript API for rendering interactive 2D and 3D graphics within any compatible web browser without the use of plug-ins. WebGL is fully integrated with other web standards, allowing GPU-accelerated usage of physics, image processing, and effects in the HTML canvas. WebGL elements can be mixed with other HTML elements and composited with other parts of the page or page background. WebGL programs consist of control code written in JavaScript, and shader code written in OpenGL ES Shading Language (GLSL ES, sometimes referred to as ESSL), a language similar to C or C++. WebGL code is executed on a computer's GPU. WebGL is designed and maintained by the non-profit Khronos Group. On February 9, 2022, Khronos Group announced WebGL 2.0 support from all major browsers. From 2024, a new graphics API, WebGPU, is being developed to supersede WebGL. WebGPU provides extended capabilities, a more modern interface, and direct GPU access, which is useful for demanding graphics as well as AI applications. == Design == WebGL 1.0 is based on OpenGL ES 2.0 and provides an API for 3D graphics. It uses the HTML5 canvas element and is accessed using Document Object Model (DOM) interfaces. WebGL 2.0 is based on OpenGL ES 3.0. It guarantees the availability of many optional extensions of WebGL 1.0, and exposes new APIs. Automatic memory management is provided implicitly by JavaScript. Like OpenGL ES 2.0, WebGL lacks the fixed-function APIs introduced in OpenGL 1.0 and deprecated in OpenGL 3.0. This functionality, if required, has to be implemented by the developer using shader code and JavaScript. Shaders in WebGL are written in GLSL and passed to the WebGL API as text strings. The WebGL implementation compiles these strings to GPU code. This code is executed for each vertex sent through the API and for each pixel rasterized to the screen. == History == WebGL evolved out of the Canvas 3D experiments started by Vladimir Vukićević at Mozilla. Vukićević first demonstrated a Canvas 3D prototype in 2006. By the end of 2007, both Mozilla and Opera had made their own separate implementations. In early 2009, the non-profit technology consortium Khronos Group started the WebGL Working Group, with initial participation from Apple, Google, Mozilla, Opera, and others. Version 1.0 of the WebGL specification was released March 2011. An early application of WebGL was Zygote Body. In November 2012 Autodesk announced that they ported most of their applications to the cloud running on local WebGL clients. These applications included Autodesk Fusion and AutoCAD. Development of the WebGL 2 specification started in 2013 and finished in January 2017. The specification is based on OpenGL ES 3.0. First implementations are in Firefox 51, Chrome 56 and Opera 43. == Implementations == === Almost Native Graphics Layer Engine === Almost Native Graphics Layer Engine (ANGLE) is an open source graphic engine which implements WebGL 1.0 (2.0 which closely conforms to ES 3.0) and OpenGL ES 2.0 and 3.0 standards. It is a default backend for both Google Chrome and Mozilla Firefox on Windows platforms and works by translating WebGL and OpenGL calls to available platform-specific APIs. ANGLE currently provides access to OpenGL ES 2.0 and 3.0 to desktop OpenGL, OpenGL ES, Direct3D 9, and Direct3D 11 APIs. ″[Google] Chrome uses ANGLE for all graphics rendering on Windows, including the accelerated Canvas2D implementation and the Native Client sandbox environment.″ == Software == WebGL is widely supported by modern browsers. However, its availability depends on other factors, too, like whether the GPU supports it. The official WebGL website offers a simple test page. More detailed information (like what renderer the browser uses, and what extensions are available) can be found at third-party websites. === Desktop browsers === Source: Google Chrome – WebGL 1.0 has been enabled on all platforms that have a capable graphics card with updated drivers since version 9, released in February 2011. By default on Windows, Chrome uses the ANGLE (Almost Native Graphics Layer Engine) renderer to translate OpenGL ES to Direct X 9.0c or 11.0, which have better driver support. However, on Linux and Mac OS X, the default renderer is OpenGL. It is also possible to force OpenGL as the renderer on Windows. Since September 2013, Chrome also has a newer Direct3D 11 renderer, which requires a newer graphics card. Chrome 56+ supports WebGL 2.0. Firefox – WebGL 1.0 has been enabled on all platforms that have a capable graphics card with updated drivers since version 4.0. Since 2013 Firefox also uses DirectX on the Windows platform via ANGLE. Firefox 51+ supports WebGL 2.0. Safari – Safari 6.0 and newer versions installed on OS X Mountain Lion, Mac OS X Lion and Safari 5.1 on Mac OS X Snow Leopard implemented support for WebGL 1.0, which was disabled by default before Safari 8.0. Safari version 12 (available in MacOS Mojave) has available support for WebGL 2.0 as an "Experimental" feature. Safari 15 enables WebGL 2.0 for all users. Opera – WebGL 1.0 has been implemented in Opera 11 and 12, but was disabled by default in 2014. Opera 43+ supports WebGL 2.0. Internet Explorer – WebGL 1.0 is partially supported in Internet Explorer 11. Internet Explorer initially failed most of the official WebGL conformance tests, but Microsoft later released several updates. The latest 0.94 WebGL engine currently passes ≈97% of Khronos tests. WebGL support can also be manually added to earlier versions of Internet Explorer using third-party plugins such as IEWebGL. Microsoft Edge – For Microsoft Edge Legacy, the initial stable release supports WebGL version 0.95 (context name: "experimental-webgl") with an open source GLSL to HLSL transpiler. Version 10240+ supports WebGL 1.0 as prefixed. Latest Chromium-based Edge supports WebGL 2.0. === Mobile browsers === Google Chrome – WebGL 1.0 is supported on Android as of Chrome 25. WebGL 2.0 is supported on Android as of Chrome 58. Chrome is used for the Android system webview as of Android 5. Firefox for mobile – WebGL 1.0 is available for Android devices since Firefox 4. Safari on iOS – WebGL 1.0 is available for mobile Safari in iOS 8. WebGL 2.0 is available for mobile Safari in iOS 15. Microsoft Edge – Prefixed WebGL 1.0 was available on Windows 10 Mobile.. Latest Chromium-based Edge supports WebGL 2.0. Opera Mobile – Opera Mobile 12 supports WebGL 1.0 (on Android only). Sailfish OS – WebGL 1.0 is supported in the default Sailfish browser. Tizen – WebGL 1.0 is supported == Tools and ecosystem == === Utilities === The low-level nature of the WebGL API, which provides little on its own to quickly create desirable 3D graphics, motivated the creation of higher-level libraries that abstract common operations (e.g. loading scene graphs and 3D objects in certain formats; applying linear transformations to shaders or view frustums). Some such libraries were ported to JavaScript from other languages. Examples of libraries that provide high-level features include A-Frame (VR), BabylonJS, PlayCanvas, three.js, OSG.JS, Google’s model-viewer and CopperLicht. Web3D also made a project called X3DOM to make X3D and VRML content run on WebGL. === Games === There has been an emergence of 2D and 3D game engines for WebGL, such as Unreal Engine 4 and Unity. The Stage3D/Flash-based Away3D high-level library also has a port to WebGL via TypeScript. A more light-weight utility library that provides just the vector and matrix math utilities for shaders is sylvester.js. It is sometimes used in conjunction with a WebGL specific extension called glUtils.js. There are also some 2D libraries built atop WebGL, like Cocos2d-x or Pixi.js, which were implemented this way for performance reasons in a move that parallels what happened with the Starling Framework over Stage3D in the Flash world. The WebGL-based 2D libraries fall back to HTML5 canvas when WebGL is not available. Removing the rendering bottleneck by giving almost direct access to the GPU has exposed performance limitations in the JavaScript implementations. Some were addressed by asm.js and WebAssembly (similarly, the introduction of Stage3D exposed performance problems within ActionScript, which were addressed by projects like CrossBridge). === Content creation === As with any other graphics API, creating content for WebGL scenes requires using a 3D content creation tool and exporting the scene to a format that is readable by the viewer or helper library. Desktop 3D authoring software such as Blender, Autodesk Maya or SimLab Composer can be used for this purpose. In particular, Blend4Web allows a WebGL scene to be authored entirely in Blender and exported to a browser with a single click, even as a standalone web page. There are also some WebGL-specific software such as CopperCube and the online WebGL-based editor Clara.io. Online platforms such as Sketchfab and Clara.io allow users to directly upload their 3D models
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.
Temporal resolution
Temporal resolution (TR) refers to the discrete resolution of a measurement with respect to time. It is defined as the amount of time needed to revisit and acquire data for the same location. When applied to remote sensing, this amount of time is influenced by the sensor platform's orbital characteristics and the features of the sensor itself. The temporal resolution is low when the revisiting delay is high and vice versa. Temporal resolution is typically expressed in days. == Physics == Often there is a trade-off between the temporal resolution of a measurement and its spatial resolution, due to Heisenberg's uncertainty principle. In some contexts, such as particle physics, this trade-off can be attributed to the finite speed of light and the fact that it takes a certain period of time for the photons carrying information to reach the observer. In this time, the system might have undergone changes itself. Thus, the longer the light has to travel, the lower the temporal resolution. == Technology == === Computing === In another context, there is often a tradeoff between temporal resolution and computer storage. A transducer may be able to record data every millisecond, but available storage may not allow this, and in the case of 4D PET imaging the resolution may be limited to several minutes. === Electronic displays === In some applications, temporal resolution may instead be equated to the sampling period, or its inverse, the refresh rate, or update frequency in Hertz, of a TV, for example. The temporal resolution is distinct from temporal uncertainty. This would be analogous to conflating image resolution with optical resolution. One is discrete, the other, continuous. The temporal resolution is a resolution somewhat the 'time' dual to the 'space' resolution of an image. In a similar way, the sample rate is equivalent to the pixel pitch on a display screen, whereas the optical resolution of a display screen is equivalent to temporal uncertainty. Note that both this form of image space and time resolutions are orthogonal to measurement resolution, even though space and time are also orthogonal to each other. Both an image or an oscilloscope capture can have a signal-to-noise ratio, since both also have measurement resolution. === Oscilloscopy === An oscilloscope is the temporal equivalent of a microscope, and it is limited by temporal uncertainty the same way a microscope is limited by optical resolution. A digital sampling oscilloscope has also a limitation analogous to image resolution, which is the sample rate. A non-digital non-sampling oscilloscope is still limited by temporal uncertainty. The temporal uncertainty can be related to the maximum frequency of continuous signal the oscilloscope could respond to, called the bandwidth and given in Hertz. But for oscilloscopes, this figure is not the temporal resolution. To reduce confusion, oscilloscope manufacturers use 'Sa/s' instead of 'Hz' to specify the temporal resolution. Two cases for oscilloscopes exist: either the probe settling time is much shorter than the real time sampling rate, or it is much larger. The case where the settling time is the same as the sampling time is usually undesirable in an oscilloscope. It is more typical to prefer a larger ratio either way, or if not, to be somewhat longer than two sample periods. In the case where it is much longer, the most typical case, it dominates the temporal resolution. The shape of the response during the settling time also has as strong effect on the temporal resolution. For this reason probe leads usually offer an arrangement to 'compensate' the leads to alter the trade off between minimal settling time, and minimal overshoot. If it is much shorter, the oscilloscope may be prone to aliasing from radio frequency interference, but this can be removed by repeatedly sampling a repetitive signal and averaging the results together. If the relationship between the 'trigger' time and the sample clock can be controlled with greater accuracy than the sampling time, then it is possible to make a measurement of a repetitive waveform with much higher temporal resolution than the sample period by upsampling each record before averaging. In this case the temporal uncertainty may be limited by clock jitter.