text
stringlengths 11
320k
| source
stringlengths 26
161
|
|---|---|
Since the early 2000s,Chinahas increased its presence in the TOP500 rankings ofsupercomputers, with systems like Tianhe-1A reaching the top position in 2010 and Sunway TaihuLight leading in 2016.[1]
By 2018, China had the highest number of supercomputers listed on the TOP500, reflecting its commitment to advancing computational capabilities across various sectors, including scientific research, industrial applications, and national defense.[1]However, this progress has been met with challenges, notably from U.S. sanctions aimed at curbing China's access to advanced computing technologies.[2]Since 2019, after the U.S. began levyingsanctionson several Chinese companies involved with supercomputing, public information on the state of supercomputing in China had become less available.[3]
The origins of these centers go back to the 1980s,[4]when theState Planning Commission, theState Science and Technology Commissionand theWorld Bankjointly launched a project to developnetworkingand supercomputer facilities in China. In addition to network facilities, the project included three supercomputer centers.[5]The progress of supercomputing inChinahas been rapid; the country's most powerful supercomputer placed 43rd in November 2002 (DeepComp 1800[6]), 11th by November 2003 (DeepComp 6800[7]), 10th by June 2004 (Dawning 4000A[8]), and by November 2010 (Tianhe-1A[9]) held top spot. China would go on to fall behind Japan in June 2011 until June 2013 when the country's most powerful supercomputer once again clocked in as the world record.[citation needed]
According to theMIT Technology Review, the Loongson processor would power theDawning supercomputersby 2012, producing a line of totally Chinese-made supercomputers that reach petaflop speeds.[10]
Prior to theSunway TaihuLight, Chinese supercomputers have used "off the shelf" processors, e.g.Tianhe-Iuses thousands ofIntelandNvidiachips, and uses theLinuxoperating system which isopen-source software. However, to avoid possible future technology embargo restrictions, the Chinese are developing their own processors such as theLoongson, aMIPS typeprocessor.[11][12]
In November 2015, China increased its number of supercomputers on theTOP500list to 109, up 196% from 37 just six months earlier. This expansion reflected growing investment in domestic innovation, with observers noting that “the Chinese government and companies want to become the creators and not just producer of products that are being designed elsewhere".[13]
In 2016, China's Sunway TaihuLight supercomputer became the world's fastest, achieving a peak performance of 93 petaflops per second. It was nearly three times faster than the next most powerful machine,Tianhe-2, and used over 10 million processor cores designed and manufactured in China. That year also marked the first time China surpassed the United States in total installed supercomputing capacity.[14]China led in the number of systems on the TOP500 list, with 167 supercomputers compared to 165 from the United States.[15]
In 2018, China extended its lead in the number of supercomputers on the TOP500 list, with 206 systems compared to 124 from the United States. While the United States regained the top spot for the fastest individual machine, the list showed that China remained the most prolific producer of supercomputers.[16]
In April 2021, seven Chinese supercomputing entities were added to theEntity Listof theUnited States Department of Commerce'sBureau of Industry and Security.[17]The U.S. government cited their involvement in supporting China's military modernization and weapons development programs. Placement on the list subjects these entities to additional license requirements for exports, re-exports, and in-country transfers of items subject to U.S. export regulations.[18]
In 2023, China'sNational Supercomputing Centerin Guangzhou launched the Tianhe Xinyi supercomputer, claiming it to be about five times more powerful than Tianhe-2A. Specific performance metrics were not disclosed.[19]
The scope of these restrictions widened in March 2025, when over 50 additional China-based firms were added to the Entity List. The U.S. government said the companies had sought advanced technologies in supercomputing,artificial intelligence, andquantum computingfor military applications.[20]The Commerce Department stated that 27 entities had acquired U.S.-origin items to support China's military modernization, while seven were involved in advancing China's quantum technology capabilities. The agency said the expanded restrictions were part of broader efforts to limit Beijing's access to sensitive technologies, includingexascale computingand high-performance AI chips.[21]
Also in March 2025, researchers at theUniversity of Science and Technology of Chinaannounced a breakthrough with the Zuchongzhi-3 quantum computer, claiming it to be a quadrillion times faster than the most powerful classical supercomputer. According to the team, the system completed a random circuit sampling task that would take the classical supercomputer Frontier approximately 5.9 billion years to simulate. The researchers attributed this performance to improvements in chip fabrication and wiring configuration.[22]
The Supercomputing Center of theChina Academy of Sciences(SCCAS) is a support service unit affiliated to the Computer Network Information Center (CNIC) of the Chinese Academy of Sciences with the origin going back to the 1980s.[4]The Supercomputing Center of theChina Academy of Sciences(SCCAS) provides academic support functions to the National Centers. SCCAS, which is located in Beijing, is the Northern main node and operation center for China National Grid (CNGrid).[23]
Yinhe-1 was independently designed and manufactured as the first leading China's supercomputer in 1983 with a performance level of 100MFLOPS.[24]
Shanghai Supercomputer Center (SSC) is a high-performance computing facility located in the Zhangjiang Hi-Tech Park in Pudong, Shanghai, China. Established in December 2000, it was the country's first public high-performance computing service platform open for general use.[25]Funded by the Shanghai Municipal Government, the SSC provides advanced supercomputing resources for scientific research and industrial applications.[25]
The Shanghai Supercomputer Center was officially founded on December 28, 2000, as an initiative to bolster China's scientific computing capabilities.[26]From the outset, its mission was to offer high-performance computing services to a broad community of users, helping to bridge a crucial gap in the country's research infrastructure. Located in Shanghai's Zhangjiang Hi-Tech Park, the center was established with funding from the Shanghai Municipal Government. Upon opening, SSC deployed its first supercomputer, the Shenwei-I, a massively parallel system with a performance peak of 384 gigaflops (billion floating-point operations per second).[26]Early applications included climate modeling and pharmaceutical research, demonstrating the practical value of shared supercomputing resources for China's scientific community.[27]
By late 2003, growing demand for HPC services led to a comprehensive upgrade, undertaken as part of Shanghai's broader “Information Port” initiative.[28]This expansion culminated in the installation of the Dawning 4000A supercomputer—capable of 10 trillion computations per second—at SSC in November 2004.[29]In June 2004, it became the first Chinese supercomputer to enter the world's top ten on the TOP500 list, ranking No. 10.[30]Ongoing government support and rapid technological advancements spurred further enhancements in the late 2000s. On June 15, 2009, SSC launched its next-generation system, nicknamed the “Magic Cube” (Dawning 5000A), reaching a maximum performance of 180.6 teraflops.[31]This milestone marked the first Chinese supercomputer to surpass 100 trillion calculations per second, positioning it as the fastest supercomputer in Asia at the time.[32]
The Shanghai Supercomputer Center has continually enhanced its infrastructure to support advanced scientific research and industrial innovation. At the core of its capabilities is the Magic Cube III supercomputer, with a peak performance of 3.3 petaflops, ranking it among China's most powerful computing systems.[33]SSC offers computational support across diverse fields—including artificial intelligence, environmental modeling, and drug discovery—enabling researchers to address complex global challenges. The center's infrastructure also features a robust suite of scientific simulation software, boosting its effectiveness in areas such as climate forecasting and public health analytics.[33]
SSC's AI platform supports the full lifecycle of artificial intelligence development, from data preprocessing to model training and deployment. This infrastructure fosters breakthroughs in domains such as autonomous technologies and machine learning. To meet increasing demands for high-performance computing, SSC continues to upgrade its systems and expand its partnerships with domestic and international research institutions. These developments not only reinforce SSC's central role in global scientific collaboration but also contribute to China's broader push for technological self-reliance, particularly in the fields of big data analytics, supercomputing, and AI.[34]
The SSC provides a suite of services tailored to meet the diverse high-performance computing needs of both scientific and industrial users. Approximately 70% of its user base consists of researchers from universities and public research institutions, while the remaining 30% includes industry clients from sectors such as automotive, aerospace, and energy[26]
The center's service model comprises three core offerings. First, Computing Resource Leasing grants users access to SSC's high-performance computing clusters to run large-scale simulations, data analysis, and modeling tasks. Second, Technical Consulting Services provide expert guidance in areas such as computational fluid dynamics, structural mechanics, and algorithm optimization. Finally, Research and Development (R&D) Collaboration allows SSC to partner with external institutions on joint projects, offering technical support and co developing computational solutions.[34]
The article effectively outlines the future goals of the Shanghai Supercomputer Center, emphasizing its strategic role in advancing China's high-performance computing capabilities. One of the primary objectives is the continued development of next-generation supercomputing systems, such as the Magic Cube III, which is designed to meet growing demands in fields like artificial intelligence, environmental modeling, and biomedicine.[34]
While these future-oriented developments are well integrated into the article, the section could benefit from greater specificity—such as naming particular research partnerships, projected technological milestones, or policy frameworks guiding these advancements.[33]
TheNational Supercomputing Center in Tianjinwas approved in May 2009 as the country’s first state-level supercomputing facility.[35]It hosts the Tianhe-1 system, developed by the National University of Defense Technology and unveiled on October 29, 2009. The project began under China’s 863 Program in 2008. Tianhe-1 achieved a peak theoretical performance of 1,206 teraflops and a sustained LINPACK result of 563.1 teraflops, ranking first on the TOP500 list.[36][37]It was the second system globally to exceed 10¹⁶ operations per second.[37]
The Tianjin Computer Institute had been active since 1984 when it developed the 16-bit TQ-0671microcomputersystem.[38]A commercial affiliate of the Tianjin center had previously made the PHPC100personal supercomputerin 2008 which was approximately twice the size of a desktop and offering forty times its performance; a second-generation model appeared in 2010.[39]
The National Supercomputing Center in Shenzhen (NSCS) was approved by the Ministry of Science and Technology in May 2009 as one of China's first national supercomputing centers in the central-southern region. It is the second national supercomputing center after the one based in Tianjin and houses the second fastest machine in China, and the third fastest in the world.[40]
Located in Xili Lake International Science & Education City, SSC Phase I occupies 43,400 square meters and is equipped with a world-class supercomputer system. In May 2010 theNebulaecomputer inShenzhenplaced second on theTop 500supercomputer list, after the Cray computer at theOak Ridge National Laboratoryin Tennessee.[41]
Phase II is located in Guangming Science City. It covers 46,000 square meters of land, has a total construction area of 116,800 square meters, and is scheduled for completion by 2025.[42]This phase aims to expand computational capacity while integrating sustainable design principles. The center will house a 2E-level supercomputer and work alongside Phase I to provide large-scale scientific computing, industrial computation, big data processing, and intelligent supercomputing services.[43]
The Shenzhen Supercomputing Pingshan Service Platform is the first regional platform established by the National Supercomputing Shenzhen Center, with support from the Pingshan District Innovation Bureau. It aims to serve the "9+2" industrial clusters in Pingshan District, promote the development of technology industries in the Pingshan High-tech Zone, and enhance the district's innovation capacity.[44]
Foundations for a new major branch of the National Supercomputing Center (国家超级计算中心Guójiā Chāojíjìsuàn Zhōngxīn) were laid inHunan University,Changshaon 28 November 2010 as the first National Supercomputing Center inCentral Chinaand the third National Supercomputing Center in China apart from the two centers which are located in Tianjin and Shenzhen.[45]The National Supercomputing Changsha Center is managed and operated by Hunan University.[46]It operates theTianhe-1A Hunan Solution – NUDT YH MPPsupercomputer which runs at 1342teraflops.[47]It was the most powerful supercomputer in the world at that time from its operation in November 2010 to November 2011.[48][49]
The National Supercomputing Center in Jinan (NSCCJN) is located in the capital ofShandongProvince in East China. It is situated inside the Jinan Supercomputing Center Science and Technology Park, which opened in May 2019.[citation needed]The CPU runs the ShenWei processor SW1600 at 975 MHz, running at 796teraflopsand using 137,200 cores in the processor.[50]
The NSCCJN built the first prototype of the Sunway E-class computer in 2018.[51]
The center has also worked on projects that strengthen the internet access across various regions in China. In May 2024, the NSCCJN launched the "Shandong Computing Network", the first supercomputing Internet project in China that covers 16 cities in the province.[52]
TheNational Supercomputer Center in Guangzhouoperates the tenth most powerful supercomputer in the world (as of November 2022)Tianhe-2 (MilkyWay-2),[53]which runs at 33,000teraflops. It also operates theTianhe-1A Guangzhou Solution – NUDT YH MPPsupercomputer that runs at 211teraflops.[54]The center has been active since 2018.[citation needed]
In December 2023, China unveiled the domestically developed supercomputing system "Tianhe Xingyi" at the National Supercomputing Center in Guangzhou. The new system is reported to outperform the previousTianhe-2in several areas, including CPU computing power, networking, storage, and applications.[55]
Located in central China'sHenanprovince, the National Supercomputing Zhengzhou Center passed the inspection for operation in December 2020, becoming the seventh national supercomputing center in China.[56]
In 2020, the National Supercomputing Kunshan Center successfully passed the acceptance of experts, becoming the second supercomputing center inJiangsuProvince and the eighth supercomputing center in China.[57]
The National Supercomputing Center inChengdu, officially launched in September 2020, serves as a pivotal node in China'shigh-performance computing(HPC) infrastructure.[58]Situated inTianfu New Areain theSichuanprovince in southwest China, the center provides a range of services encompassing computing resources, software development, and talent cultivation.[59]It collaborates with over 1,400 users across more than 30 sectors, including basic science, artificial intelligence (AI), and urban governance.[60]
Chengdu distinguishes itself as one of the few Chinese cities hosting both a national supercomputing center and an intelligent computing center. The Chengdu Intelligent Computing Center, located in Pidu, focuses on AI applications and integrates platforms for AI computing power, scientific research innovation, and urban intelligence. With an initial computing power scale of 300 petaflops, it supports over 120 enterprises in developing AI solutions across various domains such as intelligent governance, healthcare, and manufacturing.[60]
In early 2025, Chengdu was designated as the main node in a new cross-regional computing service hub, with sub-nodes in Yibin and Lushan. This initiative aims to establish unified management and scheduling of computing resources, fostering an integrated ecosystem for the computing industry.[58]
The city's commitment to AI development is further evidenced by its designation as the first national AI innovation application pilot zone in western China. By 2023, Chengdu's AI and robotics industry had grown to a scale of 78 billion RMB, with projections exceeding 100 billion RMB in 2024.[61]
|
https://en.wikipedia.org/wiki/Supercomputing_in_China
|
Several centers forsupercomputingexist acrossEurope, and distributed access to them is coordinated by European initiatives to facilitatehigh-performance computing. One such initiative, theHPC Europaproject, fits within theDistributed European Infrastructure for Supercomputing Applications(DEISA), which was formed in 2002 as a consortium of eleven supercomputing centers from seven European countries. Operating within theCORDISframework, HPC Europa aims to provide access to supercomputers across Europe.[1]
According to theTOP500list of November 2024,Italy's HPC6 is the fastest European supercomputer.[2]
In June 2011,France'sTera 100was certified the fastest supercomputer in Europe, and ranked 9th in the world at the time (has now dropped off the list).[3][4][5][6]It was the firstpetascalesupercomputer designed and built in Europe.[7]
There are several efforts to coordinate European leadership in high-performance computing. The ETP4HPC Strategic Research Agenda (SRA) outlines a technology roadmap for exascale in Europe, with a key motivation being an increase in the global market share of the HPC technology developed in Europe.[8]The Eurolab4HPC Vision provides a long-term roadmap, covering the years 2023 to 2030, with the aim of fostering academic excellence in European HPC research.[9]
There have been several projects to organise supercomputing applications within Europe. The first was theDistributed European Infrastructure for Supercomputing Applications(DEISA). This ran from 2002–2011. The organisation of supercomputing has been taken over by the Partnership for Advanced Computing in Europe (PRACE).
From 2018-2026 further supercomputer development is taking place under theEuropean High-Performance Computing Joint Undertakingwithin theHorizon 2020framework. Under Horizon 2020, European HPC Centres of Excellence are being funded to promote Exascale capabilities and scale up existing parallel codes in the domains of renewable energy, materials modelling and design, molecular and atomic modeling, climate change, global system science, and bio-molecular research.[10][11]
In addition to advances being shared with the HPC research community such as the "Putting the Ocean into the Center" visualization[12][13]and progress on the "Digital Twin" that is already being used to run in silico clinical trials,[14][15]EU countries are already beginning to directly benefit from work done by the Centres of Excellence under Horizon 2020: In summer 2021, software from a European Centre of Excellence was used to forecast ash clouds from the La Palma volcano.[16]Additionally, EU Centres of Excellence are providing support throughout the Covid19 pandemic creating models to guide policy makers, expediting the discovery of possible treatments, and generally facilitating the sharing of research data during the race to understand the corona virus.[17][18][19]
PRACE provides "access to leading-edge computing and data management resources and services for large-scale scientific and engineering applications at the highest performance level".[20]PRACE categorises European HPC facilities into 3 tiers: Tier-0 are European Centres with petaflop machines, Tier-1 are national centres, and Tier-2 are regional centres.
PRACE has 8 Tier-0 systems:[21]
The Vienna Scientific Cluster is a collaboration between several Austrian universities. The current flagship of the VSC family is VSC-4, a Linux cluster with approximately 790 compute nodes, 37,920 cores and a theoretical peak performance is 3.7 PFlop/s.[24]The VSC-4 cluster was ranked 82nd in the Top-500 list in June 2019.[24]VSC-4 was installed in summer 2019 at the Arsenal TU building in Vienna.
On 25 October 2012,Ghent University(Belgium) inaugurated the first Tier 1 supercomputer of theFlemish Supercomputer Centre(VSC). The supercomputer is part of an initiative by the Flemish government to provide the researchers in Flanders with a very powerful computing infrastructure. The new cluster was ranked 163rd in the worldwide Top500 list of supercomputers in November 2012.[25][26]In 2014, a supercomputer started operating at Cenaero inGosselies.
In 2016, VSC started operating theBrENIACsupercomputer (NEC HPC1816Rg, Xeon E5-2680v4 14C 2.4 GHz, Infiniband EDR) inLeuven. It has 16,128 cores providing 548,000 Gflops (Rmax) or 619,315 Gflops (Repack).[27]
TheNational Center for Supercomputing ApplicationsinSofiaoperates anIBMBlue Gene/P supercomputer, which offers high-performance processing to theBulgarian Academy of SciencesandSofia University, among other organizations.[28]The system was on theTOP500list until November 2009, when it ranked as number 379.[29]
A second supercomputer, the "Discoverer", was installed in 2020 and ranked 91st in theTOP500in 2021.[30]"Discoverer", Bulgaria's supercomputer was the third launched under the program on 21 October 2021. Located on the territory of the Bulgarian Science and Technology Park "Sofia Tech Park" in Sofia, Bulgaria. The cost is co-financed by Bulgaria and EuroHPC JU with a joint investment of €11.5 million completed by Atos. Discoverer has a stable performance of 4.5 petaflops and a peak performance of 6 petaflops.[31][32][33][34][35][36]
A third supercomputer "Hemus", owned by the Bulgarian Academy of Sciences and the Institute of Information and Communication Technologies was launched on 19 October 2023. The supercomputer's performance of 3 petaflops will aid in science research, data processing, application development and medical imaging. The project was completed by HP and is jointly financed by Bulgaria and theEuropean Regional Development Fundfor a total cost of €15 million.[37]
The Center for Advanced Computing and Modelling (CNRM) inRijekawas established in 2010 and conducts multidisciplinary scientific research through the use of advanced high-performance solutions based on CPU and GPGPU server technologies and technologies for data storage.[38]They operate the supercomputer "Bura" which consists of 288 computing nodes and has a total of 6912CPUcores, its peak performance is 233.6teraflopsand it ranked at 440th on the November 2015TOP500list.[39]
CSC – IT Center for Scienceoperated aCray XC30system called "Sisu" with 244 TFlop/s.[40]In September 2014 the system was upgraded toCray XC40, giving a theoretical peak of 1,688 TFLOPS. Sisu was ranked 37th in the November 2014 Top500 list,[41]but had dropped to 107th by November 2017.[42]By the end of 2023, the CSC was operating a newLUMIsystem at a sustained 380 petaflops, making it the top performing HPC in Europe[43]while awaitingJUPITER's construction as part of theEuropean High-Performance Computing Joint Undertaking.
TheCommissariat à l'énergie atomique et aux énergies alternatives(CEA) operates the Tera 100 machine in theResearch and Technology Computing CenterinEssonne,Île-de-France.[3]The Tera 100 has a peak processing speed of 1,050teraflops, making it the fastest supercomputer in Europe in 2011.[4]Built byGroupe Bull, it had 140,000 processors.[44]
TheNational Computer Center of Higher Education(French acronym: CINES) was established inMontpellierin 1999, and offers computer services for research and higher education.[45][46]In 2014 the Occigen system was installed, which was manufactured by theBull, Atos Group. It has 50,544 cores and a peak performance of 2.1Petaflops.[47][48]
In Germany, supercomputing is organized at two levels. The three national centers atGarching(LRZ),Juelich(JSC) andStuttgart(HLRS) together form the Gauss Center for Supercomputing, and provide both the European Tier 0 level of HPC and the German national Tier 1 level. A number of medium-sized centers are also organized in the Gauss Alliance.
TheJülich Supercomputing Centre(JSC) and theGauss Centre for Supercomputingjointly owned theJUGENEcomputer at theForschungszentrum JülichinNorth Rhine-Westphalia. JUGENE was based on IBM's Blue Gene/P architecture, and in June 2011 was ranked the 12th fastest computer in the world by TOP500.[49]It was replaced by the Blue Gene/Q systemJUQUEENon 31 July 2012.[50]
TheLeibniz-Rechenzentrum, a supercomputing center inMunich, houses theSuperMUCsystem, which began operations in 2012 at a processing speed of 3 petaflops. This was, at the time it entered service, the fastest supercomputer in Europe. TheHigh Performance Computing Centerin Stuttgart fastest computing system is Hawk with a peak performance of 26 petaflops,[51]replacing Hazel Hen with a peak performance of more than 7.4 petaflops. As of November 2015[update]Hazel Hen, which is based on Cray XC40 technology, was ranked the 8th fastest system worldwide.[52]
Greece's main supercomputing institution isGRNET SA, a Greek state-owned company that is supervised by the General Secretariat for Research and Technology of theMinistry of Education, Research and Religious Affairs. GRNET's high-performance computing system is called ARIS (Advanced Research Information System) and during its introduction to theTOP500list, in June 2015, it got the 467th place.[53]ARISinfrastructure consists offour computing systemsislets: thin nodes, fat nodes, GPU nodes and Phi Nodes.GRNETis the Greek member in the Partnership for Advanced Computing in Europe[54]and ARIS is a Tier-1 PRACE node.
TheIrish Centre for High-End Computing(ICHEC) is the national supercomputing centre and operates the "Kay" supercomputer, commissioned in August 2018. The system, which was provided by Intel, consists of a cluster of 336 high-performance servers with 13,440 CPU (Central Processing Unit) cores and 64 terabytes of memory for general purpose computations. Additional components aimed at more specialised requirements include 6 large memory nodes with 1.5 terabytes of memory per server, plus 32 accelerator nodes divided between Intel Xeon Phi and NVidia V100 GPUs (Graphics Processing Units). The network linking all of these components together is Intel's 100 Gbit/s Omnipath technology and DataDirect Networks are providing 1 petabyte of high-performance storage over a parallel file system. Penguin Computing has integrated this hardware and provided the software management and user interface layers.[55]
The main supercomputing institution in Italy isCINECA, a consortium of many universities and research institutions scattered throughout the country. As of June 2023, the highest CINECA supercomputer in theTOP500list (4th place) isLeonardo, an acceleratedpetascalecluster based onXeon Platinumprocessors,NVIDIA A100Tensor Core GPUs, andNVIDIA Mellanox HDR100 InfiniBandconnectivity with 1,824,768 total cores for 238.70petaFLOPS(Rmax) and 7,404 kW.[56]
Due to the involvement of theNational Institute for Nuclear Physics(INFN) in the main experiments taking place atCERN, Italy also hosts some of the largest nodes of theWorldwide LHC Computing Grid, including one Tier 1 facility and 11 Tier 2 facilities out of 151 total nodes.[57][58]
The Luxembourg supercomputer Meluxina was officially launched on 7 June 2021 and is part of theEuropean High-Performance Computing Joint Undertaking(EuroHPC JU). It is located at the LuxProvide data center inBissen,Luxembourg. It is the second supercomputer to be launched after Vega of eight planned supercomputers (EuroHPC JU). The system was completed by company Atos. Luxembourg paid for two thirds of the project. The European Commission funded the other third, with 35% of the computing power to be made available to the 32 countries taking part in the EuroHPC joint venture. The value of the joint investment is €30.4 million euros. Meluxina has a stable performance of 10 petaflops and a peak performance of 15 petaflops.[59][60][61][62]
The supercomputer Snellius is operated by the organization SURF (formerly known as SURFsara) and it is hosted in theAmsterdam Science Park. Since 1984 the organization has been operating the Dutch national supercomputing facilities for research.
Additionally, theEuropean Grid Infrastructure, a continent-widedistributed computingsystem, is also headquartered at theScience ParkinAmsterdam.[63]
UNINETT Sigma2 AS maintains the national infrastructure for large-scale computational science in Norway and provides high-performance computing and data storage for all Norwegian universities and colleges, as well as other publicly funded organizations and projects. Sigma2 and its projects are financed by theResearch Council of Norwayand the Sigma2 consortium partners (the universities ofOslo,Bergen, andTromsø, and theNorwegian University of Science and TechnologyinTrondheim) Its head office in Trondheim.[64]Sigma2 operates three systems:Stalloand Fram (located in Tromsø) and Saga (in Trondheim).[65]An additional machine (named Betzy afterElizabeth Stephansen) was inaugurated on 7 December 2020.[66][67]
TheNorwegian University of Science and Technology(NTNU) inTrondheimoperates the "Vilje" supercomputer, owned by NTNU and the Norwegian Meteorological Institute. "Vilje" is operating at 275 teraflops.[68]
Decommissioned systems include Hexagon (2008-2017) at the University of Bergen; Gardar (2012 to 2015); and Abel (2012 to 2020) at the University of Oslo.[69]The "Abel" supercomputer was named after the famous Norwegian mathematicianNiels Henrik Abel(1802–1829). It operated at 258 teraflops through over 650 nodes and over 10000 cores (CPU's), where each node typically has 64 GiB of RAM.[70]It was ranked 96th in the TOP500 list in June 2012 when it was installed.[71]
Currently, since 2015, the fastest supercomputer in Poland is "Prometheus" that belongs to theAGH University of Science and TechnologyinKraków.[72]It provides 2399 teraflops of computing power and has 10 petabytes of storage.[73]It currently holds 21st place in Europe, and was 77th in the world according to the November 2017 TOP500 list.[74]
ThePolish Grid Infrastructure PL-Gridwas built between 2009 and 2011 as a nationwide computing infrastructure, and will remain within the PLGrid Plus project until 2014. At the end of 2012, it provided 230 teraflops of computing power and 3,600 terabytes of storage for the Polish scientific community.
The Galeracomputer clusterat theGdańsk University of Technologywas ranked 299th on the TOP500 list in November 2010.[75][76]The Zeuscomputer clusterat the ACK Cyfronet AGH in Kraków was ranked 106th on the TOP500 list in November 2012, but had dropped to 386th by November 2015.[77]
In November 2011, the 33,072-processor Lomonosov supercomputer inMoscowwas ranked the 18th-fastest supercomputer in the world, and the third-fastest in Europe. The system was designed byT-Platforms, and used Xeon 2.93 GHz processors, Nvidia 2070 GPUs, and anInfinibandinterconnect.[78]In July 2011, the Russian government announced a plan to focus on constructing larger supercomputers by 2020.[79]In September 2011, T-Platforms stated that it would deliver a water-cooled supercomputer in 2013.[80]
Since 2016, Russia has had the most powerful military supercomputer in the world with a speed of 16petaflops, called theNDMC Supercomputer.[citation needed]
The Slovenian supercomputer Vega was officially launched on 20 April 2021 and is part of theEuropean High-Performance Computing Joint Undertaking(EuroHPC JU). It is located at the Institute of Information Science Maribor (IZUM) inMaribor,Slovenia. This is the first of eight planned supercomputers (EuroHPC JU). The system was completed by local company Atos. Vega supercomputer was jointly financed by EuroHPC JU through EU funds and the Institute of Information Science Maribor (IZUM). The value of the joint investment is €17.2 million euros. Vega has a stable performance of 6.9 petaflops and a peak performance of 10.1 petaflops.[81][82]
The Slovenian National Grid Initiative (NGI) provides resources to the European Grid Initiative (EGI). It is represented in the EGI Council byARNES. ARNES manages a cluster for testing computing technology where users can also submit jobs. The cluster consists of 2300 cores and is growing.[83]
Arctur also provides computer resources on itsArctur-2and previouslyArctur-1supercomputers to the Slovenian NGI and industry as the only privately owned HPC provider in the region.[84]
TheJožef Stefan Institutehas most of the HPC installations in Slovenia. They are not however a single uniform HPC system, but several dispersed systems at separate research departments (F-1,[85]F-9[86]and R-4[87]).
TheBarcelona Supercomputing Centeris located at theTechnical University of Cataloniaand was established in 2005.[88]The center operates the Tier-0 11.1 petaflopsMareNostrum 4supercomputer and other supercomputing facilities. This centre manages the Red Española de Supercomputación (RES). The BSC is a hosting member of the Partnership for Advanced Computing in Europe (PRACE) HPC initiative. InGaliciaCESGAestablished in 1993, operates theFinisTerrae II, a 328TFlopssupercomputer, which will be replaced byFinisTerrae IIIin 2021 with 1,9PFlops. TheSupercomputing and Visualization Center of Madrid(CeSViMa) at theTechnical University of Madridoperates the 182,78TFlopsMagerit 3supercomputer. TheSpanish Supercomputing Networkfurthermore provides access to several supercomputers distributed across Spain.
TheNational Supercomputer Centre in Sweden(NSC) is located inLinköpingand operates theTriolithsupercomputer which achieved 407.2 Teraflop/s on the Linpack benchmark which placed it 79th on the November 2013 TOP500 list of the fastest supercomputers in the world.[89]In mid-2018 "Triolith" will be superseded by "Tetralith", which will have an estimated maximum speed of just over 4 petaflops.[90]
Sweden'sRoyal Institute of Technologyoperates the Beskow supercomputer, which consists of 53,632 processors and has achieved sustained 1.397 Petaflops/s.[91]
TheSwiss National Supercomputing Centrewas founded in 1991 and is operated byETH Zurich. It is based inLugano,Ticino, and provides supercomputing services to national research institutions and Swiss universities, as well as the internationalCERNorganisation andMeteoSchweiz, the Swiss weather service.[92]In September 2024, the centre inaugurated theAlpsmassively parallelsupercomputer, which ranked 6th on the worldwide TOP500 list in 2024.[93][94]
Alps replaced the earlier supercomputer,Piz Daint, which had been operational since 2013. It delivers 20 times more computing power and is used for scientific research in areas such asartificial intelligence,climate modeling, andastrophysics.[95]
Developed as a prototype in 2010, the IBMAquasarsupercomputer was installed at ETH Zurich's campus. It was designed to test energy-efficient cooling technologies, using hot water to capture waste heat from computing and help warm university buildings.[96][97]
TheEPCCsupercomputer center was established at theUniversity of Edinburghin 1990.[98]TheHECToRproject at the University of Edinburgh provided supercomputing services using a 360-teraflopCray XE6system, the fastest supercomputer in the UK at the time.[99]In 2013, HECToR was replaced by ARCHER, a Cray XC30 system.[100]In 2021, ARCHER was replaced by its successor ARCHER2, an HPE Cray EX system with an estimated peak performance of 28 petaflop/s.[101]ARCHER2 is the tier one national supercomputing service for theEngineering and Physical Sciences Research Council(EPSRC) and theNatural Environment Research Council(NERC).[102]The EPCC also provides the UK's connection to PRACE.[21]
In addition to the ARCHER2 tier one facility, EPSRC supports a number of tier two facilities:[102]
TheDiRACsupercomputing facility is theScience and Technology Facilities Council(STFC)'s tier one facility for particle physics and astronomy research.[102]It comprises a data intensive service hosted by the universities ofCambridgeandLeicester, a memory intensive service (1.37 PF Rmax; 1.9 PF peak performance with 360 nodes in phase 1 (2021), upgraded with a further 168 nodes in phase 2 (2023) giving 528 TB RAM)[104][105][106]hosted by theInstitute for Computational CosmologyatDurham University, and an extreme scaling service hosted byEPCCat theUniversity of Edinburgh.[107][108]In addition to DiRAC, STFC operates the JASMIN high performance data analysis facility on behalf of NERC atRutherford Appleton Laboratory.[109]
TheEuropean Centre for Medium-Range Weather Forecasts(ECMWF) inReading, Berkshire, operates a 100-teraflop IBMpSeries-based system. TheMet Officehas a 14 PFlops computer for weather forecasting,[110]and a joint NERC and Met Office supercomputer, Monsoon2, for research.[111]TheAtomic Weapons Establishmenthas two supercomputers, a 4.3 petaflop Bull Sequana X1000 supercomputer, and a 1.8 petaflop SGI IceX supercomputer.[112]Both these platforms are used for running nuclear weaponry simulations, required after theComprehensive Nuclear-Test-Ban Treatywas signed by the UK.[112]
TheUniversity of Bristol, was chosen in 2023 to host the UK's tier one Artificial Intelligence Research Resource (AIRR), Isambard-AI, building on the success of GW4's Isambard supercomputer.[113][114]The UK government has awarded £225 million to Bristol to develop the system, which will be installed in the National Composites Centre, in collaboration with the universities ofBath,CardiffandExeter.[115]The AIRR is also planned to take in the Dawn supercomputer at theUniversity of Cambridge, which was launched in late 2023 with further development expected in 2024.[116][117]
TheUniversity of Edinburghwas also announced in 2023 as the host of the UK's first exascale supercomputer, intended to build on experience with Isambard-AI.[118]However, funding for this was cancelled by the new UK government in 2024.[119]
|
https://en.wikipedia.org/wiki/Supercomputing_in_Europe
|
Supercomputing in Indiahas a history going back to the 1980s.[1]TheGovernment of Indiacreated an indigenous development programme as they had difficulty purchasing foreignsupercomputers.[1]As of November 2024[update], the AIRAWAT supercomputer is the fastest supercomputer in India, having been ranked 136th fastest in the world in theTOP500supercomputer list.[2]AIRAWAT has been installed at theCentre for Development of Advanced Computing(C-DAC) inPune.[3]
India had faced difficulties in the 1980s when trying to purchasesupercomputersfor academic andweather forecastingpurposes.[1]In 1986 theNational Aerospace Laboratories(NAL) started the Flosolver project to develop a computer forcomputational fluid dynamicsandaerospace engineering.[4][5]The Flosolver MK1, described as aparallel processingsystem, started operations in December 1986.[4][6][5]
In 1987, theIndian governmenthad requested to purchase aCray X-MPsupercomputer; this request was denied by theUnited States governmentas the machine could have adual usein weapons development.[7]After this problem, in the same year, the Government of India decided to promote an indigenous supercomputer development programme.[8][9][10]Multiple projects were commissioned from different groups including theCentre for Development of Advanced Computing(C-DAC), theCentre for Development of Telematics(C-DOT), the National Aerospace Laboratories (NAL), theBhabha Atomic Research Centre(BARC), and theAdvanced Numerical Research and Analysis Group(ANURAG).[9][10]C-DOT created "CHIPPS": the C-DOT High-Performance Parallel Processing System. NAL had started to develop theFlosolverin 1986.[4][11]BARC created theAnupam series of supercomputers. ANURAG created the PACE series of supercomputers.[10]
The C-DAC was created at some point between November 1987 and August 1988.[8][10][9]C-DAC was given an initial 3 year budget of Rs 375 million to create a 1000MFLOPS (1GFLOPS) supercomputer by 1991.[10]C-DAC unveiled thePARAM8000 supercomputer in 1991.[1]This was followed by the PARAM 8600 in 1992/1993.[10][9]These machines demonstrated Indian technological prowess to the world and led to export success.[10][9]Param 8000 was replicated and installed at ICAD Moscow in 1991 with Russian collaboration.
The PARAM 8000 was considered a success for C-DAC in delivering a gigaFLOPS range parallel computer.[10]From 1992 C-DAC undertook its "Second Mission" to deliver a 100 GFLOPS range computer by 1997/1998.[1]The plan was to allow the computer to scale to 1 teraFLOPS.[10][12]In 1993 the PARAM 9000 series of supercomputers was released, which had a peak computing power of 5 GFLOPS.[1]In 1998 the PARAM 10000 was released; this had a sustained performance of 38 GFLOPS on the LINPACK benchmark.[1]
The C-DAC's third mission was to develop a teraFLOPS range computer.[1]ThePARAM Padmawas delivered in December 2002.[1]This was the first Indian supercomputer to feature on a list of the world's fastest supercomputers, in June 2003.[1]
By the early 2000s it was noted that only ANURAG, BARC, C-DAC and NAL were continuing development of their supercomputers.[6]NAL's Flosolver had 4 subsequent machines built in its series.[6]At the same time ANURAG continued to develop PACE, primarily based on SPARC processors.[6]
The Indian Government has proposed to commit US$2.5 billion to supercomputing research during the12th Five-Year Planperiod (2012–2017). The project will be handled byIndian Institute of Science(IISc),Bangalore.[13]Additionally, it was later revealed that India plans to develop a supercomputer with processing power in theexaflopsrange.[14]It will be developed byC-DACwithin the subsequent five years of approval.[15]
In 2015 theMinistry of Electronics and Information Technologyannounced a "National Supercomputing Mission" (NSM) to install 73 indigenous supercomputers throughout the country by 2022.[16][17][18][19]This is a seven-year program worth $730 million (Rs. 4,500 crore).[20]Whilst previously computer were assembled in India, the NSM aims to produce the components within the country.[21]The NSM is being implemented by C-DAC and theIndian Institute of Science.[19]
The aim is to create a cluster of geographically distributed high-performance computing centers linked over a high-speed network, connecting various academic and research institutions across India.[17]This has been dubbed the "National Knowledge Network" (NKN).[21]The mission involves both capacity and capability machines and includes standing up three petascale supercomputers.[22][23]
The first phase involved deployment of supercomputers which have 60% Indian components.[19]The second phase machines are intended to have an Indian designed processor,[19]with a completion date of April 2021.[21]The third and final phase intends to deploy fully indigenous supercomputers,[19]with an aimed speed of 45 petaFLOPS within the NKN.[21]
By October 2020, the first assembled in India supercomputer had been installed.[21]The NSM hopes to have the manufacturing capability for indigenous production by December 2020.[21]
A total of 24.83 petaFLOPS of High Performance Computing (HPC) machines were put into service between 2019 and 2023. In addition to 5,930 specialists from more than 100 institutes using the newly constructed facilities, 1.75lakh(175,000) people received training in HPCs. A total of 73.25 lakh (7.325 million) computational high performance queries were run. Seven systems with processing power greater than one petaFLOPS, eight systems with computational capacities between 500 teraFLOPS and 1 petaFLOPS, and thirteen systems with capacities between 50 teraFLOPS and 500 teraFLOPS were installed during this time.[24]
As of November 2024[update]there are 6 systems based in India on theTOP500supercomputer list.[2][25]
|
https://en.wikipedia.org/wiki/Supercomputing_in_India
|
Japanoperates a number of centers forsupercomputingwhich hold world records in speed, with theK computerbeing the world's fastest from June 2011 to June 2012,[1][2][3]andFugakuholding the lead from June 2020 until June 2022.
The K computer's performance was impressive, according to professorJack Dongarrawho maintains theTOP500list ofsupercomputers, and it surpassed its next 5 competitors combined.[1]The K computer cost US$10 million a year to operate.[1]
Japan's entry into supercomputing began in the early 1980s. In 1982,Osaka University's LINKS-1 Computer Graphics System used amassively parallelprocessing architecture, with 514microprocessors, including 257Zilog Z8001control processorsand 257iAPX86/20(the pairing of an 8086 with an 8087 FPU)floating-point processors. It was mainly used for rendering realistic3Dcomputer graphics.[4]It was claimed by the designers to be the world's most powerful computer, as of 1984.[5]
TheSX-3 supercomputerfamily was developed byNEC Corporationand announced in April 1989.[6]The SX-3/44R became the fastest supercomputer in the world in 1990. Fujitsu'sNumerical Wind Tunnelsupercomputer gained the top spot in 1993. Except for the Sandia National Laboratories' win in June 1994, Japanese supercomputers continued to top theTOP500lists up until 1997.[7]
The K computer's placement on the top spot was seven years after Japan held the title in 2004.[1][2]NEC'sEarth Simulatorsupercomputer built byNECat the Japan Agency for Marine-Earth Science and Technology (JAMSTEC) was the fastest in the world at that time. It used 5,120NEC SX-6iprocessors, generating a performance of 28,293,540MIPS(millioninstructionsper second).[8]It also had a peak performance of 131TFLOPS(131trillionfloating-pointoperations per second), using proprietaryvector processingchips.
TheK computerused over 60,000 commercialscalarSPARC64 VIIIfxprocessors housed in over 600 cabinets. The fact thatK computerwas over 60 times faster than the Earth Simulator, and that the Earth Simulator ranked as the 68th system in the world 7 years after holding the top spot, demonstrates both the rapid increase in top performance in Japan and the widespread growth of supercomputing technology worldwide.
The GSIC Center at theTokyo Institute of Technologyhouses theTsubame2.0 supercomputer, which has a peak of 2,288TFLOPSand in June 2011 ranked 5th in the world.[9]It was developed at the Tokyo Institute of Technology in collaboration withNECandHP, and has 1,400 nodes using both HP Proliant and NVIDIA Tesla processors.[10]
TheRIKEN MDGRAPE-3for molecular dynamics simulations of proteins is a special purpose petascale supercomputer at the Advanced Center for Computing and Communication,RIKENinWakō, Saitama, just outside Tokyo. It uses over 4,800 custom MDGRAPE-3 chips, as well asIntel Xeonprocessors.[11]However, given that it is a special purpose computer, it can not appear on theTOP500list which requiresLinpackbenchmarking.
The next significant system isJapan Atomic Energy Agency's PRIMERGY BX900Fujitsusupercomputer. It is significantly slower, reaching 200 TFLOPS and ranking as the 38th in the world in 2011.[12][13]
Historically, theGravity Pipe(GRAPE) system forastrophysicsat theUniversity of Tokyowas distinguished not by its top speed of 64 Tflops, but by its cost and energy efficiency, having won theGordon Bell Prizein 1999, at about $7 per megaflops, using special purpose processing elements.[14]
DEGIMAis a highly cost and energy-efficient computer cluster at the Nagasaki Advanced Computing Center,Nagasaki University. It is used for hierarchicalN-body simulationsand has a peak performance of 111 TFLOPS with an energy efficiency of 1376 MFLOPS/watt. The overall cost of the hardware was approximately US$500,000.[15][16]
The Computational Simulation Centre, International Fusion Energy Research Centre of theITERBroader Approach[17]/Japan Atomic Energy Agencyoperates a 1.52 PFLOPS supercomputer (currently operating at 442 TFLOPS) inRokkasho, Aomori. The system, called Helios (Roku-chan in Japanese), consists of 4,410Groupe Bullbullx B510 compute blades, and is used forfusionsimulation projects.
The University of Tokyo's Information Technology Center inKashiwa, Chiba, began operating Oakleaf-FX in April 2012. This supercomputer is a FujitsuPRIMEHPC FX10(a commercial version of theK computer) configured with 4,800 compute nodes for a peak performance of 1.13 PFLOPS. Each of the compute nodes is aSPARC64 IXfxprocessor connected to other nodes via a six-dimensional mesh/torus interconnect.[18]
In June 2012, the Numerical Prediction Division, Forecast Department of theJapan Meteorological Agencydeployed an 847 TFLOPSHitachiSR16000/M1 supercomputer, which is based on theIBMPower 775, at the Office of Computer Systems Operations and the Meteorological Satellite Center inKiyose, Tokyo.[19]The system consists of two SR16000/M1s, each a cluster of 432-logical nodes. Each node consists of four 3.83 GHz IBMPOWER7processors and 128 GB of memory. The system is used to run a high-resolution (2 km horizontally and 60 layers vertically, up to 9-hour forecast) local weather forecast model every hour.
Starting in 2003, Japan usedgrid computingin the National Research Grid Initiative (NAREGI) project to develop high-performance, scalable grids over very high-speed networks as a future computational infrastructure for scientific and engineering research.[20]
|
https://en.wikipedia.org/wiki/Supercomputing_in_Japan
|
TheSlurm Workload Manager, formerly known asSimple Linux Utility for Resource Management(SLURM), or simplySlurm, is afree and open-sourcejob schedulerforLinuxandUnix-likekernels, used by many of the world'ssupercomputersandcomputer clusters.
It provides three key functions:
Slurm is the workload manager on about 60% of theTOP500supercomputers.[1]
Slurm uses abest fit algorithmbased onHilbert curve schedulingorfat treenetwork topology in order to optimize locality of task assignments on parallel computers.[2]
Slurm began development as a collaborative effort primarily byLawrence Livermore National Laboratory,SchedMD,[3]Linux NetworX,Hewlett-Packard, andGroupe Bullas a Free Software resource manager. It was inspired by the closed sourceQuadrics RMSand shares a similar syntax. The name is a reference to thesodainFuturama.[4]Over 100 people around the world have contributed to the project. It has since evolved into a sophisticated batch scheduler capable of satisfying the requirements of many large computer centers.
As of November 2021[update],TOP500list of most powerful computers in the world indicates that Slurm is the workload manager on more than half of the top ten systems.
Slurm's design is very modular with about 100 optional plugins. In its simplest configuration, it can be installed and configured in a couple of minutes. More sophisticated configurations provide database integration for accounting, management of resource limits and workload prioritization.
Slurm features include:[citation needed]
The following features are announced for version 14.11 of Slurm, was released in November 2014:[5]
Recent Slurm releases run only onLinux. Older versions had been ported to a few otherPOSIX-basedoperating systems, includingBSDs(FreeBSD,NetBSDandOpenBSD),[6]but this is no longer
feasible as Slurm now requirescgroupsfor core operations. Clusters running operating systems other than Linux will need to use
a different batch system, such as LPJS. Slurm also supports several unique computer architectures, including:
Slurm is available under theGNU General Public License v2.
In 2010, the developers of Slurm founded SchedMD, which maintains the canonical source, provides development, level 3 commercial support and training services. Commercial support is also available fromBull,Cray, and Science + Computing (subsidiary ofAtos).
Theslurmsystem has three main parts:
The clients can issue commands to the control daemon, which would accept and divide the workload to the computing daemons.
For clients, the main commands aresrun(queue up an interactive job),sbatch(queue up a job),squeue(print the job queue) andscancel(remove a job from the queue).
Jobs can be run inbatch modeorinteractive mode. For interactive mode, a compute node would start a shell, connects the client into it, and run the job. From there the user may observe and interact with the job while it is running. Usually, interactive jobs are used for initial debugging, and after debugging, the same job would be submitted bysbatch. For a batch mode job, itsstdoutandstderroutputs are typically directed to text files for later inspection.
|
https://en.wikipedia.org/wiki/Slurm_Workload_Manager
|
High performance computingapplications run onmassively parallelsupercomputersconsist ofconcurrent programsdesigned usingmulti-threaded,multi-process models. The applications may consist of various constructs (threads, local processes, distributed processes, etc.) with varying degree of parallelism. Although high performance concurrent programs use similar design patterns, models and principles as that of sequential programs, unlike sequential programs, they typically demonstrate non-deterministic behavior. The probability of bugs increases with the number of interactions between the various parallel constructs.Race conditions, data races,deadlocks, missed signals and live lock are common error types.
Parallel programs can be divided into two general categories: explicitly andimplicitly parallel. Using parallel language constructs defined for process creation,communicationandsynchronizationmake an application explicitly parallel. Using a tool orparallelizing compilerto convert a serial program into a parallel one, makes it implicitly parallel. Both categories are equally bug-prone.
Concurrent applications should execute correctly on every possible thread schedule in the underlying operating system. However, traditional testing methods detect few bugs, chiefly because of theHeisenbugs[1]problem. A Heisenbug is an error that changes or disappears when an attempt is made to isolate and probe them viadebugger, by adding some constructs such as synchronization requests or delay statements.
Another issue is caused due to the unpredictable behavior of thescheduler. Differences in system load influence scheduler behavior. This behavior cannot be changed manually. To counter this indeterminism, the program must be executed many times under various execution environments. Still, it is not guaranteed that a bug can be reproduced. Most of the time, the program runs correctly, and the bug is visible only when specific conditions are matched. As a result, non-repeatability of the concurrent programs is a major source of roadblock for detecting error. As an example, consider the following.
Clearly, this has a problem of causing deadlocks. Yet, it may cause deadlock in some runs of the program while in others, it may run successfully.
Probe effectis seen in parallel programs when delay-statements are inserted in parallel programs facing synchronization problems. This effect, like Heisenbugs, alters behavior changes that may obscure problems. Detecting the source of a probe effect is a great challenge in testing parallel applications.The main difference between Probe effect and Heisenbugs is that Heisenbugs are seen when additional delay statements and/or synchronization requests are added to the concurrent application during testing, while probe effect is seen when the developer adds delay statements to concurrent applications with poor synchronization.
The differences between sequential and concurrent programs lead to the differences in their testing strategies. Strategies for sequential programs can be modified to make them suitable for concurrent applications. Specialized strategies have also been developed. Conventionally, testing includes designing test cases and checking that the program produces the expected results. Thus, errors in specification, functionality, etc. are detected by running the application and subjecting it to testing methods such asFunctional Testing,White Box,Black BoxandGrey Box Testing.[2]Static analysisis also used for detecting errors in high performance software using methods such asData Flow Analysis,Control Flow Analysis,Cyclomatic Complexities,Thread Escape AnalysisandStatic Slicing Analysisto find problems. Using static analysis before functionality testing can save time. It can detect ‘what the error is’ find the error source. Static analysis techniques can detect problems like lack ofsynchronization, improper synchronizations, predict occurrence ofdeadlocksandpost-waiterrors inrendezvous requests.
Details:
The indeterminacy of scheduling has two sources.[1]
To make concurrent programs repeatable, an external scheduler is used. The program under test is instrumented to add calls to this scheduler. Such calls are made at the beginning and end of each thread as well as before every synchronization request. This scheduler selectively blocks threads of execution by maintaining a semaphore associated with each thread, such that only one thread is ready for execution at any given time. Thus, it converts parallel non-deterministic application into a serial execution sequence in order to achieve repeatability. The number of scheduling decisions made by the serializing scheduler is given by –
To obtain more accurate results using deterministic scheduling, an alternate approach can be chosen. A few properly-placed pre-emptions in the concurrent program can detect bugs related to data-races.[1]Bugs are found in clusters. The existence of one bug establishes a high probability of more bugs in the same region of code. Thus each pass of the testing process identifies sections of code with bugs. The next pass more thoroughly scrutinizes those sections by adding scheduler calls around them. Allowing the problematic locations to execute in a different order can reveal unexpected behavior.
This strategy ensures that the application is not prone to the Probe Effect. Sources of errors that cause the Probe Effect can range from task creation issues to synchronization and communication problems. Requirements of timing related tests:[3]
The number of test cases per input data set is:
This equation has exponential order. In order to reduce the number of test cases, either Deterministic Execution Method (DET) orMultiple Execution Technique(MET) is used. Various issues must be handled:
This method applies the concept of define-use pair, in order to determine the paths to be tested.
Software verificationis a process that proves that software is working correctly and is performing the intended task as designed.
Input is given to the system to generate a known result. This input-result pair can be obtained from previous empirical results and/or manual calculations.[4]This is a system-level test that can be performed only when all relevant modules are integrated. Moreover, it only shows that bugs exist. It offers no detailed information regarding the number of bugs, their location or nature.
These tests are primarily used for scientific simulations. The output of the simulation often cannot be predicted. Since these simulations attempt to describe scientific laws, any symmetries in the theory must be honored by the simulation. Thus, by varying input conditions along the lines of symmetry and then comparing the obtained results with externally derived results, the existence of bugs can be detected.[4]
In scientific computing most data lies in the central region of the simulation conditions. As a result, it is difficult to perform Boundary-value testing[2]with real time experimental data. Thus, center of simulation (for example, for data value of 10 in Figure 1) is shifted to one of the boundaries so as to test the boundary condition effectively.
Parallel implementation tests are usually used for applications that usedistributed memoryprogramming models such asmessage passing. These tests are often applied to programs that use regular grids of processors.[4][clarification needed]
Many parallel databases use distributed parallel processing to execute the queries. While executing an aggregate function such as sum, the following strategy is used:[5]
The final result may contain some rounding error as each processor independently rounds-off local results. One test is to ensure that such errors do not occur. This requires showing that the aggregate sum is decomposition-independent. An alternate summation scheme is to send all of the individual values to one processor for summation. This result can be compared with the distributed result to ensure consistency.
This tool eliminates non-determinacy using deterministic scheduling. It tracks schedule paths executed previously and guarantees that each time a new schedule path is executed.[1][clarification needed]
|
https://en.wikipedia.org/wiki/Testing_high-performance_computing_applications
|
Ultra Network Technologies(previously calledUltra Corporation) was a networking company. It offered high-speed network products for the scientific computing market as well as some commercial companies. It was founded in 1986 by James N. Perdue (formerly ofNASA, Ames Research Center), Drew Berding, and Wes Meador (ofControl Data Corporation) to provide higher speed connectivity and networking forsupercomputersand their peripherals and workstations. At the time, the only other companies offering high speed networking and connectivity for the supercomputer and high-end workstation market wasNetwork Systems Corporation(NSC) andComputer Network Technology Corporation(CNT). They both offered 50megabytesper second (MB/s) bandwidth between controllers but at that time, their architecture was not implemented using standard networking protocols and their applications were generally focused on supporting connectivity at high speed between large mainframes and peripherals, often only implementing only point-to-point connections. Ethernet was available in 1986 and was used by most computer centers for general networking purposes. Its bandwidth was not high enough to manage the high data rate required by the 100 MB/s supercomputer channels and 4 MB/s VMEbus channels on workstations.[citation needed]
Ultra's first customer,Apple Computer, purchased a system to connect their Cray 1 supercomputer to a high speed graphicsframebufferso that Apple could simulate new personal computers on theCray Researchcomputer (at the hardware level) and use the framebuffer as the simulated computer display device. Although not a networking application, this first contract allowed Ultra to demonstrate the basic technologies and gave them capital to continue development on a true networking processor.
In 1988, Ultra introduced ISO TP4 (level 4 networking protocol) as part of their controllers and implemented a type of star configuration network using coax and fiber optic connections. They called this product,UltraNet. They later offered a fast version of TCP/IP in their controllers, as this protocol was most frequently encountered in an actual computer center network environment. The clock rates on the Ultra network processors provided 250Mbit/s transfer rates and four of these could be connected together to achieve one gigabit per second transfer rates for a single logical connection. Effective transfer rates betweenSilicon GraphicsandSun Microsystemsworkstationsexceeded 4 MB/s using one 250 Mbit/s physical connection, a factor of over 10 to 12 greater than then currentEthernetconnections and often exceeded the effective transfer rates of the competing NSC and CNT connections in similar applications. Customers with dual Cray computers measured the connections between Cray processors over theUltraNetthat exceeded80 MB/seffective transfer rates. Ultra Network Technologies products included network cards for workstations and mini-supercomputers usingVMEbusconnectors andfiber opticcable for the network physical connections, host network cards which resided in the network hub for Cray Supercomputers,IBMmainframes, mini-supercomputers fromConvex Computer,HIPPIstandard channel, and others. There were two sizes of high speed network hubs that contained the mainframe host cards plus the fiber opticnetwork hub to network hubcards. The network topology was in the form of connected hubs. Engineers at the Stuttgart University computer center demonstrated long distance connections using German PTT provided fiber optics of effective transfer rates over 4 MB/s up to an 800 km distance. Later products incorporatedTCP/IPnetwork protocols in their processors.
A typical network configuration of several workstations and a single mainframe host could cost $250,000. A configuration with many workstations and two or three mainframe computers could reach $1 Million.
The company grew to about 140 employees at its high point. Its headquarters was located at 101 Daggett Drive, San Jose, CA with other offices in Dallas, Los Angeles, Seattle, Washington DC, Düsseldorf, Germany, and Paris, France. In 1992, the company was abandoned by its investors and sold due to an inability to become profitable and the advent of less expensive network technologies, mainly created by the advent of the higher speed personal computers and lower cost workstations used in the scientific labs; the buyer was Computer Network Technology Corporation of Plymouth, Minnesota (NASDAQ: CMNT). The company's Chairman of the Board was M. Kenneth Oshman, formally chairman ofROLMCorporation, and President was Stan Tenold, previously the president of ROLM's Military Products division. The company's various customers included many high-end computer centers, including, severalNASAsites,NSA,US Air Force,US Navy,Aramco,France's EDF,Pittsburg Supercomputer Center,University of Stuttgart,Leibniz University Hannover,Apple Computer,Houston Chronicle, and many other such high end computer users.
|
https://en.wikipedia.org/wiki/Ultra_Network_Technologies
|
Formal scienceis abranch of sciencestudying disciplines concerned with abstract structures described byformal systems, such aslogic,mathematics,statistics,theoretical computer science,artificial intelligence,information theory,game theory,systems theory,decision theoryandtheoretical linguistics. Whereas thenatural sciencesandsocial sciencesseek to characterizephysical systemsandsocial systems, respectively, usingtheoreticalandempiricalmethods, the formal sciences use languagetoolsconcerned with characterizing abstract structures described byformal systemsand the deductions that can be made from them. The formal sciences aid the natural and social sciences by providing information about the structures used to describe the physical world, and what inferences may be made about them.[1]
One reason why mathematics enjoys special esteem, above all other sciences, is that its laws are absolutely certain and indisputable, while those of other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts.
Because of their non-empirical nature, formal sciences are construed by outlining a set ofaxiomsanddefinitionsfrom which other statements (theorems) are deduced. For this reason, inRudolf Carnap'slogical-positivistconception of theepistemology of science, theories belonging to formal sciences are understood to contain nosynthetic statements, instead containing onlyanalytic statements.[3][4]
|
https://en.wikipedia.org/wiki/Formal_science
|
Withintheoretical computer science, theSun–Ni law(orSun and Ni's law, also known asmemory-bounded speedup) is a memory-boundedspeedupmodel which states that as computing power increases the corresponding increase in problem size is constrained by the system’s memory capacity. In general, as a system grows in computational power, the problems run on the system increase in size. Analogous toAmdahl's law, which says that the problem size remains constant as system sizes grow, andGustafson's law, which proposes that the problem size should scale but be bound by a fixed amount of time, the Sun–Ni law states the problem size should scale but be bound by the memory capacity of the system. Sun–Ni law[1][2]was initially proposed by Xian-He Sun and Lionel Ni at the Proceedings ofIEEE Supercomputing Conference 1990.
With the increasing disparity between CPU speed and memory data access latency, application execution time often depends on the memory speed of the system.[3]As predicted by Sun and Ni, data access has become the premier performance bottleneck for high-end computing. From this one can see the intuition behind the law; as system resources increase, applications are often bottlenecked by memory speed and bandwidth, thus an application can achieve a larger speedup by utilizing all the memory capacity in the system. The law can be applied to different layers of amemory hierarchysystem, fromL1 cacheto main memory. Through its memory-bounded function,W=G(M), it reveals the trade-off between computing and memory in algorithm andsystem architecturedesign.
All three speedup models, Sun–Ni, Gustafson, and Amdahl, provide a metric to analyze speedup forparallel computing. Amdahl’s law focuses on the time reduction for a given fixed-size problem. Amdahl’s law states that the sequential portion of the problem (algorithm) limits the total speedup that can be achieved as system resources increase. Gustafson’s law suggests that it is beneficial to build a large-scale parallel system as the speedup can grow linearly with the system size if the problem size is scaled up to maintain a fixed execution time.[4]Yet as memory access latency often becomes the dominant factor in an application’s execution time,[5]applications may not scale up to meet the time bound constraint.[1][2]The Sun–Ni law, instead of constraining the problem size by time, constrains the problem by the memory capacity of the system, or in other words bounds based on memory. The law is a generalization of Amdahl's Law and Gustafson's Law. When the memory-bounded functionG(M)=1, it resolves to Amdahl's law; when the memory-bounded functionG(M)=m, the number of processors, it resolves to Gustafson's law.
LetW∗{\displaystyle \textstyle W^{*}}be the scaled workload under a memory space constraint. The memory bounded speedup can be defined as:
Supposef{\displaystyle \textstyle f}is the portion of the workload that can be parallelized and(1−f){\displaystyle \textstyle (1-f)}is the sequential portion of the workload.
Lety=g(x){\displaystyle \textstyle y=g(x)}be the function that reflects the parallel workload increase factor as the memory capacity increases m times.
Let:W=g(M){\displaystyle \textstyle W=g(M)}and:W∗=g(m⋅M){\displaystyle \textstyle W^{*}=g(m\cdot M)}whereM{\displaystyle \textstyle M}is the memory capacity of one node.
Thus,W∗=g(m⋅g−1(W)){\displaystyle W^{*}=g(m\cdot g^{-1}(W))}
The memory bounded speedup is then:
(1−f)W+f⋅g(m⋅g−1(W))(1−f)W+f⋅g(m⋅g−1(W))m{\displaystyle {\frac {(1-f)W+f\cdot g(m\cdot g^{-1}(W))}{(1-f)W+{\frac {f\cdot g(m\cdot g^{-1}(W))}{m}}}}}
For any power functiong(x)=axb{\displaystyle \textstyle g(x)=ax^{b}}and for any rational numbersaandb, we have:
g(mx)=a(mx)b=mb⋅axb=mbg(x)=g¯(m)g(x){\displaystyle g(mx)=a(mx)^{b}=m^{b}\cdot ax^{b}=m^{b}g(x)={\bar {g}}(m)g(x)}
whereg¯(m){\displaystyle \textstyle {\bar {g}}(m)}is the power function with the coefficient as 1.
Thus by taking the highest degree term to determine the complexity of the algorithm, one can rewrite memory bounded speedup as:
(1−f)W+f⋅g¯(m)W(1−f)W+f⋅g¯(m)Wm=(1−f)+f⋅g¯(m)(1−f)+f⋅g¯(m)m{\displaystyle {\frac {(1-f)W+f\cdot {\bar {g}}(m)W}{(1-f)W+{\frac {f\cdot {\bar {g}}(m)W}{m}}}}={\frac {(1-f)+f\cdot {\bar {g}}(m)}{(1-f)+{\frac {f\cdot {\bar {g}}(m)}{m}}}}}
In this equation,g¯(m){\displaystyle \textstyle {\bar {g}}(m)}represents the influence of memory change on the change in problem size.
Supposeg¯(m)=1{\displaystyle \textstyle {\bar {g}}(m)=1}. Then the memory-bounded speedup model reduces toAmdahl's law, since problem size is fixed or independent of resource increase.
Supposeg¯(m)=m{\displaystyle \textstyle {\bar {g}}(m)=m}, Then the memory-bounded speedup model reduces toGustafson's law, which means when memory capacity increasesmtimes and the workload also increasesmtimes all the data needed is local to every node in the system.
Often, for simplicity and for matching the notation of Amdahl's Law and Gustafson's Law, the letterGis used to represent the memory bound functiong¯(m){\displaystyle \textstyle {\bar {g}}(m)}, andnreplacesm. Using this notation one gets:
Speedupmemory-bounded=(1−f)+f⋅G(n)(1−f)+f⋅G(n)n{\displaystyle Speedup_{\text{memory-bounded}}={\frac {(1-f)+f\cdot G(n)}{(1-f)+{\frac {f\cdot G(n)}{n}}}}}
Suppose one would like to determine the memory-bounded speedup of matrix multiplication. The memory requirement of matrix multiplication is roughlyx=3N2{\displaystyle \textstyle x=3N^{2}}whereNis the dimension of the twoN X Nsource matrices. And the computation requirement is2N3{\displaystyle 2N^{3}}
Thus we have:
g(x)=2(x/3)3/2=233/2x3/2{\displaystyle g(x)=2(x/3)^{3/2}={\frac {2}{3^{3/2}}}x^{3/2}}andg¯(x)=x3/2{\displaystyle {\bar {g}}(x)=x^{3/2}}
Thus the memory-bounded speedup is for matrix multiplication is:
(1−f)+f⋅g¯(m)(1−f)+f⋅g¯(m)m=(1−f)+f⋅m3/2(1−f)+f⋅m1/2{\displaystyle {\frac {(1-f)+f\cdot {\bar {g}}(m)}{(1-f)+{\frac {f\cdot {\bar {g}}(m)}{m}}}}={\frac {(1-f)+f\cdot m^{3/2}}{(1-f)+f\cdot m^{1/2}}}}
The following is another matrix multiplication example which illustrates the rapid increase in parallel execution time.[6]The execution time of aN X Nmatrix for a uniprocessor is:O(n3){\displaystyle O(n^{3})}. While the memory usage is:O(n2){\displaystyle O(n^{2})}
Suppose a10000-by-10000matrix takes800 MBof memory and can be factorized in1hour on a uniprocessor. Now for the scaled workload suppose is possible to factorize a320,000-by-320,000matrix in32hours. The time increase is quite large, but the increase in problem size may be more valuable for someones whose premier goal is accuracy. For example, an astrophysicist may be more interested in simulating anN-body problemwith as the number of particles as large as possible.[6]This example shows for computation intensive applications, memory capacity does not need to proportionally scale up with computing power.
The memory-bounded speedup model is the first work to reveal that memory is the performance constraint for high-end computing and presents a quantitative mathematical formulation for the trade-off between memory and computing. It is based on the memory-bounded function,W=G(n), where W is the work and thus also the computation for most applications.Mis the memory requirement in terms of capacity, andGis the reuse rate.W=G(M)gives a very simple, but effective, description of the relation between computation and memory requirement. From an architecture viewpoint, the memory-bounded model suggests the size, as well as speed, of the cache(s) should match the CPU performance. Today, modern microprocessors such as thePentium Pro,Alpha 21164,Strong Arm SA110, and Longson-3A use 80% or more of their transistors for the on-chip cache rather than computing components. From an algorithm design viewpoint, we should reduce the number of memory accesses. That is, reuse the data when it is possible. The functionG()gives the reuse rate. Today, the termmemory bound functionshas become a general term which refers to functions which involve extensive memory access.[7]Memory complexity analysis has become a discipline of computer algorithm analysis.
|
https://en.wikipedia.org/wiki/Sun%E2%80%93Ni_law
|
Network computingrefers to computers or nodes working together over anetwork.
Network computingmay also mean:
|
https://en.wikipedia.org/wiki/Network_computing_(disambiguation)
|
TheWDR paper computerorKnow-how Computeris an educational model of a computer consisting only of a pen, a sheet of paper, and individual matches in the most simple case.[1]This allows anyone interested to learn how to program without having anelectronic computerat their disposal.
The paper computer was created in the early 1980s when computer access was not yet widespread in Germany, to allow people to familiarize themselves with basic computer operation andassembly-like programming languages. It was distributed in over400000copies and at its time belonged to the computers with the widest circulation.
The Know-how Computer was developed byWolfgang Back[de]and Ulrich Rohde and was first presented in the television programWDR Computerclub(broadcast byWestdeutscher Rundfunk) in 1983. It was also published in German computer magazinesmcandPC Magazin[de].[2]
The original printed version of the paper computer has up to 21 lines of code on the left and eightregisterson the right, which are represented as boxes that contain as manymatchesas the value in the corresponding register.[3]A pen is used to indicate the line of code which is about to be executed. The user steps through the program, adding and subtracting matches from the appropriate registers and following program flow until the stop instruction is encountered.
The instruction set of five commands is small butTuring completeand therefore enough to represent all mathematical functions:
In the original newspaper article about this computer, it was written slightly differently (translation):
[4]
An emulator forWindowsis available on Wolfgang Back's website,[5]but a JavaScript emulator also exists.[6]Emulators place fewer restrictions on line count or the number of registers, allowing longer and more complex programs.
The paper computer's method of operation is nominally based on aregister machineby Elmar Cohors-Fresenborg,[2][7]but follows more the approach ofJohn Cedric ShepherdsonandHoward E. Sturgisin theirShepherdson–Sturgis register machinemodel.[8]
A derived version of the paper computer is used as a "Know-How Computer" inNamibianschool education.[9]
|
https://en.wikipedia.org/wiki/WDR_paper_computer
|
ThePhillips Machine, also known as theMONIAC(Monetary National Income Analogue Computer),Phillips Hydraulic Computerand theFinancephalograph, is ananalogue computerwhich usesfluidic logicto model the workings of an economy. The name "MONIAC" is suggested by associatingmoneyandENIAC, an early electronicdigital computer.
It was created in 1949 by theNew ZealandeconomistBill Phillipsto model the national economic processes of theUnited Kingdom, while Phillips was a student at theLondon School of Economics(LSE). While designed as a teaching tool, it was discovered to be quite accurate, and thus an effective economic simulator.
At least twelve machines were built, donated to or purchased by various organisations around the world. As of 2023[update], several are in working order.
Phillips scrounged materials to create his prototype computer, including bits and pieces of war surplus parts from oldLancaster bombers.[1]The first MONIAC was created in his landlady's garage inCroydonat a cost of£400 (equivalent to £18,000 in 2023).
According to the Anna Corkhill:
Phillips discussed the idea with Walter Newlyn, a junior academic at Leeds University who had studied with Phillips at the LSE, and proceeded to build a prototype (with Newlyn’s assistance) over one summer in a garage in Croydon. Newlyn persuaded the head of department at Leeds to advance
£100 towards building the prototype. Newlyn helped as a craftsman’s mate—sanding and gluing together pieces of acrylic and supplementing Phillips’ economic knowledge.[2]
Phillips first demonstrated the machine to leading economists at theLondon School of Economics(LSE), of which Phillips was a student, in 1949. It was very well received and Phillips was soon offered a teaching position at the LSE.
The machine had been designed as a teaching aid but was also discovered to be an effective economic simulator.[3]When the machine was created, electronic digital computers that could run complex economic simulations were unavailable. In 1949, the few computers in existence were restricted to government and military use and their lack of adequate visual displays made them unable to illustrate the operation of complex models. Observing the machine in operation made it much easier for students to understand the interrelated processes of a national economy. The range of organisations that acquired a machine showed that it was used in both capacities.[original research?]
The machine is approximately 2 m (6 ft 7 in) high, 1.2 m (3 ft 11 in) wide and almost 1 m (3 ft 3 in) deep, and consisted of a series of transparent plastic tanks and pipes which were fastened to a wooden board. Each tank represented some aspect of the UK nationaleconomyand the flow of money around the economy was illustrated by coloured water. At the top of the board was a large tank called the treasury. Water (representing money) flowed from the treasury to other tanks representing the various ways in which a country could spend its money. For example, there were tanks for health and education. To increase spending on health care a tap could be opened to drain water from the treasury to the tank which represented health spending. Water then ran further down the model to other tanks, representing other interactions in the economy. Water could be pumped back to the treasury from some of the tanks to representtaxation. Changes in tax rates were modeled by increasing or decreasing pumping speeds.
Savingsreduce the funds available to consumers andinvestmentincome increases those funds.[citation needed]The machine showed it by draining water (savings) from the expenditure stream and by injecting water (investment income) into that stream. When the savings flow exceeds the investment flow, the level of water in the savings and investment tank (the surplus-balances tank) would rise to reflect the accumulated balance. When the investment flow exceeds the savings flow for any length of time, the surplus-balances tank would run dry. Import and export were represented by water draining from the model and by additional water being poured into the model.
The flow of the water was automatically controlled through a series of floats, counterweights, electrodes, and cords. When the level of water reached a certain level in a tank, pumps and drains would be activated. To their surprise, Phillips and his associate Walter Newlyn found that machine could be calibrated to an accuracy of 2%.
The flow of water between the tanks was determined by economic principles and the settings for various parameters. Different economic parameters, such as tax rates and investment rates, could be entered by setting the valves which controlled the flow of water about the computer. Users could experiment with different settings and note their effects. The machine's ability to model the subtle interaction of a number of variables made it a powerful tool for its time.[citation needed]When a set of parameters resulted in a viable economy the model would stabilise and the results could be read from scales. The output from the computer could also be sent to a rudimentaryplotter.
It is thought that twelve to fourteen machines were built:
TheTerry PratchettnovelMaking Moneycontains a similar device as a major plot point. However, after the device is fully perfected, itmagically becomes directly coupledto the economy it was intended to simulate, with the result that the machine cannot then be adjusted without causing a change in the actual economy (in parodic resemblance toGoodhart's law).[improper synthesis?]
EconomistKate Raworth's bookDonut Economicscritiques the use of an electric pump as the power source, claiming that because its power consumption was not considered, it left out an important component out of the economic model it was portraying:[11][12]
"This is where Bill Phillips’s MONIAC machine was fundamentally flawed. While brilliantly demonstrating the economy’s circular flow of income, it completely overlooked its throughflow of energy. To make his hydraulic computer start up, Phillips had to flip a switch on the back of it to turn on its electric pump. Like any real economy it relied upon an external source of energy to make it run, but neither Phillips nor his contemporaries spotted that the machine’s power source was a critical part of what made the model work. That lesson from the MONIAC applies to all of macroeconomics: the role of energy deserves a far more prominent place in economic theories that hope to explain what drives economic activity."
|
https://en.wikipedia.org/wiki/MONIAC
|
Digital Enhanced Cordless Telecommunications(DECT) is acordless telephonystandard maintained byETSI. It originated inEurope, where it is the common standard, replacing earlier standards, such asCT1andCT2.[1]Since the DECT-2020 standard onwards, it also includesIoTcommunication.
Beyond Europe, it has been adopted byAustraliaand most countries inAsiaandSouth America. North American adoption was delayed byUnited Statesradio-frequency regulations. This forced development of a variation of DECT calledDECT 6.0, using a slightly different frequency range, which makes these units incompatible with systems intended for use in other areas, even from the same manufacturer. DECT has almost completely replaced other standards in most countries where it is used, with the exception of North America.
DECT was originally intended for fast roaming between networked base stations, and the first DECT product wasNet3wireless LAN. However, its most popular application is single-cell cordless phones connected totraditional analog telephone, primarily in home and small-office systems, though gateways with multi-cell DECT and/or DECT repeaters are also available in manyprivate branch exchange(PBX) systems for medium and large businesses, produced byPanasonic,Mitel,Gigaset,Ascom,Cisco,Grandstream,Snom,Spectralink, and RTX. DECT can also be used for purposes other than cordless phones, such asbaby monitors,wireless microphonesand industrial sensors. TheULE Alliance'sDECT ULEand its "HAN FUN" protocol[2]are variants tailored for home security, automation, and theinternet of things(IoT).
The DECT standard includes thegeneric access profile(GAP), a common interoperability profile for simple telephone capabilities, which most manufacturers implement. GAP-conformance enables DECT handsets and bases from different manufacturers to interoperate at the most basic level of functionality, that of making and receiving calls. Japan uses its own DECT variant, J-DECT, which is supported by the DECT forum.[3]
The New Generation DECT (NG-DECT) standard, marketed asCAT-iqby the DECT Forum, provides a common set of advanced capabilities for handsets and base stations. CAT-iq allows interchangeability acrossIP-DECTbase stations and handsets from different manufacturers, while maintaining backward compatibility with GAP equipment. It also requires mandatory support forwideband audio.
DECT-2020New Radio, marketed as NR+ (New Radio plus), is a5Gdata transmission protocol which meets ITU-RIMT-2020requirements for ultra-reliable low-latency and massive machine-type communications, and can co-exist with earlier DECT devices.[4][5][6]
The DECT standard was developed byETSIin several phases, the first of which took place between 1988 and 1992 when the first round of standards were published. These were the ETS 300-175 series in nine parts defining the air interface, and ETS 300-176 defining how the units should be type approved. A technical report, ETR-178, was also published to explain the standard.[7]Subsequent standards were developed and published by ETSI to cover interoperability profiles and standards for testing.
Named Digital European Cordless Telephone at its launch by CEPT in November 1987; its name was soon changed to Digital European Cordless Telecommunications, following a suggestion by Enrico Tosato of Italy, to reflect its broader range of application including data services. In 1995, due to its more global usage, the name was changed from European to Enhanced. DECT is recognized by theITUas fulfilling theIMT-2000requirements and thus qualifies as a3Gsystem. Within the IMT-2000 group of technologies, DECT is referred to as IMT-2000 Frequency Time (IMT-FT).
DECT was developed by ETSI but has since been adopted by many countries all over the World. The original DECT frequency band (1880–1900 MHz) is used in all countries inEurope. Outside Europe, it is used in most ofAsia,AustraliaandSouth America. In theUnited States, theFederal Communications Commissionin 2005 changed channelization and licensing costs in a nearby band (1920–1930 MHz, or 1.9GHz), known asUnlicensed Personal Communications Services(UPCS), allowing DECT devices to be sold in the U.S. with only minimal changes. These channels are reserved exclusively for voice communication applications and therefore are less likely to experience interference from other wireless devices such asbaby monitorsandwireless networks.
The New Generation DECT (NG-DECT) standard was first published in 2007;[8]it was developed by ETSI with guidance from theHome Gateway Initiativethrough the DECT Forum[9]to supportIP-DECTfunctions inhome gateway/IP-PBXequipment. The ETSI TS 102 527 series comes in five parts and covers wideband audio and mandatory interoperability features between handsets and base stations. They were preceded by an explanatory technical report, ETSI TR 102 570.[10]The DECT Forum maintains theCAT-iqtrademark and certification program; CAT-iq wideband voice profile 1.0 and interoperability profiles 2.0/2.1 are based on the relevant parts of ETSI TS 102 527.
TheDECT Ultra Low Energy(DECT ULE) standard was announced in January 2011 and the first commercial products were launched later that year byDialog Semiconductor. The standard was created to enablehome automation, security, healthcare and energy monitoring applications that are battery powered. Like DECT, DECT ULE standard uses the 1.9 GHz band, and so suffers less interference thanZigbee,Bluetooth, orWi-Fifrom microwave ovens, which all operate in the unlicensed 2.4 GHzISM band. DECT ULE uses a simple star network topology, so many devices in the home are connected to a single control unit.
A new low-complexity audio codec,LC3plus, has been added as an option to the 2019 revision of the DECT standard. This codec is designed for high-quality voice and music applications such as wireless speakers, headphones, headsets, and microphones. LC3plus supports scalable 16-bit narrowband, wideband, super wideband, fullband, and 24-bit high-resolution fullband and ultra-band coding, with sample rates of 8, 16, 24, 32, 48 and 96 kHz and audio bandwidth of up to 48 kHz.[11][12]
DECT-2020 New Radio protocol was published in July 2020; it defines a new physical interface based oncyclic prefixorthogonal frequency-division multiplexing (CP-OFDM) capable of up to 1.2Gbit/s transfer rate withQAM-1024 modulation. The updated standard supports multi-antennaMIMOandbeamforming, FECchannel coding, and hybridautomatic repeat request. There are 17 radio channel frequencies in the range from 450MHz up to 5,875MHz, and channel bandwidths of 1,728, 3,456, or 6,912kHz. Direct communication between end devices is possible with amesh networktopology. In October 2021, DECT-2020 NR was approved for theIMT-2020standard,[4]for use in Massive Machine Type Communications (MMTC) industry automation, Ultra-Reliable Low-Latency Communications (URLLC), and professionalwireless audioapplications with point-to-point ormulticastcommunications;[13][14][15]the proposal was fast-tracked by ITU-R following real-world evaluations.[5][16]The new protocol will be marketed as NR+ (New Radio plus) by the DECT Forum.[6]OFDMAandSC-FDMAmodulations were also considered by the ESTI DECT committee.[17][18]
OpenD is an open-source framework designed to provide a complete software implementation of DECT ULE protocols on reference hardware fromDialog SemiconductorandDSP Group; the project is maintained by the DECT forum.[19][20]
The DECT standard originally envisaged three major areas of application:[7]
Of these, the domestic application (cordless home telephones) has been extremely successful. The enterprisePABXmarket, albeit much smaller than the cordless home market, has been very successful as well, and all the major PABX vendors have advanced DECT access options available. The public access application did not succeed, since public cellular networks rapidly out-competed DECT by coupling their ubiquitous coverage with large increases in capacity and continuously falling costs. There has been only one major installation of DECT for public access: in early 1998Telecom Italialaunched a wide-area DECT network known as "Fido" after much regulatory delay, covering major cities in Italy.[21]The service was promoted for only a few months and, having peaked at 142,000 subscribers, was shut down in 2001.[22]
DECT has been used forwireless local loopas a substitute for copper pairs in the "last mile" in countries such as India and South Africa. By using directional antennas and sacrificing some traffic capacity, cell coverage could extend to over 10 kilometres (6.2 mi). One example is thecorDECTstandard.
The first data application for DECT wasNet3wireless LAN system by Olivetti, launched in 1993 and discontinued in 1995. A precursor to Wi-Fi, Net3was a micro-cellular data-only network with fast roaming between base stations and 520 kbit/s transmission rates.
Data applications such as electronic cash terminals, traffic lights, and remote door openers[23]also exist, but have been eclipsed byWi-Fi,3Gand4Gwhich compete with DECT for both voice and data.
The DECT standard specifies a means for aportable phoneor "Portable Part" to access a fixed telephone network via radio.Base stationor "Fixed Part" is used to terminate the radio link and provide access to a fixed line. Agatewayis then used to connect calls to the fixed network, such aspublic switched telephone network(telephone jack), office PBX, ISDN, or VoIP over Ethernet connection.
Typical abilities of a domestic DECTGeneric Access Profile(GAP) system include multiple handsets to one base station and one phone line socket. This allows several cordless telephones to be placed around the house, all operating from the same telephone line. Additional handsets have a battery charger station that does not plug into the telephone system. Handsets can in many cases be used asintercoms, communicating between each other, and sometimes aswalkie-talkies, intercommunicating without telephone line connection.
DECT operates in the 1880–1900 MHz band and defines ten frequency channels from 1881.792 MHz to 1897.344 MHz with a band gap of 1728 kHz.
DECT operates as a multicarrierfrequency-division multiple access(FDMA) andtime-division multiple access(TDMA) system. This means that theradio spectrumis divided into physical carriers in two dimensions: frequency and time. FDMA access provides up to 10 frequency channels, and TDMA access provides 24 time slots per every frame of 10ms. DECT usestime-division duplex(TDD), which means that down- and uplink use the same frequency but different time slots. Thus a base station provides 12 duplex speech channels in each frame, with each time slot occupying any available channel – thus 10 × 12 = 120 carriers are available, each carrying 32 kbit/s.
DECT also providesfrequency-hopping spread spectrumoverTDMA/TDD structure for ISM band applications. If frequency-hopping is avoided, each base station can provide up to 120 channels in the DECT spectrum before frequency reuse. Each timeslot can be assigned to a different channel in order to exploit advantages of frequency hopping and to avoid interference from other users in asynchronous fashion.[24]
DECT allows interference-free wireless operation to around 100 metres (110 yd) outdoors. Indoor performance is reduced when interior spaces are constrained by walls.
DECT performs with fidelity in common congested domestic radio traffic situations. It is generally immune to interference from other DECT systems,Wi-Finetworks,video senders,Bluetoothtechnology, baby monitors and other wireless devices.
ETSI standards documentation ETSI EN 300 175 parts 1–8 (DECT), ETSI EN 300 444 (GAP) and ETSI TS 102 527 parts 1–5 (NG-DECT) prescribe the following technical properties:
The DECTphysical layeruses FDMA/TDMA access with TDD.
Gaussian frequency-shift keying(GFSK) modulation is used: the binary one is coded with a frequency increase by 288 kHz, and the binary zero with frequency decrease of 288 kHz. With high quality connections, 2-, 4- or 8-level differential PSK modulation (DBPSK, DQPSK or D8PSK), which is similar to QAM-2, QAM-4 and QAM-8, can be used to transmit 1, 2, or 3 bits per each symbol. QAM-16 and QAM-64 modulations with 4 and 6 bits per symbol can be used for user data (B-field) only, with resulting transmission speeds of up to 5,068Mbit/s.
DECT provides dynamic channel selection and assignment; the choice of transmission frequency and time slot is always made by the mobile terminal. In case of interference in the selected frequency channel, the mobile terminal (possibly from suggestion by the base station) can initiate either intracell handover, selecting another channel/transmitter on the same base, or intercell handover, selecting a different base station altogether. For this purpose, DECT devices scan all idle channels at regular 30s intervals to generate a received signal strength indication (RSSI) list. When a new channel is required, the mobile terminal (PP) or base station (FP) selects a channel with the minimum interference from the RSSI list.
The maximum allowed power for portable equipment as well as base stations is 250 mW. A portable device radiates an average of about 10 mW during a call as it is only using one of 24 time slots to transmit. In Europe, the power limit was expressed aseffective radiated power(ERP), rather than the more commonly usedequivalent isotropically radiated power(EIRP), permitting the use of high-gain directional antennas to produce much higher EIRP and hence long ranges.
The DECTmedia access controllayer controls the physical layer and providesconnection oriented,connectionlessandbroadcastservices to the higher layers.
The DECTdata link layeruses Link Access Protocol Control (LAPC), a specially designed variant of theISDNdata link protocol called LAPD. They are based onHDLC.
GFSK modulation uses a bit rate of 1152 kbit/s, with a frame of 10ms (11520bits) which contains 24 time slots. Each slots contains 480 bits, some of which are reserved for physical packets and the rest is guard space. Slots 0–11 are always used for downlink (FP to PP) and slots 12–23 are used for uplink (PP to FP).
There are several combinations of slots and corresponding types of physical packets with GFSK modulation:
The 420/424 bits of a GFSK basic packet (P32) contain the following fields:
The resulting full data rate is 32 kbit/s, available in both directions.
The DECTnetwork layeralways contains the following protocol entities:
Optionally it may also contain others:
All these communicate through a Link Control Entity (LCE).
The call control protocol is derived fromISDNDSS1, which is aQ.931-derived protocol. Many DECT-specific changes have been made.[specify]
The mobility management protocol includes the management of identities, authentication, location updating, on-air subscription and key allocation. It includes many elements similar to the GSM protocol, but also includes elements unique to DECT.
Unlike the GSM protocol, the DECT network specifications do not define cross-linkages between the operation of the entities (for example, Mobility Management and Call Control). The architecture presumes that such linkages will be designed into the interworking unit that connects the DECT access network to whatever mobility-enabled fixed network is involved. By keeping the entities separate, the handset is capable of responding to any combination of entity traffic, and this creates great flexibility in fixed network design without breaking full interoperability.
DECTGAPis an interoperability profile for DECT. The intent is that two different products from different manufacturers that both conform not only to the DECT standard, but also to the GAP profile defined within the DECT standard, are able to interoperate for basic calling. The DECT standard includes full testing suites for GAP, and GAP products on the market from different manufacturers are in practice interoperable for the basic functions.
The DECT media access control layer includes authentication of handsets to the base station using the DECT Standard Authentication Algorithm (DSAA). When registering the handset on the base, both record a shared 128-bit Unique Authentication Key (UAK). The base can request authentication by sending two random numbers to the handset, which calculates the response using the shared 128-bit key. The handset can also request authentication by sending a 64-bit random number to the base, which chooses a second random number, calculates the response using the shared key, and sends it back with the second random number.
The standard also providesencryptionservices with the DECT Standard Cipher (DSC). The encryption isfairly weak, using a 35-bitinitialization vectorand encrypting the voice stream with 64-bit encryption. While most of the DECT standard is publicly available, the part describing the DECT Standard Cipher was only available under anon-disclosure agreementto the phones' manufacturers fromETSI.
The properties of the DECT protocol make it hard to intercept a frame, modify it and send it later again, as DECT frames are based on time-division multiplexing and need to be transmitted at a specific point in time.[26]Unfortunately very few DECT devices on the market implemented authentication and encryption procedures[26][27]– and even when encryption was used by the phone, it was possible to implement aman-in-the-middle attackimpersonating a DECT base station and revert to unencrypted mode – which allows calls to be listened to, recorded, and re-routed to a different destination.[27][28][29]
After an unverified report of a successful attack in 2002,[30][31]members of the deDECTed.org project actually did reverse engineer the DECT Standard Cipher in 2008,[27]and as of 2010 there has been a viable attack on it that can recover the key.[32]
In 2012, an improved authentication algorithm, the DECT Standard Authentication Algorithm 2 (DSAA2), and improved version of the encryption algorithm, the DECT Standard Cipher 2 (DSC2), both based onAES128-bit encryption, were included as optional in the NG-DECT/CAT-iq suite.
DECT Forum also launched the DECT Security certification program which mandates the use of previously optional security features in the GAP profile, such as early encryption and base authentication.
Various access profiles have been defined in the DECT standard:
DECT 6.0 is a North American marketing term for DECT devices manufactured for the United States and Canada operating at 1.9 GHz. The "6.0" does not equate to a spectrum band; it was decided the term DECT 1.9 might have confused customers who equate larger numbers (such as the 2.4 and 5.8 in existing 2.4 GHz and 5.8 GHz cordless telephones) with later products. The term was coined by Rick Krupka, marketing director at Siemens and the DECT USA Working Group / Siemens ICM.
In North America, DECT suffers from deficiencies in comparison to DECT elsewhere, since theUPCS band(1920–1930 MHz) is not free from heavy interference.[34]Bandwidth is half as wide as that used in Europe (1880–1900 MHz), the 4 mW average transmission power reduces range compared to the 10 mW permitted in Europe, and the commonplace lack of GAP compatibility among US vendors binds customers to a single vendor.
Before 1.9 GHz band was approved by the FCC in 2005, DECT could only operate in unlicensed2.4 GHzand 900 MHz Region 2ISM bands; some users ofUnidenWDECT 2.4 GHz phones reported interoperability issues withWi-Fiequipment.[35][36][unreliable source?]
North-AmericanDECT 6.0products may not be used in Europe, Pakistan,[37]Sri Lanka,[38]and Africa, as they cause and suffer from interference with the local cellular networks. Use of such products is prohibited by European Telecommunications Authorities,PTA, Telecommunications Regulatory Commission of Sri Lanka[39]and the Independent Communication Authority of South Africa. European DECT products may not be used in the United States and Canada, as they likewise cause and suffer from interference with American and Canadian cellular networks, and use is prohibited by theFederal Communications CommissionandInnovation, Science and Economic Development Canada.
DECT 8.0 HD is a marketing designation for North American DECT devices certified withCAT-iq 2.0"Multi Line" profile.[40]
Cordless Advanced Technology—internet and quality (CAT-iq) is a certification program maintained by the DECT Forum. It is based on New Generation DECT (NG-DECT) series of standards from ETSI.
NG-DECT/CAT-iq contains features that expand the generic GAP profile with mandatory support for high quality wideband voice, enhanced security, calling party identification, multiple lines, parallel calls, and similar functions to facilitateVoIPcalls throughSIPandH.323protocols.
There are several CAT-iq profiles which define supported voice features:
CAT-iq allows any DECT handset to communicate with a DECT base from a different vendor, providing full interoperability. CAT-iq 2.0/2.1 feature set is designed to supportIP-DECTbase stations found in officeIP-PBXandhome gateways.
DECT-2020, also called NR+, is a new radio standard byETSIfor the DECT bands worldwide.[41][42]The standard was designed to meet a subset of theITUIMT-20205Grequirements that are applicable toIOTandIndustrial internet of things.[43]DECT-2020 is compliant with the requirements for Ultra Reliable Low Latency CommunicationsURLLCand massive Machine Type Communication (mMTC) of IMT-2020.
DECT-2020 NR has new capabilities[44]compared to DECT and DECT Evolution:
The DECT-2020 standard has been designed to co-exist in the DECT radio band with existing DECT deployments. It uses the same Time Division slot timing and Frequency Division center frequencies and uses pre-transmit scanning to minimize co-channel interference.
Other interoperability profiles exist in the DECT suite of standards, and in particular the DPRS (DECT Packet Radio Services) bring together a number of prior interoperability profiles for the use of DECT as a wireless LAN and wireless internet access service. With good range (up to 200 metres (660 ft) indoors and 6 kilometres (3.7 mi) using directional antennae outdoors), dedicated spectrum, high interference immunity, open interoperability and data speeds of around 500 kbit/s, DECT appeared at one time to be a superior alternative toWi-Fi.[45]The protocol capabilities built into the DECT networking protocol standards were particularly good at supporting fast roaming in the public space, between hotspots operated by competing but connected providers. The first DECT product to reach the market, Olivetti'sNet3, was a wireless LAN, and German firmsDosch & AmandandHoeft & Wesselbuilt niche businesses on the supply of data transmission systems based on DECT.
However, the timing of the availability of DECT, in the mid-1990s, was too early to find wide application for wireless data outside niche industrial applications. Whilst contemporary providers of Wi-Fi struggled with the same issues, providers of DECT retreated to the more immediately lucrative market for cordless telephones. A key weakness was also the inaccessibility of the U.S. market, due to FCC spectrum restrictions at that time. By the time mass applications for wireless Internet had emerged, and the U.S. had opened up to DECT, well into the new century, the industry had moved far ahead in terms of performance and DECT's time as a technically competitive wireless data transport had passed.
DECT usesUHFradio, similar to mobile phones, baby monitors, Wi-Fi, and other cordless telephone technologies.
In North America, the 4 mW average transmission power reduces range compared to the 10 mW permitted in Europe.
The UKHealth Protection Agency(HPA) claims that due to a mobile phone's adaptive power ability, a European DECT cordless phone's radiation could actually exceed the radiation of a mobile phone. A European DECT cordless phone's radiation has an average output power of 10 mW but is in the form of 100 bursts per second of 250 mW, a strength comparable to some mobile phones.[46]
Most studies have been unable to demonstrate any link to health effects, or have been inconclusive.Electromagnetic fieldsmay have an effect on protein expression in laboratory settings[47]but have not yet been demonstrated to have clinically significant effects in real-world settings. The World Health Organization has issued a statement on medical effects of mobile phones which acknowledges that the longer term effects (over several decades) require further research.[48]
|
https://en.wikipedia.org/wiki/DECT
|
TheGeneric Access Profile(GAP) (ETSIstandard EN 300 444[1]) describes a set of mandatory requirements to allow any conformingDECTFixed Part (base) to interoperate with any conforming DECT Portable Part (handset) to provide basic telephony services when attached to a 3.1 kHz telephone network (as defined by EN 300 176-2).
The objective of GAP is to ensure interoperation at the air interface (i.e., the radio connection) and at the level of procedures to establish, maintain and release telephone calls (Call Control). GAP also mandates procedures for registering Portable Parts to a Fixed Part (Mobility Management). A GAP-compliant handset from one manufacturer should work, at the basic level of making calls, with a GAP-compliant base from another manufacturer, although it may be unable to access advanced features of the base station such as phone book synchronization or remote operation of ananswering machine. Most consumer-level DECT phones and base stations support the GAP profile, even those that do not publicize the feature, and thus can be used together. However some manufacturers lock their systems to prevent interoperability, or supply bases that cannot register new handsets.
The GAP does not describe how the Fixed Part is connected to the external telephone network.
This article aboutwireless technologyis astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Generic_access_profile
|
Apicocellis a small cellularbase stationtypically covering a small area, such as in-building (offices, shopping malls, train stations, stock exchanges, etc.), or more recently in-aircraft. In cellular networks, picocells are typically used to extend coverage to indoor areas where outdoor signals do not reach well, or to addnetworkcapacity in areas with very dense phone usage, such as train stations or stadiums. Picocells provide coverage and capacity in areas difficult or expensive to reach using the more traditionalmacrocellapproach.[1]
In cellular wireless networks, such asGSM, the picocell base station[2]is typically a low-cost, small (typically the size of areamofA4 paper), reasonably simple unit that connects to abase station controller(BSC). Multiple picocell 'heads' connect to each BSC: the BSC performs radio resource management and hand-over functions, and aggregates data to be passed to themobile switching centre(MSC) or the gatewayGPRSsupport node (GGSN).
Connectivity between the picocell heads and the BSC typically consists of in-building wiring. Although originally deployed systems (1990s) usedplesiochronous digital hierarchy(PDH) links such as E1/T1 links, more recent systems useEthernetcabling. Aircraft use satellite links.[3]
More recent work has developed the concept towards a head unit containing not only a picocell, but also many of the functions of the BSC and some of the MSC. This form of picocell is sometimes called anaccess point base stationor 'enterprisefemtocell'. In this case, the unit contains all the capability required to connect directly to the Internet, without the need for the BSC/MSC infrastructure. This is a potentially more cost-effective approach.
Picocells offer many of the benefits of "small cells" (similar to femtocells) in that they improve data throughput for mobile users and increase capacity in the mobile network. In particular, the integration of picocells with macrocells through a heterogeneous network can be useful in seamless hand-offs and increased mobile data capacity.[4]
Picocells are available for most cellular technologies including GSM, CDMA, UMTS and LTE from manufacturers includingip.access, ZTE, Huawei and Airwalk.[5]
Typically the range of amicrocellis less than two kilometers wide, a picocell is 200 meters or less, and a femtocell is on the order of 10 meters,[6]although AT&T calls its product, with a range of 40 feet (12 m), a "microcell".[7]AT&T uses "AT&T 3G MicroCell" as a trademark and not necessarily the "microcell" technology, however.[8]
|
https://en.wikipedia.org/wiki/Picocell
|
Cordless Advanced Technology—internet and quality(CAT-iq) is a technology initiative from theDigital Enhanced Cordless Telecommunications(DECT) Forum, based on ETSI TS 102 527 New Generation DECT (NG-DECT) European standard series.
NG-DECT contains backward compatible extensions to basic DECTGAP(Generic access profile) functionality which allow bases and handsets from different vendors to work together with full feature richness expected from SIP terminals and VoIP gateways.
CAT-iq defines several profiles for high quality wideband voice services with multiple lines, as well as low bit-rate data applications.
The CAT-iq profiles are split between voice and data service, with the following mandatory features:[1]
The CAT-iq profiles require a stricter set of mandatory features than the relevant ETSI NG-DECT standards, which make many advanced features optional.
Optional voice codecs include 64 kbit/sG.711μ-law/A-law PCM (narrow band), 32 kbit/sG.729.1(wideband), 32 kbit/sMPEG-4ER AAC-LD(wideband), and 64 kbit/s MPEG-4 ER AAC-LD (super-wideband).
Within the voice profiles, the revisions are sequential: CAT-iq 1.0 provides basic wideband voice, while CAT-iq 2.0 and 2.1 add new functions which expand and supersede parts of the lower profiles.
The Data and IOT profiles can either be considered in isolation for data only devices or as a complementary service to the voice enabled devices.
CAT-iq allowsIP-DECTgateways with integrated NG-DECT base stations operate in full-feature mode with any certified CAT-iq 2.0/2.1 handset, regardless of different vendors and silicon or software protocol stack implementations. The base stations will also work with GAP handsets, though supporting only the basic GAP features.
The DECT Forum maintains CAT-iq 2.0 Certification program, which opened in December 2010 for member companies.[2]Certification ensures feature compatibility between devices from different vendors with specific focus on the over-the-air protocol, RF emissions and wideband audio requirements. Certified products are allowed to carry the CAT-iq logo and the HD Voice logo from theGSM Association.
This technology-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/CAT-iq
|
Wi-Fi calling, also calledVoWiFi,[1]refers tomobile phonevoice calls and data that are made overIPnetworks usingWi-Fi, instead of thecell towersprovided bycellular networks.[2]Using this feature, compatible handsets are able to route regular cellular calls through a wireless LAN (Wi-Fi) network withbroadband Internet, while seamlessly changing connections between the two where necessary.[3]This feature makes use of theGeneric Access Network(GAN) protocol, also known asUnlicensed Mobile Access(UMA).[4][5]
Voice over wireless LAN(VoWLAN), alsovoice over Wi‑Fi(VoWiFi[6]), is the use of awirelessbroadband network according to theIEEE 802.11standards for the purpose of vocal conversation. In essence, it isvoice over IP(VoIP) over aWi-Finetwork.
Essentially, GAN/UMA allows cell phone packets to be forwarded to a network access point over the internet, rather than over-the-air usingGSM/GPRS,UMTSor similar. A separate device known as a "GAN Controller" (GANC)[5]receives this data from the Internet and feeds it into the phone network as if it were coming from an antenna on a tower. Calls can be placed from or received to the handset as if it were connected over-the-air directly to the GANC'spoint of presence, making the call invisible to the network as a whole.[7]This can be useful in locations with poor cell coverage where some other form ofinternet accessis available,[2]especially at the home or office. The system offers seamlesshandoff, so the user can move from cell to Wi-Fi and back again with the same invisibility that the cell network offers when moving from tower to tower.[3]
Since the GAN system works over the internet, a UMA-capable handset can connect to its service provider from any location with internet access. This is particularly useful for travelers, who can connect to their provider's GANC and make calls into their home service area from anywhere in the world.[citation needed]This is subject to the quality of the internet connection, however, and may not work well over limited bandwidth or long-latency connection. To improvequality of service(QoS) in the home or office, some providers also supply a specially programmedwireless access pointthat prioritizes UMA packets.[8]Another benefit of Wi-Fi calling is that mobile calls can be made through the internet using the same native calling client; it does not require third-partyVoice over IP(VoIP) closed services likeWhatsApporSkype, relying instead on the mobile cellular operator.[9]
The GAN protocol that extends mobile voice, data and multimedia (IP Multimedia Subsystem/Session Initiation Protocol(IMS/SIP)) applications over IP networks. The latest generation system is named orVoWiFiby a number of handset manufacturers, includingAppleandSamsung, a move that is being mirrored by carriers likeT-Mobile USandVodafone.[citation needed]The service is dependent on IMS, IPsec,IWLANandePDG.
The original Release 6 GAN specification supported a 2G (A/Gb) connection from the GANC into the mobile core network (MSC/GSN). Today[when?]all commercial GAN dual-mode handset deployments are based on a 2G connection and all GAN enabled devices are dual-mode 2G/Wi-Fi. The specification, though, defined support for multimode handset operation. Therefore, 3G/2G/Wi-Fi handsets are supported in the standard. The first 3G/UMA devices were announced in the second half of 2008.
A typical UMA/GAN handset will have four modes of operation:
In all cases, the handset scans for GSM cells when it first turns on, to determine its location area. This allows the carrier to route the call to the nearest GANC, set the correct rate plan, and comply with existing roaming agreements.
At the end of 2007, the GAN specification was enhanced to support 3G (Iu) interfaces from the GANC to the mobile core network (MSC/GSN). This native 3G interface can be used for dual-mode handset as well as 3Gfemtocellservice delivery. The GAN release 8 documentation describes these new capabilities.
While UMA is nearly always associated with dual-mode GSM/Wi-Fi services, it is actually a ‘generic’ access network technology that provides a generic method for extending the services and applications in an operator's mobile core (voice, data, IMS) over IP and the public Internet.
GAN defines a secure, managed connection from the mobile core (GANC) to different devices/access points over IP.
A Wi-Fi network that supports voice telephony must be carefully designed in a way that maximizes performance and is able to support the applicable call density.[12]A voice network includes call gateways in addition to the Wi-Fi access points. The gateways provide call handling among wireless IP phones and connections to traditional telephone systems. The Wi-Fi network supporting voice applications must provide much stronger signal coverage than what's needed for most data-only applications. In addition, the Wi-Fi network must provide seamless roaming between access points.
UMA was developed by a group of operator and vendor companies.[13]The initial specifications were published on 2 September 2004. The companies then contributed the specifications to the3rd Generation Partnership Project(3GPP) as part of 3GPP work item "Generic Access to A/Gb interfaces". On 8 April 2005, 3GPP approved specifications for Generic Access to A/Gb interfaces for 3GPP Release 6 and renamed the system to GAN.[14][15]But the termGANis little known outside the 3GPP community, and the termUMAis more common in marketing.[citation needed]
For carriers:
For subscribers:
The first service launch was BT withBT Fusionin the autumn of 2005. The service is based on pre-3GPP GAN standard technology. Initially, BT Fusion used UMA over Bluetooth with phones fromMotorola. From January 2007, it used UMA over 802.11 with phones from Nokia, Motorola and Samsung[18]and was branded as a "Wi-Fi mobile service". BT has since discontinued the service.
On August 28, 2006,TeliaSonerawas the first to launch an 802.11 based UMA service called "Home Free".[19]The service started in Denmark but is no longer offered.
On September 25, 2006Orangeannounced its "Unik service", also known as Signal Boost in the UK.[20][21]However this service is no longer available to new customers in the UK.[22]The announcement, the largest to date, covers more than 60m of Orange's mobile subscribers in the UK, France, Poland, Spain and the Netherlands.
Cincinnati Bellannounced the first UMA deployment in the United States.[23]The service, originally called CB Home Run, allows users to transfer seamlessly from the Cincinnati Bell cellular network to a home wireless network or to Cincinnati Bell's WiFi HotSpots. It has since been rebranded as Fusion WiFi.
This was followed shortly byT-Mobile USon June 27, 2007.[24]T-Mobile's service, originally named "Hotspot Calling", and rebranded to "Wi-Fi Calling" in 2009, allows users to seamlessly transfer from the T-Mobile cellular network to an 802.11x wireless network or T-Mobile HotSpot in the United States.
In Canada, bothFidoandRogers Wirelesslaunched UMA plans under the names UNO and Rogers Home Calling Zone (later rebranded Talkspot, and subsequently rebranded again as Wi-Fi Calling), respectively, on May 6, 2008.[25]
In Australia, GAN has been implemented by Vodafone, Optus and Telstra.[26]
Since 10 April 2015, Wi-Fi Calling has been available for customers ofEEin the UK initially on theNokia Lumia 640andSamsung Galaxy S6andSamsung Galaxy S6 Edgehandsets.[27]
In March 2016,Vodafone Netherlandslaunched Wi-Fi Calling support along withVoLTE.[28]
Since the Autumn of 2016, Wifi Calling / Voice over Wifi has been available for customers of Telenor Denmark, including the ability to do handover to and from the 4G (VoLTE) network. This is available for several Samsung and Apple handsets.
AT&T[29]andVerizon[30]are going to launch Wi-Fi calling in 2015.
Industry organisationUMA Todaytracks all operator activities and handset development.
In September 2015, South African cellular network Cell C launched WiFi Calling on its South African network.[31]
In November 2024, Belgian cellular network Voo launched WiFi Calling on its Belgian network.[32]
GAN/UMA is not the first system to allow the use of unlicensed spectrum to connect handsets to a GSM network. TheGIP/IWPstandard forDECTprovides similar functionality, but requires a more direct connection to the GSM network from the base station. While dual-mode DECT/GSM phones have appeared, these have generally been functionally cordless phones with a GSM handset built-in (or vice versa, depending on your point of view), rather than phones implementing DECT/GIP, due to the lack of suitable infrastructure to hook DECT base-stations supporting GIP to GSM networks on an ad-hoc basis.[33]
GAN/UMA's ability to use the Internet to provide the "last mile" connection to the GSM network solves the major issue that DECT/GIP has faced. Had GIP emerged as a practical standard, the low power usage of DECT technology when idle would have been an advantage compared to GAN.[citation needed]
There is nothing preventing an operator from deploying micro- and pico-cells that use towers that connect with the home network over the Internet. Several companies have developed femtocell systems that do precisely that, broadcasting a "real" GSM or UMTS signal, bypassing the need for special handsets that require 802.11 technology. In theory, such systems are more universal, and again require lower power than 802.11, but their legality will vary depending on the jurisdiction, and will require the cooperation of the operator. Further, users may be charged at higher cell phone rates, even though they are paying for the DSL or other network that ultimately carries their traffic; in contrast, GAN/UMA providers charge reduced rates when making calls off the providers cellular phone network.[citation needed]
|
https://en.wikipedia.org/wiki/Voice_over_WLAN
|
ThePersonal Handy-phone System(PHS), also known as thePersonal Communication Telephone(PCT) in Thailand, and thePersonal Access System(PAS) and commercially branded asXiaolingtong(Chinese:小灵通) inChina, was amobile networksystem operating in the 1880–1930MHzfrequency band. InJapan, it was introduced as a low-cost wireless service with smaller coverage areas than standard cellular networks. Its affordability made it popular in China,Taiwan, and other parts ofAsia, as both the handsets and network infrastructure were relatively inexpensive to maintain.[1]
Developed in the 1990s, PHS used amicrocellarchitecture with low-power base stations covering 100 to 500 metres (330 to 1,640 ft). unlike conventionalcellular networksthat relied on large cell sites for extensive coverage, PHS’s design was better suited for dense urban environments and reduced infrastructure costs.
PHS was overtaken in the marketplace byGSM(3G) andUMTS(4G), with the last retail network decommissioned in 2021 and the last commercial network terminated in 2023.[2]
PHS is essentially acordless telephonelikeDECT, with the capability tohandoverfrom onecellto another. PHS cells are small, with transmission power ofbase stationa maximum of 500 mW and range typically measures in tens or at most hundreds of metres (some can range up to about 2 kilometres in line-of-sight), contrary to the multi-kilometre ranges ofCDMAandGSM. This makes PHS suitable for dense urban areas, but impractical for rural areas, and the small cell size also makes it difficult if not impossible to make calls from rapidly moving vehicles.
PHS usesTDMA/TDDfor its radiochannel access method, and 32 kbit/sADPCMfor its voicecodec. Modern PHS phone can also support manyvalue-added servicessuch as high speed wirelessdata/Internetconnection (64 kbit/s and higher),WWWaccess,e-mailing, and text messaging.
PHS technology is also a popular option for providing awireless local loop, where it is used for bridging the "last mile" gap between thePOTSnetwork and the subscriber's home. It was developed under the concept of providing a wirelessfront-endof anISDNnetwork. Thus a PHS base station is compatible with ISDN and is often connected directly to ISDNtelephone exchangeequipment e.g. a digital switch.
In spite of its low-costbase station,micro-cellularsystem and "Dynamic Cell Assignment" system, PHS offers higher number-of-digits frequency use efficiency with lower cost (throughput per area basis), compared with typical3Gcellular telephonesystems. It enables flat-rate wireless service such asAIR-EDGE, throughout Japan.
The speed of an AIR-EDGE data connection is accelerated by combining lines, each of which basically is 32 kbit/s. The first version of AIR-EDGE, introduced in 2001, provided 32 kbit/s service. In 2002, 128 kbit/s service (AIR-EDGE 4×) started and in 2005, 256 kbit/s (8×) service started. In 2006, the speed of each line was also upgraded to 1.6 times with the introduction of "W-OAM" technology. The speed of AIR-EDGE 8× is up to 402 kbit/s with the latest "W-OAM" capable instrument.
In April 2007, "W-OAM typeG" was introduced allowing data speeds of 512 kbit/s for AIR-EDGE 8x users. Furthermore, the "W-OAM typeG" 8× service was planned to be upgraded to a maximum throughput of 800 kbit/s, when the upgrading for access points (mainly switching lines from ISDN tofibre optic) in its system are completed. Thus it was expected to exceed the speeds of popularW-CDMA3G service likeNTT DoCoMo'sFOMAin Japan.
Developed byNTT Laboratoryin Japan in 1989 and far simpler to implement and deploy than competing systems likePDCorGSM, the commercial services were started by three PHS operators (NTT-Personal, DDI-Pocket, and ASTEL) in Japan in 1995, forming the PIAF (PHS Internet Access Forum). However, the service was pejoratively dubbed as the "poor man's cellular", due to its limited range and roaming abilities. NTT DoCoMo, which absorbed NTT Personal, and ASTEL terminated the PHS service in January 2008.
In Thailand, TelecomAsia (nowTrue Corporation) integrated the PHS system withIntelligent Networkand marketed the service as Personal Communication Telephone (PCT).[3]The integrated system was the world's first that allowed thefixed linetelephone subscribers of thepublic switched telephone networkto use PHS as a value added service with the same telephone number and shared the same voice mailbox.[4][5]The PCT service was commercially launched in November 1999 with the peak of 670,000 subscribers in 2001. However, the number of subscribers had declined to 470,000 in 2005 before the breakeven in 2006 after six years of heavy investment up to 15 billion THB. With the popularity of other cellular phone services, the company shifted the focus of the PCT to a niche market segment of youths ages 10-18.[6]
Wireless local loop(WLL) systems based on PHS technology are in use in some of the above-mentioned countries.WILLCOM, formerly DDI-Pocket, introduced flat-ratewireless networkand flat-rate calling in Japan, which reversed the local fate of PHS up to an extent. In China, there was an explosive expansion of subscribers until around 2005. In Chile,Telefónica del Surlaunched a PHS-based telephony service in some cities of the southern part of the country in March 2006. In Brazil,Suporte Tecnologiahas a PHS-based telephony service inBetim, state of Minas Gerais[needs update], andTransit Telecomannounced a rollout of a PHS network in 2007[needs update].
China Telecomoperated a PAS system in China, although technically it was not regarded as allowed to provide mobile services, because of some particularities of the Chinese governance.China Netcom, the other fixed-line operator in China, also provides Xiaolingtong service. The system was a runaway hit, with over 90 million subscribers signed up as of 2007[update]; the largest equipment vendors wereUTStarcomandZTE. However, low priced mobile phones rapidly replaced PHS. TheMinistry of Industry and Information Technology of the People's Republic of Chinaissued notices on 13 February 2009 that both registration of new users and expansion of the network were to be discontinued, with the service to be ended by the end of 2011.[7]
A PHS global roaming service was available between Japan (WILLCOM), Taiwan, and Thailand.
This is a list of commercial PHS deployments around the world, all of which are now defunct:[8]
Apr 2023(commercial)[2]
PHS-enabledPCMCIA/CompactFlashcards include:
|
https://en.wikipedia.org/wiki/Personal_Access_System
|
Amateur radio, also known asham radio, is the use of theradio frequencyspectrumfor purposes ofnon-commercialexchange of messages,wirelessexperimentation, self-training, private recreation,Radiosport,contesting, andemergency communications.[1]The term"radio amateur"is used to specify"a duly authorized person interested in radioelectric practice with a purely personal aim and withoutpecuniaryinterest"[2](either direct monetary or other similar reward); and to differentiate it fromcommercial broadcasting, public safety (police and fire), ortwo-way radioprofessional services (maritime, aviation, taxis, etc.).
The amateur radio service (amateur serviceandamateur-satellite service) is established by theInternational Telecommunication Union(ITU) through their recommendedRadio Regulations. National governments regulate technical and operational characteristics of transmissions and issue individual station licenses with a unique identifyingcall sign, which must be used in all transmissions.Amateur operatorsmust hold anamateur radio licenseobtained by successfully passing an official examination that demonstrates adequate technical and theoretical knowledge of amateur radio, electronics, and related topics essential for the hobby; it also assesses sufficient understanding of the laws and regulations governing amateur radio within the country issuing the license.
Radio amateurs are privileged to transmit on a limited specific set of frequency bands – theamateur radio bands– allocated internationally, throughout theradio spectrum, but within these bands are allowed to transmit on anyfrequency; although on some of those frequencies they are limited to one or a few of a variety of modes of voice, text, image, anddata communications. This enables communication across a city, region, country, continent, the world, or even into space. In many countries, amateur radio operators may also send, receive, or relay radio communications betweencomputersortransceiversconnected to securevirtual private networkson theInternet.
Amateur radio is officially represented and coordinated by theInternational Amateur Radio Union(IARU), which is organized in three regions and has as its members the national amateur radio societies which exist in most countries. According to a 2011 estimate by theARRL(theU.S.national amateur radio society), two million people throughout the world are regularly involved with amateur radio.[3]About830000amateur radio stationsare located inIARU Region 2(the Americas) followed byIARU Region 3(South and East Asia and the Pacific Ocean) with about750000stations.Significantly fewer, about400000stations,are located inIARU Region 1(Europe, Middle East,CIS, Africa).
The origins of amateur radio can be traced to the late 19th century, but amateur radio as practised today began in the early 20th century. TheFirst Annual Official Wireless Blue Book of the Wireless Association of America, produced in 1909, contains a list of amateur radio stations.[4]This radiocallbooklistswireless telegraphstations in Canada and the United States, including 89 amateur radio stations. As with radio in general, amateur radio was associated with various amateur experimenters and hobbyists. Amateur radio enthusiasts have significantly contributed toscience,engineering, industry, andsocial services. Research by amateur operators has founded new industries,[5]built economies,[6]empowered nations,[7]and saved lives in times of emergency.[8][9]Ham radio can also be used in the classroom to teach English, map skills, geography, math, science, and computer skills.[10]
The term"ham"was first apejorativeterm used in professionalwired telegraphyduring the 19th century, to mock operators with poorMorse code-sending skills ("ham-fisted").[11][12][13][14]This term continued to be used after the invention of radio, and the proliferation of amateur experimentation with wireless telegraphy; among land- and sea-based professional radio telegraphers,"ham"amateurs were considered a nuisance. The use of"ham"meaning"amateurish or unskilled"survives today sparsely in other disciplines (e.g."ham actor").
The amateur radio community subsequently reclaimed the word as a label of pride,[15]and by the mid-20th century it had lost its pejorative meaning. Although not an acronym or initialism, it is occasionally written as "HAM" in capital letters.
The many facets of amateur radio attract practitioners with a wide range of interests. Many amateurs begin with a fascination with radio communication and then combine other personal interests to make pursuit of the hobby rewarding. Some of the focal areas amateurs pursue includeradio contesting,radio propagationstudy,public service communication,technical experimentation, andcomputer networking.
Hobbyist radio enthusiasts employ avariety of transmission methods for interaction. The primary modes for vocal communications arefrequency modulation(FM) andsingle sideband(SSB). FM is recognized for its superior audio quality, whereas SSB is more efficient both for long-range communication and for limitedbandwidthconditions.[16]The most efficient for both distance and limited bandwidth remains CW, and lately, some digital modes.
RadiotelegraphyusingInternational Morse code, also known as "CW" from "continuous wave", is the wireless extension of landline (wired)telegraphyfirst developed bySamuel Morse, and greatly revised byAlfred Vail,Friedrich Gerke, and a comittee of theITU; in one revision or another, it dates to the earliest days of radio. Although computer-based (digital) modes and methods have largely replaced CW for commercial and military applications, many amateur radio operators still use the CW mode – particularly on theshortwavebands and for experimental work, such asEarth–Moon–Earth communication– because of its inherent advantage insignal-to-noise ratio. Morse, using internationally agreed message encodings such as theQ code, enables communication between amateurs who speak different languages. It is also popular withhomebrewersand in particular with"QRP"or very-low-power enthusiasts, as CW-only transmitters are simpler to construct, and the human ear-brain signal processing system can pull weak CW signals out of the noise where voice signals would be effectively inaudible. Similarly, the "legacy"amplitude modulation(AM) mode is popular with some home constructors because of its simpler modulation-demodulation circuitry; it is also pursued by manyvintage amateur radioenthusiasts and aficionados ofvacuum tubetechnology.
Demonstrating a proficiency in Morse code was for many years a requirement to obtain an amateur license to transmit on frequencies below 30 MHz. Following changes in international regulations in 2003, countries are no longer required to demand proficiency.[17]The United StatesFederal Communications Commission, for example, phased out this requirement for all license classes on 23 February 2007.[18][19]
Modern personal computers have encouraged the use ofdigitalmodes such asradioteletype(RTTY) which previously required cumbersome mechanical equipment.[20]Hams led the development ofpacket radioin the 1970s, which has employed protocols such asAX.25andTCP/IP. Specialized digital modes such asPSK31allow real-time, low-power communications on the shortwave bands but have been losing favor in place of newer digital modes such asFT8.
Radio over IP, or RoIP, is similar toVoice over IP(VoIP), but augments two-way radio communications rather than telephone calls.EchoLinkusing VoIP technology has enabled amateurs to communicate through local Internet-connected repeaters and radio nodes,[21]whileIRLPhas allowed the linking of repeaters to provide greater coverage area.
Automatic link establishment (ALE) has enabled continuous amateur radio networks to operate on thehigh frequencybands with global coverage. Other modes, such as FSK441 using software such asWSJT, are used for weak signal modes includingmeteor scatterandmoonbouncecommunications.[22]
Fast scanamateur televisionhas gained popularity as hobbyists adapt inexpensive consumer video electronics like camcorders and video cards inPCs. Because of the widebandwidthand stable signals required, amateur television is typically found in the70 cm(420–450 MHz) wavelength range, though there is also limited use on33 cm(902–928 MHz),23 cm(1240–1300 MHz) and shorter. These requirements also effectively limit the signal range to between 20 and 60 miles (30–100 km).
Linkedrepeatersystems, however, can allow transmissions ofVHFand higher frequencies across hundreds of miles.[23]Repeaters are usually located on heights of land or on tall structures, and allow operators to communicate over hundreds of miles using hand-held or mobiletransceivers. Repeaters can also be linked together by using otheramateur radio bands,landline, or theInternet.
Amateur radio satellitescan be accessed, some using a hand-held transceiver (HT), even, at times, using the factory "rubber duck" antenna.[24]Hams also use themoon, theaurora borealis, and the ionized trails ofmeteorsas reflectors of radio waves.[25]Hams can also contact theInternational Space Station(ISS) because manyastronautsare licensed as amateur radio operators.[26][27]
Amateur radio operators use theiramateur radio stationto make contacts with individual hams as well as participate in round-table discussion groups or "rag chew sessions" on the air. Some join in regularly scheduled on-air meetings with other amateur radio operators, called "nets" (as in "networks"), which are moderated by a station referred to as "Net Control".[28]Nets can allow operators to learn procedures for emergencies, be an informal round table, or cover specific interests shared by a group.[29]
Amateur radio operators, using battery- or generator-powered equipment, often provide essential communications services when regular channels are unavailable due to natural disaster or other disruptive events .[30]
Many amateur radio operators participate in radio contests, during which an individual or team of operators typically seek to contact and exchange information with as many other amateur radio stations as possible in a given period of time. In addition to contests, a number ofamateur radio operating awardschemes exist, sometimes suffixed with "on the Air", such asSummits on the Air, Islands on the Air,Worked All StatesandJamboree on the Air.
Amateur radio operators may also act ascitizen scientistsfor propagation research andatmospheric science.[31]
Radio transmission permits are closely controlled by nations' governments because radio waves propagate beyond national boundaries, and therefore radio is of international concern.[32]
Both the requirements for and privileges granted to a licensee vary from country to country, but generally follow the international regulations and standards established by theInternational Telecommunication Union[33]andWorld Radio Conferences.
All countries that license citizens to use amateur radio require operators to display knowledge and understanding of key concepts, usually by passing an exam.[34]The licenses grant hams the privilege to operate in larger segments of theradio frequencyspectrum, with a wider variety of communication techniques, and with higher power levels relative to unlicensed personal radio services (such asCB radio,FRS, andPMR446), which require type-approved equipment restricted in mode, range, and power.[35]
Amateur licensing is a routine civil administrative matter in many countries. Amateurs therein must pass an examination to demonstrate technical knowledge, operating competence, and awareness of legal and regulatory requirements, in order to avoid interfering with other amateurs and other radio services.[36]A series of exams are often available, each progressively more challenging and granting more privileges: greater frequency availability, higher power output, permitted experimentation, and, in some countries, distinctive call signs.[37][38]Some countries, such as the United Kingdom and Australia, have begun requiring a practical assessment in addition to the written exams in order to obtain a beginner's license, which they call a Foundation License.[39]
In most countries, an operator will be assigned acall signwith their license.[40]In some countries, a separate "station license" is required for any station used by an amateur radio operator. Amateur radio licenses may also be granted to organizations or clubs. In some countries, hams were allowed to operate only club stations.[41]
An amateur radio license is valid only in the country where it is issued or in another country that has a reciprocal licensing agreement with the issuing country.[42][43]
In some countries, an amateur radio license is necessary in order to purchase or possess amateur radio equipment.[44]
Amateur radio licensing in the United Statesexemplifies the way in which some countries[which?]award different levels of amateur radio licenses based on technical knowledge: three sequential levels of licensing exams (Technician Class, General Class, and Amateur Extra Class) are currently offered, which allow operators who pass them access to larger portions of the Amateur Radio spectrum and more desirable (shorter) call signs. An exam, authorized by the Federal Communications Commission (FCC), is required for all levels of the Amateur Radio license. These exams are administered by Volunteer Examiners, accredited by the FCC-recognized Volunteer Examiner Coordinator (VEC) system. The Technician Class and General Class exams consist of 35 multiple-choice questions, drawn randomly from a pool of at least 350. To pass, 26 of the 35 questions must be answered correctly.[45]The Extra Class exam has 50 multiple choice questions (drawn randomly from a pool of at least 500), 37 of which must be answered correctly.[45]The tests cover regulations, customs, and technical knowledge, such as FCC provisions, operating practices, advanced electronics theory, radio equipment design, and safety. Morse Code is no longer tested in the U.S. Once the exam is passed, the FCC issues an Amateur Radio license which is valid for ten years. Studying for the exam is made easier because the entire question pools for all license classes are posted in advance. The question pools are updated every four years by the National Conference of VECs.[45]
Prospective amateur radio operators are examined on understanding of the key concepts of electronics, radio equipment, antennas,radio propagation,RFsafety, and the radio regulations of the government granting the license.[1]These examinations are sets of questions typically posed in either a short answer or multiple-choice format. Examinations can be administered bybureaucrats, non-paid certified examiners, or previously licensed amateur radio operators.[1]
The ease with which an individual can acquire an amateur radio license varies from country to country. In some countries, examinations may be offered only once or twice a year in the national capital and can be inordinately bureaucratic (for example in India) or challenging because some amateurs must undergo difficult security approval (as inIran). Currently, onlyYemenandNorth Koreado not issue amateur radio licenses to their citizens.[46][47]Some developing countries, especially those in Africa, Asia, andLatin America, require the payment of annual license fees that can be prohibitively expensive for most of their citizens. A few small countries may not have a national licensing process and may instead require prospective amateur radio operators to take the licensing examinations of a foreign country. In countries with the largest numbers of amateur radio licensees, such as Japan, the United States, Thailand, Canada, and most of the countries in Europe, there are frequent license examinations opportunities in major cities.
Granting a separate license to a club or organization generally requires that an individual with a current and valid amateur radio license who is in good standing with the telecommunications authority assumes responsibility for any operations conducted under the club license or club call sign.[38]A few countries may issue special licenses to novices or beginners that do not assign the individual a call sign but instead require the newly licensed individual to operate from stations licensed to a club or organization for a period of time before a higher class of license can be acquired.[1]
A reciprocal licensing agreement between two countries allows bearers of an amateur radio license in one country under certain conditions to legally operate an amateur radio station in the other country without having to obtain an amateur radio license from the country being visited, or the bearer of a valid license in one country can receive a separate license and a call sign in another country, both of which have a mutually-agreed reciprocal licensing approvals. Reciprocal licensing requirements vary from country to country. Some countries have bilateral or multilateral reciprocal operating agreements allowing hams to operate within their borders with a single set of requirements. Some countries lack reciprocal licensing systems. Others use international bodies such as the Organization of American States to facilitate licensing reciprocity.[48]
When traveling abroad, visiting amateur operators must follow the rules of the country in which they wish to operate. Some countries have reciprocalinternational operatingagreements allowing hams from other countries to operate within their borders with just their home country license. Other host countries require that the visiting ham apply for a formal permit, or even a new host country-issued license, in advance.
The reciprocal recognition of licenses frequently not only depends on the involved licensing authorities, but also on the nationality of the bearer. As an example, in the US, foreign licenses are recognized only if the bearer does not have US citizenship and holds no US license (which may differ in terms of operating privileges and restrictions). Conversely, a US citizen may operate under reciprocal agreements in Canada, but not a non-US citizen holding a US license.
Many people start their involvement in amateur radio on social media or by finding a local club. Clubs often provide information about licensing, local operating practices, and technical advice. Newcomers also often study independently by purchasing books or other materials, sometimes with the help of a mentor, teacher, or friend. In North America, established amateurs who help newcomers are often referred to as "Elmers", as coined by Rodney Newkirk (W9BRD),[49]within the ham community.[50][51]In addition, many countries have national amateur radio societies which encourage newcomers and work with government communications regulation authorities for the benefit of all radio amateurs. The oldest of these societies is theWireless Institute of Australia, formed in 1910; other notable societies are theRadio Society of Great Britain, theAmerican Radio Relay League,Radio Amateurs of Canada,Bangladesh NGOs Network for Radio and Communication, theNew Zealand Association of Radio TransmittersandSouth African Radio League. (SeeCategory:Amateur radio organizations)
An amateur radio operator uses acall signon the air to legally identify the operator or station.[52]In some countries, the call sign assigned to the station must always be used, whereas in other countries, the call sign of either the operator or the station may be used.[53]In certain jurisdictions, an operator may also select a"vanity"call sign although these must also conform to the issuing government's allocation and structure used for amateur radio call signs.[54]Some jurisdictions require a fee to obtain a vanity call sign; in others, such as the UK, a fee is not required and the vanity call sign may be selected when the license is applied for. The FCC in the U.S. discontinued its fee for vanity call sign applications in September 2015, but reinstated it at $35 in 2022.[55]
Call sign structure as prescribed by the ITU consists of three parts which break down as follows, using the call signZS1NATas an example:
The combination of the three parts identifies the specific transmitting station, and the station's identification (its call sign) is determined by the license held by its operator. In the case of commercial stations and amateur club stations, the operator is a corporation; in the case of amateur radio operators, the license-holder is a resident of the country identified by the first part of the call sign.
Many countries do not follow the ITU convention for the second-part digit. In the United Kingdom the original callsG0xxx,G2xxx,G3xxx,G4xxx, were Full (A) License holders along with the lastM0xxxfull call signs issued by theCity & Guildsexamination authority in December 2003. Additional Full Licenses were originally granted to (B) Licenses withG1xxx,G6xxx,G7xxx,G8xxxand 1991 onward withM1xxxcall signs. The newer three-level Intermediate License holders are assigned2E0xxxand2E1xxx, and the basic Foundation License holders are granted call signsM3xxx,M6xxxorM7xxx.[56]
Instead of using numbers, in the U.K. the second letter after the initial 'G' or 'M' identifies the station's location; for example, a call signG7OOEbecomesGM7OOEandM0RDMbecomesMM0RDMwhen the license holder is operating their station in Scotland. PrefixGM&MMare Scotland,GW&MWare Wales,GI&MIare Northern Ireland,GD&MDare the Isle of Man,GJ&MJare Jersey andGU&MUare Guernsey. Intermediate licence call signs are slightly different. They begin2z0and2z1where thezis replaced with one of the country letters, as above. For example2M0and2M1are Scotland,2W0and2W1are Wales and so on. The exception however is for England, whose letter would be 'E'; however, letter 'E'isused, butonlyin intermediate-level call signs, and perplexingly never by the advanced licenses. For example2E0&2E1are used whereas the call signs beginning 'G' or 'M' for foundation and full licenses never use the 'E'.[57]
In the United States, for non-vanity licenses, the numeral indicates the geographical district the holder resided in when the license was first issued. Prior to 1978, US hams were required to obtain a new call sign if they moved out of their geographic district.
In Canada, call signs start withVA,VE,VY,VO, andCY. Call signs starting with 'V' end with a number after to indicate the political region; whereas the prefixCYindicates geographic islands. PrefixesVA1andVE1are used forNova Scotia;VA2&VE2forQuebec;VA3&VE3forOntario;VA4&VE4forManitoba;VA5&VE5forSaskatchewan;VA6&VE6forAlberta;VA7&VE7forBritish Columbia;VE8for theNorthwest Territories;VE9forNew Brunswick;VY0forNunavut;VY1for theYukon;VY2forPrince Edward Island;VO1forNewfoundland; andVO2forLabrador.CYis for amateurs operating fromSable Island(CY0) orSt. Paul Island(CY9). Special permission is required to access either of these: fromParks Canadafor Sable andCoast Guardfor St. Paul. The last two or three letters of the call signs are typically the operator's choice (upon completing the licensing test, the ham writes three most-preferred options). Two-letter call sign suffixes require a ham to have already been licensed for 5 years. Call signs in Canada can be requested with a fee.
Also, for smaller geopolitical entities, the digit at the second or third character might be part of the country identification. For example,VP2xxxis in the British West Indies, which is subdivided intoVP2ExxAnguilla,VP2MxxMontserrat, andVP2VxxBritish Virgin Islands.VP5xxxis in the Turks and Caicos Islands,VP6xxxis on Pitcairn Island,VP8xxxis in the Falklands, andVP9xxxis in Bermuda.
Onlinecallbooksor call sign databases can be browsed or searched to find out who holds a specific call sign.[58]An example of an online callbook isQRZ.com. Various partial lists of famous people who hold or held amateur radio call signs have been compiled and published.[59]
Many jurisdictions (but not in the U.K. nor Europe) may issue specialtyvehicle registration platesto licensed amateur radio operators.[60][61]The fees for application and renewal are usually less than the standard rate for specialty plates.[60][62]
In most administrations, unlike other RF spectrum users, radio amateurs may build or modify transmitting equipment for their own use within the amateur spectrum without the need to obtain government certification of the equipment.[63][a][64][b]Licensed amateurs can also use any frequency in their bands (rather than being allocated fixed frequencies or channels) and can operate medium-to-high-powered equipment on a wide range of frequencies[65]so long as they meet certain technical parameters including occupied bandwidth, power, and prevention ofspurious emission.
Radio amateurs have access to frequency allocations throughout the RF spectrum, usually allowing choice of an effective frequency for communications across a local, regional, or worldwide path. The shortwave bands, orHF, are suitable for worldwide communication, and theVHFandUHFbands normally provide local or regional communication, while themicrowavebands have enough space, orbandwidth, for amateur television transmissions and high-speedcomputer networks.
In most countries, an amateur radio license grants permission to the license holder to own, modify, and operate equipment that is not certified by a governmental regulatory agency. This encourages amateur radio operators to experiment with home-constructed or modified equipment. The use of such equipment must still satisfy national and international standards onspurious emissions.
Amateur radio operators are encouraged both by regulations and tradition of respectful use of the spectrum to use as little power as possible to accomplish the communication.[66]This is to minimise interference orEMCto any other device. Although allowablepowerlevels are moderate by commercial standards, they are sufficient to enable global communication. Lower license classes usually have lower power limits; for example, the lowest license class in the UK (Foundation licence) has a limit of 25 W.[67]
Power limits vary from country to country and between license classes within a country. For example, thepeak envelope powerlimits for the highest available license classes in a few selected countries are: 2.25kWin Canada;[68]1.5 kW in the United States; 1.0 kW in Belgium,Luxembourg, Switzerland, South Africa, the United Kingdom, and New Zealand; 750 W in Germany; 500 W in Italy; 400 W in Australia and India; and 150 W inOman.
Output power limits may also depend on the mode of transmission. In Australia, for example, 400 W may be used forSSBtransmissions, but FM and other modes are limited to 120 W.
The point at which power output is measured may also affect transmissions: The United Kingdom measures at the point the antenna is connected to the signal feed cable, which means the radio system may transmit more than 400 W to overcome signal loss in the cable; conversely, the U.S. and Germany measure power at the output of the final amplification stage, which results in a loss in radiated power with longer cable feeds.[citation needed]
Certain countries permit amateur radio licence holders to hold a Notice of Variation that allows higher power to be used than normally allowed for certain specific purposes. E.g. in the UK some amateur radio licence holders are allowed to transmit using (33 dBw) 2.0 kW for experiments entailing using the moon as a passive radio reflector (known asEarth–Moon–Earth communication) (EME).
TheInternational Telecommunication Union(ITU) governs the allocation of communications frequencies worldwide, with participation by each nation's communications regulation authority. National communications regulators have some liberty to restrict access to thesebandplanfrequencies or to award additional allocations as long as radio services in other countries do not suffer interference. In some countries, specificemission typesare restricted to certain parts of the radio spectrum, and in most other countries,International Amateur Radio Union(IARU) member societies adopt voluntary plans to ensure the most effective use of spectrum.
In a few cases, a national telecommunication agency may also allow hams to use frequencies outside of the internationally allocated amateur radio bands. InTrinidad and Tobago, hams are allowed to use a repeater which is located on 148.800 MHz. This repeater is used and maintained by theNational Emergency Management Agency(NEMA), but may be used by radio amateurs in times of emergency or during normal times to test their capability and conduct emergency drills. This repeater can also be used by non-ham NEMA staff andREACTmembers. In Australia and New Zealand, ham operators are authorized to use one of the UHF TV channels. In the U.S., amateur radio operators providing essential communication needs in connection with the immediate safety of human life and immediate protection of property when normal communication systems are not available may use any frequency including those of other radio services such as police and fire and in cases of disaster in Alaska may use the statewide emergency frequency of 5.1675 MHz with restrictions upon emissions.[69]
Similarly, amateurs in the United States may apply to be registered with theMilitary Auxiliary Radio System(MARS). Once approved and trained, these amateurs also operate on US government military frequencies to provide contingency communications and morale message traffic support to the military services.
Amateurs use a variety of voice, text, image, and data communication modes the over radio. Generally new modes can be tested in the amateur radio service, although national regulations may require disclosure of a new mode to permit radio licensing authorities to monitor the transmissions.Encryption, for example, is not generally permitted in the Amateur Radio service except for the special purpose of satellite vehicle control uplinks. The following is a partial list of the modes of communication used, where the mode includes bothmodulationtypes and operating protocols.
In former times, most amateur digital modes were transmitted by inserting audio into the microphone input of a radio and using an analog scheme, such asamplitude modulation(AM),frequency modulation(FM), orsingle-sideband modulation(SSB). Beginning in 2017, increased use of several digital modes, particularlyFT8, became popular within the amateur radio community.[70]
The following "modes" use no one specific modulation scheme but rather are classified by the activity of the communication.
|
https://en.wikipedia.org/wiki/Amateur_radio
|
Citizens band radio(CB radio) is aland mobile radio system, a system allowing short-distance one-to-many bidirectional voice communication among individuals, usingtwo-way radiosoperating near 27MHz(or the 11-m wavelength) in thehigh frequencyorshortwaveband. Citizens band is distinct from otherpersonal radio serviceallocations such asFRS,GMRS,MURS,UHF CBand theAmateur Radio Service("ham"radio). In many countries, CB operation does not require a license and may be used for business or personal communications.
Like many other land mobile radio services, multiple radios in a local area share a single frequency channel, but only one can transmit at a time. The radio is normally in receive mode to receive transmissions of other radios on the channel; when users want to communicate they press a "push to talk" button on their radio, which turns on their transmitter. Users on a channel must take turns transmitting. In theUSandCanada, and in theEUand theUK, transmitter power is limited to 4wattswhen usingAMandFMand 12 WPEPwhen usingSSB. Illegal amplifiers to increase range are common.[citation needed]
CB radios using an omni-directional vertical antenna typically have a range of about 5 km to 30 km depending on terrain, for line of sight communication; however, various radio propagation conditions may intermittently allow communication over much greater distances. Base stations however may be connected to a directionalYagi–Uda antennacommonly called a Beam or a Yagi.
Multiple countries have created similar radio services, with varying technical standards and requirements for licensing. While they may be known by other names, such as the General Radio Service in Canada,[1]they often use similar frequencies (26–28 MHz) and have similar uses, and similar technical standards. Although licenses may be required, eligibility is generally simple. Some countries also have personal radio services in the UHF band, such as the EuropeanPMR446and the Australian UHF CB.
The citizens band radio service originated in the United States as one of several personal radio services regulated by theFederal Communications Commission(FCC). These services began in 1945 to permit citizens a radio band for personal communication (e.g., radio-controlled model airplanes and family and business communications). In 1948, the original CB radios were designed for operation on the 460–470 MHz UHF band.[2]There were two classes of CB radio: "A" and "B". Class B radios had simpler technical requirements, and were limited to a smaller frequency range.Al Grossestablished the Citizens Radio Corporation during the late 1940s to manufacture class B handhelds for the general public.[3]: 13Originally designed for use by the public sector, the Citizens Radio Corporation sold over 100,000 units, primarily to farmers and the US Coast Guard.[4]
Ultra-high frequency(UHF) radios, at the time, were neither practical nor affordable for the average consumer. On September 11, 1958[3]: 14CB service class D was created on 27 MHz, and this band became what is commonly known today as "Citizens Band". Only 23 channels were available at the time; the first 22 were taken from the formeramateur radio service11-meter band, and channel 23 was shared with radio-controlled devices. Some hobbyists continue to use the designation "11 meters" to refer to the Citizens Band and adjoining frequencies. Part 95 of the Code of Federal Regulations regulates the class D CB service, on the 27 MHz band, since the 1970s and continuing today.[5]Most of the 460–470 MHz band was reassigned for business and public-safety use; CB Class A is the forerunner of theGeneral Mobile Radio Service(GMRS). CB Class B is a more distant ancestor of theFamily Radio Service. TheMulti-Use Radio Serviceis another two-way radio service in theVHF high band. An unsuccessful petition was filed in 1973 to create a CB Class E service at 220 MHz, (part of the amateur radio1.25-meter bandat the time) which was opposed by amateur radio organizations and others.[6]There are several other classes of personal radio services for specialized purposes (such asremote controldevices).
During the 1960s, the service was used by small businesses (e.g., electricians, plumbers, carpenters),truck driversand radio hobbyists. By the late 1960s, advances insolid-stateelectronics allowed the weight, size, and cost of the radios to fall, giving the public access to a communications medium previously only available to specialists.[7]CB clubs were formed; aCB slanglanguage evolved alongsideten-code, similar to those used inemergency services.
After the1973 oil crisis, the U.S. government imposed anationwide 55 mph speed limit, and fuel shortages and rationing were widespread. Drivers (especially commercialtruckers) used CB radios to locate service stations with better supplies of fuel, to notify other drivers ofspeed traps, and to organize blockades and convoys in a 1974 strike protesting the new speed limit and other trucking regulations.[8]The radios were crucial for independent truckers; many were paid by the mile, and the 55 mph speed limit lowered their productivity.[7]
Their use spread further into the general population in the US in the middle of the 1970s. Originally, CB radios required the use of acallsignin addition to a purchased license ($20 in the early 1970s, reduced to $4 on March 1, 1975, or $23.4 in 2024 dollars). However, when the CB craze was at its peak, many people ignored the requirement and invented their own nicknames (known as "handles"). Lax enforcement of the rules on authorized use of CB radio led to further widespread disregard of regulations (such as forantennaheight, distance communications, licensing, call signs, and transmitter power). Individual licensing came to an end on April 28, 1983.[9]
The growing popularity of CB radios in the 1970s was frequently depicted in film, television, music and books. Movies such asSmokey and the Bandit(1977),Breaker! Breaker!(1977),Handle with Care(also released asCitizens Band; 1977), andConvoy(1978), made heavy reference to the phenomenon, as did television series such asMovin' On(debuted in 1974),The Dukes of Hazzard(debuted in 1979) and the animated seriesCB Bears(debuted in 1977) helped cement CB radio's status as a nationwide craze in the United States over the mid- to late-1970s. InStephen King's 1978 novelThe Stand, Ralph Brentner's radio signal fromBoulder, Coloradowas able to be received over CB.[10]The phenomenon also inspired several country and rock music songs in 1975 and 1976.
Betty Ford, the formerFirst Lady of the United States, used the CB handle "First Mama".[14]
Voice actorMel Blancwas also an active CB operator, often using "Bugs" or "Daffy" as his handle and talking on the air in theLos Angelesarea in one of his many voice characters. He appeared in an interview (with clips having fun talking to children on his home CB radio station) in theNBC Knowledgetelevision episode about CB radio in 1978. Similar to Internetchat roomsa quarter-century later, CB allowed people to get to know one another in a quasi-anonymous manner.
Originally, the U.S. had 23 CB channels; the 40 channel band plan was implemented in 1977. Two more channels between 22 & 23, commonly referred to as 22A & 22B, were available. Several people had 22A, but few had 22B. In the early 1970s Radio Shack sold a "base station" CB radio that contained a crystal for each of the 23 channels, two extra slots existed, and one could order the 22A & 22B crystals for an easy plug-in. Channel 9 was officially reserved for emergency use by the FCC in 1969.[15]: 12Channel 10 was originally often used for highway travel communications east of theMississippi River, and channel 19 west of the Mississippi; channel 19 then became the preferred highway channel in most areas, as it did not haveadjacent-channel interferenceproblems with channel 9. Many CBers called channel 19 "the trucker's channel". The FCC originally restricted channel 11 for use as the calling channel.
The original FCC output power limitation for CB radios was "5 watts DC input to the final amplifier stage", which was a reference to the earlier radios equipped with tubes. With solid state radios becoming more common in the 1970s, the FCC revised this specification at the same time the authorized channels were increased to 40. The current specification is simply "4 watts output (AM) or 12 watts output (SSB)" as measured at the antenna connector on the back of the radio. The old specification was often used in false advertising by some manufacturers who would claim their CB radios had "5 watts" long after the specification had changed to 4 watts output. The older 23 channel radios built under the old specifications typically had an output of around 3.5 to 3.8 watts output when measured at the antenna connector. The FCC simply rounded up the old "5 watts DC input to the final amplifier stage" specification to the new "4 watts output as measured at the antenna connector on the back of the radio", resulting in a far simpler and easier specification.
Initially, the FCC intended for CB to be the "poor man's business-band radio", and CB regulations were structured similarly to those regulating thebusiness bandradio service. Until 1975,[15]: 14only channels 9–14 and 23[a]could be used for "interstation" calls (to other licensees). Channels 1–8 and 15–22 were reserved for "intrastation" communications (among units with the same license).[b]After the inter-station/intra-station rule was dropped, channel 11 was reserved as a calling frequency (for the purpose of establishing communications); however, this was withdrawn in 1977.[15]: 120During this early period, many CB radios had "inter-station" channels colored on their dials, while the other channels were clear or normally colored.[c]It was common for a town to adopt an inter-station channel as its "home" channel. This helped prevent overcrowding on channel 11, enabling a CBer to monitor a town's home channel to contact another CBer from that town instead of a making a general call on channel 11.
Since the price of CB was dropping andVHF Marine Bandwas still expensive, many boaters installed CB radios. Business caught on to this market, and introduced marine CBs containing a weather band (WX). There was much controversy over whether the Coast Guard should monitor CB radio, but for safety they did so, usingMotorolabase stations at their search and rescue stations. The Coast Guard stopped this practice in the late 1980s and recommends VHF Marine Band radios for boaters.[16]
CB has lost much of its original appeal due to development ofmobile phones, theinternetand theFamily Radio Service. Changingradio propagationfor long-distance communications due to the 11–yearsunspot cycleis a factor at these frequencies. In addition, CB may have become a victim of its own popularity; with millions of users on a finite number of frequencies during the mid-to-late 1970s and early 1980s, channels often were noisy and communication difficult, which reduced interest among hobbyists. Business users (such as tow-truck operators, plumbers, and electricians) moved to the VHF and UHF business band frequencies. The business band requires an FCC license, and usually results in an assignment to a single frequency. The advantages of fewer users sharing a frequency, greater authorized output power, clarity ofFMtransmission, lack of interference by distant stations due toskip propagation, and consistent communications made the VHF (Very High Frequency) radio an attractive alternative to the overcrowded CB channels.
With these factors in play, CB radio has once again gained popularity in recent years, an uptick not seen since the '90s. Manufacturers report an increase in sales, while social media sites like YouTube show a growing popularity in CB radio content, mainly as a hobby. The technology has also given way to more compact CB radios with far more features afforded in older models.[citation needed]
The FCC restricts channel 9 to emergency communications and roadside assistance.[17]Most highway travelers monitor channel 19. Truck drivers still use CB, which is an effective means of obtaining information about road construction, accidents andpolice speed traps.
Before CB was authorized in Australia, hand-held 27-MHz "walkie-talkies" were available, which used several frequencies between the present CB channels, such as 27.240 MHz.[19][20]By the mid-1970s, hobbyists were experimenting with handheld radios and unauthorized 23 channel American CB radios. At that time in Australia, licensed ham operators and Emergency Services still used the 11 meter band[21]which was not yet available for CB use. Multiple CB clubs had formed by this time, which assigned call signs to members, exchangedQSL cards, and lobbied for the legalisation of CB. In late 1977, having legalised Australian CB and allowed the import / sale of American and Japanese 23 channel sets, the Federal Government drafted new interim regulations for Australian 18 channel transceivers. The new RB249 regulations came into effect on January 1, 1978, and the last official registration date for 23 channel sets was January 31, 1978. After this date, use of unregistered 23 channel CB sets was deemed illegal and unlicensed sets were no longer eligible to be licensed. The 18 channel band plan used 16 channels of the 23 channel CB radios plus 2 extra channels at 27.095 and 27.195 MHz, to make up the 18 channels. The original channels 1, 2, 3, 4, 10, 21 and 23 were deleted from the 18 channel band plan. So channel 1 on an 18 channel was actually channel 5 on a 23 channel radio. These roughly corresponded to the present channels 5–22, except for the two unique frequencies that are known as 11A (Channel 7 on an 18 channel Australian CB) and 19A (Channel 16 on an 18 channel Australian CB) or remote control frequencies but are no longer part of the Australian 27 MHz CB band since 40 Channels were introduced.[22]On January 1, 1982, the American 40 channel band plan was adopted.
From the outset, the government attempted to regulate CB radio with license fees and call signs, but eventually they abandoned this approach. Enthusiasts rushed for licences when the doors opened at post offices around Australia in mid-1977 and by the end of the first quarter of 1978 an estimated 200,000 licences were issued (Australia's Population in 1978 was 14.36 million). The regulations called for one licence per CB radio. The price for a licence in 1977 was AU$25 per year (In mid 1977 the Australian Dollar exchange rate was AU$0.90 to US$1.00), a not insubstantial amount for the average Australian wage-earner. Australian CB radio uses AM, USB, and LSB modes (no FM) on 27 MHz, allowed output power being 4 Watts AM and 12 Watts SSB. WhenUHF CBwas first legalised the 27 MHz CB Band was intended to be closed to Australian CBers in 1982 and only the 477 MHz UHF band was to continue, however this did not eventuate. The first 477 MHz CB radio in 1977 was designed and made in Australia by Philips TMC and was a 40 channel CB called the FM320.
The first CB club in Australia was theCharlie Brown Touring Car Club(CBTCC),[23]which formed inMorwell, Victoriain 1967 and consisted mainly offour-wheel driveenthusiasts. The club used the prefix "GL" (forGippsland), since "CB" could not be used.[24]After July 1, 1977, the club changed its name to Citizens Band Two Way Communication Club (CBTCC).[citation needed]Other early clubs were "LV" (Latrobe Valley) and "WB" (named after Wayne Britain). Members of these clubs are still active, and have also becomeamateur radio operators. Other Australian cities which became CB radio "hotspots" were Seymour, Benalla, Holbrook and Gundagai, all located on the busy Hume Highway between Melbourne and Sydney. Other regional cities such as Bendigo, Mildura, Mount Gambier and Port Augusta, developed lively, colourful CB radio communities.
With the introduction ofUHF CBradios in 1977, many operators used both UHF and HF radios and formed groups to own and operate local FM repeaters. Members of the CBTCC formed what became known as Australian Citizens Radio Movement (ACRM) in the early 1970s; this organization became the voice for legalization of CB radio throughout Australia. After peaking in the 1970s and early 1980s, the use of 27 MHz CB in Australia has fallen dramatically due to the introduction of 477 MHz UHF CB (with FM and repeaters) and the proliferation of cheap, compact handheld UHF transceivers. Technology such as mobile telephones and the internet have provided people with other choices for communications. The Australian government has changed the allocation of channels available for UHF CB Radio from 40 to 80, and doubled the number of repeater channels from 8 to 16.[25]
Several channels are allocated for maritime use in Australia. Australia also permits the use ofmarine VHF radio. 27 MHz radios have the advantage of not requiring a certificate of proficiency to use,[26]however they may not be monitored by rescue organisations or larger vessels. VHF radios are recommended by state agencies, such as Marine Safety Victoria andMarine Rescue NSW.[18][27][28]
InCanada, theCRTCofficially calls the CB band the "General Radio Service", though regulations also note that this service is commonly called "CB".[29]Canada's GRS uses the identical frequencies and modes as the United States citizens band, and no special provisions are required for either Canadians or Americans using CB gear while traveling across the border.
The General Radio Service was authorized in 1962. Initially, CB channels 1–3 remained allocated to amateur radio and channel 23 was used bypaging services. American CB licensees were initially required to apply for a temporary license to operate in Canada.[30]In April 1977, the service was expanded to the same 40 channels as the American service.[31]
InIndonesia, CB radios were first introduced about 1977 when some transceivers were imported illegally from Australia, Japan and the United States. The dates are hard to confirm accurately, but early use was known around large cities such asJakarta,Bandung,Yogyakarta,SurabayaandMedan. The Indonesian government legalized CB on 6 October 1980 with a decision by the Minister of Communications, the "Ministerial Decree on the Licensing for the Operation of Inter-Citizens Radio Communication". Because many people were already using 40 channel radios prior to legalization, the American band plan (with AM and SSB) was adopted; aVHF bandwas added in 1994, along with allowing use of the AustralianUHF CBchannel plan at 476-477 MHz On 10 November 1980, the Indonesian Directorate General of Posts and Telecommunications issued another decree establishing RAPI (Radio Antar Penduduk Indonesia) as the official citizens band radio organization in Indonesia.[32]
In Malaysia, citizens band radios became legal when the "Notification of Issuance Of Class Assignments" by the Malaysian Communication and Multimedia was published on 1 April 2000. Under this class assignment, a CB radio is classified as a "Personal Radio Service device". The frequency band is HF, 26.9650 MHz to 27.4050 MHz (40 channels), power output is 4 watts for AM and FM and 12 watts PEP for SSB. Channel 9 is reserved for emergencies, and channel 11 is a calling channel. On UHF 477 MHz, citizens band PRS radio devices are allowed 5 watts power output on FM on 39 assigned channels spaced at 12.5 kHz intervals from 477.0125 to 477.4875 MHz. Channel 9 is reserved for emergencies, and channel 11 for calling. A short-rangesimplexradio communications service for recreational use is from 477.5250 to 477.9875 MHz FM mode with 38 channels and a power output of 500 mW. A CB radio or Personal Radio Service Device under Class Assignment does not need an individual license to operate in Malaysia if it adheres to the rules of theWarta Kerajaan Malaysia[33]
On 1 April 2010 the MCMC released a new regulation[34]and later on 2017.[35]This includes a new UHF PMR 446 MHz allocation: an eight-channelanalogPersonal Mobile Radio 446 MHz (Analog PMR446) with frequencies from 446.00625 to 446.09375 MHz (12.5 kHz spacing) FM with 0.5 watt power output, and 16 channels forDigitalPersonal Mobile Radio 446 MHz (Digital PMR 446). Frequencies for Digital PMR 446 are from 446.103125 to 446.196875 MHz with 6.25 kHzchannel spacingin 4FSK mode and a power output of 0.5 watt. An unofficial citizens band radio club in Malaysia is the "Malaysia Boleh Citizen Radio Group", known as "Mike Bravo" (Malaysia Boleh).[36]Subsequently, the MCMC revoked 477 MHz personal radio service as a class assignment on 1 January 2022.[37]
In the UK, a small but growing number of people were illegally using American CB radios during the late 1970s and early 1980s. The prominence of CB radio grew in Britain partly due to the success of novelty songs like C.W. McCall's "Convoy" and Laurie Lingo & The Dipsticks' "Convoy GB" in 1976 (both of which were Top 5 hits). By 1980, CB radio was becoming a popular pastime in Britain; as late as the summer of 1981 the British government was still saying that CB would never be legalized on 27 MHz, proposing a UHF service around 860 MHz called "Open Channel" instead. However, in November 1981 (after high-profile public demonstrations) 40 frequencies unique to the UK, known as the27/81 Bandplanusing FM were allocated at 27 MHz plus 20 channels on 934 MHz (934.0125–934.9625 MHz with 50 kHz spacing). CB's inventor,Al Gross, made the ceremonial first legal British CB call fromTrafalgar Squarein London.
The maximum power allowable on the MPT 1320 27/81 system was 4 watts (in common with the American system), although initially radios were equipped to reduce output power by 10dB(to 0.4 watts) if the antenna was mounted more than 7 meters (23') above ground level. The power-reduction switch is also useful in reducingTV interference. MPT 1320 also restricted antennas to a maximum length of 1.5 meters (5'), with base loading being the only type permitted for 27 MHz operation. Over the next several years antenna regulations were relaxed, with antenna length increasing to 1.65 meters (5'5") and centre- or top-loading of the main radiating element permitted. On 1 September 1987 the UK added the usual 40 frequencies (26.965–27.405 MHz) used worldwide, for a total of 80 channels at 27 MHz; antenna regulations were further relaxed, and the 934 MHz band was withdrawn in 1998.
CB radio in the UK was deregulated in December 2006 by the regulatory bodyOfcom, and CB radio in the UK is now license-free. The old MPT 1320 27/81 band will continue to be available for the foreseeable future. On 27 June 2014, changes were made by Ofcom to allow the use of AM & SSB modes on CB in the UK legally for the first time. The rules regarding non-approved radios and power levels above 4 Watts on AM/FM and 12 Watts on SSB still apply, despite deregulation. Persons using illegal equipment or accessories still risk prosecution, fines or confiscation of equipment, although this is rarely enforced. AM and SSB on the freeband and amplifier use are common among enthusiasts.Packet radiois legal in the UK, although not widely used. Internet gateway stations are also beginning to appear; although illegal on 27 MHz, these units are connected to other CB stations around the world.
Although the use of CB radios in the UK is limited they are still in use, especially with the farming community, truckers, off-roaders and mini-cab services.[38]The widely used channel for theYoung Farmers' Clubis channel 11. The normal calling and truckers' channel is channel 19, although many truck organisations and groups use other channels to avoid abuse.
CB radio is not a worldwide, standardized radio service. Each country decides if it wants to authorize such a radio service from its domestic frequency authorizations, and what its standards will be; however, similar radio services exist in many countries. Frequencies, power levels and modes (such asfrequency modulation(FM),amplitude modulation(AM), andsingle-sideband modulation(SSB), often vary from country to country; use of foreign equipment may be illegal. However, many countries have adopted the American channels and their associated frequencies, which is generally in AM mode except some higher channels which are sometimes in SSB mode. In September 2021 the FCC approved the use of FM on CB radio.[39]
The standard channel numbering is harmonized through theFCC(USA) and theCEPT(Europe).[d][40]
See alsochannel assignments for CB use in the United States.
When looking at the FCC/CEPT channel list there are some channels with a spacing of 20 kHz instead of the regular 10 kHz step. These intermediate frequencies are reserved for the Radio Control Radio Service (RCRS).[e]The RCRS service is commonly used for remote control of model aircraft and boats. It is an unofficial practice to name these channels by their next lower standard channel number along with a suffix "A" (after). For example, channel "11A" is 27.095 MHz, spaced 10 kHzafterstandard channel 11 (at 27.085 MHz) is used to provide for part of European railroad'sEurobaliseradio communication with trains.
Single-sideband(SSB) operation involves the selection of either the Lower Side Band (LSB) or the Upper Side Band (USB) mode for transmit and receive. SSB radios also have the standard AM mode for communicating with standard CB radio models. With the original 23 CB channels SSB stations commonly used channel 16, to avoid interference to those usingAM(SSB stations are authorized to use 12 watts, as opposed to 4 watts for AM stations) and to more easily locate other SSB stations. With the FCC authorization of 40 channels, SSB operation shifted to channels 36–40. Channel 36 (or 38 for LSB) became the unofficial SSB "calling channels" for stations seeking contacts, with the subsequent conversation moving to channels 37–40. CBers with AM-only radios are asked to not use channels 36 through 40. In return, SSB stations stay off the remaining 35 channels so they could be used by AM stations. This agreement provides interference-free operation for all operators by separating the far more powerful SSB stations from the AM stations. This solution also resolves the confusion created by the false advertising that SSB radios have 120 channels compared to only 40 for AM radios.
While a SSB radio has three possible "modes" (AM, LSB, USB) it can operate in, operation is still limited to the same 40 channels. Some manufacturers tried to sell more radios by claiming that with three different modes possible for each channel, it was the equivalent to 120 channels.[citation needed]Reality is far different: Attempting an SSB conversation while an AM conversation is in progress results in jammed communications for everyone. In general, each channel can only support one AM conversation and no others; if no AM conversation is in progress, two SSB conversations can share one channel without interference if one is in LSB and the other in USB mode. For a particular conversation, everyone must be tuned to the same channel and same mode in order to talk with each other.
Starting in October 2021 the FCC has approved FM for CB in the US.
TheEuropean Conference of Postal and Telecommunications Administrations(CEPT) adopted the North American channel assignments, except channel 23, frequency 27.255 MHz; channel 24, frequency 27.235 MHz; and channel 25, frequency 27.245 MHz.[40]However, legal CB equipment sold in Europe does follow the North American channel designation. Some member countries permit additional modes and frequencies; for example,Germanyhas 40 additional channels at 26 MHz for a total of 80. TheUnited Kingdomhas an additional 40 channels between 27.60125 and 27.99125 MHz, also making 80 channels in total. Before CEPT, most member countries used a subset of the 40 U.S. channels.
InRussiaandPoland, the channels are shifted 5 kHz down; for example, channel 30 is 27.300 MHz.[citation needed]Many operators add a switch to change between the "zeroes" (the Russian/Polish channel assignment) and the "fives" (the international/European assignment). Most contemporary radios built for those markets can do "fives" as well as "zeroes" out of the box. Since roughly 2005–2006, Russia and Poland have adopted use of the standard US channel offset as well as the older channel plan, for two overlapping "grids" of channels.
Russia uses an alphanumeric designation for their CB channel plans, because several "grids" or "bands" of 40 channels each are used, along with both AM and FM mode. Russian CB allocations follow the CB band 26.965–27.405 MHz (designated as band C), as well as 26.515–26.955 MHz (designated as band B) and 27.415–27.855 MHz (designated as band D). Some radios refer to the "mid band" (standard CB band) as "band D" which shifts the letters up one (making 26.515–26.955 MHz "band C" and 27.415–27.855 MHz "band E".
For the convenience of users of the grid were marked by letters. Classic is considered the marking when the main range is designated letter "C". The most common description of the channel is considered to be similar to the following: (C9FM or C9EFM or C9EF or 9EF).
In it:
An example of correct designations: C9EF, C9EA, C9RF, C9RA
The 25–30 MHz band (including the CB allocations and frequencies above and below the 26.5–27.860 MHz band) is heavily used for taxi cab and other mobile two-way communications systems in Russia, Ukraine and other former USSR country states.
New ZealandandJapanhave unique allocations compared to any other country. New Zealand authorizes use of their New Zealand specific 40 channel 26.330–26.770 MHzfrequency planin addition to the "standard" 40 channel 26.965–27.405 MHz frequency plan for a total of 80 HF CB channels. New Zealand has adopted the Australian UHF CB System as well.[42]
Japan's CB allocation consists of 8 voice and 2 radio control channels with a maximum power output of 500 mW. AM mode is the only mode permitted and antennas must be non-removable and less than 199 cm (78 inches) long. In Japan, the 26–28 MHz range is allocated to fishery radio services and these frequencies are heavily used for marine communications. Japan's "double side band fishery radio" or "DSB Fishery Radio Service" covers 26.760 MHz-26.944 MHz, 27.016 MHz, and 27.524-27.988 MHz. However, frequencies such as 27.005 MHz AM are widely pirated in Japan with very high power transmitters. This causes interference to the authorized low-power 1 Watt DSB (1 Watt AM) fishery radio service. Instead of 26–27 MHz, Japan has authorized several UHF FM CB-typepersonal radio servicesin the 348 MHz, 420–422 MHz and 903–904 MHz bands.
Indonesiahas the usual 40 channels at 27 MHz, plus a unique 60-channel VHF allocation from 142.050 to 143.525 MHz. Unlike most other countries, CB operators in Indonesia are required to have a CB-specific license (similar to GMRS licensing in the United States, albeit it cannot be used by other immediate family members and only valid for 5 years). An alphanumeric callsign, like it's amateur radio counterpart, would then be issued.[43]
InBrazil, CB channels were upgraded from 23 to 60 channels starting in 1979 and again in 1980 to80 channels[pt](from 26.965 MHz to 27.855 MHz).
CB Radio Channels (ANATEL)
South Africa, like New Zealand and the UK, permits the use of two HF CB bands. South Africa has a 23 channel AM / SSB 29 MHz CB allocation (called "29 Megs" or "29 MHz CB") from 29.710 to 29.985 MHz in 12.5 kHz steps. South Africa also permits use of standard CB channels 19–27 (27.185–27.275 MHz) with AM / SSB permitted. Many radios sold in South Africa feature both the 27 MHz and 29 MHz bands. A license fromICASAis no longer required to purchase or use a CB radio in South Africa.
Hungaryallows use of the "low channels" for a total of 80 channels (26.515 MHz to 27.405 MHz).
Germanyauthorizes a similar allocation, with 40 channels from 26.965 to 27.405 MHz and another 40 channels from 26.565 to 26.955 MHz in regular 10-kHz steps.
TheCzech Republicauthorizes 80 channels as well (same as the German 80-channel plan). As in Germany, digital modes are allowed on certain frequencies.[f]Internet gatewaysandradio repeatersare allowed on channels 18 [27.175 MHz] and 23 [27.255 MHz]. Paging is permitted on channel 1 (26.965 MHz) and channel 80 (26.955 MHz) is the recommended call channel for Czech CB radio operators.
Using radios outside their intended market can be dangerous, as well as illegal, as frequencies used by Citizen's Band radios from other countries may operate on frequencies close to, or be used by, emergency services (for example, the Indonesian service around 142 MHz operates on frequencies allocated to a public safety network shared with police, fire and EMS services in Ontario, Canada).
In thePhilippines, up to present time, the use of 27 MHz CB is still banned since the Marcos regime banned it in 1980s. Before it was banned, they use the same allocation as the US. A few operators still illegally utilize the 40 CB channels.
CB was the only practical two‑way radio system for the individual consumer, and served several subsets of users such as truck drivers, radio hobbyists, and those in need of short‑range radio communications, such as electricians, plumbers, and carpenters, who needed to communicate between job site and main office. While some users have moved on to other radio services, CB is still a popularhobbyin many countries. The 27 MHz frequencies used by CB, which require a relatively long aerial antennas and tend to propagate poorly indoors, discourage the use of handheld radios. Many users of handheld radios (families, hunters and hikers) have moved on to 49 MHz and the UHFFamily Radio Service; those needing a simple radio for professional use (e.g., tradesmen) have moved on[citation needed]to "dot-color" Business Band radios and the VHFMulti-Use Radio Service.
CB is still commonly used by long-haul truck drivers to communicate directions, traffic problems and other relevant matters.[44]The unofficial "travelers channel" in most of the world is channel 19; in Australia it is channel 8 (27.055 MHz) and UHF channel 40 (477.400 MHz). In Russia, it is channel 15 (in addition to traditional emergency channel 9 and truckers' channel 19), in Greece it is channel 13, in Italy it is channel 5, all AM. These frequencies may have evolved because tuned circuits (particularly antennas) work best in the middle of the band; the frequency for channel 19 (not channel 20) is the center of the 40 channel US band and other things being equal, signals will be transmitted and heard the farthest. Since less standardization exists in Europe, CB there is more associated with hobbyists than with truckers.
Legal (short‑range) use of CB radio is sometimes impeded by users of illegal high‑power transmitters, which can be heard hundreds of miles away. The other problem with short‑range CB use is propagation; during long‑range"skip"conditions local signals are inaudible due to reception of multiple distant signals.
In the United States, the number of users and law enforcement financing by theFederal Communications Commissionmean that only the worst offenders are sanctioned, which makes legitimate operation on the citizens band unreliable. Most offenders are not caught for interfering with other CB users; often, their self‑modified equipment generatesharmonicsandspurswhich cause interference to services outside the citizens band and to consumer equipment.
The maximum legal CB power output level in the U.S. is 4 watts for AM (un-modulated carrier; modulation can be four times the carrier power, or 16 wattsPEP) and 12 watts forSSB, as measured at the transmitter antenna connection. However, externallinear amplifiersare often used illegally.
During the 1970s, the FCC banned the sale of linear amplifiers capable of operation from 24 to 35 MHz to discourage their use on the CB band, although the use of high‑power amplifiers continued. Late in 2006, the FCC amended the regulation to exclude only 26–28 MHz to facilitate amateur 10 meter operation.[45]Lax enforcement enables manufacturers of illegal linear amplifiers to openly advertise their products; many CB dealers include these amplifiers in their catalogs. Due to their rampant, unchecked use of linear amplifiers, American CB Radio operators are often referred to as "Alligators", by operators in other countries (suggesting American operators are "All Mouth and No Ears"). Attempts by law-abiding CB users to increase regulatory oversight have been ineffective.
At the beginning of the CB radio service, transmitters and receivers usedvacuum tubes; solid-state transmitters were not widely available until 1965, after the introduction of RF power-transistors.[46]Walkie-talkiehand-held units became affordable with the use of transistors. Early receivers did not cover all the channels of the service; channels were controlled by plug-in quartzcrystals, with one of several operating frequencies selected by a panel control in more expensive units.Superheterodynereceivers (using one or two conversion stages) were the norm in good-quality equipment, although low-cost toy-type units usedsuper-regenerativereceivers. With the earliest sets two quartz crystals were needed for transmitting and receiving on each channel, which was costly. By the mid-1960s "mixer" circuits made frequency-synthesized radios possible, which reduced cost and allowed full coverage of all 23 channels with a smaller number of crystals (typically 14). The next improvement came during the mid-1970s; crystal synthesis was replaced byPLLtechnology using ICs, enabling 40 channel sets with only one crystal (10.240 MHz). Almost all wereAM-only, although there were a fewsingle sidebandsets.
Most CB radios sold in the United States have the following features:
Microphonechoices include:
27 MHz is a relatively long wavelength for mobile communications, and the choice of antenna has a considerable impact on the performance of a CB radio. A common mobile antenna is a quarter-wave vertical whip. This is roughly 9 feet (2.7 m) tall; it is mounted low on the vehicle body, and often has a spring-and-ball mount to enhance its flexibility when scraping or striking overhead objects. Where a nine-foot whip is undesirable, shorter antennas includeloading coilsto make the antenna impedance the same as a physically longer antenna. The loading coil may be on the bottom, middle, or top of the antenna, while some antennas are wound in a continuously loaded helix.
Many truckers use two co-phased antennas, mounted on their outside mirrors. Such an array is intended to enhance performance to the front and back, while reducing it to the sides (a desirable pattern for long-haul truckers). To achieve this effect, the antennas must be separated by about eight feet, only practical on large trucks. Two antennas may be installed for symmetrical appearance, with only one connected.
Another mobile antenna is the continuously loaded half-wave version. They do not require aground planeto present a near-50ohmload to the radio, and are often used on fiberglass vehicles such as snowmobiles or boats. They are also useful in base stations where circumstances preclude the use of a ground-plane antenna. Handheld CBs may use either a telescoping center-loaded whip or a continuously loaded "rubber ducky" antenna.
Base CB antennas may be vertical for omnidirectional coverage, or directional "beam" antennas may be used to direct communications to a particular region. Ground-plane kits exist as mounting bases for mobile whips, and have several wire terminals or hardwired ground radials attached. These kits are designed to have a mobile whip screwed on top (a full-length, quarter-wave steel whip is preferred) and mounted on a mast. The ground radials replace the vehicle body (which is thecounterpoisefor a mobile whip in a typical vehicle installation).
All frequencies in theHFspectrum (3–30 MHz) can be refracted by charged ions in theionosphere. Refracting signals off the ionosphere is calledskywavepropagation, and the operator is said to be "shooting skip". CB operators have communicated across thousands of miles and sometimes around the world. Even low-power 27 MHz signals can sometimes propagate over long distances.
In times of highsunspotactivity, the band can remain open to much of the world for long periods of time. During low sunspot activity it may be impossible to use skywave at all, except during periods ofSporadic-E propagation(from late spring through mid-summer). Skip contributes to noise on CB frequencies. In the United States, it is no longer illegal to engage in (or attempt to engage in) CB communications with any station more than 250 km (160 mi) from an operator's location.[48]This restriction used to exist to keep CB as a local (line-of-sight) radio service; however, in the United States the restriction has been dropped. The legality of shooting skip is not an issue in most other countries. A recent FCC decision now allows shooting skip in the United States.[49]
Operation on frequencies above or below the citizens band (on the "uppers" or "lowers") is called "freebanding" or "outbanding".[g]While frequencies just below the CB segment (or between the CB segment and the amateur radio10-meter band) seem quiet and under-utilized, they are allocated to other radio services (including government agencies) and unauthorized operation on them is illegal. Furthermore, illegal transmitters and amplifiers may not meet good engineering practice for harmonic distortion or "splatter", which may disrupt other communications and make the unapproved equipment obvious to regulators. Freebanding is done with modified CB or amateur equipment, foreign CB radios which may offer different channels, or with radios intended for export. Legal operation in one country may be illegal in another; for example, in the UK until June 2014 only 80 FM channels were legal.
Unlike amateur radios with continuous frequency tuning, CBs manufactured for export are channelized. Frequency selection resembles that of modified American CBs more than any foreign frequency plan. They typically have a knob and display that reads up to channel 40, but include an extra band selector that shifts all 40 channels above or below the band and a "+10 kHz" button to reach the model control "A" channels. These radios may have 6 or even 12 bands, establishing a set of quasi-CB channels on many unauthorized frequencies. The bands are typically lettered A through F, with the normal citizens band as D.
For example, a freebander with an export radio who wants to use 27.635 MHz would choose channel 19 (27.185 MHz) and then shift the radio up one band (+450 kHz). It requires arithmetic on the part of the operator to determine the actual frequency, although more expensive radios include afrequency counteror a frequency display — two different components, providing an identical result. Illegal operations may unintentionally end up on frequencies very much in use. For instance, channel 19 shifted two bands up is 28.085 MHz, which is in a Morse code / data-only part of the 10 meter ham band. Voice transmissions in a Morse code-only segment are easily detectable by authorities. Amateur Radio Service operators record, locate, and report to the FCC frequency trespassing and intrusions of their frequency allocations by pirate transmissions or illegal operators for enforcement action.[50]
Many freeband operators useamateur radiosmodified to transmit out of band, which is illegal in some countries. Older amateur radios may require component changes; for instance, the 1970sYaesu FT-101was modified for CB by replacing a set of crystals used to tune portions of the 10 meter band, although some variants of the FT-101 were sold with the US FCC channels standard and were capable of transmitting above and below the legal 40 channels by another 10 or more channels.[3]: 174On some newer radios, the modification may be as simple as disconnecting a jumper wire or a diode. Many types of amateur transceivers may be found on CB and freeband, ranging from full-coverage HF transceivers to simpler10 metermobile radios. In the United States, the FCC bans the importation and marketing of radios it deems easily modifiable for CB;[51]it is illegal to transmit on CB frequencies with a ham radio except in emergencies where no other method of communication is available.
Agray markettrade in imported CB gear exists in many countries. In some instances, the sale or ownership of foreign-specification CB gear is not illegal but its use is. With the FCC's minimal enforcement of its CB rules, enthusiasts in the US use "export radios" or Europeanfrequency modulation(FM) CB gear to escape the crowded AM channels. American AM gear has also been exported to Europe.
"Export radios" are sold in the United States as10 meterAmateur Radio transceivers. Marketing, import and sale of such radios is illegal if they are distributed as anythingotherthan Amateur Radio transceivers. It is also illegal to use these radios outside of the Amateur Radio bands by anyone in the US, since they are not type-certified for other radio services and usually exceed authorized power limits. The use of these radios within the Amateur Radio Service by a licensed Amateur Radio operator within his / her license privileges is legal, as long as all FCC regulations for Amateur Radio are followed.
A callbook is a directory of radio station call signs. Originally a bound book that resembled a telephone directory, it contains the name and addresses of radio stations in a given jurisdiction (country).Modern Electricspublished the first callbook in the United States in 1909. Today, the primary purpose of a callbook is to allow radio operators to send a confirmation post card, called aQSL card, to an operator with whom they have communicated via radio. Callbooks have evolved to include online databases that are accessible via the Internet to instantly obtain the address of another amateur radio operator and their QSL managers. The most well known and used on-line QSL database for the 11 meter / freebander community is QRZ11.COM, designed after its "big brother"QRZ.comfor amateur radio.[52][53][54][55][56][57]
During the 1970s and 1980s peak years of CB radio, many citizens band-themed magazines appeared on newsstands. Two magazines that dominated the time period wereS9 CB RadioandCB Radio Magazine.S9’s successor wasPopular Communications, which had the same editor under a different publisher beginning in 1982. It covered hobby radio as well as CB. The same publisher produced a magazine calledRADIO!forRadioShackstores in the mid-1990s. In Australia,CB Action Magazinewas produced monthly from mid-1977 and continuing publication through until the early 1990s.CB Actionspawned several other popular publications, including a communications and scanning magazine andAmateur Radio Actionmagazine, produced over several decades and running to some 18 volumes.
In the early 2000s,National Communications Magazineadded CB radio coverage to its coverage ofscanner radiosand still remains the only magazine inNorth Americacovering CB radio.
|
https://en.wikipedia.org/wiki/Citizens_band_radio
|
TheFamily Radio Service(FRS) is an improvedwalkie-talkieradio system authorized in the United States since 1996. Thispersonal radio serviceuses channelized frequencies around 462 and 467 MHz in theultra high frequency(UHF) band. It does not suffer the interference effects found oncitizens' band(CB) at 27MHz, or the 49 MHz band also used bycordless telephones, toys, andbaby monitors. FRS usesfrequency modulation(FM) instead ofamplitude modulation(AM). Since the UHF band has differentradio propagationcharacteristics, short-range use of FRS may be more predictable than the more powerful license-free radios operating in theHFCB band.
Initially proposed byRadioShackin 1994 for use by families, FRS has also seen significant adoption by business interests, as an unlicensed, low-cost alternative to thebusiness band. New rules issued by the FCC in May 2017 clarify and simplify the overlap between FRS andGeneral Mobile Radio Service(GMRS) radio services.[1]
Worldwide, a number of similar personal radio services exist; these share the characteristics of low power operation in the UHF (or upper VHF) band using FM, and simplified or no end-user licenses. Exact frequency allocations differ, so equipment legal to operate in one country may cause unacceptable interference in another. Radios approved for FRS are not legal to operate anywhere in Europe.
FRS radios use narrow-band frequency modulation (NBFM) with a maximum deviation of 2.5 kilohertz. The channels are spaced at 12.5 kilohertz intervals.
All 22channelsare shared with GMRS radios. Initially, the FRS radios were limited to 500milliwattsacross all channels. However, after May 18, 2017, the limit is increased to 2 watts on channels 1-7 and 15–22.[1]
FRS radios frequently have provisions for using sub-audible tonesquelch(CTCSSandDCS) codes, filtering out unwanted chatter from other users on the same frequency. Although these codes are sometimes called "privacy codes" or "private line codes" (PL codes), they offer no protection from eavesdropping and are intended only to help reduce unwanted audio when sharing busy channels. Tone codes also do nothing to prevent desired transmissions from being swamped by stronger signals having a different code.
All equipment used on FRS must be certified according to FCC regulations. Radios are not certified for use in this service if they exceed limits on power output, have a detachable antenna, allow for unauthorized selection of transmitting frequencies outside of the 22 frequencies designated for FRS, or for other reasons.[2][3]After December 2017, the FCC no longer accepts applications to certify hand-held FRS units providing for transmission in any other radio band.
FRS radios must use only permanently attached antennas;[2]there are also table-top FRS "base station" radios that have whip antennas. This limitation intentionally restricts the range of communications, allowing greatest use of the available channels by the community. The use of duplexradio repeatersand interconnects to thetelephone networkare prohibited under FRS rules.
The range advertised on specific devices might not apply in real-world situations, since large buildings, trees, etc., can interfere with the signal and reduce range. Under exceptional conditions, (such as hilltop to hilltop, or over open water) communication is possible at 60 km (37 mi) or more, but that is rare. Under normal conditions, with line of sight blocked by a few buildings or trees, FRS has an actual range of about 0.5 to 1.5 km (0.3 to 1 mile).
In May 2017, the FCC significantly revised the rules for combination FRS/GMRS radios. Combination radios will be permitted to radiate up to 2 watts on 15 of the 22 channels (as opposed to 0.5 watts), and all FRS channels are now considered shared with the GMRS service. Operation over 2 watts, or operation on GMRS repeater input channels, will still require GMRS licensing. The FCC will not certify combination FRS/GMRS radios that exceed the current power limits for the FRS service.[1]
Hybrid FRS/GMRS consumer radios have been introduced that have 22 channels. Before May 2017, radios had been certified for unlicensed operation on the 7 FRS frequencies, channels 8–14, under FRS rules.[4]
Prior to the 2017 revision, FCC rules required a GMRS license to operate on channels 1–7 using more than 0.5 watts.[2]Many hybrid radios have anERPthat is lower than 0.5 watts on channels 1–7, or can be set by the user to operate at low power on these channels. This allows hybrid radios to be used under the license-free FRS rules if the ERP is less than 0.5 watts and the unit is certified for FRS operation on these frequencies.[2]Beginning September 28, 2017, FRS operation is permitted at up to 2 watts on these channels.
Interference to licensed services may be investigated by the FCC.[5]
Channels 8–14, formerly exclusive to FRS, since 28 September 2017 can be used by GMRS at 0.5 watts. Channels 15–22, formerly reserved exclusively for GMRS, can be used at up to 2 watts in the FRS.
Effective September 30, 2019, it became unlawful in the US to import, manufacture, sell or offer to sell radio equipment capable of operating under both GMRS and FRS. This does not includeamateurand other radio equipment that are not certified under Part 95, such as many handheld radios that are marketed for amateur use but are also able to transmit on FRS and GMRS frequencies.[1]
Personal UHF radio services similar to the American FRS exist in other countries, although since technical standards and frequency bands will differ, usually FCC-approved FRS equipment may not be used in other jurisdictions.
American-standard FRS radios have been approved for use inCanadasince April 2000. As of 2016,[update]only low-power (2 WERP),half duplexGMRS operation is permitted, but a license is not required.[6]Repeater and high-power operations are not permitted. This allows the use of dual-mode FRS/GMRS walkie-talkies, but precludes the use of higher-powered GMRS devices designed for vehicle and base-station purposes.
Sincetouristsoften bring their FRS radios with them, and since trade between the U.S., Canada, andMexicois of great value to all three countries, the MexicanSecretary of Communication and Transportationhas authorized use of the FRS frequencies and equipment similar to that in the US. However, dual-mode FRS/GMRS equipment is not approved in Mexico, so caution should be exercised in operating hybrid FRS/GMRS devices purchased elsewhere.[7]
Dual-mode GMRS/FRS equipment is also approved inBrazil(GMRS only in simplex mode, GMRS frequencies 462.550, 467.550, 462.725, 467.725 are not allowed)[8]and most otherSouth Americancountries. Portable radios are heavily used in private communications, mainly by security staff in nightclubs and malls, but also in private parking, maintenance, and delivery services.
|
https://en.wikipedia.org/wiki/Family_Radio_Service
|
TheGeneral Mobile Radio Service(GMRS) is aland-mobileFMUHFradioservice designed for short-range two-way voice communication and authorized under part 95 of the USFCCcode. It requires a license in theUnited States, but some GMRS compatible equipment can be used license-free inCanada. The US GMRS license is issued for a period of 10 years. The United States permits use by adult individuals who possess a valid GMRS license, as well as their immediate family members.[a]Immediate relatives of the GMRS system licensee are entitled to communicate among themselves for personal or business[1]purposes, but employees of the licensee who are not family members are not covered by the license. Non-family members must be licensed separately.
GMRS radios are typically handheld portable (walkie-talkies) much likeFamily Radio Service(FRS) radios, and they share a frequency band with FRS near 462 and 467 MHz.Mobileandbase station-style radios are available as well, but these are normally commercial UHF radios as often used in public service and commercial land mobile bands. These are legal for use in this service as long as they are certified for GMRS under USC 47 Part 95. Older radios that are certified under USC 47 Part 90 are "grandfathered in", and can also be used legally for GMRS.[citation needed]
GMRS licensees are allowed to establishrepeatersto extend their communications range. However repeaters cannot be linked together over the internet nor connected to thepublic switched telephone network.[2]
Any individual in the United States who is at least 18 years of age and not a representative of a foreign government may apply for a GMRS license by completing the application form, online through the FCC's Universal Licensing System. No exam is required. A GMRS license is issued for a 10–year term.[3]The fee was reduced to $35 for all applicants on April 19, 2022.[4]
A GMRS individual license extends to immediate family members and authorizes them to use the licensed system.[5]GMRS license holders are allowed to communicate with FRS users on those frequencies that are shared between the two services.[6]GMRS individual licenses do not extend to employees.[5]
New GMRS licenses are being issued only to individuals. Prior to July 31, 1987, the FCC issued GMRS licenses to non-individuals (corporations, partnerships, government entities, etc.). These licensees aregrandfatheredand may renew but not make major modifications to their existing licenses.[7]
In any case, each GMRS station must be identified by transmission of its FCC-assigned call sign at the end of a transmission or a series of transmissions, and at least once every 15 minutes for a series lasting more than 15 minutes. The call sign may be spoken or sent with audible tones using Morse code. A repeater station handling properly identified transmissions of others is not required to send its own station identification.[8]
As with other UHF radio services, reliable range is considered to beline-of-sightand the distance to theradio horizoncan be estimated based on antenna height. Theoretically, the range between two hand-held units on flat ground would be about one or two miles (about 1.5–3 km). Mobile units might have a slightly farther range when mounted on a car roof. A GMRS repeater with an antenna mounted high above the surrounding terrain can extend the usable range to 20 miles or more (30+ km) depending on height.[9]Obstructions such as hills, trees, and buildings will reduce range. Higher power does not give much increase in range although it may improve the reliability of communication at the limits of line-of-sight distance.
GMRS is allotted 30 frequency channels in the vicinity of 462 MHz and 467 MHz. They are divided into 16 main channels and 14 interstitial channels.[10]
Licensees may use the eight main 462 MHz channels forsimplex communicationorrepeateroutputs.[11]
The eight main 467 MHz channels may only be used as repeater inputs,[12]in conjunction with the 462 MHz channels as outputs. The repeater input frequencies are exclusive to GMRS, and may be used only by licensed GMRS operators.
GMRS operators are permitted to transmit at up to 50wattstransmitter power output, on the 16 main channels,[13]but transmitting 1 to 5 watts is more common in practice.
The interstitial frequencies are in-between the main channels, and the 462 MHz interstitial frequencies may be used for simplex as long as theeffective radiated power(ERP) does not exceed 5 watts.[14]The 467 MHz interstitial frequencies have a power limit of 500 milliwatts ERP,[15]and only hand-held portable units may transmit on these channels.[16]
All 22Family Radio Service(FRS) frequencies are shared with GMRS, and users of the two services may communicate with each other. With the exception of FRS channels 8 through 14, GMRS licensees may use higher power radios with detachable or external antennas.
Note:Some inexpensive GMRS mobiles and portables do not fully comply with FCC permissible modulation bandwidth for GMRS and thus have weak transmitter audio and reduced range.
Conditions:Effective February 16, 1999, the GMRS rules have been amended and one may operate on any of the primary or interstitial channels shown in section 95.1763. Exception: Licensees who operate North of Line A and East of Line C may not operate on channels 462.6500 MHz (channel 19), 467.6500 MHz (channel 19R), 462.7000 MHz (channel 21) and 467.7000 MHz (channel 21R), unless one's previous license authorized such operations.
The FCC stipulates a specific channel bandwidth for FRS and GMRS. The bandwidth is constrained by the modulation which is FM deviation (GMRS = +/- 5.0 kHz, FRS = +/- 2.5 kHz) plus the uncertainties of the filtering of the transmitter and receiver. Additionally receivers and transmitters may drift over time or temperature so the bandwidth is further constrained to prevent interference to the adjacent channel. Channel spacing is 25 kHz for GMRS and so a 20 kHz bandwidth fits into that channel with protection on each side. FRS channels are spaced within a 12.5 kHz space directly between two GMRS channels. FRS radios generally utilize an 11 kHz transmitter bandwidth and a power lower than GMRS so the interference to an adjacent GMRS channel is minimized.
The predecessor to GMRS was namedClass A Citizens Radio Servicewhen it was commissioned in the 1940s. Tube-type transceivers were used, and transmitter power was limited to 60 watts (plate input power to the final amplifier tube). The original service ran wideband FM with ±15 kHz transmitter deviation and 50 kHz channel spacing. At the time, this was the norm for all U.S. land mobile services. There was also a Class B Citizens Radio Service which used a different set of 461 MHz channels and was limited to five watts output. Business users were permitted to license in this radio service. Radios were built by consumer electronics firms and commercial two-way radio vendors.
In the 1960s, the UHF 450–470 MHz band was re-allocated to 25 kHz channels. This meant transmitter deviation was reduced to ±5 kHz. This doubled the number of channels available across the entire 450–470 MHz band. Class B Citizens Radio Service channels were re-allocated to other radio services.
In the 1970s, allowed power was again changed to 50 watts across the output terminals of the transmitter. In 1987, licensing of business users was discontinued and businesses were allowed to continue operating until their licenses expired. There was congestion on all channels in largermetropolitan statistical areasand moving businesses toBusiness Radio Servicechannels would provide some relief. The radio service was changed to its present name; General Mobile Radio Service or GMRS.
In 2010 theU.S. Federal Communications Commission(FCC) proposed removing the individual licensing requirement. In 2015, the FCC ruled to keep the license requirement, but to remove the regulator fee for licensing.[17]Adopted on May 20, 2015, the ruling would be in effect after a 90–day notification period to Congress; the fee will not be eliminated before August 18, 2015.[18]The fee for a 5–year license was $90, with the regulatory fee portion of the license at $5 per year (or $25 for the 5–year life of the license). After the notification period, the fee for a 5–year license was to become $65. The change became effective on September 3, 2015.
Effective September 28, 2017, FCC revised the definition of the FRS service. FRS operation is now permitted with up to 2 watts on the shared FRS/GMRS channels. The FCC will not grant certification for hybrid radios that would exceed the limits for the FRS service on the FRS channels. Current "hybrid" FRS/GMRS radios will not require a GMRS license for power up to 2 watts, but FRS radios will still not be permitted to use the input frequencies of GMRS repeaters. Any radio exceeding the limits of the new FRS service will be classified as a GMRS radio.[19]
On September 30, 2019, it became unlawful in the United States to import, manufacture, sell, or offer to sell radio equipment capable of operating under both GMRS and FRS.[20]
The use of radio transmitters is regulated by national laws and international agreements. Often radio equipment accepted for use in one part of the world may not be operated in other parts due to conflicts with frequency assignments and technical standards. Some of the roles that the licensed GMRS service fills in the United States are, in other countries, filled by unlicensed or class-licensed services. Generally these services have strict technical standards for equipment to prevent interference with licensed transmitters and systems.
InCanada, hand-held GMRS radios up to 2 watts have been approved for use without a license since September 2004.[21]Typically these are dual FRS and GMRS units, with fixed antennas, and operating at 2 watts on some GMRS channels and 0.5 watts on the FRS-only channels. Mobile units (permanently mounted in vehicles),base stationsandrepeatersare not currently permitted on the GMRS channels in Canada.
Other countries have licensed and unlicensed personal radio services with somewhat similar characteristics, but technical details and operating conditions vary according to national rules. Many European countries use a similar 16–channel system near 446 MHz known asPMR446, as well as a 69–channel low-powerLPD433which is shared with the 433.92 MHzISM band. GMRS equipment that is approved for use in the United States will not communicate with PMR446 radios due to using different frequency ranges.
Currently, the application fee for a GMRS license is $35. An FCC Report and Order released December 23, 2020, and in a subsequent notice issued by the FCC on March 23, 2022, the fee dropped from $70 to $35 starting on April 19, 2022. The license is still valid for 10 years and covers an entire family.
GMRS used to be regularly linked to otherrepeaters. The common practice for this linking was done via internet. This was commonly referred toGMRS LinkingusingNodesrunning a modifiedHamVoipversion ofAsterisk.
Around January 24, 2024 the FCC changed the wording on GMRs operations page that said " You cannot directly interconnect a GMRS station with the telephone network." to now include (among other things) "or any other network".[22]As of November 1, 2024 it now has this added to the operations tab: "In other words, repeaters may not be linked via the internet—an example of an “other network” in the rules—to extend the range of the communications across a large geographic area. Linking multiple repeaters to enable a repeater outside the communications range of the handheld or mobile device to retransmit messages violates sections 95.1733(a)(8) and 95.1749 of the Commission's rules, and potentially other rules in 47 C.F.R. Repeaters may be connected to the telephone network or other networks only for purposes of remote control of a GMRS station, not for carrying communication signals. "[23]Wording changes did not affect local repeaters. This change did shut down several major nodes.[24][25][26]This change was not an official act; it was instead done through interpretation set forth by theChevron Defense. The changes and shutdowns of nodes has drawn mixed reactions.[27][28]
|
https://en.wikipedia.org/wiki/General_Mobile_Radio_Service
|
In theUnited States, theMulti-Use Radio Service(MURS) is a licensed by rule (i.e. under part 95, subpart J, of title 47, Code of Federal Regulations[1])two-way radioservice similar to theCitizens band(CB). Established by theU.S.Federal Communications Commissionin the fall of 2000, MURS created a radio service allowing for licensed by rule (Part 95) operation in a narrow selection of theVHF band, with a power limit of 2watts. The FCC formally defines MURS as "a private, two-way, short-distance voice or data communications service for personal or business activities of the general public." MURS stations may not be connected to the public telephone network, may not be used forstore and forwardoperations, andradio repeatersare not permitted.
In 2009,Industry Canada(IC) established a five-year transition plan, which would have permitted the use of MURS in Canada starting June 2014.[2]In August 2014 IC announced a deferral of MURS introduction, as "the Department does not feel that the introduction of MURS devices in Canada is warranted at this time, and has decided to defer the introduction of MURS devices in Canada until a clearer indication of actual need is provided by Canadian MURS advocates and/or stakeholders ..."[3]
No licenses are required or issued for MURS within the United States.
MURS comprises the following fivefrequencies:
Channels 1–3 must use "narrowband"frequency modulation(2.5 kHzdeviation; 11.25 kHz bandwidth). Channels 4 and 5 may use either "wideband" FM (5 kHz deviation; 20 kHz bandwidth) or "narrowband" FM.[5]All five channels may useamplitude modulationwith a bandwidth up to 8 kHz.[6]MURS falls under part 95 and was not mandated for narrow-banding, such as those of Part 90 in the public service bands by January 2013.
Because previous business band licensees who have maintained their active license remaingrandfatheredwith their existing operating privileges, it is possible to find repeaters or other operations not authorized by Part 95 taking place. These are not necessarily illegal. If legal, such operations may enjoy primary status on their licensed frequency and as such are legally protected from harmful interference by MURS users.[7]
MURS range will vary, depending on antenna size and placement. With an external antenna, ranges of 10 miles (16 km) or more can be expected.[8]Since MURS radios use frequencies in the VHF business band, they are subject to obstructions in line of sight, which includes the curvature of the Earth. The higher you can place the antennas on both transmit and receive sides (within legal limits), the further you can transmit and receive. Some antenna manufacturers claim an external antenna can increase the effective radiated power of a transmitter by a factor of 4.[9]
MURS operation is authorized anywhere a CB radio station is authorized and within or over any area of the world where radio services are regulated by the FCC. Those areas are within the territorial limits of:
There are a wide variety of radio products that use MURS frequencies. MURS devices include wireless base station intercoms, handheld two-way radios, wireless dog training collars, wireless public address units, customer service callboxes, wireless remote switches, and wireless callboxes with or without gate opening ability.
Since MURS uses standard frequencies, most devices that use MURS are compatible with each other.
Most analog two-way radios utilize a technology calledCTCSSorDCSthat helps block out unwanted transmissions. To make MURS two-way radios work together, they must have matchingCTCSSorDCStones. This can usually be done via basic programming which almost all MURS two-way radios support.
ThegoTenna, a digital radio product, operates on the MURS band and pairs withsmartphonesto enable users to send texts and share locations on apeer-to-peerbasis. goTenna is not interoperable with other MURS devices, even though they operate on the same spectrum, employing "listen-before-talk" to reduce interference in the band's five channels.[11][12][13]
According toBill Fawcett's Spaniel Journal,Spanielpro-handler Dan Langhans was given a set of VHF business-band radios on the frequency of 154.57 MHz which became known by the trade as "blue dot" radios.[14]
Costco Wholesale use Motorola DTR600, DLR1020, and Motorola Curve on Frequency 1 for general use among employees and Frequency 2 for communication with major sales departments.WalmartandSam's Clubuse aMotorola Solutionsmodel Motorola RDM2070D, which is exclusive to Walmart and Sam's Club. The Motorola RDM2070D is preprogrammed on MURS frequencies with most channels usingCTCSS tone 21/4Z/136.5Hz.[15]
|
https://en.wikipedia.org/wiki/Multi-Use_Radio_Service
|
Intelecommunications, afemtocellis a small, low-powercellular base station, typically designed for use in a home or small business. A broader term which is more widespread in the industry issmall cell, withfemtocellas a subset. It typically connects to the service provider's network via the Internet through a wired broadband link (such asDSLorcable); current designs typically support four to eight simultaneously active mobile phones in a residential setting depending on version number and femtocell hardware, and eight to sixteen mobile phones in enterprise settings. A femtocell allows service providers to extend service coverage indoors or at the cell edge, especially where access would otherwise be limited or unavailable. Although much attention is focused onWCDMA, the concept is applicable to all standards, includingGSM,CDMA2000,TD-SCDMA,WiMAXandLTEsolutions.
The use of femtocells allows network coverage in places where the signal to the main network cells might be too weak. Furthermore, femtocells lower contention on the main network cells, by forming a connection from the end user, through an internet connection, to the operator's private network infrastructure elsewhere. The lowering of contention to the main cells plays a part inbreathing, where connections are offloaded based on physical distance to cell towers.
Consumers and small businesses benefit from greatly improved coverage and signal strength since they have ade factobase station inside their premises. As a result of being relatively close to the femtocell, the mobile phone (user equipment) expends significantly less power for communication with it, thus increasing battery life. They may also get better voice quality (viaHD voice) depending on a number of factors such as operator/network support, customer contract/price plan, phone and operating system support. Some carriers may also offer more attractive tariffs, for example discounted calls from home.
Femtocells are an alternative way to deliver the benefits offixed–mobile convergence(FMC). The distinction is that most FMC architectures require a new dual-mode handset which works with existing unlicensed spectrum home/enterprisewireless access points, while a femtocell-based deployment will work with existing handsets but requires the installation of a new access point that uses licensed spectrum.
Many operators worldwide offer a femtocell service, mainly targeted at businesses but also offered to individual customers (often for a one-off fee) when they complain to the operator regarding a poor or non-existent signal at their location. Operators who have launched a femtocell service includeSFR,AT&T,C Spire,Sprint Nextel,Verizon,Zain,Mobile TeleSystems,T-Mobile US,Orange,Vodafone,EE,O2,Three, and others.
In3GPPterminology, aHome NodeB(HNB) is a 3G femtocell. AHome eNodeB(HeNB) is anLTE 4Gfemtocell.
Theoretically the range of astandard base stationmay be up to 35 kilometres (22 mi), and in practice could be 5–10 km (3–6 mi), amicrocellis less than two kilometers wide, apicocellis 200 meters or less, and a femtocell is in the order of 10 meters,[1]although AT&T calls its product, with a range of 40 feet (12 m), a "microcell".[2]AT&T uses "AT&T 3G MicroCell" as a trademark and not necessarily the "microcell" technology, however.[3]
Femtocells are sold or loaned by amobile network operator(MNO) to its residential or enterprise customers. A femtocell is typically the size of aresidential gatewayor smaller, and connects to the user'sbroadbandline. Integrated femtocells (which include both a DSL router and femtocell) also exist. Once plugged in, the femtocell connects to the MNO's mobile network, and provides extra coverage. From a user's perspective, it isplug and play, there is no specific installation or technical knowledge required—anyone can install a femtocell at home.
In most cases,[4]the user must then declare which mobile phone numbers are allowed to connect to their femtocell, usually via a web interface provided by the MNO.[5]This needs to be done only once. When these mobile phones arrive under coverage of the femtocell, they switch over from themacrocell(outdoor) to the femtocell automatically. Most MNOs provide a way for the user to know this has happened, for example by having a different network name appear on the mobile phone. All communications will then automatically go through the femtocell. When the user leaves the femtocell coverage (whether in a call or not) area, their phone hands over seamlessly to the macro network. Femtocells require specific hardware, so existing WiFi or DSL routers cannot be upgraded to a femtocell.
Once installed in a specific location, most femtocells have protection mechanisms so that a location change will be reported to the MNO. Whether the MNO allows femtocells to operate in a different location depends on the MNO's policy. International location change of a femtocell is not permitted because the femtocell transmits licensed frequencies which belong to different network operators in different countries.
The main benefits for an end user are the following:
Femtocells can be used to give coverage in rural areas.
The standards bodies have published formal specifications for femtocells for the most popular technologies, namelyWCDMA,CDMA2000,LTEandWiMAX. These all broadly conform to an architecture with three major elements:
The key interface in these architectures is that between the femtocell access points and the femtocell gateway. Standardisation enables a wider choice of femtocell products to be used with any gateway, increasing competitive pressure and driving costs down. For the common WCDMA femtocells, this is defined as the Iuh interface. In the Iuh architecture, the femtocell gateway sits between the femtocell and the core network and performs the necessary translations to ensure the femtocells appear as a radio network controller to existingmobile switching centres(MSCs). Each femtocell talks to the femtocell gateway and femtocell gateways talk to the Core Network Elements (CNE) (MSC forcircuit-switchedcalls,SGSNforpacket-switchedcalls). This model was proposed by 3GPP and the Femto Forum.[8]New protocols (HNBAPand RUA[9][RANAP User Adaptation]) have been derived; HNBAP is used for the control signaling between the HNB and HNB-GW[10]while RUA[9]is a lightweight mechanism to replace theSignalling Connection Control Part(SCCP) andM3UAprotocols in theRadio Network Controller(RNC); its primary function is transparent transfer of RANAP messages.[11]
In March 2010, the Femto Forum and ETSI conducted the firstPlugfestto promote interoperability of the Iuh standard.[12]
The CDMA2000 standard released in March 2010[13]differs slightly by adopting theSession Initiation Protocol(SIP) to set up a connection between the femtocell and a femtocell convergence server (FCS). Voice calls are routed through the FCS which emulates an MSC. SIP is not required or used by the mobile device itself. In the SIP architecture, the femtocell connects to a core network of the mobile operator that is based on the SIP/IMS architecture. This is achieved by having the femtocells behave toward the SIP/IMS network like a SIP/IMS client by converting the circuit-switched 3G signaling to SIP/IMS signaling, and by transporting the voice traffic over RTP as defined in theIETFstandards.
Although much of the commercial focus seems to have been on theUniversal Mobile Telecommunications System(UMTS), the concept is equally applicable to all air-interfaces. Indeed, the first commercial deployment was theCDMA2000Airave in 2007 by Sprint.
Femtocells are also under development or commercially available forGSM,TD-SCDMA,WiMAXandLTE.
The H(e)NB functionality and interfaces are basically the same as for regularHigh Speed Packet Access(HSPA) or LTE base stations except few additional functions. The differences are mostly to support differences in access control to support closed access for residential deployment or open access for enterprise deployment, as well as handover functionality for active subscribers and cell selection procedures for idle subscribers. For LTE additional functionality was added in 3GPP Release 9 which is summarized in.[14]
The placement of a femtocell has a critical effect on the performance of the wider network, and this is the key issue to be addressed for successful deployment. Because femtocells can use the same frequency bands as the conventional cellular network, there has been the worry that rather than improving the situation they could potentially cause problems.
Femtocells incorporate interference mitigation techniques—detecting macrocells, adjusting power[15]and scrambling codes accordingly. Ralph de la Vega, AT&T President, reported in June 2011 they recommended against using femtocells where signal strength was middle or strong because of interference problems they discovered after widescale deployment.[16]This differs from previous opinions expressed by AT&T and others.
A good example is the comments made by Gordon Mansfield, executive director of RAN Delivery, AT&T, speaking at the Femtozone at CTIA March 2010:
We have deployed femtocells co-carrier with both the hopping channels for GSM macrocells and with UMTS macrocells. Interference isn't a problem. We have tested femtocells extensively in real customer deployments of many thousands of femtocells, and we find that the mitigation techniques implemented successfully minimise and avoid interference. The more femtocells you deploy, the more uplink interference is reduced.
The Femto Forum has some extensive reports on this subject, which have been produced together with 3GPP and 3GPP2.[17][18]
To quote from theSummary Paper — Summary of Findings:
The simulations performed in the Femto Forum WG2 and 3GPP RAN4 encompass a wide spectrum of
possible deployment scenarios including shared channel and dedicated channel deployments. In addition, the
studies looked at the impact in different morphologies, as well as in closed versus open access. The following
are broad conclusions from the studies:
The conclusions are common to the 850 MHz and 2100 MHz bands that were simulated in the studies, and can be
extrapolated to other mobile bands. With interference mitigation techniques successfully implemented, simulations show
that femtocell deployments can enable very high capacity networks by providing between a 10 and 100 times
increase in capacity with minimal deadzone impact and acceptable noise rise.
Femtocells can also create a much better user experience by enabling substantially higher data rates than can be obtained with a macro network and net throughputs that will be ultimately limited by backhaul in most cases (over 20 Mbps in 5 MHz).
Access point base stations, in common with all other public communications systems, are required to comply withlawful interceptionrequirements in most countries.
Other regulatory issues[19]relate to the requirement in most countries for the operator of a network to be able to show exactly where each base-station is located, and forE911requirements to provide the registered location of the equipment to the emergency services. There are issues in this regard for access point base stations sold to consumers for home installation, for example. Further, a consumer might try to carry their base station with them to a country where it is not licensed. Some manufacturers are usingGPSwithin the equipment to lock the femtocell when it is moved to a different country;[20]this approach is disputed[citation needed], as GPS is often unable to obtain position indoors because of weak signal.
Access Point Base Stations are also required, since carrying voice calls, to provide a911(or999,112,etc.) emergency service, as is the case forVoIPphone providers in some jurisdictions.[19]This service must meet the same requirements for availability as current wired telephone systems, such as functioning during a power failure. There are several ways to achieve this, such as alternative power sources or fallback to existing telephone infrastructure.
When using anEthernetorADSLhome backhaul connection, an Access Point Base Station must either share the backhaul bandwidth with other services, such as Internet browsing, gaming consoles,set-top boxesandtriple-playequipment in general, or alternatively directly replace these functions within an integrated unit. In shared-bandwidth approaches, which are the majority of designs currently being developed, the effect onquality of servicemay be an issue.
The uptake of femtocell services will depend on the reliability and quality of both the cellular operator's network and the third-party broadband connection, and the broadband connection's subscriber understanding the concept of bandwidth utilization by different applications a subscriber may use. When things go wrong, subscribers will turn to cellular operators for support even if the root cause of the problem lies with the broadband connection to the home or workplace. Hence, the effects of any third-party ISP broadband network issues or traffic management policies need to be very closely monitored and the ramifications quickly communicated to subscribers.
A key issue recently identified is activetraffic shapingby many ISPs on the underlying transport protocolIPSec.[citation needed]
To meetFederal Communications Commission(FCC) /Ofcomspectrum maskrequirements, femtocells must generate theradio frequencysignal with a high degree of precision. To do this over a long period of time is a major technical challenge. The solution to this problem is to use an external, accurate signal to constantly calibrate the oscillator to ensure it maintains its accuracy. This is not simple (broadband backhaul introduces issues of network jitter/wander and recovered clock accuracy), but technologies such as theIEEE 1588time synchronisation standard may address the issue. Also,Network Time Protocol(NTP) is being pursued by some developers as a possible solution to provide frequency stability. Conventional (macrocell) base stations often use GPS timing for synchronization and this could be used,[20]although there are concerns on cost and the difficulty of ensuring good GPS coverage.
Standards bodies have recognized the challenge of this and the implications on device cost. For example, 3GPP has relaxed the 50ppbparts per billionprecision to 100ppb for indoor base stations in Release 6 and a further loosening to 250ppb for Home Node B in Release 8.
At the 2013 Black Hat hacker conference in Las Vegas, NV, a trio of security researchers detailed their ability to use a Verizon femtocell to secretly intercept the voice calls, data, and SMS text messages of any handset that connects to the device.
During a demonstration of their exploit, they showed how they could begin recording audio from a cell phone even before the call began. The recording included both sides of the conversation. They also demonstrated how it could trick Apple's iMessage–which encrypts texts sent over its network using SSL to render them unreadable to snoopers, to SMS—allowing the femtocell to intercept the messages.
They also demonstrated it was possible to "clone" a cell phone that runs on a CDMA network by remotely collecting its device ID number through the femtocell, in spite of added security measures to prevent against cloning of CDMA phones.[21]
The impact of a femtocell is most often to improve cellular coverage, without the cellular carrier needing to improve their infrastructure (cell towers, etc.). This is net gain for the cellular carrier. However, the user must provide and pay for an internet connection to route the femtocell traffic, and then (usually) pay an additional one-off or monthly fee to the cellular carrier. Some have objected to the idea that consumers are being asked to pay to help relieve network shortcomings.[22]On the other hand, residential femtocells normally provide a ‘personal cell’ which provides benefits only to the owner's family and friends.[23]
The difference is also that while mobile coverage is provided through subscriptions from an operator with one business model, a fixed fibre or cable may work with a completely different business model. For example, mobile operators may imply restrictions on services which an operator on a fixed may not. Also, WiFi connects to a local network such as home servers and media players. This network should possibly not be within reach of the mobile operator.
According to market research firm Informa and the Femto Forum,[24]as of December 2010 18 operators have launched commercial femtocell services, with a total of 30 committed to deployment.
At the end of 2011, femtocell shipments had reached roughly 2 million units deployed annually, and the market is expected to grow rapidly with distinct segments for consumer, enterprise, and carrier-grade femtocell deployments.[25]Femtocell shipments are estimated to have reached almost 2 million at the end of 2010.[26]Research firm Berg Insight estimates that the shipments will grow to 12 million units worldwide in 2014.[27]
Within the United States, Cellcom (Wisconsin), was the first CDMA carrier in the U.S. to be a member of the non-profit organization founded in 2007 to promote worldwide femtocell deployment. In 2009, Cellcom received the first Femtocell Industry Award for significant progress or commercial launch by a small carrier at the Femtocells World Summit in London. Additional significant deployments within the United States were bySprint Nextel,Verizon WirelessandAT&T Wireless. Sprint started in the third quarter of 2007 as a limited rollout (DenverandIndianapolis) of a home-based femtocell built by Samsung Electronics called the Sprint Airave that works with any Sprint handset.[28]From 17 August 2008, the Airave was rolled out on a nationwide basis. Other operators in the United States have followed suit. In January 2009, Verizon rolled out its Wireless Network Extender, based on the same design as the Sprint/Samsung system.[29]In late March 2010, AT&T announced nationwide roll-out of its 3G MicroCell, which commenced in April. The equipment is made byCisco Systemsandip.access, and was the first 3G femtocell in US, supporting both voice and dataHSPA.[30]Both Sprint[31]and Verizon[32]upgraded to 3G CDMA femtocells during 2010, with capacity for more concurrent calls and much higher data rates. In November 2015, T-Mobile US began deployment of 4G LTE femtocells manufactured byAlcatel Lucent.
In Asia, several service providers have rolled out femtocell networks. In Japan,SoftBanklaunched its residential 3G femtocell service in January 2009[33]with devices provided by Ubiquisys. In the same year, the operator launched a project to deploy femtocells to deliver outdoor services in rural environments where existing coverage is limited. In May 2010, SoftBank Mobile launched the first free femtocell offer, providing open access femtocells free of charge to its residential and business customers. In Singapore, Starhub rolled out its first nationwide commercial 3G femtocell services with devices provided byHuawei Technologies, though the uptake is low, while Singtel's offering is targeted at small medium enterprises. In 2009, China Unicom announced its own femtocell network.[34]NTT DoCoMo in Japan launched their own femtocell service on 10 November 2009.
In July 2009, Vodafone released the first femtocell network in Europe,[35]the Vodafone Access Gateway provided by Alcatel-Lucent.[36]This was rebranded as SureSignal in January 2010,[37]after which Vodafone also launched service in Spain, Greece, New Zealand,[38][39]Italy, Ireland,[40]Hungary[41]and The Netherlands.[42]Other operators in Europe have followed since then.
From 2019 onwards, all 3 French carriers still proposing Femtocell retired their offering, focusing instead on using theVoice Over Wifitechnology when a better 3G/4G covering is impractical to deploy.
Ref. 8 can be found at :https://www.etsi.org/deliver/etsi_ts/125400_125499/125467/08.02.00_60/ts_125467v080200p.pdfa new release exists :https://www.etsi.org/deliver/etsi_ts/125400_125499/125467/10.06.00_60/ts_125467v100600p.pdf
|
https://en.wikipedia.org/wiki/Femtocell
|
Small cellsare low-powered cellularradio access nodesthat have a ranges of around 10 meters to a few kilometers. They are base stations with low power consumption and cost. They can provide high data rates by being deployed densely to achieve high spatial spectrum efficiency.[1]
In the United States, recent FCC orders have provided size and elevation guidelines to help more clearly define small cell equipment.[2][3]They are "small" compared to a mobilemacrocell, partly because they have a shorter range and partly because they typically handle fewer concurrent calls or sessions. As wireless carriers seek to 'densify' existing wireless networks to provide for the data capacity demands of 5G, small cells are currently viewed as a solution to allow re-using the same frequencies,[4][5][6]and as an important method of increasing cellular network capacity, quality, and resilience with a growing focus usingLTE Advanced.
Small cells may encompassfemtocells,picocells, andmicrocells. Small-cell networks can also be realized by means of distributed radio technology using centralized baseband units andremote radio heads.Beamformingtechnology (focusing a radio signal on a very specific area) can further enhance or focus small cell coverage. These approaches to small cells all feature central management bymobile network operators.
Small cells provide a small radio footprint, which can range from 10 meters within urban and in-building locations to 2 km for a rural location. Picocells and microcells can also have a range of a few hundred meters to a few kilometers, but they differ from femtocells in that they do not always haveself-organisingand self-management capabilities.
Small cells are available for a wide range of air interfaces includingGSM,CDMA2000,TD-SCDMA,W-CDMA,LTEand5G. In3GPPterminology, aHome Node B(HNB) is a 3G femtocell. A Home eNode B (HeNB) is an LTE femtocell.Wi-Fiis a small cell but does not operate inlicensed spectrumand therefore cannot be managed as effectively as small cells utilising licensed spectrum. Small cell deployments vary according to the use case and radio technology employed.
The most common form of small cells are femtocells. They were initially designed for residential and small business use, with a short range and a limited number of channels. Femtocells with increased range and capacity spawned a proliferation of terms: metrocells, metro femtocells, public access femtocells, enterprise femtocells, super femtos, Class 3 femto, greater femtos and microcells. The term "small cells" is frequently used by analysts and the industry as an umbrella to describe the different implementations of femtocells, and to clear up any confusion that femtocells are limited to residential uses. Small cells are sometimes, incorrectly, also used to describedistributed-antenna systems(DAS) which are not low-powered access nodes.
Small cells can be used to provide in-building and outdoor wireless service. Mobile operators use them to extend their service coverage and/or increasenetwork capacity.
ABI Research argues that small cells also help service providers discover new revenue opportunities through their location andpresence information. If a registered user enters a femtozone, the network is notified of their location. The service provider, with the user's permission, could share this location information to update user's social media status, for instance. Opening up small-cellAPIsto the wider mobile ecosystem could enable along-taileffect.
Rural coverage is also a key market that has developed as mobile operators have started to install public access metrocells in remote and rural areas that either have only 2G coverage or no coverage at all. The cost advantages of small cells compared with macro cells make it economically feasible to provide coverage of much smaller communities – from a few ten to a few hundred. The Small Cell Forum have published a white paper outlining the technology and business case aspects.[7]Mobile operators in both developing and developed countries are either trialing or installing such systems. The pioneer in providing rural coverage using small cells wasSoftBank Mobile– the Japanese mobile operator – who have installed more than 3000 public access 3G small cells on post offices throughout rural Japan. In the UK, Vodafone's Rural Open Sure Signal program and EE's rural 3G/4G scheme increase geographic coverage.
Small cells are an integral part of LTE networks. In 3G networks, small cells are viewed as an offload technique.[8]In 4G networks, the principle of heterogeneous network (HetNet) is introduced where the mobile network is constructed with layers of small and large cells.[9]In LTE, all cells will be self-organizing, drawing upon the principles laid down in current Home NodeB (HNB), the 3GPP term for residential femtocells.
Future innovations in radio access design introduce the idea of an almost flat architecture where the difference between a small cell and a macrocell depends on how many cubes are stacked together.
The transmitting signal from Macro Base Station (MBS) weakens quickly once the MBS signal reaches indoors. Femtocells provide a solution to the difficulties present in macrocell-based system. Thus, Femto Base Station (FBS) network coverage is one of the prime concerns in indoor environment to get good quality of service (QoS).[10]
By December 2017 a total of over 12 million small cells have been deployed worldwide, with forecasts as high as 70 million by 2025.[11]
Backhaulis needed to connect the small cells to the core network, internet and other services. For in-building use, existing broadband internet can be used. In urban outdoors, mobile operators consider this more challenging than macrocell backhaul because a) small cells are typically in hard-to-reach, near-street-level locations rather than in more open, above-rooftop locations and b)carrier gradeconnectivity must be provided at much lower cost per bit. Many different wireless and wired technologies have been proposed as solutions, and it is agreed that a toolbox of these will be needed to address a range of deployment scenarios. An industry consensus view of how the different solution characteristics match with requirements is published by the Small Cell Forum.[12]The backhaul solution is influenced by a number of factors, including the operator's original motivation to deploy small cells, which could be for targeted capacity, indoor or outdoor coverage.[13]
|
https://en.wikipedia.org/wiki/Small_Cells
|
Amodular connectoris a type ofelectrical connectorfor cords and cables of electronic devices and appliances, such as incomputer networking, telecommunication equipment, and audio headsets.
Modular connectors were originally developed for use on specificBell Systemtelephone sets in the 1960s, and similar types found use for simple interconnection of customer-provided telephone subscriber premises equipment to the telephone network. TheFederal Communications Commission(FCC) mandated in 1976 an interface registration system, in which they became known asregistered jacks. The convenience of prior existence for designers and ease of use led to a proliferation of modular connectors for many other applications. Many applications that originally used bulkier, more expensive connectors have converted to modular connectors. Probably the best-known applications of modular connectors are fortelephoneandEthernet.
Accordingly, various electronic interface specifications exist for applications using modular connectors, which prescribe physical characteristics and assign electrical signals to their contacts.
Modular connectors are often referred to asmodular phone jack and plug,RJ connector, andWestern jack and plug. The termmodular connectorarose from its original use in modular wiring components of telephone equipment by theWestern Electric Companyin the 1960s.[1]This includes the6P2Cused fortelephone line connectionsand4P4Cused forhandsetconnectors.
Registered jackdesignations describe the signals and wiring used for voice and data communication at customer-facing interfaces of thepublic switched telephone network(PSTN). It is common to use a registered jack number to refer to the physical connector itself; for instance, the regular 8P8C modular connector type is often labeledRJ45because the registered jack standard of the similar nameRJ45Sspecified a similar, but modified, 8P8C modular connector. Similarly, various six-position modular connectors may be calledRJ11. Likewise, the 4P4C connector is sometimes calledRJ9orRJ22though no such official designations exist.[citation needed]
The first types of small modular telephone connectors were created by AT&T in the mid-1960s for the plug-in handset and line cords of theTrimlinetelephone.[1]Driven by demand for multiple sets in residences with various lengths of cords, the Bell System introduced customer-connectable part kits and telephones, sold throughPhoneCenterstores in the early 1970s.[2]For this purpose, Illinois Bell started installing modular telephone sets on a limited scale in June 1972. The patents by Edwin C. Hardesty and coworkers,US 3699498(1972) andUS 3860316(1975), followed by other improvements, were the basis for the modular molded-plastic connectors that became commonplace for telephone cords by the 1980s. In 1976, these connectors were standardized nationally in the United States by the Registration Interface program of theFederal Communications Commission(FCC), which designated a series of Registered Jack (RJ) specifications for interconnection of customer-premises equipment to the PSTN.[3][4]
Modular connectors havegender:plugsare considered to bemale, whilejacksorsocketsare considered to befemale. Plugs are used to terminate cables and cords, while jacks are used for fixed locations on surfaces of walls, panels, and equipment. Other than telephone extension cables, cables with a modular plug on one end and a jack on the other are rare. Instead, cables are usually connected using a female-to-female coupler, having two jacks wired back-to-back.
Most modular connectors are designed with a latching mechanism that secures the physical connection. As a plug is inserted into a jack, a plastic tab on the plug locks against a ridge in the socket so that the plug cannot be removed without disengaging the tab by pressing it against the plug body. The standard orientation for installing a jack in a vertical surface is with the tab down.
The modular plug is often installed with aboot, a plastic covering over the tab and body, to prevent the latching tab from hooking into other cords or edges, which may cause excessive bending or breaking of the tab. Suchsnaglesscords are usually constructed by installing the protective boot before the modular plug is crimped.
Modular connectors are designated using two numbers that represent the maximum number of contact positions and the number of installed contacts, with each number followed byPandC, respectively. For example,6P2Cis a connector having six positions and two installed contacts. Alternate designations omit the letters while separating the position and contact quantities with either anx(6x2) or a slash (6/2).
When not installed, contacts are usually omitted from the outer positions inward, such that the number of contacts is almost always even. The connector body positions with omitted or unconnected contacts are unused for the electrical connection but ensure that the plug fits correctly. For instance, inexpensive telephone cords often have connectors with six positions and four contacts, to which are attached just two wires, carrying only line 1 from a one-, two-, or three-linejack.
The contact positions are numbered sequentially starting from 1. When viewed head-on with the retention mechanism on the bottom, jacks will have contact position number 1 on the left and plugs will have it on the right. Contacts are numbered by the contact position. For example, on a six-position, two-contact plug, where the outermost four positions do not have contacts, the two contacts are numbered 3 and 4.
Modular connectors are manufactured in four sizes, with four, six, eight, and ten positions. The insulating plastic bodies of 4P and 6P connectors have different widths, whereas 8P or 10P connectors share an even larger body width.
Internally, the contacts in the plugs have sharp prongs that, whencrimped, displace the wire insulation and connect with the conductors inside—a mechanism known asinsulation displacement. Cables have either solid or stranded (tinsel wire) conductors, and a given plug is designed for only one type. The sharp prongs are different in the connectors made for each type of wire, and a mismatch between plug type and wire type results in unreliable connections. A modular plug for solid (single-strand) wire often has three slightly splayed prongs on each contact to securely surround and grip the conductor while scraping along the outside, and a plug for stranded wire has prongs that are designed to pierce the insulation and go straight through to contact multiple wire strands.
Some modular connectors areindexed, meaning their dimensions are intentionally non-standard, preventing connections with connectors of standard dimensions. The means of indexing may be non-standard cross-sectional dimensions or shapes, retention mechanism dimensions or configuration. For example, aModified Modular Jackusing an offset latching tab was developed byDigital Equipment Corporationto prevent accidental interchange of data and telephone cables.
The dimensions of modular connectors are such that a narrower plug can be inserted into a wider jack that has more positions than the plug, leaving the jack's outermost contacts unconnected. The height of the plug's insertion area is 0.260 inches (6.60 mm) and the contacts are 0.040 inches (1.02 mm) apart (contact pitch), so the width is dependent on the number of pin positions.[7][8]However, not all plugs from all manufacturers have this capability, and some jack manufacturers warn that their jacks are not designed to accept smaller plugs without damage. If an inserted plug lacks slots to accommodate the jack's contacts at the outermost extremes, it may permanently deform the outermost contacts of an incompatible jack. Excessive resistance may be encountered when inserting an incompatible plug, as the outermost contacts in the jack are forcibly deformed.
Special modular plugs have been manufactured (for example, theSiemonUP-2468[9]) which have extra slots beyond their standard contacts, to accommodate the wider jack's outermost contacts without damage. These special plug connectors can be visually identified by carefully looking for the extra slots molded into the plug. The molded plastic bodies of the special plugs may also be colored with a light blueish tinge to aid in quick recognition. The special plugs are preferred for test equipment and adapters, which may be rapidly connected to a large number of corresponding connectors in quick succession for testing purposes. The use of the special plugs avoids inadvertent damage to the equipment under test, even when a narrower plug is inserted into a nominally incompatible wider jack.
Termination of cables with modular connectors is similar across the various number of positions and contacts in the plug. The crimping tool contains a die that is often exchangeable and is closely matched to the shape and pin count of the modular plug.
A crimping die-set looks similar to an 8P8C jack, except for the eight teeth lining the top portion of the die. When the tool is operated, the die compresses around the 8P8C plug. As the die compresses, these teeth force the plug contacts into the conductors of the cable being terminated. The crimper may also permanently deform part of the plastic plug body in such a way that it grips the outer sheath of the cable for secure fastening andstrain relief. These actions permanently attach the plug to the cable.
The contact assignments (pinout) of modular connectors vary by application. Telephone network connections are standardized by registered jack designations, andEthernet over twisted pairis specified by theANSI/TIA-568standard. For other applications, standardization may be lacking; for example, multiple conventions exist for the use of 8P8C connectors inRS-232applications. For this reason,D-sub-to-modular adapters are typically shipped with the D-sub contacts (pins or sockets) terminated but not inserted into the connector body so that the D-sub-to-modular contact pairing can be assigned as needed.
The four-position four-conductor (4P4C) connector is the standard modular connector used on both ends of telephonehandsetcords and is therefore often called ahandset connector.[10]
This handset connector is not a registered jack, because it was not intended to connect directly to telephone lines. However, it is often referred to asRJ9,RJ10, orRJ22.
Handsets and oftenheadsetsfor use with telephones commonly use a 4P4C connector. The two center pins are commonly used for the receiver, and the outer pins connect the transmitter so that a reversal of conductors between the ends of a cord does not affect the signal routing. This may differ for other equipment, including hands-free headsets.
TheMacintosh 128K,Macintosh 512KandMacintosh PlusfromAppleas well as theAmiga 1000fromCommodoreuse 4P4C connectors to connect the keyboard to the main computer housing. The connector provides power to the keyboard on the outer two contacts and receives data signals on the inner pair. The cable between the computer and the keyboard is a coiled cord with an appearance very similar to a telephone handset cable.[11]The connector on the Amiga 1000 uses crossover wiring, similar to a telephone handset. The connector wiring on the Apple computers, however, requires a polarized straight-through pinout. Using a telephone handset cable instead of the supplied cable could short out the +5 volt DC supply and damage the Apple computer or the keyboard.[12]
Modular connectors are often used for data links, such as serial line connections, because of their compact dimensions. For example, someDirecTVset-top boxes include a 4P4C data port with an adapter cord to connect to a computer serial port to control the set-top box.[13]
Modular plugs are described by the maximum number of physicalcontactpositions and the actual number of contacts installed in these positions. The6P2C,6P4C, and6P6Cmodular connectors are probably best known for their use asRJ11,RJ14, andRJ25non-powered registered jacks, respectively (and 6P4C and 6P6C for powered RJ11 and RJ14, power being delivered on the outer pairs). These interfaces use the same six-position modular connector body but have different numbers of pins installed.
RJ11 is a jack, a physical interface, by definition used for terminating a single telephone line. RJ14 is similar, but for two lines, and RJ25 is for three lines.RJ61is a similar registered jack for four lines, but uses 8P8C connectors.
Cables sold asRJ11(the name of a single-jack, not a cable) often actually use 6P4C connectors (six positions, four contacts). Two of its six possible contact positions connecttip and ringof a single telephone line, and the other two contact positions may be unused, carry a second line, or provide low-voltage power for night light or other features on the telephone set. In some installations, an extra contact was also required for thegroundconnection forselective ringers.
The pins of the 6P6C connector are numbered 1 to 6, counting left to right when holding the connector tab side down with the opening for the cable facing the viewer.
However, with German domestic telephone equipment, and that in some neighboring countries,6P4Cplugs and sockets are typically only used to connect the telephone cord to the phone base unit, whereas the mechanically differentTAE connectoris used at the other end to connect to a service provider interface. Older base units may accommodate the additional connectors of TAE (E, W, a2, b2) and may feature non-RJ standard sockets that can be connected directly to TAE plugs. Further, flat DIN 47100 cables typically place the wires in ascending order. When used directly with 6P4C plugs, the color coding may be undetermined.
In the powered version of the RJ11 interface, pins 2 and 5 (black and yellow) may carry low-voltage AC or DC power. While the telephone line on pins 3 and 4 (red and green) supplies enough power for most telephone terminals, old telephone terminals with incandescent lights, such as the Western ElectricPrincessandTrimline telephones, need more power than the phone line can supply. Typically, the power on pins 2 and 5 is supplied by anAC adapterplugged into a nearby power outlet which potentially even supplies power to all of the jacks in the house.
Structured cablingnetworks adhering toANSI/TIA-568,ISO/IEC 11801(or ISO/IEC 15018 for home networks) are widely used for both computer networking and analog telephony. These standards specify the T568A or T568B wiring arrangements compatible withEthernet. The 8P8C jack used by structured cabling physically accepts the 6-position connector that fits RJ11, RJ14 and RJ25. Only lines 1 and 2 have electrical compatibility, with T568A wiring, and only line 1 with T568B wiring, because Ethernet-compatible pin assignments split the third pair of RJ25 across two separate cable pairs, rendering that pair unusable by an analog phone. (With T568B wiring, a telephone may connect to line 3 asline 2.) Both the third and fourth pairs of RJ61 are similarly split. The incompatible T568A and T568B layouts were necessary to preserve the electrical properties of the third and fourth pairs for Ethernet, which operates at much higher frequencies than analog telephony. Because of these incompatibilities, and because RJ25 and RJ61 were never very common, the T568A and T568B conventions have largely displaced RJ25 and RJ61 for telephones with more than two lines.
The8 position 8 contact(8P8C) connector is a modular connector commonly used to terminatetwisted pairand multi-conductorflat cable. These connectors are commonly used forEthernet over twisted pair, registered jacks and other telephone applications,RS-232serial communication using theANSI/TIA-568(formerly TIA/EIA-568) andYoststandards, and other applications involvingunshielded twisted pair,shielded twisted pair, and multi-conductor flat cable.
An 8P8C modular connection consists of a male plug and a female jack, each with eight equally spaced contacts. On the plug, the contacts are flat metal bars positioned parallel to the connector body. Inside the jack, the contacts are metal spring wires angled away from the insertion interface. When the plug is mated with the jack, the contacts meet and create an electrical connection. The spring force of the jack contacts ensures a good interface.
Although commonly referred to asRJ45in the context of Ethernet andstructured cabling,RJ45originally referred to a specific wiring configuration of an 8P8C female connector.[14][15][16]The original telephone-system-standard RJ45 plug has a key that excludes insertion in an un-keyed 8P8C socket.[17]
The original RJ45S[a]was intended for high-speed modems and is obsolete. The RJ45S jack mates with a keyed 8P2C modular plug,[18][19]and has pins 4 and 5 (the middle positions) wired for the ring and tip conductors of a single telephone line and pins 7 and 8 shorting a programming resistor. This is a different mechanical interface and wiring scheme than ANSI/TIA-568 T568A and T568B schemes with the 8P8C connector in Ethernet and telephone applications. Generic 8P8C modular connectors are similar to those used for the RJ45S variant, although the RJ45S plug is keyed and not compatible with non-keyed 8P8C modular jacks.
Telephone installers who wired RJ45S modem jacks or RJ61X telephone jacks were familiar with the pin assignments of the standard. However, the standard unkeyed modular connectors became ubiquitous for computer networking and informally inherited the nameRJ45.
The shape and dimensions of an 8P8C modular connector are specified for US telephone applications by the Administrative Council for Terminal Attachment (ACTA) in national standardANSI/TIA-1096-A and international standard ISO-8877. This standard does not use the short term 8P8C and covers more than just 8P8C modular connectors, but the 8P8C modular connector type is the eight-position connector type described therein, with eight contacts installed.
Fordata communicationapplications (LAN,structured cabling), International StandardIEC60603 specifies in parts 7-1, 7-2, 7-4, 7-5, and 7-7 not only the same physical dimensions but also high-frequency performance requirements for shielded and unshielded versions of this connector for carrying frequencies up to 100, 250 and 600MHz.
8P8C connectors are frequently terminated using the T568A or T568B assignments that are defined in ANSI/TIA-568. The drawings to the right show that the copper connections and pairing are the same, the only difference is that the orange and green pairs (colors) are swapped. A cable wired as T568A at one end and wired as T568B at the other end (Tx and Rx pairs reversed) is anEthernet crossover cable. Before the widespread acceptance ofauto MDI-Xcapabilities, a crossover cable was needed to interconnect similar network equipment (such asEthernet hubsto Ethernet hubs). Crossover cables are sometimes still used to connect two computers together without a switch or hub, however, most network interface cards (NIC) in use today implement auto MDI-X to automatically configure themselves based on the type of cable plugged into them. A cable wired the same at both ends is called apatchorstraight-throughcable, because no pin/pair assignments are swapped. If apatchorstraightcable is used to connect two computers with auto MDI-X capable NICs, one NIC will configure itself to swap the functions of its Tx and Rx wire pairs.
Pin numbering on plug face. Connected pins on plug and jack have the same number.
Two types of 8P8C plugs andcrimpingtools for installing the plug onto a cable are commonly available: Western Electric/Stewart Stamping (WE/SS) and Tyco/AMP. While the two types are similar, the tooling and plug types cannot be interchanged.[b]WE/SS compatible plugs are available from a large number of manufacturers, whereas Tyco/AMP plugs are produced exclusively byTyco Electronics.[citation needed]Both types of modular plugs can be mated with a standard 8P8C modular jack.
Both types of 8P8C plugs are available in shielded and unshielded varieties for differentattenuationtolerances as needed. Shielded plugs are more expensive and require shielded cable, but have a lower attenuation, and may reduceelectromagnetic interference.
Although a narrower 4-pin and 6-pin plug fits into the wider 8-pin jack and makes a connection with the available contacts on the plug, because the body of the smaller connector may stress the remaining contacts,[c]the smaller connector can potentially damage the springs of the larger jack.
8P8C connectors are commonly used in computer networking applications, where interconnecting cables are terminated at each end with an 8P8C modular plug wired according to TIA/EIA standards. Most wired Ethernet communications are carried over Category 5e or Category 6 cable terminated with 8P8C modular plugs. The connector is also used in other telecommunications connections, including ISDN andT1.
Where building network and telephone wiring is pre-installed, the center (blue) pair is often used to carrytelephonysignals. While this allows an RJ11 plug to connect, it may damage the modular jack; an approved converter prevents damage. In landline telephony, an 8P8C jack is used at the point a line enters the building to allow the line to be broken to insert automatic dialing equipment, includingintrusion alarmpanels.
TheEIA/TIA-561 standard describes the use of 8P8C connectors for RS-232 serial interfaces.[23]This application is common as a console interface fornetwork equipment, such asswitches,routers, andheadless computers.
8P8C modular connectors are also commonly used as a microphone connector forPMR,LMR, andamateur radiotransceivers. Frequently the pinout is different, usually mirrored (i.e. what would be pins 1 to 8 in the ANSI/TIA-568 standard might be pins 8 to 1 in the radio and its manual).
In analog mobile telephony, the 8P8C connector was used to connect anAMPScellular handset to its (separate) base unit; this usage is now obsolete.
The physical connector is standardized as the IEC 60603-7 8P8C modular connector with differentcategoriesof performance. The physical dimensions of the male and female connectors are specified in ANSI/TIA-1096-A and ISO-8877 standards and normally wired to the T568A and T568B pinouts specified in the ANSI/TIA-568 standard to be compatible with both telephone and Ethernet.
A similar standard jack once used for modem and data connections, the RJ45S, used akeyedvariety of the 8P8C body with an extra tab that prevents it from mating with other connectors; the visual difference compared to the more common 8P8C is subtle, but it is a different connector. The original RJ45S[18][24]keyed 8P2C modular connector, obsolete today, had pins 5 and 4 wired for tip and ring of a single telephone line and pins 7 and 8 shorting a programming resistor.
Electronics catalogs commonly advertise 8P8C modular connectors asRJ45. An installer can wire the jack to any pin-out or use it as part of a genericstructured cablingsystem such as ISO/IEC 15018 orISO/IEC 11801using 8P8C patch panels for both phone and data.
A router-to-routercrossover cableuses two 8-position connectors and aunshielded twisted pair(UTP) cable with differently wired connectors at each end.
The10P10Cconnector is commonly referred to as anRJ50connector,[25]although this was never a standard registered jack. The 10P10C has 10 contact positions and 10 contacts.
The most common uses of the 10P10C connector are in proprietary data transfer systems.[26]
|
https://en.wikipedia.org/wiki/8P8C
|
Inradio-frequency engineering, anantenna(American English) oraerial(British English) is an electronic device that converts analternating electric currentintoradio waves(transmitting), or radio waves into an electric current (receiving).[1][2]It is the interface between radio wavespropagatingthrough space and electric currents moving in metalconductors, used with atransmitterorreceiver.[1]Intransmission, a radio transmitter supplies an electric current to the antenna'sterminals, and the antenna radiates the energy from the current aselectromagnetic waves(radio waves). Inreception, an antenna intercepts some of the power of a radio wave in order to produce an electric current at its terminals, that is applied to a receiver to beamplified. Antennas are essential components of allradioequipment.[3]
An antenna is an array ofconductor segments(elements), electrically connected to the receiver or transmitter. Antennas can be designed to transmit and receive radio waves in all horizontal directions equally (omnidirectional antennas), or preferentially in a particular direction (directional, or high-gain, or "beam" antennas). An antenna may include components not connected to the transmitter,parabolic reflectors,horns, orparasitic elements, which serve to direct the radio waves into a beam or other desiredradiation pattern. Strongdirectivityand good efficiency when transmitting are hard to achieve with antennas with dimensions that are much smaller than a halfwavelength.
The first antennas were built in 1886 by German physicistHeinrich Hertzin his pioneering experiments to prove the existence of electromagnetic waves predicted by the 1867 electromagnetic theory ofJames Clerk Maxwell. Hertz placeddipole antennasat the focal point ofparabolic reflectorsfor both transmitting and receiving.[4]Starting in 1895,Guglielmo Marconibegan development of antennas practical for long-distancewireless telegraphyand opened a factory inChelmsford, England, to manufacture his invention in 1898.[5]
The wordsantennaandaerialare used interchangeably. Occasionally the equivalent term "aerial" is used to specifically mean an elevated horizontal wire antenna. The origin of the wordantennarelative to wireless apparatus is attributed to Italian radio pioneerGuglielmo Marconi. In the summer of 1895, Marconi began testing his wireless system outdoors on his father's estate nearBolognaand soon began to experiment with long wire "aerials" suspended from a pole.[6]InItaliana tent pole is known asl'antenna centrale, and the pole with the wire was simply calledl'antenna. Until then wireless radiating transmitting and receiving elements were known simply as "terminals". Because of his prominence, Marconi's use of the wordantennaspread among wireless researchers and enthusiasts, and later to the general public.[7][8][9]
Antennamay refer broadly to an entire assembly including support structure, enclosure (if any), etc., in addition to the actualRFcurrent-carrying components. A receiving antenna may include not only the passive metal receiving elements, but also an integrated preamplifier ormixer, especially at and abovemicrowavefrequencies.
Antennas are required by any radio receiver or transmitter to couple its electrical connection to the electromagnetic field.[11]Radiowaves areelectromagnetic waveswhich carry signals through space at thespeed of lightwith almost notransmission loss.
Antennas can be classified asomnidirectional, radiating energy approximately equally in all horizontal directions, ordirectional, where radio waves are concentrated in some direction(s). A so-calledbeam antennais unidirectional, designed for maximum response in the direction of the other station, whereas many other antennas are intended to accommodate stations in various directions but are not truly omnidirectional. Since antennas obeyreciprocitythe sameradiation patternapplies to transmission as well as reception of radio waves. A hypothetical antenna that radiates equally in all directions (vertical as well as all horizontal angles) is called anisotropic radiator; however, these cannot exist in practice nor would they be particularly desired. For most terrestrial communications, rather, there is an advantage inreducingradiation toward the sky or ground in favor of horizontal direction(s).
Adipole antennaoriented horizontally sends no energy in the direction of the conductor – this is called the antenna null – but is usable in most other directions. A number of such dipole elements can be combined into anantenna arraysuch as theYagi–Udain order to favor a single horizontal direction, thus termed a beam antenna.
The dipole antenna, which is the basis for most antenna designs, is abalancedcomponent, with equal but opposite voltages and currents applied at its two terminals. The vertical antenna is amonopoleantenna, not balanced with respect to ground. The ground (or any large conductive surface) plays the role of the second conductor of a monopole. Since monopole antennas rely on a conductive surface, they may be mounted with aground planeto approximate the effect of being mounted on the Earth's surface.
More complex antennas increase the directivity of the antenna. Additional elements in the antenna structure, which need not be directly connected to the receiver or transmitter, increase its directionality. Antenna "gain" describes the concentration of radiated power into a particular solid angle of space. "Gain" is perhaps an unfortunately chosen term, by comparison with amplifier "gain" which implies a net increase in power. In contrast, for antenna "gain", the power increased in the desired direction is at the expense of power reduced in undesired directions. Unlike amplifiers, antennas are electrically "passive" devices which conserve total power, and there is no increase in total power above that delivered from the power source (the transmitter), only improved distribution of that fixed total.
Aphased arrayconsists of two or more simple antennas which are connected together through an electrical network. This often involves a number of parallel dipole antennas with a certain spacing. Depending on the relativephaseintroduced by the network, the same combination of dipole antennas can operate as a "broadside array" (directional normal to a line connecting the elements) or as an "end-fire array" (directional along the line connecting the elements). Antenna arrays may employ any basic (omnidirectional or weakly directional) antenna type, such as dipole, loop or slot antennas. These elements are often identical.
Log-periodic and frequency-independent antennas employself-similarityin order to be operational over a wide range ofbandwidths. The most familiar example is thelog-periodic dipole arraywhich can be seen as a number (typically 10 to 20) of connected dipole elements with progressive lengths in anendfire arraymaking it rather directional; it finds use especially as a rooftop antenna for television reception. On the other hand, aYagi–Uda antenna(or simply "Yagi"), with a somewhat similar appearance, has only one dipole element with an electrical connection; the otherparasitic elementsinteract with the electromagnetic field in order to realize a highly directional antenna but with a narrow bandwidth.
Even greater directionality can be obtained usingaperture antennassuch as theparabolic reflectororhorn antenna. Since high directivity in an antenna depends on it being large compared to the wavelength, highly directional antennas (thus with highantenna gain) become more practical at higher frequencies (UHFand above).
At low frequencies (such asAM broadcast), arrays of vertical towers are used to achieve directionality[12]and they will occupy large areas of land. For reception, a longBeverage antennacan have significant directivity. For non directional portable use, a short vertical antenna or smallloop antennaworks well, with the main design challenge being that ofimpedance matching. With a vertical antenna aloading coilat the base of the antenna may be employed to cancel thereactive component of impedance;small loop antennasare tuned with parallel capacitors for this purpose.
An antenna lead-in is thetransmission line, orfeed line, which connects the antenna to a transmitter or receiver. The "antenna feed" may refer to all components connecting the antenna to the transmitter or receiver, such as animpedance matchingnetwork in addition to the transmission line. In a so-called "aperture antenna", such as a horn or parabolic dish, the "feed" may also refer to a basic radiating antenna embedded in the entire system of reflecting elements (normally at the focus of the parabolic dish or at the throat of a horn) which could be considered the one active element in that antenna system. A microwave antenna may also be fed directly from awaveguidein place of a (conductive)transmission line.
An antennacounterpoise, orground plane, is a structure of conductive material which improves or substitutes for the ground. It may be connected to or insulated from the natural ground. In a monopole antenna, this aids in the function of the natural ground, particularly where variations (or limitations) of the characteristics of the natural ground interfere with its proper function. Such a structure is normally connected to the return connection of an unbalanced transmission line such as the shield of acoaxial cable.
An electromagnetic wave refractor in some aperture antennas is a component which due to its shape and position functions to selectively delay or advance portions of the electromagnetic wavefront passing through it. The refractor alters the spatial characteristics of the wave on one side relative to the other side. It can, for instance, bring the wave to a focus or alter the wave front in other ways, generally in order to maximize the directivity of the antenna system. This is the radio equivalent of anoptical lens.
Anantenna coupling networkis a passive network (generally a combination ofinductiveandcapacitivecircuit elements) used forimpedance matchingin between the antenna and the transmitter or receiver. This may be used to minimize losses on the feed line, by reducing transmission line'sstanding wave ratio, and to present the transmitter or receiver with a standard resistive impedance needed for its optimum operation. The feed point location(s) is selected, and antenna elements electrically similar totunercomponents may be incorporated in the antenna structure itself, to improvethe match.
It is a fundamental property of antennas that most of the electrical characteristics of an antenna, such as those described in the next section (e.g.gain,radiation pattern,impedance,bandwidth,resonant frequencyandpolarization), are the same whether the antenna istransmittingorreceiving.[13][14]For example, the"receiving pattern"(sensitivity to incoming signals as a function of direction) of an antenna when used for reception is identical to theradiation patternof the antenna when it isdrivenand functions as a radiator, even though the current and voltage distributions on the antenna itself are different for receiving and sending.[15]This is a consequence of thereciprocity theoremof electromagnetics.[14]Therefore, in discussions of antenna properties no distinction is usually made between receiving and transmitting terminology, and the antenna can be viewed as either transmitting or receiving, whichever is more convenient.
A necessary condition for the aforementioned reciprocity property is that the materials in the antenna and transmission medium arelinearand reciprocal.Reciprocal(orbilateral) means that the material has the same response to an electric current or magnetic field in one direction, as it has to the field or current in the opposite direction. Most materials used in antennas meet these conditions, but some microwave antennas use high-tech components such asisolatorsandcirculators, made of nonreciprocal materials such asferrite.[13][14]These can be used to give the antenna a different behavior on receiving than it has on transmitting,[13]which can be useful in applications likeradar.
The majority of antenna designs are based on theresonanceprinciple. This relies on the behaviour of moving electrons, which reflect off surfaces where thedielectric constantchanges, in a fashion similar to the way light reflects when optical properties change. In these designs, the reflective surface is created by the end of a conductor, normally a thin metal wire or rod, which in the simplest case has afeed pointat one end where it is connected to atransmission line. The conductor, orelement, is aligned with the electrical field of the desired signal, normally meaning it is perpendicular to the line from the antenna to the source (or receiver in the case of a broadcast antenna).[16]
The radio signal's electric component induces a voltage in the conductor. This causes an electrical current to begin flowing in the direction of the signal's instantaneous field. When the resulting current reaches the end of the conductor, it reflects, which is equivalent to a 180 degree change in phase. If the conductor is1/4of a wavelength long, current from the feed point will undergo 90 degree phase change by the time it reaches the end of the conductor, reflect through 180 degrees, and then another 90 degrees as it travels back. That means it has undergone a total 360 degree phase change, returning it to the original signal. The current in the element thus adds to the current being created from the source at that instant. This process creates astanding wavein the conductor, with the maximum current at the feed.[17]
The ordinaryhalf-wave dipoleis probably the most widely used antenna design. This consists of two1/4wavelength elements arranged end-to-end, and lying along essentially the same axis (orcollinear), each feeding one side of a two-conductor transmission wire. The physical arrangement of the two elements places them 180 degrees out of phase, which means that at any given instant one of the elements is driving current into the transmission line while the other is pulling it out. Themonopole antennais essentially one half of the half-wave dipole, a single1/4wavelength element with the other side connected togroundor an equivalentground plane(orcounterpoise). Monopoles, which are one-half the size of a dipole, are common for long-wavelength radio signals where a dipole would be impractically large. Another common design is thefolded dipolewhich consists of two (or more) half-wave dipoles placed side by side and connected at their ends but only one of which is driven.
The standing wave forms with this desired pattern at the design operating frequency,fo, and antennas are normally designed to be this size. However, feeding that element with 3fo(whose wavelength is1/3that offo) will also lead to a standing wave pattern. Thus, an antenna element isalsoresonant when its length is3/4of a wavelength. This is true for all odd multiples of1/4wavelength. This allows some flexibility of design in terms of antenna lengths and feed points. Antennas used in such a fashion are known to beharmonically operated.[18]Resonant antennas usually use a linear conductor (orelement), or pair of such elements, each of which is about a quarter of the wavelength in length (an odd multiple of quarter wavelengths will also be resonant). Antennas that are required to be small compared to the wavelength sacrifice efficiency and cannot be very directional. Since wavelengths are so small at higher frequencies (UHF,microwaves) trading off performance to obtain a smaller physical size is usually not required.
The quarter-wave elements imitate aseries-resonantelectrical element due to the standing wave present along the conductor. At the resonant frequency, the standing wave has a current peak and voltage node (minimum) at the feed. In electrical terms, this means that at that position, the element has minimumimpedance magnitude, generating the maximum current for minimum voltage. This is the ideal situation, because it produces the maximum output for the minimum input, producing the highest possible efficiency. Contrary to an ideal (lossless) series-resonant circuit, a finite resistance remains (corresponding to the relatively small voltage at the feed-point) due to the antenna'sresistance to radiating, as well as any conventionalelectrical lossesfrom producing heat.
Recall that a current will reflect when there are changes in the electrical properties of the material. In order to efficiently transfer the received signal into the transmission line, it is important that the transmission line has the sameimpedanceas its connection point on the antenna, otherwise some of the signal will be reflected backwards into the body of the antenna; likewise part of the transmitter's signal power will be reflected back to transmitter, if there is a change in electrical impedance where the feedline joins the antenna. This leads to the concept ofimpedance matching, the design of the overall system of antenna and transmission line so the impedance is as close as possible, thereby reducing these losses. Impedance matching is accomplished by a circuit called anantenna tunerorimpedance matching networkbetween the transmitter and antenna. The impedance match between the feedline and antenna is measured by a parameter called thestanding wave ratio(SWR) on the feedline.
Consider a half-wave dipole designed to work with signals with wavelength 1 m, meaning the antenna would be approximately 50 cm from tip to tip. If the element has a length-to-diameter ratio of 1000, it will have an inherent impedance of about 63 ohms resistive. Using the appropriate transmission wire or balun, we match that resistance to ensure minimum signal reflection. Feeding that antenna with a current of 1 Ampere will require 63 Volts, and the antenna will radiate 63 Watts (ignoring losses) of radio frequency power. Now consider the case when the antenna is fed a signal with a wavelength of 1.25 m; in this case the current induced by the signal would arrive at the antenna's feedpoint out-of-phase with the signal, causing the net current to drop while the voltage remains the same. Electrically this appears to be a very high impedance. The antenna and transmission line no longer have the same impedance, and the signal will be reflected back into the antenna, reducing output. This could be addressed by changing the matching system between the antenna and transmission line, but that solution only works well at the new design frequency.
The result is that the resonant antenna will efficiently feed a signal into the transmission line only when the source signal's frequency is close to that of the design frequency of the antenna, or one of the resonant multiples. This makes resonant antenna designs inherently narrow-band: Only useful for a small range of frequencies centered around the resonance(s).
It is possible to use simpleimpedance matchingtechniques to allow the use of monopole or dipole antennas substantially shorter than the1/4or1/2wave, respectively, at which they are resonant. As these antennas are made shorter (for a given frequency) their impedance becomes dominated by a series capacitive (negative) reactance; by adding an appropriate size"loading coil"– a series inductance with equal and opposite (positive) reactance – the antenna's capacitive reactance may be cancelled leaving only a pure resistance.
Sometimes the resulting (lower) electrical resonant frequency of such a system (antenna plus matching network) is described using the concept ofelectrical length, so an antenna used at a lower frequency than its resonant frequency is called anelectrically short antenna[19]
For example, at 30 MHz (10 m wavelength) a true resonant1/4wave monopole would be almost 2.5 meters long, and using an antenna only 1.5 meters tall would require the addition of a loading coil. Then it may be said that the coil has lengthened the antenna to achieve an electrical length of 2.5 meters. However, the resulting resistive impedance achieved will be quite a bit lower than that of a true1/4wave (resonant) monopole, often requiring further impedance matching (a transformer) to the desired transmission line. For ever shorter antennas (requiring greater "electrical lengthening") the radiation resistance plummets (approximately according to the square of the antenna length), so that the mismatch due to a net reactance away from the electrical resonance worsens. Or one could as well say that the equivalent resonant circuit of the antenna system has a higherQ factorand thus a reduced bandwidth,[19]which can even become inadequate for the transmitted signal's spectrum.Resistive lossesdue to the loading coil, relative to the decreased radiation resistance, entail a reducedelectrical efficiency, which can be of great concern for a transmitting antenna, but bandwidth is the major factor[dubious–discuss][dubious–discuss]that sets the size of antennas at 1 MHz and lower frequencies.
Theradiant fluxas a function of the distance from the transmitting antenna varies according to theinverse-square law, since that describes the geometrical divergence of the transmitted wave. For a given incoming flux, the power acquired by a receiving antenna is proportional to itseffective area. This parameter compares the amount ofpowercaptured by a receiving antenna in comparison to the flux of an incoming wave (measured in terms of the signal's power density in watts per square metre). A half-wave dipole has an effective area of about 0.13λ2seen from the broadside direction. If higher gain is needed one cannot simply make the antenna larger. Due to the constraint on the effective area of a receiving antenna detailedbelow, one sees that for an already-efficient antenna design, the only way to increase gain (effective area) is byreducingthe antenna's gain in another direction.
If a half-wave dipole is not connected to an external circuit but rather shorted out at the feedpoint, then it becomes a resonant half-wave element which efficiently produces a standing wave in response to an impinging radio wave. Because there is no load to absorb that power, it retransmits all of that power, possibly with a phase shift which is critically dependent on the element's exact length. Thus such a conductor can be arranged in order to transmit a second copy of a transmitter's signal in order to affect the radiation pattern (and feedpoint impedance) of the element electrically connected to the transmitter. Antenna elements used in this way are known aspassive radiators.
AYagi–Udaarray uses passive elements to greatly increase gain in one direction (at the expense of other directions). A number of parallel approximately half-wave elements (of very specific lengths) are situated parallel to each other, at specific positions, along a boom; the boom is only for support and not involved electrically. Only one of the elements is electrically connected to the transmitter or receiver, while the remaining elements are passive. The Yagi produces a fairly large gain (depending on the number of passive elements) and is widely used as a directional antenna with anantenna rotorto control the direction of its beam. It suffers from having a rather limited bandwidth, restricting its use to certain applications.
Rather than using one driven antenna element along with passive radiators, one can build anarray antennain which multiple elements arealldriven by the transmitter through a system of power splitters and transmission lines in relative phases so as to concentrate the RF power in a single direction. What's more, aphased arraycan be made "steerable", that is, by changing the phases applied to each element the radiation pattern can be shiftedwithoutphysically moving the antenna elements. Another common array antenna is thelog-periodic dipole arraywhich has an appearance similar to the Yagi (with a number of parallel elements along a boom) but is totally dissimilar in operation as all elements are connected electrically to the adjacent element with a phase reversal; using the log-periodic principle it obtains the unique property of maintaining its performance characteristics (gain and impedance) over a very large bandwidth.
When a radio wave hits a large conducting sheet it is reflected (with the phase of the electric field reversed) just as a mirror reflects light. Placing such a reflector behind an otherwise non-directional antenna will insure that the power that would have gone in its direction is redirected toward the desired direction, increasing the antenna's gain by a factor of at least 2. Likewise, acorner reflectorcan insure that all of the antenna's power is concentrated in only one quadrant of space (or less) with a consequent increase in gain. Practically speaking, the reflector need not be a solid metal sheet, but can consist of a curtain of rods aligned with the antenna's polarization; this greatly reduces the reflector's weight andwind load. Specular reflection of radio waves is also employed in aparabolic reflectorantenna, in which acurvedreflecting surface effectsfocussingof an incoming wave toward a so-calledfeed antenna; this results in an antenna system with an effective area comparable to the size of the reflector itself. Other concepts fromgeometrical opticsare also employed in antenna technology, such as with thelens antenna.
The antenna'spower gain(or simply "gain") also takes into account the antenna's efficiency, and is often the primary figure of merit. Antennas are characterized by a number of performance measures which a user would be concerned with in selecting or designing an antenna for a particular application. A plot of the directional characteristics in the space surrounding the antenna is itsradiation pattern.
The frequency range orbandwidthover which an antenna functions well can be very wide (as in a log-periodic antenna) or narrow (as in a small loop antenna); outside this range the antenna impedance becomes a poor match to the transmission line and transmitter (or receiver). Use of the antenna well away from its design frequency affects itsradiation pattern, reducing its directive gain.
Generally an antenna will not have a feed-point impedance that matches that of a transmission line; a matching network between antenna terminals and the transmission line will improve power transfer to the antenna. A non-adjustable matching network will most likely place further limits the usable bandwidth of the antenna system. It may be desirable to use tubular elements, instead of thin wires, to make an antenna; these will allow a greater bandwidth. Or, several thin wires can be grouped in acageto simulate a thicker element. This widens the bandwidth of the resonance.
Amateur radioantennas that operate at several frequency bands which are widely separated from each other may connect elements resonant at those different frequencies in parallel. Most of the transmitter's power will flow into the resonant element while the others present a high impedance. Another solution usestraps, parallel resonant circuits which are strategically placed in breaks created in long antenna elements. When used at the trap's particular resonant frequency the trap presents a very high impedance (parallel resonance) effectively truncating the element at the location of the trap; if positioned correctly, the truncated element makes a proper resonant antenna at the trap frequency. At substantially higher or lower frequencies the trap allows the full length of the broken element to be employed, but with a resonant frequency shifted by the net reactance added by the trap.
The bandwidth characteristics of a resonant antenna element can be characterized according to itsQwhere the resistance involved is theradiation resistance, which represents the emission of energy from the resonant antenna to free space.
TheQof a narrow band antenna can be as high as 15. On the other hand, the reactance at the same off-resonant frequency of one using thick elements is much less, consequently resulting in aQas low as 5. These two antennas may perform equivalently at the resonant frequency, but the second antenna will perform over a bandwidth 3 times as wide as the antenna consisting of a thin conductor.
Antennas for use over much broader frequency ranges are achieved using further techniques. Adjustment of a matching network can, in principle, allow for any antenna to be matched at any frequency. Thus thesmall loop antennabuilt into most AM broadcast (medium wave) receivers has a very narrow bandwidth, but is tuned using a parallel capacitance which is adjusted according to the receiver tuning. On the other hand,log-periodic antennasarenotresonant at any single frequency but can (in principle) be built to attain similar characteristics (including feedpoint impedance) over any frequency range. These are therefore commonly used (in the form of directionallog-periodic dipole arrays) as television antennas.
Gainis a parameter which measures the degree ofdirectivityof the antenna'sradiation pattern. A high-gain antenna will radiate most of its power in a particular direction, while a low-gain antenna will radiate over a wide angle. Theantenna gain, orpower gainof an antenna is defined as the ratio of theintensity(power per unit surface area)I{\displaystyle I}radiated by the antenna in the direction of its maximum output, at an arbitrary distance, divided by the intensityIiso{\displaystyle I_{\text{iso}}}radiated at the same distance by a hypotheticalisotropic antennawhich radiates equal power in all directions. This dimensionless ratio is usually expressedlogarithmicallyindecibels, these units are calleddecibels-isotropic(dBi)
A second unit used to measure gain is the ratio of the power radiated by the antenna to the power radiated by ahalf-wave dipoleantennaIdipole{\displaystyle I_{\text{dipole}}}; these units are calleddecibels-dipole(dBd)
Since the gain of a half-wave dipole is 2.15 dBi and the logarithm of a product is additive, the gain in dBi is just 2.15 decibels greater than the gain in dBd
High-gain antennas have the advantage of longer range and better signal quality, but must be aimed carefully at the other antenna. An example of a high-gain antenna is aparabolic dishsuch as asatellite televisionantenna. Low-gain antennas have shorter range, but the orientation of the antenna is relatively unimportant. An example of a low-gain antenna is thewhip antennafound on portable radios andcordless phones. Antenna gain should not be confused withamplifier gain, a separate parameter measuring the increase in signal power due to an amplifying device placed at the front-end of the system, such as alow-noise amplifier.
Theeffective areaor effective aperture of a receiving antenna expresses the portion of the power of a passing electromagnetic wave which the antenna delivers to its terminals, expressed in terms of an equivalent area. For instance, if a radio wave passing a given location has a flux of 1 pW / m2(10−12Watts per square meter) and an antenna has an effective area of 12 m2, then the antenna would deliver 12 pW ofRFpower to the receiver (30 microvoltsRMSat 75 ohms). Since the receiving antenna is not equally sensitive to signals received from all directions, the effective area is a function of the direction to the source.
Due toreciprocity(discussed above) the gain of an antenna used for transmitting must be proportional to its effective area when used for receiving. Consider an antenna with noloss, that is, one whoseelectrical efficiencyis 100%. It can be shown that its effective area averaged over all directions must be equal toλ2/4π, the wavelength squared divided by4π. Gain is defined such that the average gain over all directions for an antenna with 100%electrical efficiencyis equal to 1. Therefore, the effective areaAeffin terms of the gainGin a given direction is given by:
For an antenna with anefficiencyof less than 100%, both the effective area and gain are reduced by that same amount. Therefore, the above relationship between gain and effective area still holds. These are thus two different ways of expressing the same quantity.Aeffis especially convenient when computing the power that would be received by an antenna of a specified gain, as illustrated by the above example.
Theradiation patternof an antenna is a plot of the relative field strength of the radio waves emitted by the antenna at different angles in the far field. It is typically represented by a three-dimensional graph, or polar plots of the horizontal and vertical cross sections. The pattern of an idealisotropic antenna, which radiates equally in all directions, would look like asphere. Many nondirectional antennas, such asmonopolesanddipoles, emit equal power in all horizontal directions, with the power dropping off at higher and lower angles; this is called anomnidirectional patternand when plotted looks like atorusor donut.
The radiation of many antennas shows a pattern of maxima or "lobes" at various angles, separated by "nulls", angles where the radiation falls to zero. This is because the radio waves emitted by different parts of the antenna typicallyinterfere, causing maxima at angles where the radio waves arrive at distant pointsin phase, and zero radiation at other angles where the radio waves arriveout of phase. In adirectional antennadesigned to project radio waves in a particular direction, the lobe in that direction is designed larger than the others and is called the "main lobe". The other lobes usually represent unwanted radiation and are called "sidelobes". The axis through the main lobe is called the "principal axis" or "boresightaxis".
The polar diagrams (and therefore the efficiency and gain) of Yagi antennas are tighter if the antenna is tuned for a narrower frequency range, e.g. the grouped antenna compared to the wideband. Similarly, the polar plots of horizontally polarized yagis are tighter than for those vertically polarized.[20]
The space surrounding an antenna can be divided into three concentric regions: The reactive near-field (also called the inductive near-field), the radiating near-field (Fresnel region) and the far-field (Fraunhofer) regions. These regions are useful to identify the field structure in each, although the transitions between them are gradual; there are no clear boundaries.
The far-field region is far enough from the antenna to ignore its size and shape: It can be assumed that the electromagnetic wave is purely a radiating plane wave (electric and magnetic fields are in phase and perpendicular to each other and to the direction of propagation). This simplifies the mathematical analysis of the radiated field.
Efficiencyof a transmitting antenna is the ratio of power actually radiated (in all directions) to the power absorbed by the antenna terminals. The power supplied to the antenna terminals which is not radiated is converted into heat. This is usually throughloss resistancein the antenna's conductors, or loss between the reflector and feed horn of a parabolic antenna.
Antenna efficiency is separate fromimpedance matching, which may also reduce the amount of power radiated using a given transmitter. If anSWRmeter reads 150 W of incident power and 50 W of reflected power, that means 100 W have actually been absorbed by the antenna (ignoring transmission line losses). How much of that power has actually been radiated cannot be directly determined through electrical measurements at (or before) the antenna terminals, but would require (for instance) careful measurement offield strength. The loss resistance and efficiency of an antenna can be calculated once the field strength is known, by comparing it to the power supplied to the antenna.
Theloss resistancewill generally affect the feedpoint impedance, adding to its resistive component. That resistance will consist of the sum of theradiation resistanceRradand the loss resistanceRloss. If a currentIis delivered to the terminals of an antenna, then a power ofI2Rradwill be radiated and a power ofI2Rlosswill be lost as heat. Therefore, the efficiency of an antenna is equal toRrad/(Rrad+Rloss). Only the total resistanceRrad+Rlosscan be directly measured.
According toreciprocity, the efficiency of an antenna used as a receiving antenna is identical to its efficiency as a transmitting antenna, described above. The power that an antenna will deliver to a receiver (with a properimpedance match) is reduced by the same amount. In some receiving applications, the very inefficient antennas may have little impact on performance. At low frequencies, for example, atmospheric or man-made noise can mask antenna inefficiency. For example, CCIR Rep. 258-3 indicates man-made noise in a residential setting at 40 MHz is about 28 dB above the thermal noise floor. Consequently, an antenna with a 20 dB loss (due to inefficiency) would have little impact on system noise performance. The loss within the antenna will affect the intended signal and the noise/interference identically, leading to no reduction in signal to noise ratio (SNR).
Antennas which are not a significant fraction of a wavelength in size are inevitably inefficient due to their small radiation resistance. AM broadcast radios include a smallloop antennafor reception which has an extremely poor efficiency. This has little effect on the receiver's performance, but simply requires greater amplification by the receiver's electronics. Contrast this tiny component to the massive and very tall towers used at AM broadcast stations for transmitting at the very same frequency, where every percentage point of reduced antenna efficiency entails a substantial cost.
The definition ofantenna gainorpower gainalready includes the effect of the antenna's efficiency. Therefore, if one is trying to radiate a signal toward a receiver using a transmitter of a given power, one need only compare the gain of various antennas rather than considering the efficiency as well. This is likewise true for a receiving antenna at very high (especially microwave) frequencies, where the point is to receive a signal which is strong compared to the receiver's noise temperature. However, in the case of a directional antenna used for receiving signals with the intention ofrejectinginterference from different directions, one is no longer concerned with the antenna efficiency, as discussed above. In this case, rather than quoting theantenna gain, one would be more concerned with thedirective gain, or simplydirectivitywhich doesnotinclude the effect of antenna (in)efficiency. The directive gain of an antenna can be computed from the published gain divided by the antenna's efficiency. In equation form, gain = directivity × efficiency.
The orientation and physical structure of an antenna determine thepolarizationof the electric field of the radio wave transmitted by it. For instance, an antenna composed of a linear conductor (such as adipoleorwhip antenna) oriented vertically will result in vertical polarization; if turned on its side the same antenna's polarization will be horizontal.
Reflections generally affect polarization. Radio waves reflected off theionospherecan change the wave's polarization. Forline-of-sight communicationsorground wavepropagation, horizontally or vertically polarized transmissions generally remain in about the same polarization state at the receiving location. Using a vertically polarized antenna to receive a horizontally polarized wave (or visa-versa) results in relatively poor reception.
An antenna's polarization can sometimes be inferred directly from its geometry. When the antenna's conductorsviewed from a reference locationappear along one line, then the antenna's polarization will be linear in that very direction. In the more general case, the antenna's polarization must be determined throughanalysis. For instance, aturnstile antennamounted horizontally (as is usual), from a distant location on Earth, appears as a horizontal line segment, so its radiation received there is horizontally polarized. But viewed at a downward angle from an airplane, the same antenna doesnotmeet this requirement; in fact its radiation is elliptically polarized when viewed from that direction. In some antennas the state of polarization will change with the frequency of transmission. The polarization of a commercial antenna is an essentialspecification.
In the most general case, polarization iselliptical, meaning that over each cycle the electric field vector traces out anellipse. Two special cases arelinear polarization(the ellipse collapses into a line) as discussed above, andcircular polarization(in which the two axes of the ellipse are equal). In linear polarization the electric field of the radio wave oscillates along one direction. In circular polarization, the electric field of the radio wave rotates around the axis of propagation. Circular or elliptically polarized radio waves aredesignated as right-handed or left-handedusing the "thumb in the direction of the propagation" rule. Note that for circular polarization, optical researchers use the oppositeright-hand rule[citation needed]from the one used by radio engineers.
It is best for the receiving antenna to match the polarization of the transmitted wave for optimum reception. Otherwise there will be a loss of signal strength: when a linearly polarized antenna receives linearly polarized radiation at a relative angle of θ, then there will be a power loss of cos2θ[citation needed]. A circularly polarized antenna can be used to equally well match vertical or horizontal linear polarizations, suffering a 3dBsignal reduction. However it will be blind to a circularly polarized signal of the opposite orientation.
Maximum power transfer requires matching the impedance of an antenna system (as seen looking into the transmission line) to thecomplex conjugateof the impedance of the receiver or transmitter. In the case of a transmitter, however, the desired matching impedance might not exactly correspond to the dynamic output impedance of the transmitter as analyzed as asource impedancebut rather the design value (typically 50 Ohms) required for efficient and safe operation of the transmitting circuitry. The intended impedance is normally resistive, but a transmitter (and some receivers) may have limited additional adjustments to cancel a certain amount of reactance, in order to "tweak" the match.
When a transmission line is used in between the antenna and the transmitter (or receiver) one generally would like an antenna system whose impedance is resistive and nearly the same as thecharacteristic impedanceof that transmission line, in addition to matching the impedance that the transmitter (or receiver) expects. The match is sought to minimize the amplitude ofstanding waves(measured via thestanding wave ratio; SWR) that a mismatch raises on the line, and the increase in transmission line losses it entails.
Antenna tuning, in thestrict senseof modifying the antenna itself, generally refers only to cancellation of any reactance seen at the antenna terminals, leaving only a resistive impedance which might or might not be exactly the desired impedance (that of the available transmission line).
Although an antenna may be designed to have a purely resistive feedpoint impedance (such as a dipole 97% of a half wavelength long) at just one frequency, this will very likely not be exactly true at other frequencies that the antenna is eventually used for. In most cases, in principle the physical length of the antenna can be "trimmed" to obtain a pure resistance, although this is rarely convenient. On the other hand, the addition of a contrary inductance or capacitance can be used to cancel a residual capacitive or inductive reactance, respectively, and may be more convenient than lowering and trimming or extending the antenna, then hoisting it back.
Antennareactancemay be removed using lumped elements, such ascapacitorsorinductorsin the main path of current traversing the antenna, often near the feedpoint, or by incorporating capacitive or inductive structures into the conducting body of the antenna to cancel the feedpoint reactance – such as open-ended "spoke" radial wires, or looped parallel wires – hencegenuinelytune the antenna to resonance. In addition to those reactance-neutralizing add-ons, antennas of any kind may include atransformerand / or transformerbalunat their feedpoint, to change the resistive part of the impedance to more nearly match the feedline'scharacteristic impedance.
Antenna tuningin the loose sense, performed by animpedance matchingdevice (somewhat inappropriately named an "antenna tuner", or the older, more appropriate termtransmatch) goes beyond merely removing reactance and includes transforming the remaining resistance to match the feedline and radio.
An additional problem is matching the remaining resistive impedance to thecharacteristic impedanceof the transmission line: A generalimpedance matchingnetwork (an "antenna tuner" or ATU) will have at least two adjustable elements to correct both components of impedance. Anymatching networkwill have both power losses and power restrictions when used for transmitting.
Commercial antennas are generally designed to approximately match standard 50Ohmcoaxial cables, at standard frequencies; the design expectation is that a matching network will be merely used to 'tweak' any residual mismatch.
In some cases matching is done in a more extreme manner, not simply to cancel a small amount of residual reactance, but to resonate an antenna whose resonance frequency is quite different from the intended frequency of operation.
Ground reflections is one of the common types of multipath.[21][22][23]
The radiation pattern and even the driving point impedance of an antenna can be influenced by the dielectric constant and especiallyconductivityof nearby objects. For a terrestrial antenna, the ground is usually one such object of importance. The antenna's height above the ground, as well as the electrical properties (permittivityand conductivity) of the ground, can then be important. Also, in the particular case of a monopole antenna, the ground (or an artificialground plane) serves as the return connection for the antenna current thus having an additional effect, particularly on the impedance seen by the feed line.
When an electromagnetic wave strikes a plane surface such as the ground, part of the wave is transmitted into the ground and part of it is reflected, according to theFresnel coefficients. If the ground is a very good conductor then almost all of the wave is reflected (180° out of phase), whereas a ground modeled as a (lossy) dielectric can absorb a large amount of the wave's power. The power remaining in the reflected wave, and the phase shift upon reflection, strongly depend on the wave'sangle of incidenceandpolarization. The dielectric constant and conductivity (or simply the complex dielectric constant) is dependent on the soil type and is a function of frequency.
Forvery low frequenciestohigh frequencies(< 30 MHz), the ground behaves as a lossydielectric,[24]thus the ground is characterized both by aconductivity[25]andpermittivity(dielectric constant) which can be measured for a given soil (but is influenced by fluctuating moisture levels) or can be estimated from certain maps. At lowermediumwavefrequencies the ground acts mainly as a good conductor, whichAM broadcast(0.5–1.7 MHz) antennas depend on.
At frequencies between 3–30 MHz, a large portion of the energy from a horizontally polarized antenna reflects off the ground, with almost total reflection at the grazing angles important forground wavepropagation. That reflected wave, with its phase reversed, can either cancel or reinforce the direct wave, depending on the antenna height in wavelengths and elevation angle (for asky wave).
On the other hand, vertically polarized radiation is not well reflected by the ground except at grazing incidence or over very highly conducting surfaces such as sea water.[26]However the grazing angle reflection important for ground wave propagation, using vertical polarization, isin phasewith the direct wave, providing a boost of up to 6 dB, as is detailed below.
At VHF and above (> 30 MHz) the ground becomes a poorer reflector. However, forshortwavefrequencies, especially below ~15 MHz, it remains a good reflector especially for horizontal polarization and grazing angles of incidence. That is important as these higher frequencies usually depend on horizontalline-of-sight propagation(except for satellite communications), the ground then behaving almost as a mirror.
The net quality of a ground reflection depends on the topography of the surface. When the irregularities of the surface are much smaller than the wavelength, the dominant regime is that ofspecular reflection, and the receiver sees both the real antenna and an image of the antenna under the ground due to reflection. But if the ground has irregularities not small compared to the wavelength, reflections will not be coherent but shifted by random phases. With shorter wavelengths (higher frequencies), this is generally the case.
Whenever both the receiving or transmitting antenna are placed at significant heights above the ground (relative to the wavelength), waves reflectedspecularlyby the ground will travel a longer distance than direct waves, inducing a phase shift which can sometimes be significant. When asky waveis launched by such an antenna, that phase shift is always significant unless the antenna is very close to the ground (compared to the wavelength).
The phase of reflection of electromagnetic waves depends on thepolarizationof the incident wave. Given the largerrefractive indexof the ground (typicallyn≈ 2) compared to air (n= 1), the phase of horizontally polarized radiation is reversed upon reflection (a phase shift ofπradians, or 180°). On the other hand, the vertical component of the wave's electric field is reflected at grazing angles of incidence approximatelyin phase. These phase shifts apply as well to a ground modeled as a good electrical conductor.
This means that a receiving antenna "sees" an image of the emitting antenna but with 'reversed' currents (opposite in direction and phase) if the emitting antenna is horizontally oriented (and thus horizontally polarized). However, the received current will be in the same absolute direction and phase if the emitting antenna is vertically polarized.
The actual antenna which istransmittingthe original wave then also mayreceivea strong signal from its own image from the ground. This will induce an additional current in the antenna element, changing the current at the feedpoint for a given feedpoint voltage. Thus the antenna's impedance, given by the ratio of feedpoint voltage to current, is altered due to the antenna's proximity to the ground. This can be quite a significant effect when the antenna is within a wavelength or two of the ground. But as the antenna height is increased, the reduced power of the reflected wave (due to theinverse square law) allows the antenna to approach its asymptotic feedpoint impedance given by theory. At lower heights, the effect on the antenna's impedance isverysensitive to the exact distance from the ground, as this affects the phase of the reflected wave relative to the currents in the antenna. Changing the antenna's height by a quarter wavelength, then changes the phase of the reflection by 180°, with a completely different effect on the antenna's impedance.
The ground reflection has an important effect on the net far fieldradiation patternin the vertical plane, that is, as a function of elevation angle, which is thus different between a vertically and horizontally polarized antenna. Consider an antenna at a heighthabove the ground, transmitting a wave considered at the elevation angleθ. For a vertically polarized transmission the magnitude of the electric field of the electromagnetic wave produced by the direct ray plus the reflected ray is:
Thus thepowerreceived can be as high as 4 times that due to the direct wave alone (such as whenθ= 0), following thesquareof the cosine. The sign inversion for the reflection of horizontally polarized emission instead results in:
where:
For horizontal propagation between transmitting and receiving antennas situated near the ground reasonably far from each other, the distances traveled by the direct and reflected rays are nearly the same. There is almost no relative phase shift. If the emission is polarized vertically, the two fields (direct and reflected) add and there is maximum of received signal. If the signal is polarized horizontally, the two signals subtract and the received signal is largely cancelled. The vertical plane radiation patterns are shown in the image at right. With vertical polarization there is always a maximum forθ= 0, horizontal propagation (left pattern). For horizontal polarization, there is cancellation at that angle. The above formulae and these plots assume the ground as a perfect conductor. These plots of the radiation pattern correspond to a distance between the antenna and its image of 2.5λ. As the antenna height is increased, the number of lobes increases as well.
The difference in the above factors for the case ofθ= 0 is the reason that most broadcasting (transmissions intended for the public) uses vertical polarization. For receivers near the ground, horizontally polarized transmissions suffer cancellation. For best reception the receiving antennas for these signals are likewise vertically polarized. In some applications where the receiving antenna must work in any position, as inmobile phones, thebase stationantennas use mixed polarization, such as linear polarization at an angle (with both vertical and horizontal components) orcircular polarization.
On the other hand, analog television transmissions are usually horizontally polarized, because in urban areas buildings can reflect the electromagnetic waves and createghost imagesdue tomultipath propagation. Using horizontal polarization, ghosting is reduced because the amount of reflection in the horizontal polarization off the side of a building is generally less than in the vertical direction. Vertically polarized analog television have been used in some rural areas. Indigital terrestrial televisionsuch reflections are less problematic, due to robustness of binary transmissions anderror correction.
The flow of current in wire antennas is identical to the solution of counter-propagating waves in asingle conductor transmissionline, which can be solved using thetelegrapher's equations.
Solutions of currents along antenna elements are more conveniently and accurately obtained bynumerical methods, so transmission-line techniques have largely been abandoned for precision modelling, but they continue to be a widely used source of useful, simple approximations that describe well the impedance profiles of antennas.[28](pp 7–10)[27](p 232)
Unlike transmission lines, currents in antennas contribute power to the radiated part electromagnetic field, which can be modeled usingradiation resistance.[a]
The end of an antenna element corresponds to an unterminated (open) end of a single-conductor transmission line, resulting in a reflected wave identical to the incident wave, with its voltageinphase with the incident wave
and its current in theoppositephase (thus net zero current, where there is, after all, no further conductor). The combination of the incident and reflected wave, just as in a transmission line, forms astanding wavewith a current node at the conductor's end, and a voltage node one-quarter wavelength from the end (if the element is at least that long).[28][27]
In aresonant antenna, the feedpoint of the antenna is at one of those voltage nodes.[citation needed]Due to discrepancies from the simplified version of the transmission line model, the voltage one quarter wavelength from the current node is not exactly zero, but it is near a minimum, and small compared to the much large voltage at the conductor's end. Hence, a feed pointmatching the antennaat that spot requires a relatively small voltage but large current (the currents from the two waves add in-phase there), thus a relatively low feedpoint impedance.
Feeding the antenna at other points involves a large voltage, thus a large impedance,[citation needed]and usually one that is primarily reactive (lowpower factor), which is a terrible impedance match to available transmission lines. Therefore, it is usually desired for an antenna to operate as a resonant element with each conductor having a length of one quarter wavelength (or any other odd multiples of a quarter wavelength).
For instance, a half-wave dipole has two such elements (one connected to each conductor of a balanced transmission line) about one quarter wavelength long. Depending on the conductors' diameters, a small deviation from this lengthis adoptedin order to reach the point where the antenna current and the (small) feedpoint voltage are exactly in phase. Then the antenna presents a purely resistive impedance, and ideally one close to thecharacteristic impedanceof an available transmission line.
Despite these useful properties, resonant antennas have the disadvantage that they achieve resonance (purely resistive feedpoint impedance) only at a fundamental frequency, and perhaps[citation needed]some of itsharmonics, and the feedpoint resistance is larger at higher-order resonances. Therefore, resonant antennas can only achieve their good performance within a limited bandwidth, depending on theQat the resonance.
The electric and magnetic fields emanating from a driven antenna element will generally affect the voltages and currents in nearby antennas, antenna elements, or other conductors. This is particularly true when the affected conductor is a resonant element (multiple of half-wavelengths in length) at about the same frequency, as is the case where the conductors are all part of the same active or passiveantenna array.
Because the affected conductors are in the near-field, one cannotjust treat two antennas as transmitting and receiving a signal according to theFriis transmission formulafor instance, but must calculate themutual impedancematrix which takes into account both voltages and currents (interactions through both the electric and magnetic fields). Thus using the mutual impedances calculated for a specific geometry, one can solve for the radiation pattern of aYagi–Uda antennaor the currents and voltages for each element of aphased array. Such an analysis can also describe in detail reflection of radio waves by aground planeor by acorner reflectorand their effect on the impedance (and radiation pattern) of an antenna in its vicinity.
Often such near-field interactions are undesired and pernicious. Currents in random metal objects near a transmitting antenna will often be in poor conductors, causing loss of RF power in addition to unpredictably altering the characteristics of the antenna. By careful design, it is possible to reduce the electrical interaction between nearby conductors. For instance, the 90 degree angle in between the two dipoles composing theturnstile antennainsures no interaction between these, allowing these to be driven independently (but actually with the same signal in quadrature phases in the turnstile antenna design).
Antennas can be classified by operating principles or by their application. Different authorities placed antennas in narrower or broader categories. Generally these include
These antenna types and others are summarized in greater detail in the overview article,Antenna types, as well as in each of the linked articles in the list above, and in even more detail in articles which those link to.
The dictionary definition ofantennaat Wiktionary
|
https://en.wikipedia.org/wiki/Antenna_(radio)
|
Thebasic exchange telephone radio serviceorBETRSis a fixed radio service where amultiplexed,digital radiolink is used as the last segment of thelocal loopto providewirelesstelephone service to subscribers in remote areas. BETRS technology was developed in the mid-1980s and allows up to four subscribers to use a single radiochannelpair,simultaneously, withoutinterferingwith one another.
In the US, this service may operate in the paired 152/158 and 454/459MHzbands and on 10channel blocksin the 816-820/861-865 MHz bands. BETRS may be licensed only to state-certified carriers in the area where the service is provided and is considered a part of thepublic switched telephone network(PSTN) by state regulators.
Regulation of this service currently resides in parts 1 and 22 of theCode of Federal Regulations(CFR), Subtitle 47 onTelecommunications, and may be researched or ordered through theGovernment Printing Office(GPO).
|
https://en.wikipedia.org/wiki/Basic_exchange_telephone_radio_service
|
Anelectrical cableis an assembly of one or morewiresrunning side by side or bundled, which is used as anelectrical conductorto carryelectric current.
Electrical cables are used to connect two or more devices, enabling the transfer ofelectrical signals,power, or both from one device to the other. Physically, an electrical cable is an assembly consisting of one or more conductors with their own insulations and optional screens, individual coverings, assembly protection and protective covering.
One or more electrical cables and their correspondingconnectorsmay be formed into a cable assembly,[1]which is not necessarily suitable for connecting two devices but can be a partial product (e.g. to be soldered onto aprinted circuit boardwith a connector mounted to the housing). Cable assemblies can also take the form of acable treeorcable harness, used to connect many terminals together.
Electrical cables are used to connect two or more devices, enabling the transfer of electrical signals or power from one device to the other. Long-distance communication takes place overundersea communication cables.Power cablesare used for bulk transmission of alternating and direct current power, especially usinghigh-voltage cable. Electrical cables are extensively used inbuilding wiringfor lighting, power and control circuits permanently installed in buildings. Since all the circuit conductors required can be installed in a cable at one time, installation labor is saved compared to certain other wiring methods.
Physically, an electrical cable is an assembly consisting of one or more conductors with their own insulations and optional screens, individual coverings, assembly protection and protective coverings. Electrical cables may be made more flexible by stranding the wires. In this process, smaller individual wires are twisted or braided together to produce larger wires that are more flexible than solid wires of similar size. Bunching small wires before concentric stranding adds the most flexibility.Copper wires in a cablemay be bare, or they may be plated with a thin layer of another metal, most oftentinbut sometimesgold,silveror some other material. Tin, gold, and silver are much less prone tooxidationthan copper, which may lengthen wire life, and makessolderingeasier. Tinning is also used to provide lubrication between strands. Tinning was used to help removal of rubber insulation. Tight lays during stranding makes the cable extensible (CBA – as in telephone handset cords).[further explanation needed]
In the 19th century and early 20th century, electrical cable was ofteninsulatedusing cloth, rubber or paper. Plastic materials are generally used today, except for high-reliability[clarification needed]power cables. The firstthermoplasticused wasgutta-percha(a naturallatex) which was found useful for underwater cables in the 19th century. The first, and still very common, man-made plastic used for cable insulation waspolyethylene. This was invented in 1930, but not available outside military use until afterWorld War 2during which a telegraph cable using it was laid across the English Channel to support troops followingD-Day.[2]
Cables can be securely fastened and organized, such as by using trunking,cable trays,cable tiesorcable lacing. Continuous-flex orflexible cablesused in moving applications withincable carrierscan be secured usingstrain relief devicesor cable ties.
Anycurrent-carrying conductor, including a cable, radiates anelectromagnetic field. Likewise, any conductor or cable will pick up energy from any existing electromagnetic field around it. These effects are often undesirable, in the first case amounting to unwanted transmission of energy which may adversely affect nearby equipment or other parts of the same piece of equipment; and in the second case, unwanted pickup ofnoisewhich may mask the desired signal being carried by the cable, or, if the cable is carryingpower supplyor control voltages, pollute them to such an extent as to cause equipment malfunction.
The first solution to these problems is to keep cable lengths in buildings short since pick up and transmission are essentially proportional to the length of the cable. The second solution is to route cables away from trouble. Beyond this, there are particular cable designs that minimize electromagnetic pickup and transmission. Three of the principal design techniques areshielding,coaxialgeometry, andtwisted-pairgeometry.
Shielding makes use of the electrical principle of theFaraday cage. The cable is encased for its entire length in foil or wire mesh. All wires running inside this shielding layer will be to a large extent decoupled from externalelectricalfields, particularly if the shield is connected to a point of constant voltage, such asearth or ground. Simple shielding of this type is not greatly effective against low-frequencymagneticfields, however - such as magnetic "hum" from a nearby powertransformer. A grounded shield on cables operating at 2.5 kV or more gathers leakage current and capacitive current, protecting people from electric shock and equalizing stress on the cable insulation.
Coaxial design helps to further reduce low-frequency magnetic transmission and pickup. In this design the foil or mesh shield has a circular cross section and the inner conductor is exactly at its center. This causes the voltages induced by a magnetic field between the shield and the core conductor to consist of two nearly equal magnitudes which cancel each other.
A twisted pair has two wires of a cable twisted around each other. This can be demonstrated by putting one end of a pair of wires in a hand drill and turning while maintaining moderate tension on the line. Where the interfering signal has a wavelength that is long compared to the pitch of the twisted pair, alternate lengths of wires develop opposing voltages, tending to cancel the effect of the interference.
Electrical cable jacket material is usually constructed of flexible plastic which will burn. The fire hazard of grouped cables can be significant.[3]Cables jacketing materials can be formulated to prevent fire spread[4](seeMineral-insulated copper-clad cable). Alternately, fire spread amongst combustible cables can be prevented by the application of fire retardant coatings directly on the cable exterior,[5]or the fire threat can be isolated by the installation of boxes constructed of noncombustible materials around the bulk cable installation.
CENELECHD 361 is a ratified standard published by CENELEC, which relates to wire and cable marking type, whose goal is to harmonize cables.Deutsches Institut für Normung(DIN, VDE) has released a similar standard (DIN VDE 0292).
|
https://en.wikipedia.org/wiki/Electrical_cable
|
AnRF connector(radio frequency connector) is anelectrical connectordesigned to work atradio frequenciesin the multi-megahertz range.
RF connectors are typically used withcoaxial cablesand are designed to maintain the shielding that the coaxial design offers. Better models also minimize the change in transmission lineimpedanceat the connection in order to reducesignal reflectionand power loss.[1]As the frequency increases,transmission lineeffects become more important, with small impedance variations from connectors causing the signal to reflect rather than pass through. An RF connector must not allow external signals into the circuit throughelectromagnetic interferenceand capacitive pickup.
Mechanically, RF connectors may provide a fastening mechanism (thread,bayonet, braces,blind mate) andspringsfor a low ohmic electric contact while sparing the gold surface, thus allowing very high mating cycles and reducing theinsertion force. Research activity in the area of radio-frequency circuit design has surged in the 2000s in direct response to the enormous market demand for inexpensive, high-data-rate wireless transceivers.[2]
Common types of RF connectors are used fortelevisionreceivers,two-way radio,Wi-FiPCIe cards with removable antennas, and industrial or scientific measurement instruments using radio frequencies.
This technology-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/RF_connector
|
Microwave Bypass, Inc.launched the world's first fixed wireless internet access technology in 1987, a decade beforeWi-Fi. It enabled local and remote networks to connect at the then full Ethernet (802.3) data rate of 10 megabits per second, and for up to 4.3 miles.[1]
The company was founded in March 1986 by David Theodore (25), operating from OneKendall Square,Cambridge, Massachusetts. Its wireless solution consisted of a modified broadcast quality video radio (23 GHz) coupled with Microwave Bypass' EtherWave Transceiver. The system met the then highest Ethernet throughput and could transmit 4.3 miles (6.9 km), in keeping with Ethernet's propagation delay allowance (46.4 μs).
Beta testing occurred at Massachusetts General Hospital in March 1987, at the invitation of network manager, David Murphy, and with Network World's Laura DiDio and representatives ofHarvard UniversityandBoston University's Dr. Mikhail Orlov in attendance. After a successful demo the first two production links were installed in parallel betweenMassachusetts General HospitalinBostonand Harvard's Cardiac Computer Center, 2.5 miles (4.0 km) across theCharles River. This also marked the first wireless transmission of MR images.
In 1988, Microwave Bypass collaborated withCisco Systemson a full-duplex EtherWave Transceiver to eliminate 802.3 collision detection and permit longer distance connections as far as the microwave could reach. This first full-duplex design was developed for an application atMIT, between its main campus and Lincoln Laboratories. Later that year Microwave Bypass completed an exclusive deal, announced byMotorola,[2]for the transfer of its EtherWave Transceiver and LAN-LINK 1000 Bridge technologies.
|
https://en.wikipedia.org/wiki/Microwave_Bypass
|
Awireless Internet service provider(WISP) is anInternet service providerwith a network based onwireless networking. Technology may include commonplaceWi-Fiwireless mesh networking, or proprietary equipment designed to operate over open900 MHz,2.4 GHz, 4.9, 5, 24, and 60 GHz bands or licensed frequencies in theUHFband (including theMMDSfrequency band),LMDS, and other bands from 6 GHz to 80 GHz.
In the US, theFederal Communications Commission(FCC) released a Report and Order, FCC 05-56 in 2005 that revised the FCC’s rules to open the 3650 MHz band for terrestrial wireless broadband operations.[1]On November 14, 2007 the Commission released a Public Notice (DA 07-4605) in which the Wireless Telecommunications Bureau announced the start date for the licensing and registration process for the 3650-3700 MHz band.[2]
As of July 2015, over 2,000fixed wirelessbroadband providers operate in the US, servicing nearly 4 million customers.[3]
Initially, WISPs were only found inruralareas not covered bycable televisionorDSL.[4]There were 879Wi-Fibased WISPs in theCzech Republicas of May 2008,[5][6]making it the country with mostWi-Fiaccess points in the wholeEU.;[7][8]which was a consequence of the then de facto monopoly of the former telecom operator on fixed data networks. The providing of wireless Internet has a big potential of lower the "digital gap" or "Internet gap" in the developing countries.Geekcorpsactively help in Africa with among others wireless network building. An example of a typical WISP system is such as the one deployed by Gaiacom Wireless Networks which is based on Wi-Fi standards. TheOne Laptop per Childproject strongly relies on good Internet connectivity, which can most likely be provided in rural areas only with satellite or wireless network Internet access. In high internet cost countries such as South Africa, prices have been drastically reduced by the government allocating spectrum to smaller WISPs, who are able to deliver high speed broadband at a much lower cost.[9]
Some WISP networks have been started in rural parts of theUnited Kingdom, to address issues with poor broadbandDSLservice (bandwidth) in rural areas ("notspots"), including slow rollout of fibre based services which could improve service (usuallyFibre to the cabinetto groups of rural buildings, potentiallyFibre to the premisesfor isolated buildings). A number of these WISPs[10][11]have been set up via theCommunity Broadband Network, using funds from theEuropean Agricultural Fund for Rural Development
WISPs often offer additional services like location-based content,Virtual Private Networking(VPN) andVoice over IP. Isolated municipal ISPs and larger statewide initiatives alike are tightly focused on wireless networking.[citation needed]
WISPs have a large market share in rural environments wherecableanddigital subscriber linesare not available; further, with technology available, they can meet or beat speeds of legacy cable and telephone systems.[12]In urban environments,gigabit wirelesslinks are common and provide levels of bandwidth previously only available through expensivefiber opticconnections.[13]
Typically, the way that a WISP operates is to order a fiber circuit to the center of the area they wish to serve. From there, the WISP builds backhauls (gigabit wireless or fiber) to elevated points in the region, such as radio towers, tall buildings, grain silos, or water towers. Those locations haveaccess pointsto provide service to individual customers, or backhauls to other towers where they have more equipment. The WISP may also use gigabit wireless links to connect a PoP (Point of Presence) to several towers, reducing the need to pay for fiber circuits to the tower. For fixed wireless connections, a smalldishor other antenna is mounted to the roof of the customer's building and aligned to the WISP's nearest antenna site. Where a WISP operates over the tightly limited range of the heavily populated2.4 GHz band, as nearly all802.11-based Wi‑Fi providers do, it is not uncommon to also see access points mounted on light posts and customer buildings.
Roamingbetween service providers is possible with the draft protocolWISPr, a set of recommendations which facilitate inter-network and inter-operator roaming of Wi-Fi users.
|
https://en.wikipedia.org/wiki/Wireless_Internet_service_provider
|
Acellular networkormobile networkis atelecommunications networkwhere the link to and from end nodes iswirelessand the network is distributed over land areas calledcells, each served by at least one fixed-locationtransceiver(such as abase station). These base stations provide the cell with the network coverage which can be used for transmission of voice, data, and other types of content viaradio waves. Each cell's coverage area is determined by factors such as the power of the transceiver, the terrain, and the frequency band being used. A cell typically uses a different set of frequencies from neighboring cells, to avoid interference and provide guaranteed service quality within each cell.[1][2]
When joined together, these cells provide radio coverage over a wide geographic area. This enables numerousdevices, includingmobile phones,tablets,laptopsequipped withmobile broadband modems, andwearable devicessuch assmartwatches, to communicate with each other and with fixed transceivers and telephones anywhere in the network, via base stations, even if some of the devices are moving through more than one cell during transmission. The design of cellular networks allows for seamlesshandover, enabling uninterrupted communication when a device moves from one cell to another.
Modern cellular networks utilize advanced technologies such asMultiple Input Multiple Output(MIMO),beamforming, and small cells to enhance network capacity and efficiency.
Cellular networks offer a number of desirable features:[2]
Major telecommunications providers have deployed voice and data cellular networks over most of the inhabited land area ofEarth. This allows mobile phones and other devices to be connected to thepublic switched telephone networkand publicInternet access. In addition to traditional voice and data services, cellular networks now supportInternet of Things(IoT) applications, connecting devices such assmart meters, vehicles, and industrial sensors.
The evolution of cellular networks from1Gto5Ghas progressively introduced faster speeds, lower latency, and support for a larger number of devices, enabling advanced applications in fields such as healthcare, transportation, andsmart cities.
Private cellular networks can be used for research[3]or for large organizations and fleets, such as dispatch for local public safety agencies or a taxicab company, as well as for local wireless communications in enterprise and industrial settings such as factories, warehouses, mines, power plants, substations, oil and gas facilities and ports.[4]
In acellular radiosystem, a land area to be supplied with radio service is divided into cells in a pattern dependent on terrain and reception characteristics. These cell patterns roughly take the form of regular shapes, such as hexagons, squares, or circles although hexagonal cells are conventional. Each of these cells is assigned with multiple frequencies (f1–f6) which have correspondingradio base stations. The group of frequencies can be reused in other cells, provided that the same frequencies are not reused in adjacent cells, which would causeco-channel interference.
The increasedcapacityin a cellular network, compared with a network with a single transmitter, comes from the mobile communication switching system developed byAmos Joelof Bell Labs[5]that permitted multiple callers in a given area to use the same frequency by switching calls to the nearest available cellular tower having that frequency available. This strategy is viable because a given radio frequency can be reused in a different area for an unrelated transmission. In contrast, a single transmitter can only handle one transmission for a given frequency. Inevitably, there is some level ofinterferencefrom the signal from the other cells which use the same frequency. Consequently, there must be at least one cell gap between cells which reuse the same frequency in a standardfrequency-division multiple access(FDMA) system.
Consider the case of a taxi company, where each radio has a manually operated channel selector knob to tune to different frequencies. As drivers move around, they change from channel to channel. The drivers are aware of whichfrequencyapproximately covers some area. When they do not receive a signal from the transmitter, they try other channels until finding one that works. The taxi drivers only speak one at a time when invited by the base station operator. This is a form oftime-division multiple access(TDMA).
The idea to establish a standard cellular phone network was first proposed on December 11, 1947. This proposal was put forward byDouglas H. Ring, aBell Labsengineer, in an internal memo suggesting the development of a cellular telephone system byAT&T.[6][7]
The first commercial cellular network, the1Ggeneration, was launched in Japan byNippon Telegraph and Telephone(NTT) in 1979, initially in the metropolitan area ofTokyo. However, NTT did not initially commercialize the system; the early launch was motivated by an effort to understand a practical cellular system rather than by an interest to profit from it.[8][9]In 1981, theNordic Mobile Telephonesystem was created as the first network to cover an entire country. The network was released in 1981 in Sweden and Norway, then in early 1982 in Finland and Denmark.Televerket, a state-owned corporation responsible for telecommunications in Sweden, launched the system.[8][10][11]
In September 1981,Jan Stenbeck, a financier and businessman, launchedComvik, a new Swedish telecommunications company. Comvik was the first European telecommunications firm to challenge the state's telephone monopoly on the industry.[12][13][14]According to some sources, Comvik was the first to launch a commercial automatic cellular system before Televerket launched its own in October 1981. However, at the time of the new network’s release, theSwedish Post and Telecom Authoritythreatened to shut down the system after claiming that the company had used an unlicensed automatic gear that could interfere with its own networks.[14][15]In December 1981, Sweden awarded Comvik with a license to operate its own automatic cellular network in the spirit of market competition.[14][15][16]
TheBell Systemhad developed cellular technology since 1947, and had cellular networks in operation inChicago, Illinois,[17]andDallas, Texas, prior to 1979; however, regulatory battles delayed AT&T's deployment of cellular service to 1983,[18]when itsRegional Holding CompanyIllinois Bellfirst provided cellular service.[19]
First-generation cellular network technology continued to expand its reach to the rest of the world. In 1990,Millicom Inc., a telecommunications service provider, strategically partnered with Comvik’s international cellular operations to become Millicom International Cellular SA.[20]The company went on to establish a 1G systems foothold in Ghana, Africa under the brand name Mobitel.[21]In 2006, the company’s Ghana operations were renamed to Tigo.[22]
Thewireless revolutionbegan in the early 1990s,[23][24][25]leading to the transition from analog todigital networks.[26]The MOSFET invented atBell Labsbetween 1955 and 1960,[27][28][29][30][31]was adapted for cellular networks by the early 1990s, with the wide adoption ofpower MOSFET,LDMOS(RF amplifier), andRF CMOS(RF circuit) devices leading to the development and proliferation of digital wireless mobile networks.[26][32][33]
The first commercial digital cellular network, the2Ggeneration, was launched in 1991. This sparked competition in the sector as the new operators challenged the incumbent 1G analog network operators.
To distinguish signals from several different transmitters, a number ofchannel access methodshave been developed, includingfrequency-division multiple access(FDMA, used by analog andD-AMPS[citation needed]systems),time-division multiple access(TDMA, used byGSM) andcode-division multiple access(CDMA, first used forPCS, and the basis of3G).[2]
With FDMA, the transmitting and receiving frequencies used by different users in each cell are different from each other. Each cellular call was assigned a pair of frequencies (one for base to mobile, the other for mobile to base) to providefull-duplexoperation. The originalAMPSsystems had 666 channel pairs, 333 each for theCLEC"A" system andILEC"B" system. The number of channels was expanded to 416 pairs per carrier, but ultimately the number of RF channels limits the number of calls that a cell site could handle. FDMA is a familiar technology to telephone companies, which usedfrequency-division multiplexingto add channels to their point-to-point wireline plants beforetime-division multiplexingrendered FDM obsolete.
With TDMA, the transmitting and receiving time slots used by different users in each cell are different from each other. TDMA typically usesdigitalsignaling tostore and forwardbursts of voice data that are fit into time slices for transmission, and expanded at the receiving end to produce a somewhat normal-sounding voice at the receiver. TDMA must introducelatency(time delay) into the audio signal. As long as the latency time is short enough that the delayed audio is not heard as an echo, it is not problematic. TDMA is a familiar technology for telephone companies, which usedtime-division multiplexingto add channels to their point-to-point wireline plants beforepacket switchingrendered FDM obsolete.
The principle of CDMA is based onspread spectrumtechnology developed for military use duringWorld War IIand improved during theCold Warintodirect-sequence spread spectrumthat was used for early CDMA cellular systems andWi-Fi. DSSS allows multiple simultaneous phone conversations to take place on a single wideband RF channel, without needing to channelize them in time or frequency. Although more sophisticated than older multiple access schemes (and unfamiliar to legacy telephone companies because it was not developed byBell Labs), CDMA has scaled well to become the basis for 3G cellular radio systems.
Other available methods of multiplexing such asMIMO, a more sophisticated version ofantenna diversity, combined with activebeamformingprovides much greaterspatial multiplexingability compared to original AMPS cells, that typically only addressed one to three unique spaces. Massive MIMO deployment allows much greater channel reuse, thus increasing the number of subscribers per cell site, greater data throughput per user, or some combination thereof.Quadrature Amplitude Modulation(QAM) modems offer an increasing number of bits per symbol, allowing more users per megahertz of bandwidth (and decibels of SNR), greater data throughput per user, or some combination thereof.
The key characteristic of a cellular network is the ability to reuse frequencies to increase both coverage and capacity. As described above, adjacent cells must use different frequencies, however, there is no problem with two cells sufficiently far apart operating on the same frequency, provided the masts and cellular network users' equipment do not transmit with too much power.[2]
The elements that determine frequency reuse are the reuse distance and the reuse factor. The reuse distance,Dis calculated as
whereRis the cell radius andNis the number of cells per cluster. Cells may vary in radius from 1 to 30 kilometres (0.62 to 18.64 mi). The boundaries of the cells can also overlap between adjacent cells and large cells can be divided into smaller cells.[34]
The frequency reuse factor is the rate at which the same frequency can be used in the network. It is1/K(orKaccording to some books) whereKis the number of cells which cannot use the same frequencies for transmission. Common values for the frequency reuse factor are 1/3, 1/4, 1/7, 1/9 and 1/12 (or 3, 4, 7, 9 and 12, depending on notation).[35]
In case ofNsector antennas on the same base station site, each with different direction, the base station site can serve N different sectors.Nis typically 3. Areuse patternofN/Kdenotes a further division in frequency amongNsector antennas per site. Some current and historical reuse patterns are 3/7 (North American AMPS), 6/4 (Motorola NAMPS), and 3/4 (GSM).
If the total availablebandwidthisB, each cell can only use a number of frequency channels corresponding to a bandwidth ofB/K, and each sector can use a bandwidth ofB/NK.
Code-division multiple access-based systems use a wider frequency band to achieve the same rate of transmission as FDMA, but this is compensated for by the ability to use a frequency reuse factor of 1, for example using a reuse pattern of 1/1. In other words, adjacent base station sites use the same frequencies, and the different base stations and users are separated by codes rather than frequencies. WhileNis shown as 1 in this example, that does not mean the CDMA cell has only one sector, but rather that the entire cell bandwidth is also available to each sector individually.
Recently alsoorthogonal frequency-division multiple accessbased systems such asLTEare being deployed with a frequency reuse of 1. Since such systems do not spread the signal across the frequency band,
inter-cell radio resource management is important to coordinate resource allocation between different cell sites and to limit the inter-cell interference. There are various means ofinter-cell interference coordination(ICIC) already defined in the standard.[36]Coordinated scheduling, multi-site MIMO or multi-site beamforming are other examples for inter-cell radio resource management that might be standardized in the future.
Cell towers frequently use adirectional signalto improve reception in higher-traffic areas. In theUnited States, theFederal Communications Commission(FCC) limits omnidirectional cell tower signals to 100 watts of power. If the tower has directional antennas, the FCC allows the cell operator to emit up to 500 watts ofeffective radiated power(ERP).[37]
Although the original cell towers created an even, omnidirectional signal, were at the centers of the cells and were omnidirectional, a cellular map can be redrawn with the cellular telephone towers located at the corners of the hexagons where three cells converge.[38]Each tower has three sets of directional antennas aimed in three different directions with 120 degrees for each cell (totaling 360 degrees) and receiving/transmitting into three different cells at different frequencies. This provides a minimum of three channels, and three towers for each cell and greatly increases the chances of receiving a usable signal from at least one direction.
The numbers in the illustration are channel numbers, which repeat every 3 cells. Large cells can be subdivided into smaller cells for high volume areas.[39]
Cell phone companies also use this directional signal to improve reception along highways and inside buildings like stadiums and arenas.[37]
Practically every cellular system has some kind of broadcast mechanism. This can be used directly for distributing information to multiple mobiles. Commonly, for example inmobile telephonysystems, the most important use of broadcast information is to set up channels for one-to-one communication between the mobile transceiver and the base station. This is calledpaging. The three different paging procedures generally adopted are sequential, parallel and selective paging.
The details of the process of paging vary somewhat from network to network, but normally we know a limited number of cells where the phone is located (this group of cells is called a Location Area in theGSMorUMTSsystem, or Routing Area if a data packet session is involved; inLTE, cells are grouped into Tracking Areas). Paging takes place by sending the broadcast message to all of those cells. Paging messages can be used for information transfer. This happens inpagers, inCDMAsystems for sendingSMSmessages, and in theUMTSsystem where it allows for low downlink latency in packet-based connections.
In LTE/4G, the Paging procedure is initiated by the MME when data packets need to be delivered to the UE.
Paging types supported by the MME are:
In a primitive taxi system, when the taxi moved away from a first tower and closer to a second tower, the taxi driver manually switched from one frequency to another as needed. If communication was interrupted due to a loss of a signal, the taxi driver asked the base station operator to repeat the message on a different frequency.
In a cellular system, as the distributed mobile transceivers move from cell to cell during an ongoing continuous communication, switching from one cell frequency to a different cell frequency is done electronically without interruption and without a base station operator or manual switching. This is called thehandoveror handoff. Typically, a new channel is automatically selected for the mobile unit on the new base station which will serve it. The mobile unit then automatically switches from the current channel to the new channel and communication continues.
The exact details of the mobile system's move from one base station to the other vary considerably from system to system (see the example below for how a mobile phone network manages handover).
The most common example of a cellular network is a mobile phone (cell phone) network. Amobile phoneis a portable telephone which receives or makes calls through acell site(base station) or transmitting tower.Radio wavesare used to transfer signals to and from the cell phone.
Modern mobile phone networks use cells because radio frequencies are a limited, shared resource. Cell-sites and handsets change frequency under computer control and use low power transmitters so that the usually limited number of radio frequencies can be simultaneously used by many callers with less interference.
A cellular network is used by themobile phone operatorto achieve both coverage and capacity for their subscribers. Large geographic areas are split into smaller cells to avoid line-of-sight signal loss and to support a large number of active phones in that area. All of the cell sites are connected totelephone exchanges(or switches), which in turn connect to thepublic telephone network.
In cities, each cell site may have a range of up to approximately1⁄2mile (0.80 km), while in rural areas, the range could be as much as 5 miles (8.0 km). It is possible that in clear open areas, a user may receive signals from a cell site 25 miles (40 km) away. In rural areas with low-band coverage and tall towers, basic voice and messaging service may reach 50 miles (80 km), with limitations on bandwidth and number of simultaneous calls.[citation needed]
Since almost all mobile phones usecellular technology, includingGSM,CDMA, andAMPS(analog), the term "cell phone" is in some regions, notably the US, used interchangeably with "mobile phone". However,satellite phonesare mobile phones that do not communicate directly with a ground-based cellular tower but may do so indirectly by way of a satellite.
There are a number of different digital cellular technologies, including:Global System for Mobile Communications(GSM),General Packet Radio Service(GPRS),cdmaOne,CDMA2000,Evolution-Data Optimized(EV-DO),Enhanced Data Rates for GSM Evolution(EDGE),Universal Mobile Telecommunications System(UMTS),Digital Enhanced Cordless Telecommunications(DECT),Digital AMPS(IS-136/TDMA), andIntegrated Digital Enhanced Network(iDEN). The transition from existing analog to the digital standard followed a very different path in Europe and theUS.[40]As a consequence, multiple digital standards surfaced in the US, whileEuropeand many countries converged towards theGSMstandard.
A simple view of the cellular mobile-radio network consists of the following:
This network is the foundation of theGSMsystem network. There are many functions that are performed by this network in order to make sure customers get the desired service including mobility management, registration, call set-up, andhandover.
Any phone connects to the network via an RBS (Radio Base Station) at a corner of the corresponding cell which in turn connects to theMobile switching center(MSC). The MSC provides a connection to thepublic switched telephone network(PSTN). The link from a phone to the RBS is called anuplinkwhile the other way is termeddownlink.
Radio channels effectively use the transmission medium through the use of the following multiplexing and access schemes:frequency-division multiple access(FDMA),time-division multiple access(TDMA),code-division multiple access(CDMA), andspace-division multiple access(SDMA).
Small cells, which have a smaller coverage area than base stations, are categorised as follows:
As the phone user moves from one cell area to another cell while a call is in progress, the mobile station will search for a new channel to attach to in order not to drop the call. Once a new channel is found, the network will command the mobile unit to switch to the new channel and at the same time switch the call onto the new channel.
WithCDMA, multiple CDMA handsets share a specific radio channel. The signals are separated by using apseudonoisecode (PN code) that is specific to each phone. As the user moves from one cell to another, the handset sets up radio links with multiple cell sites (or sectors of the same site) simultaneously. This is known as "soft handoff" because, unlike with traditionalcellular technology, there is no one defined point where the phone switches to the new cell.
InIS-95inter-frequency handovers and older analog systems such asNMTit will typically be impossible to test the target channel directly while communicating. In this case, other techniques have to be used such as pilot beacons in IS-95. This means that there is almost always a brief break in the communication while searching for the new channel followed by the risk of an unexpected return to the old channel.
If there is no ongoing communication or the communication can be interrupted, it is possible for the mobile unit to spontaneously move from one cell to another and then notify the base station with the strongest signal.
The effect of frequency on cell coverage means that different frequencies serve better for different uses. Low frequencies, such as 450 MHz NMT, serve very well for countryside coverage.GSM900 (900 MHz) is suitable for light urban coverage.GSM1800 (1.8 GHz) starts to be limited by structural walls.UMTS, at 2.1 GHz is quite similar in coverage toGSM1800.
Higher frequencies are a disadvantage when it comes to coverage, but it is a decided advantage when it comes to capacity. Picocells, covering e.g. one floor of a building, become possible, and the same frequency can be used for cells which are practically neighbors.
Cell service area may also vary due to interference from transmitting systems, both within and around that cell. This is true especially in CDMA based systems. The receiver requires a certainsignal-to-noise ratio, and the transmitter should not send with too high transmission power in view to not cause interference with other transmitters. As the receiver moves away from the transmitter, the power received decreases, so thepower controlalgorithm of the transmitter increases the power it transmits to restore the level of received power. As the interference (noise) rises above the received power from the transmitter, and the power of the transmitter cannot be increased anymore, the signal becomes corrupted and eventually unusable. InCDMA-based systems, the effect of interference from other mobile transmitters in the same cell on coverage area is very marked and has a special name,cell breathing.
One can see examples of cell coverage by studying some of the coverage maps provided by real operators on their web sites or by looking at independently crowdsourced maps such asOpensignalorCellMapper. In certain cases they may mark the site of the transmitter; in others, it can be calculated by working out the point of strongest coverage.
Acellular repeateris used to extend cell coverage into larger areas. They range from wideband repeaters for consumer use in homes and offices to smart or digital repeaters for industrial needs.
The following table shows the dependency of the coverage area of one cell on the frequency of aCDMA2000network:[41]
Lists and technical information:
Starting with EVDO the following techniques can also be used to improve performance:
Equipment:
Other:
|
https://en.wikipedia.org/wiki/Frequency_reuse
|
This article contains terminology related toCDMAInternational Roaming. To quickly find a term, click on the first letter of the term below:
#|A|B|C|D|E|F|G|H|I|J|K|L|M|N|O|P|Q|R|S|T|U|V|W|X|Y|Z
1x– See1xRTT
1xEV-DO– cdma2000 Evolution, Data Optimized
1xRTT– cdma2000 Radio Transmission Technology
2G Authentication– SeeCAVE-based Authentication
3G Authentication– SeeAKA
3GPP2– Third Generation Partnership Project 2
A12 Authentication
AAA– Authentication, Authorization, and Accounting
AC– Authentication Center – SeeCAVE-based Authentication
Access Authentication
Acquisition_Table– SeePRL
Active Pilot– base station(s) currently serving a call. A base station usually has 3 pilot numbers. Also SeePN Offset.
AKA– Authentication and Key Agreement
A-key– Authentication Key – SeeCAVE-based Authentication
AMPS– Advanced Mobile Phone System
AN– Access Network
ANI– Automatic Number Identification
ANID– Access Network Identifiers
ANSI– American National Standards Institute
ANSI-41– SeeIS-41
ARP– Authorized Receipt Point
ARPU–Average revenue per user
AT– Access Terminal
Authentication
Authorization
Automatic Call Delivery
Automatic Roaming
Autonomous Registration
Band
Bandclass
BID– Billing Identification
Bilateral Roaming
BILLID– BillingID
Border System
Call Disconnect
Caller ID
Call Release
Call termination
Carrier
CAVE– Cellular Authentication and Voice Encryption
CAVE-based Authentication
CDG– CDMA Development Group
CDMA– Code Division Multiple Access
CDR– Call Detail Record
Cell site
CIBER– Cellular Intercarrier Billing Exchange Roamer
Cibernet
CHAP– Challenge- Handshake Authentication Protocol aka (HDR–High Data Rate)
Clearing
Clearinghouse
CLI– Calling Line Identification – SeeCaller ID
CLIP– Calling Line Identification Presentation – SeeCaller ID
CLLI– Common Language Location Identifier
Clone
Closed PRL– SeePRL
CoA– Care-of-Address – SeeMobile IP
CND– Caller Number Display – SeeCaller ID
CNID– Calling Number Identification – SeeCaller ID
CRX– CDMA Packet Data Roaming Exchange
CSCF– Call Session Control Function – SeeIMS
CTIA– Cellular Telecom. & Internet Association
D-AMPS– Digital Analog Mobile Phone Service
DES– Data Encryption Standard
Diameter
DO– See1xEV-DO
DRRR– Direct Routing for Roamer to Roamer
Dual-mode handset(i.e.dual-mode mobilephones)
eHRPD– Enhanced HRPD
EDI– Electronic Data Interchange
EDT– Electronic Data Transfer
Encryption
Enhanced PRL
ERI– Enhanced Roaming Indicator – SeeRoaming Indicator
ESA– Enhanced Subscriber Authentication – SeeAKA
ESN– Electronic Serial Number
ESPM– Extended System Parameters Message
EV-DO– See1xEV-DO
FA– Foreign Agent – SeeMobile IP
FCC– U.S.Federal Communications Commission
Financial Settlement
FOTA– Firmware Over-the-Air – SeeOTA
Frequency Block
Global Challenge– SeeCAVE-based Authentication
Global Title
GTT– Global Title Translation
HA– Home Agent – SeeMobile IP
Handoff (data)
Handoff (voice)
HLR– Home Location Register
Home Address– SeeMobile IP
Home System
HNI– Home Network Identifier – SeeIMSI
Home SID/NID List
HRPD– High Rate Packet Data – See1xEV-DO
HRPD Session
HSS– Home Subscriber Server – SeeIMS
Hybrid Device
ICCID– Integrated Circuit Card IDentifier (sim cardNumber)
IETF– Internet Engineering Task Force
IFAST– International Forum on ANSI-41 Standards Technology
IIF– Interworking and Interoperability Function
IMEI– International mobile equipment Identity
IMS– IP Multimedia Subsystem
IMSI– International Mobile Subscriber Identity
IMSI 11 12– Same asMNC(Mobile Network Code)
IMSI S– Short IMSI, Mobile Identification Number
Inbound Roamer
Industry Organizations
INF– Industry Negative File
Interconnection
Inter standard roaming
IRM– International roaming MIN
IS-2000– Superseded byTIA-2000
IS-41– Superseded byTIA-41
IS-835
IS-856– Superseded byTIA-856
IS-95
ISG– International Signaling Gateway
ISUP– Integrated Services User Part
ITU– International Telecommunication Union
J-STD-038
Key
L2TP– Layer 2 Tunneling Protocol
LAC– L2TP Access Concentrator – SeeL2TP
Line Range
LNS– L2TP Network Server – SeeL2TP
MABEL– Major Account Billing Exchange Logistical
Main Service Instance– SeeService Instance
MAP– Mobile Application Part – SeeTIA-41
MBI– MIN Block Identifier
MC– Message Center – SeeSMS
MCC– Mobile Country Code
MDN– MobileDirectory Number
ME– Mobile Equipment
MEID– Mobile Equipment Identifier
MIN– Mobile Identification Number
MIP– Mobile IP – SeeMobile IP
MMD– Multimedia Domain
MMS– Multimedia Messaging Service
MN– Mobile Node
MNC– Mobile Network Code
MN ID– Mobile Node Identifier – SeeA12 Authentication
Mobile IP
MS– Mobile Station
MSC– Mobile Switching Center
MSCID– Mobile Switching Center Identification
MSCIN– MSC Identification Number
MSID– Mobile Station Identity
MSIN– Mobile Subscription Identification Number, same asMIN
MSISDN– Mobile Station Integrated Services Digital Network Number
MSL– Master Subsidy Lock
MTSO–Mobile Telephone Switching Office– SeeMSC
Multi-Band Handset
Multi-Mode Handset
NAI– Network Access Identifier
NAM– Number Assignment Module
NANP– North American Numbering Plan
Negative System– SeePRL
Net Settlement
NID– Network Identification Number
NMSI– National Mobile Station Identity
NMSID– National Mobile Station IDentity, Same asNMSI
NPA-NXX– SeeNANP
OMA– Open Mobile Alliance
Open PRL– SeePRL
OTAPA– Over The Air Parameter Administration
OTASP– Over The Air Service Provisioning
OTA– Over-The-Air Programming
Outbound Roamer
PAP– Password Authentication Protocol
Packet Data Service
Packet Data Service Option
Packet Data Session
PCS– Personal Communications Services
PDSN– packet data serving node
Permissive Mode– SeePRL
PIN– Personal Identification Number – SeeRVR
Plus Code Dialing
PN Offset– Identifies a base station. As base station usually has 3 pilot numbers. Also SeeActive Pilot.
Point of Attachment– SeeMobile IP
PPP–Point-to-Point Protocol
PPP Service Instance– SeeService Instance
PPP Session– Point-to-Point Protocol Session
Preferred System– SeePRL
PRL– Preferred Roaming List
Profiling
PUZL– Preferred User Zone List
PZID– Packet Zone Identification
RADIUS– Remote Authentication Dial In User Service
RADIUS Server– SeeAAA
RAN– Radio Access Network – SeeAN
Restrictive Mode– SeePRL
RFC– Request For Comments
RN– Radio Network – SeeAN
RoamEx
Roaming
Roaming Agreement
Roaming Indicator
RSP– Roaming Service Provider
RUIM– Removable User Identity Module
RVR– Roamer Verification and Reinstatement
Sector ID
Service Instance
Service Option
Serving System
SID– System ID
SID/NID Lockout List
SIP– Session Initiation Protocol
SMS– Short Message Service
SMSC– Short Message Service Centre – SeeSMS
Soft Handoff
SO33– Service Option 33 – SeeService Option
SO59– Service Option 59 – SeeService Option
SPC– Service Programming Code, same asMSL(Master Subsidy Lock)
SPASM– Subscriber Parameter Administration Security Mechanism
SPC– Service Programming Code
Subnet ID
Supplementary Services
SSPR– System Selection for Preferred Roaming
System table– SeePRL
TDS– Technical Data Sheet
Telcordia
TIA– Telecommunications Industry Association
TIA-2000
TIA-41– Cellular Radio-Telecommunications Intersystem Operations
TIA-856
TIA-878
TLDN– Temporary Local Directory Number
TMSI– Temporary Mobile Station Identity
Triple DES– Triple Data Encryption Standard
Trading Partner Agreements
UDR– Usage Data Records
UIM– User Identity Module – SeeRUIM
UIMID– UIM Identifier – SeeRUIM
Unique Challenge– SeeCAVE-based Authentication
Verification– SeeRVR
Visited System
VLR–Visitor Location Register
WIN– Wireless Intelligent Network
WCDMA– Wideband Code Division Multiple Access
X0 Records– SeeCIBER
X2 Records– SeeCIBER
|
https://en.wikipedia.org/wiki/List_of_CDMA_terminology
|
Thenear–far problemorhearability problemis the effect of a strong signal from a near signal source in making it hard for a receiver to hear a weaker signal from a further source due toadjacent-channel interference,co-channel interference,distortion,capture effect,dynamic rangelimitation, or the like. Such a situation is common in wireless communication systems, in particularCDMA. In somesignal jammingtechniques, the near–far problem is exploited to disrupt ("jam") communications.
Consider a receiver and two transmitters, one close to the receiver, the other far away. If both transmitters transmit simultaneously and at equal powers, then due to theinverse square lawthe receiver will receive more power from the nearer transmitter. Since one transmission'ssignalis the other'snoise, thesignal-to-noise ratio(SNR) for the further transmitter is much lower. This makes the farther transmitter more difficult, if not impossible, to understand. In short, the near–far problem is one of detecting or filtering out a weaker signal amongst stronger signals.[1]
To place this problem in more common terms, imagine you are talking to someone 6 meters away. If the two of you are in a quiet, empty room then a conversation is quite easy to hold at normal voice levels. In a loud, crowded bar, it would be impossible to hear the same voice level, and the only solution (for that distance) is for both you and your friend to speak louder. Of course, this increases the overall noise level in the bar, and every other patron has to talk louder too (this is equivalent to power control runaway). Eventually, everyone has to shout to make themselves heard by a person standing right beside them, and it is impossible to communicate with anyone more than half a meter away. In general, however, a human is very capable of filtering out loud sounds; similar techniques can be deployed in signal processing where suitable criteria for distinguishing between signals can be established (seesignal processingand notablyadaptive signal processing).
Taking this analogy back to wireless communications, the far transmitter would have to drastically increase transmission power which simply may not be possible.
InCDMAsystems and similarcellular phone-like networks, the problem is commonly solved by dynamic output power adjustment of the transmitters. That is, the closer transmitters use less power so that the SNR for all transmitters at the receiver is roughly the same. This sometimes can have a noticeable impact on battery life, which can be dramatically different depending on distance from the base station. In high-noise situations, however, closer transmitters may boost their output power, which forces distant transmitters to boost their output to maintain a good SNR. Other transmitters react to the rising noise floor by increasing their output. This process continues, and eventually distant transmitters lose their ability to maintain a usable SNR and drop from the network. This process is calledpower control runaway. This principle may be used to explain why an area with low signal is perfectly usable when the cell isn't heavily loaded, but when load is higher, service quality degrades significantly, sometimes to the point of unusability.
Other possible solutions to the near–far problem:
|
https://en.wikipedia.org/wiki/Near%E2%80%93far_problem
|
Incryptography,pseudorandom noise(PRN[1]) is asignalsimilar tonoisewhich satisfies one or more of the standard tests forstatistical randomness. Although it seems to lack any definitepattern, pseudorandom noise consists of a deterministicsequenceofpulsesthat will repeat itself after its period.[2]
Incryptographic devices, the pseudorandom noise pattern is determined by akeyand the repetition period can be very long, even millions of digits.
Pseudorandom noise is used in someelectronic musical instruments, either by itself or as an input tosubtractive synthesis, and in manywhite noise machines.
Inspread-spectrumsystems, the receivercorrelatesa locally generated signal with the receivedsignal. Such spread-spectrum systems require a set of one or more "codes" or "sequences" such that
In adirect-sequence spread spectrumsystem, each bit in thepseudorandom binary sequenceis known as achipand theinverseof its period aschip rate;comparebit rateandsymbol rate.
In afrequency-hopping spread spectrumsequence, each value in the pseudorandom sequence is known as achannel numberand theinverseof its period as thehop rate.FCC Part 15mandates at least 50 different channels and at least a 2.5 Hz hop rate for narrow band frequency-hopping systems.
GPS satellites broadcast data at a rate of 50 data bits per second – each satellite modulates its data with one PN bit stream at 1.023 millionchips per secondand the same data with another PN bit stream at 10.23 million chips per second.GPSreceivers correlate the received PN bit stream with a local reference to measure distance. GPS is a receive-only system that uses relative timing measurements from several satellites (and the known positions of the satellites) to determine receiver position.
Otherrange-findingapplications involve two-way transmissions. A local station generates a pseudorandom bit sequence and transmits it to the remote location (using any modulation technique). Some object at the remote location echoes this PN signal back to the location station – either passively, as in some kinds of radar and sonar systems, or using an active transponder at the remote location, as in the ApolloUnified S-bandsystem.[3]By correlating a (delayed version of) the transmitted signal with the received signal, a precise round trip time to the remote location can be determined and thus the distance.
Apseudo-noise code(PN code) orpseudo-random-noise code(PRN code) is one that has a spectrum similar to arandom sequenceof bits but isdeterministicallygenerated. The most commonly used sequences indirect-sequence spread spectrumsystems aremaximal length sequences,Gold codes,Kasami codes, andBarker codes.[4]
|
https://en.wikipedia.org/wiki/PN_code
|
Intelecommunicationsandelectronics,baud(/bɔːd/; symbol:Bd) is a commonunit of measurementofsymbol rate, which is one of the components that determine thespeed of communicationover adata channel.
It is the unit for symbol rate ormodulationrate insymbols per secondorpulses per second. It is the number of distinctsymbolchanges (signalling events) made to thetransmission mediumper second in a digitally modulated signal or a bd rateline code.
Baud is related togross bit rate, which can be expressed inbits per second(bit/s).[1]If there are precisely two symbols in the system (typically 0 and 1), then baud and bits per second are equivalent.
The baud unit is named afterÉmile Baudot, the inventor of theBaudot codefortelegraphy, and is represented according to the rules forSI units.
That is, the first letter of its symbol is uppercase (Bd), but when the unit is spelled out, it should be written in lowercase (baud) except when it begins a sentence or is capitalized for another reason, such as in title case.
It was defined by the CCITT (now theITU-T) in November 1926. The earlier standard had been the number of words per minute, which was a less robust measure since word length can vary.[2]
The symbol duration time, also known as theunit interval, can be directly measured as the time between transitions by looking at aneye diagramof the signal on anoscilloscope. The symbol duration timeTscan be calculated as:
wherefsis the symbol rate.
There is also a chance of miscommunication which leads to ambiguity.
The baud is scaled using standardmetric prefixes, so that for example
The symbol rate is related togross bit rateexpressed in bit/s.
The term baud has sometimes incorrectly been used to meanbit rate,[3]since these rates are the same in oldmodemsas well as in the simplest digital communication links using only one bit per symbol, such that binary digit "0" is represented by one symbol, and binary digit "1" by another symbol. In more advanced modems and data transmission techniques, a symbol may have more than two states, so it may represent more than onebit. A bit (binary digit) always represents one of two states.
IfNbits are conveyed per symbol, and the gross bit rate isR, inclusive of channel coding overhead, the symbol ratefscan be calculated as
By taking information per pulseNin bit/pulse to be the base-2-logarithmof the number of distinct messagesMthat could be sent,Hartley[4]constructed a measure of thegross bit rateRas
Here, the⌈x⌉{\displaystyle \left\lceil x\right\rceil }denotes the ceiling function ofx{\displaystyle x}, wherex{\displaystyle x}is taken to be any real number greater than zero, then the ceiling function rounds up to the nearest natural number (e.g.⌈2.11⌉=3{\displaystyle \left\lceil 2.11\right\rceil =3}).
In that case,M= 2Ndifferent symbols are used. In a modem, these may be time-limited sinewave tones with unique combinations of amplitude, phase and/or frequency. For example, in a64QAMmodem,M= 64, and so the bit rate isN= log2(64) = 6times the baud rate. In a line code, these may beMdifferent voltage levels.
The ratio is not necessarily an integer; in4B3Tcoding, the bit rate is4/3of the baud rate. (A typicalbasic rate interfacewith a 160 kbit/s raw data rate operates at 120 kBd.)
Codes with many symbols, and thus a bit rate higher than the symbol rate, are most useful on channels such as telephone lines with a limitedbandwidthbut a highsignal-to-noise ratiowithin that bandwidth. In other applications, the bit rate is less than the symbol rate.Eight-to-fourteen modulationas used on audio CDs has bit rate8/17[a]of the baud rate.
|
https://en.wikipedia.org/wiki/Baud
|
GPS signalsare broadcast byGlobal Positioning Systemsatellites to enablesatellite navigation. Using these signals, receivers on or near the Earth's surface can determine their Position,Velocityand Time (PVT). The GPSsatellite constellationis operated by the2nd Space Operations Squadron(2SOPS) ofSpace Delta 8,United States Space Force.
GPS signals include ranging signals, which are used to measure the distance to the satellite, and navigation messages. The navigation messages includeephemerisdata which are used both intrilaterationto calculate the position of each satellite in orbit and also to provide information about the time and status of the entire satellite constellation, called thealmanac.
There are four GPS signal specifications designed for civilian use. In order of date of introduction, these are:L1 C/A,L2C,L5andL1C.[1]L1 C/A is also called thelegacy signaland is broadcast by all currently operational satellites. L2C, L5 and L1C aremodernized signalsand are only broadcast by newer satellites (or not yet at all). Furthermore, as of January 2021[update], none of these three signals are yet considered to be fully operational for civilian use. In addition to the four aforementioned signals, there arerestricted signalswith published frequencies andchiprates, but the signals use encrypted coding, restricting use to authorized parties. Some limited use of restricted signals can still be made by civilians without decryption; this is calledcodelessandsemi-codelessaccess, and this is officially supported.[2][3]
The interface to the User Segment (GPS receivers) is described in theInterface Control Documents (ICD). The format of civilian signals is described in theInterface Specification (IS)which is a subset of the ICD.
TheGPS satellites(calledspace vehiclesin the GPS interface specification documents) transmit simultaneously several ranging codes and navigation data usingbinary phase-shift keying(BPSK).
Only a limited number of central frequencies are used. Satellites using the same frequency are distinguished by using different ranging codes. In other words, GPS usescode-division multiple access. The ranging codes are also calledchipping codes(in reference to CDMA/DSSS),pseudorandom noiseandpseudorandom binary sequences(in reference to the fact that the sequences are predictable yet that they statistically resemble noise).
Some satellites transmit several BPSK streams at the same frequency in quadrature, in a form ofquadrature amplitude modulation. However, unlike typical QAM systems where a single bit stream is split into two, half-symbol-rate bit streams to improvespectral efficiency, the in-phase and quadrature components of GPS signals are modulated by separate (but functionally related) bit streams.
Satellites are uniquely identified by a serial number calledspace vehicle number(SVN) which does not change during its lifetime. In addition, all operating satellites are numbered with aspace vehicle identifier(SV ID) andpseudorandom noise number(PRN number) which uniquely identifies the ranging codes that a satellite uses. There is a fixedone-to-one correspondencebetween SV identifiers and PRN numbers described in the interface specification.[4]Unlike SVNs, the SV ID/PRN number of a satellite may be changed (resulting in a change to the ranging codes it uses). That is, no two active satellites can share any one active SV ID/PRN number. The current SVNs and PRN numbers for the GPS constellation are published atNAVCEN.
The original GPS design contains two ranging codes: thecoarse/acquisition(C/A) code, which is freely available to the public, and the restrictedprecision(P) code, usually reserved for military applications.
For the ranging codes and navigation message to travel from the satellite to the receiver, they must bemodulatedonto acarrier wave. In the case of the original GPS design, two frequencies are utilized; one at 1575.42MHz(10.23 MHz × 154) called L1; and a second at 1227.60 MHz (10.23 MHz × 120), called L2.
The C/A code is transmitted on the L1 frequency as a 1.023 MHz signal using a bi-phase shift keying (BPSK) modulation technique. The P(Y)-code is transmitted on both the L1 and L2 frequencies as a 10.23 MHz signal using the same BPSK modulation, however the P(Y)-code carrier is inquadraturewith the C/A carrier (meaning it is 90° out ofphase).
Besides redundancy and increased resistance to jamming, a critical benefit of having two frequencies transmitted from one satellite is the ability to measure directly, and therefore remove, theionospheric delayerror for that satellite. Without such a measurement, a GPS receiver must use a generic model or receive ionospheric corrections from another source (such as theWide Area Augmentation SystemorWAAS). Advances in the technology used on both the GPS satellites and the GPS receivers has made ionospheric delay the largest remaining source of error in the signal. A receiver capable of performing this measurement can be significantly more accurate and is typically referred to as adual frequency receiver.
The C/A PRN codes areGold codeswith a period of 1023 chips transmitted at 1.023 Mchip/s, causing the code to repeat every 1 millisecond. They areexclusive-oredwith a 50 bit/snavigation messageand the result phase modulates the carrier aspreviously described. These codes only match up, or stronglyautocorrelatewhen they are almost exactly aligned. Each satellite uses a unique PRN code, which does notcorrelatewell with any other satellite's PRN code. In other words, the PRN codes are highlyorthogonalto one another. The 1 ms period of the C/A code corresponds to 299.8 km of distance, and each chip corresponds to a distance of 293 m. Receivers track these codes well within one chip of accuracy, so measurement errors are considerably smaller than 293 m.[by how much?]
The C/A codes are generated by combining (using "exclusive or") two bit streams, each generated by two different maximal period 10 stagelinear-feedback shift registers(LFSR). Different codes are obtained by selectively delaying one of those bit streams. Thus:
where:
The arguments of the functions therein are the number ofbitsorchipssince their epochs, starting at 0. The epoch of the LFSRs is the point at which they are at the initial state; and for the overall C/A codes it is the start of any UTC second plus any integer number of milliseconds. The output of LFSRs at negative arguments is defined consistent with the period which is 1,023 chips (this provision is necessary becauseBmay have a negative argument using the above equation).
The delay for PRN numbers 34 and 37 is the same; therefore their C/A codes are identical and are not transmitted at the same time[5](it may make one or both of those signals unusable due to mutual interference depending on the relative power levels received on each GPS receiver).
The P-code is a PRN sequence much longer than the C/A code: 6.187104 x 1012chips. Even though the P-code chip rate (10.23 Mchip/s) is ten times that of the C/A code, it repeats only once per week, eliminating range ambiguity. It was assumed that receivers could not directly acquire such a long and fast code so they would first "bootstrap" themselves with the C/A code to acquire the spacecraftephemerides, produce an approximate time and position fix, and then acquire the P-code to refine the fix.
Whereas the C/A PRNs are unique for each satellite, each satellite transmits a different segment of a master P-code sequence approximately 2.35 x 1014chips long (235,000,000,000,000 chips). Each satellite repeatedly transmits its assigned segment of the master code, restarting every Sunday at 00:00:00 GPS time. For reference, the GPS epoch was Sunday January 6, 1980 at 00:00:00 UTC, but GPS does not follow UTC exactly because GPS time does not incorporate leap seconds. Thus, GPS time is ahead of UTC by an integer (whole) number of seconds.
The P code is public, so to prevent unauthorized users from using or potentially interfering with it throughspoofing, the P-code is XORed withW-code, a cryptographically generated sequence, to produce theY-code. The Y-code is what the satellites have been transmitting since theanti-spoofing modulewas enabled. The encrypted signal is referred to as theP(Y)-code.
The details of the W-code are secret, but it is known that it is applied to the P-code at approximately 500 kHz,[6]about 20 times slower than the P-code chip rate. This has led to semi-codeless approaches for tracking the P(Y) signal without knowing the W-code.
In addition to the PRN ranging codes, a receiver needs to know the time and position of each active satellite. GPS encodes this information into thenavigation messageandmodulatesit onto both the C/A and P(Y) ranging codes at 50 bit/s. The navigation message format described in this section is called LNAV data (forlegacy navigation).
The navigation message conveys information of three types:
An ephemeris is valid for only four hours, while an almanac is valid–with little dilution of precision–for up to two weeks.[7]The receiver uses the almanac to acquire a set of satellites based on stored time and location. As the receiver acquires each satellite, each satellite’s ephemeris is decoded so that the satellite can be used for navigation.
The navigation message consists of 30-secondframes1,500 bits long, divided into five 6-secondsubframesof ten 30-bit words each. Each subframe has the GPS time in 6-second increments. Subframe 1 contains the GPS date (week number), satellite clock correction information, satellite status and satellite health. Subframes 2 and 3 together contain the transmitting satellite's ephemeris data. Subframes 4 and 5 containpage1 through 25 of the 25-page almanac. The almanac is 15,000 bits long and takes 12.5 minutes to transmit.
A frame begins at the start of the GPS week and every 30 seconds thereafter. Each week begins with the transmission of almanac page 1.[8]
There are two navigation message types: LNAV-L is used by satellites with PRN numbers 1 to 32 (calledlower PRN numbers) and LNAV-U is used by satellites with PRN numbers 33 to 63 (calledupper PRN numbers).[9]The two types use very similar formats. Subframes 1 to 3 are the same,[10]while subframes 4 and 5 are almost the same. Each message type contains almanac data for all satellites using the same navigation message type but not the other.
Each subframe begins with a Telemetry Word (TLM), which enables the receiver to detect the beginning of a subframe and determine the receiver clock time at which the navigation subframe begins. Next is the handover word (HOW) giving the GPS time (as the time for when the first bit of the next subframe will be transmitted) and identifies the specific subframe within a complete frame.[11][12]The remaining eight words of the subframe contain the actual data specific to that subframe. Each word includes 6 bits of parity generated using an algorithm based on Hamming codes, which take into account the 24 non-parity bits of that word and the last 2 bits of the previous word.
After a subframe has been read and interpreted, the time the next subframe was sent can be calculated through the use of the clock correction data and HOW. The receiver knows the receiver clock time of when the beginning of the next subframe was received from detection of the Telemetry Word thereby enabling computation of the transit time and thus the pseudorange.
GPS time is expressed with a resolution of 1.5 seconds as a week number and a time of week count (TOW).[13]Its zero point (week 0, TOW 0) is defined to be 1980-01-06T00:00Z. The TOW count is a value ranging from 0 to 403,199 whose meaning is the number of 1.5 second periods elapsed since the beginning of the GPS week. Expressing TOW count thus requires 19 bits (219= 524,288). GPS time is a continuous time scale in that it does not include leap seconds; therefore the start/end of GPS weeks may differ from that of the corresponding UTC day by an integer (whole) number of seconds.
In each subframe, each hand-over word (HOW) contains the most significant 17 bits of the TOW count corresponding to the start of the next following subframe.[14]Note that the 2 least significant bits can be safely omitted because one HOW occurs in the navigation message every 6 seconds, which is equal to the resolution of the truncated TOW count thereof. Equivalently, the truncated TOW count is the time duration since the last GPS week start/end to the beginning of the next frame in units of 6 seconds.
Each frame contains (in subframe 1) the 10 least significant bits of the corresponding GPS week number.[15]Note that each frame is entirely within one GPS week because GPS frames do not cross GPS week boundaries.[16]Sincerolloveroccurs every 1,024 GPS weeks (approximately every 19.6 years; 1,024 is 210), a receiver that computes current calendar dates needs to deduce the upper week number bits or obtain them from a different source. One possible method is for the receiver to save its current date in memory when shut down, and when powered on, assume that the newly decoded truncated week number corresponds to the period of 1,024 weeks that starts at the last saved date. This method correctly deduces the full week number if the receiver is never allowed to remain shut down (or without a time and position fix) for more than 1,024 weeks (~19.6 years).
Thealmanacconsists of coarse orbit and status information for each satellite in the constellation, anionospheric model, and information to relate GPS derived time toCoordinated Universal Time(UTC). Each frame contains a part of the almanac (in subframes 4 and 5) and the complete almanac is transmitted by each satellite in 25 frames total (requiring 12.5 minutes).[17]The almanac serves several purposes. The first is to assist in the acquisition of satellites at power-up by allowing the receiver to generate a list of visible satellites based on stored position and time, while an ephemeris from each satellite is needed to compute position fixes using that satellite. In older hardware, lack of an almanac in a new receiver would cause long delays before providing a valid position, because the search for each satellite was a slow process. Advances in hardware have made the acquisition process much faster, so not having an almanac is no longer an issue. The second purpose is for relating time derived from the GPS (called GPS time) to the international time standard ofUTC. Finally, the almanac allows a single-frequency receiver to correct forionospheric delayerror by using a global ionospheric model. The corrections are not as accurate asGNSS augmentationsystems likeWAASor dual-frequency receivers. However, it is often better than no correction, since ionospheric error is the largest error source for a single-frequency GPS receiver.
Satellite data is updated typically every 24 hours, with up to 60 days data loaded in case there is a disruption in the ability to make updates regularly. Typically the updates contain new ephemerides, with new almanacs uploaded less frequently.The Control Segmentguarantees that during normal operations a new almanac will be uploaded at least every 6 days.
Satellites broadcast a new ephemeris every two hours. The ephemeris is generally valid for 4 hours, with provisions for updates every 4 hours or longer in non-nominal conditions. The time needed to acquire the ephemeris is becoming a significant element of the delay to first position fix, because as the receiver hardware becomes more capable, the time to lock onto the satellite signals shrinks; however, the ephemeris data requires 18 to 36 seconds before it is received, due to the low data transmission rate.
Having reached full operational capability on July 17, 1995[20]the GPS system had completed its original design goals. However, additional advances in technology and new demands on the existing system led to the effort to "modernize" the GPS system. Announcements from the Vice President and the White House in 1998 heralded the beginning of these changes, and in 2000, the U.S. Congress reaffirmed the effort, referred to asGPS III.
The project involves new ground stations and new satellites, with additional navigation signals for both civilian and military users. It aims to improve the accuracy and availability for all users. The implementation goal of 2013 was established, and contractors were offered incentives if they could complete it by 2011.
Modernized GPS civilian signals have two general improvements over their legacy counterparts: a dataless acquisition aid and forward error correction (FEC) coding of the NAV message.
A dataless acquisition aid is an additional signal, called a pilot carrier in some cases, broadcast alongside the data signal. This dataless signal is designed to be easier to acquire than the data encoded and, upon successful acquisition, can be used to acquire the data signal. This technique improves acquisition of the GPS signal and boosts power levels at the correlator.
The second advancement is to use forward error correction (FEC) coding on the NAV message itself. Due to the relatively slow transmission rate of NAV data (usually 50 bits per second), small interruptions can have potentially large impacts. Therefore, FEC on the NAV message is a significant improvement in overall signal robustness.
One of the first announcements was the addition of a new civilian-use signal, to be transmitted on a frequency other than the L1 frequency used for the coarse/acquisition (C/A) signal. Ultimately, this became the L2C signal, so called because it is broadcast on the L2 frequency. Because it requires new hardware on board the satellite, it is only transmitted by the so-called Block IIR-M and later design satellites. The L2C signal is tasked with improving accuracy of navigation, providing an easy to track signal, and acting as a redundant signal in case of localized interference. L2C signals have been broadcast beginning in April 2014 on satellites capable of broadcasting it, but are still considered pre-operational.[1]As of July 2023[update], L2C is broadcast on 25 satellites.[1]
Unlike the C/A code, L2C contains two distinct PRN code sequences to provide ranging information; thecivil-moderatecode (called CM), and thecivil-longlength code (called CL). The CM code is 10,230 chips long, repeating every 20 ms. The CL code is 767,250 chips long, repeating every 1,500 ms. Each signal is transmitted at 511,500 chips per second (chip/s); however, they aremultiplexedtogether to form a 1,023,000-chip/s signal.
CM ismodulatedwith the CNAV Navigation Message (see below), whereas CL does not contain any modulated data and is called adataless sequence. The long, dataless sequence provides for approximately 24 dB greater correlation (~250 times stronger) than L1 C/A-code.
When compared to the C/A signal, L2C has 2.7 dB greater data recovery and 0.7 dB greater carrier-tracking, although its transmission power is 2.3 dB weaker.
The current status of the L2C signal as of July 3, 2023[21]is:
The civil-moderate and civil-long ranging codes are generated by amodular LFSRwhich is reset periodically to a predetermined initial state. The period of the CM and CL is determined by this resetting and not by the natural period of the LFSR (as is the case with the C/A code). The initial states are designated in the interface specification and are different for different PRN numbers and for CM/CL. The feedback polynomial/mask is the same for CM and CL. The ranging codes are thus given by:
where:
The initial states are described in the GPS interface specification as numbers expressed in octal following the convention that the LFSR state is interpreted as the binary representation of a number where the output bit is the least significant bit, and the bit where new bits are shifted in is the most significant bit. Using this convention, the LFSR shifts from most significant bit to least significant bit and when seen in big endian order, it shifts to the right. The states calledfinal statein the IS are obtained after10229cycles for CM and after767249cycles for LM (just before reset in both cases).
The CNAV data is an upgraded version of the original NAV navigation message. It contains higher precision representation and nominally more accurate data than the NAV data. The same type of information (time, status, ephemeris, and almanac) is still transmitted using the new CNAV format; however, instead of using a frame / subframe architecture, it uses a newpseudo-packetizedformat made of 12-second 300-bitmessagesanalogous to LNAV frames. While LNAV frames have a fixed information content, CNAV messages may be of one of several defined types. The type of a frame determines its information content. Messages do not follow a fixed schedule regarding which message types will be used, allowing the Control Segment some versatility. However, for some message types there are lower bounds on how often they will be transmitted.
In CNAV, at least 1 out of every 4 packets are ephemeris data and the same lower bound applies for clock data packets.[25]The design allows for a wide variety of packet types to be transmitted. With a 32-satellite constellation, and the current requirements of what needs to be sent, less than 75% of the bandwidth is used. Only a small fraction of the available packet types have been defined; this enables the system to grow and incorporate advances without breaking compatibility.
There are many important changes in the new CNAV message:
CNAV messages begin and end at start/end of GPS week plus an integer multiple of 12 seconds.[26]Specifically, the beginning of the first bit (with convolution encoding already applied) to contain information about a message matches the aforesaid synchronization. CNAV messages begin with an 8-bit preamble which is a fixed bit pattern and whose purpose is to enable the receiver to detect the beginning of a message.
Theconvolutional codeused to encode CNAV is described by:
where:
Since the FEC encoded bit stream runs at 2 times the rate than the non FEC encoded bit as already described, thent=⌊t′2⌋{\displaystyle t=\left\lfloor {\tfrac {t'}{2}}\right\rfloor }. FEC encoding is performed independently of navigation message boundaries;[27]this follows from the above equations.
An immediate effect of having two civilian frequencies being transmitted is the civilian receivers can now directly measure the ionospheric error in the same way as dual frequency P(Y)-code receivers. However, users utilizing the L2C signal alone, can expect 65% more position uncertainty due to ionospheric error than with the L1 signal alone.[28]
A major component of the modernization process is a new military signal (on L1M and L2M). Called the Military code, or M-code, it was designed to further improve the anti-jamming and secure access of the military GPS signals.
Very little has been published about this new, restricted code. It contains a PRN code of unknown length transmitted at 5.115 MHz. Unlike the P(Y)-code, the M-code is designed to be autonomous, meaning that a user can calculate their position using only the M-code signal. From the P(Y)-code's original design, users had to first lock onto the C/A code and then transfer the lock to the P(Y)-code. Later, direct-acquisition techniques were developed that allowed some users to operate autonomously with the P(Y)-code.
A little more is known about the new navigation message, which is calledMNAV. Similar to the new CNAV, this new MNAV is packeted instead of framed, allowing for very flexible data payloads. Also like CNAV it can utilize Forward Error Correction (FEC) and advanced error detection (such as aCRC).
The M-code is transmitted in the same L1 and L2 frequencies already in use by the previous military code, the P(Y)-code. The new signal is shaped to place most of its energy at the edges (away from the existing P(Y) and C/A carriers). It does not work at every satellite, and M-code was switched off for SVN62/PRN25 on 5 April 2011.[29]
In a major departure from previous GPS designs, the M-code is intended to be broadcast from a high-gain directional antenna, in addition to a full-Earth antenna. This directional antenna's signal, called a spot beam, is intended to be aimed at a specific region (several hundred kilometers in diameter) and increase the local signal strength by 20 dB, or approximately 100 times stronger. A side effect of having two antennas is that the GPS satellite will appear to be two GPS satellites occupying the same position to those inside the spot beam. While the whole Earth M-code signal is available on the Block IIR-M satellites, the spot beam antennas will not be deployed until theBlock III satellitesare deployed, which began in December 2018.
An interesting side effect of having each satellite transmit four separate signals is that the MNAV can potentially transmit four different data channels, offering increased data bandwidth.
The modulation method isbinary offset carrier, using a 10.23 MHz subcarrier against the 5.115 MHz code. This signal will have an overall bandwidth of approximately 24 MHz, with significantly separated sideband lobes. The sidebands can be used to improve signal reception.
The L5 signal provides a means of radionavigation secure and robust enough for life critical applications, such as aircraft precision approach guidance. The signal is broadcast in a frequency band protected by theITUforaeronautical radionavigation services. It was first demonstrated from satelliteUSA-203(Block IIR-M), and is available on all satellites fromGPS IIFandGPS III. L5 signals have been broadcast beginning in April 2014 on satellites that support it.[1]
The status of the L5 signal as of July 3, 2023[update]is:[30]
The L5 band provides additional robustness in the form of interference mitigation, the band being internationally protected, redundancy with existing bands, geostationary satellite augmentation, and ground-based augmentation. The added robustness of this band also benefits terrestrial applications.[31]
Two PRN ranging codes are transmitted on L5 in quadrature: the in-phase code (calledI5-code) and thequadrature-phasecode (calledQ5-code). Both codes are 10,230 chips long, transmitted at 10.23 Mchip/s (1 ms repetition period), and are generated identically (differing only in initial states). Then, I5 is modulated (by exclusive-or) with navigation data (called L5 CNAV) and a 10-bitNeuman-Hofman codeclocked at 1 kHz. Similarly, the Q5-code is then modulated but with only a 20-bit Neuman-Hofman code that is also clocked at 1 kHz.
Compared to L1 C/A and L2, these are some of the changes in L5:
The I5-code and Q5-code are generated using the same structure but with different parameters. These codes are the combination (by exclusive-or) of the output of 2 differing linear-feedback shift registers (LFSRs) which are selectively reset.
where:
AandBare maximal length LFSRs. The modulo operations correspond to resets. Note that both are reset each millisecond (synchronized withC/A codeepochs). In addition, the extra modulo operation in the description ofAis due to the fact it is reset 1 cycle before its natural period (which is 8,191) so that the next repetition becomes offset by 1 cycle with respect toB[33](otherwise, since both sequences would repeat, I5 and Q5 would repeat within any 1 ms period as well, degrading correlation characteristics).
The L5 CNAV data includes SV ephemerides, system time, SV clock behavior data, status messages and time information, etc. The 50 bit/s data is coded in a rate 1/2 convolution coder. The resulting 100 symbols per second (sps) symbol stream is modulo-2 added to the I5-code only; the resultant bit-train is used to modulate the L5 in-phase (I5) carrier. This combined signal is called the L5 Data signal. The L5 quadrature-phase (Q5) carrier has no data and is called the L5 Pilot signal. The format used for L5 CNAV is very similar to that of L2 CNAV. One difference is that it uses 2 times the data rate. The bit fields within each message,[34]message types, and forward error correction code algorithm are the same as those ofL2 CNAV. L5 CNAV messages begin and end at start/end of GPS week plus an integer multiple of 6 seconds (this applies to the beginning of the first bit to contain information about a message, as is the case for L2 CNAV).[35]
Broadcast on the L5 frequency (1176.45 MHz, 10.23 MHz × 115), which is anaeronauticalnavigation band. The frequency was chosen so that the aviation community can manage interference to L5 more effectively than L2.[35]
L1C is a civilian-use signal, broadcast on the L1 frequency (1575.42 MHz), which contains the C/A signal used by all current GPS users. The L1C signals broadcast from GPS III and later satellites, the first of which was launched in December 2018.[1]As of 2024[update], L1C signals are broadcast, and only four operational satellites are capable of broadcasting them. L1C is expected on 24 GPS satellites in the late 2020s.[1]Mediatek devices support decoding it.
L1C consists of a pilot (called L1CP) and a data (called L1CD) component.[36]These components use carriers with the same phase (within a margin of error of 100milliradians), instead of carriers in quadrature as with L5.[37]The PRN codes are 10,230 chips long and transmitted at 1.023 Mchip/s, thus repeating in 10 ms. The pilot component is also modulated by an overlay code called L1CO(a secondary code that has a lower rate than the ranging code and is also predefined, like the ranging code).[36]Of the total L1C signal power, 25% is allocated to the data and 75% to the pilot. The modulation technique used isBOC(1,1) for the data signal and TMBOC for the pilot. The timemultiplexed binary offset carrier(TMBOC) is BOC(1,1) for all except 4 of 33 cycles, when it switches to BOC(6,1).
The current status of the L1C signal as of July 3, 2023[21]is:
The L1C pilot and data ranging codes are based on aLegendre sequencewith length10223used to build an intermediate code (called aWeil code) which is expanded with a fixed 7-bit sequence to the required 10,230 bits. This 10,230-bit sequence is the ranging code and varies between PRN numbers and between the pilot and data components. The ranging codes are described by:[38]
where:
According to the formula above and the GPS IS, the firstwi{\displaystyle w_{i}}bits (equivalently, up to the insertion point ofS{\displaystyle S}) ofL1Ci′{\displaystyle {\text{L1C}}'_{i}}andL1C{\displaystyle {\text{L1C}}}are the first bits the corresponding Weil code; the next 7 bits areS{\displaystyle S}; the remaining bits are the remaining bits of the Weil code.
The IS asserts that0≤pi′≤10222{\displaystyle 0\leq p'_{i}\leq 10\,222}.[40]For clarity, the formula forL1Ci′{\displaystyle {\text{L1C}}'_{i}}does not account for the hypothetical case in whichpi′>10222{\displaystyle p'_{i}>10\,222}, which would cause the instance ofS{\displaystyle S}inserted intoL1Ci′{\displaystyle {\text{L1C}}'_{i}}to wrap from index10229to 0.
The overlay codes are 1,800 bits long and is transmitted at 100 bit/s, synchronized with the navigation message encoded in L1CD.
For PRN numbers 1 to 63 they are the truncated outputs of maximal period LFSRs which vary in initial conditions and feedback polynomials.[41]
For PRN numbers 64 to 210 they are truncated Gold codes generated by combining 2 LFSR outputs (S1i{\displaystyle {\text{S1}}_{i}}andS2i{\displaystyle {\text{S2}}_{i}}, wherei{\displaystyle i}is the PRN number) whose initial state varies.S1i{\displaystyle {\text{S1}}_{i}}has one of the 4 feedback polynomials used overall (among PRN numbers 64–210).S2i{\displaystyle {\text{S2}}_{i}}has the same feedback polynomial for all PRN numbers in the range 64–210.[42]
The L1C navigation data (called CNAV-2) is broadcast in 1,800 bits long (including FEC) frames and is transmitted at 100 bit/s.
The frames of L1C are analogous to the messages of L2C and L5. WhileL2 CNAVandL5 CNAVuse a dedicated message type for ephemeris data, all CNAV-2 frames include that information.
The common structure of all messages consists of 3 frames, as listed in the adjacent table. The content of subframe 3 varies according to its page number which is analogous to the type number of L2 CNAV and L5 CNAV messages. Pages are broadcast in an arbitrary order.[43]
The time of messages (not to be confused with clock correction parameters) is expressed in a different format thanthe formatof the previous civilian signals. Instead it consists of 3 components:
TOI is the only content of subframe 1. The week number and ITOW are contained in subframe 2 along with other information.
Subframe 1 is encoded by a modifiedBCH code. Specifically, the 8 least significant bits are BCH encoded to generate 51 bits, then combined usingexclusive orwith the most significant bit and finally the most significant bit is appended as the most significant bit of the previous result to obtain the final 52 bits.[44]Subframes 2 and 3 are individually expanded with a 24-bitCRC, then individually encoded using alow-density parity-check code, and theninterleavedas a single unit using a block interleaver.[45]
All satellites broadcast at the same two frequencies, 1.57542 GHz (L1 signal) and 1.2276 GHz (L2 signal). The satellite network uses a CDMA spread-spectrum technique where the low-bitrate message data is encoded with a high-ratepseudo-random noise(PRN) sequence that is different for each satellite. The receiver must be aware of the PRN codes for each satellite to reconstruct the actual message data. The C/A code, for civilian use, transmits data at 1.023 millionchipsper second, whereas the P code, for U.S. military use, transmits at 10.23 million chips per second. The L1 carrier is modulated by both the C/A and P codes, while the L2 carrier is only modulated by the P code.[47]The P code can be encrypted as a so-called P(Y) code which is only available to military equipment with a proper decryption key. Both the C/A and P(Y) codes impart the precise time-of-day to the user.
Each composite signal (in-phase and quadrature phase) becomes:
wherePI{\displaystyle \scriptstyle P_{\operatorname {I} }}andPQ{\displaystyle \scriptstyle P_{\operatorname {Q} }}represent signal powers;XI(t){\displaystyle \scriptstyle X_{\operatorname {I} }(t)}andXQ(t){\displaystyle \scriptstyle X_{\operatorname {Q} }(t)}represent codes with/without data(=±1){\displaystyle \scriptstyle (=\;\pm 1)}. This is a formula for the ideal case (which is not attained in practice) as it does not model timing errors, noise, amplitude mismatch between components or quadrature error (when components are not exactly in quadrature).
A GPS receiver processes the GPS signals received on its antenna to determine position, velocity and/or timing. The signal at antenna is amplified, down converted to baseband or intermediate frequency, filtered (to remove frequencies outside the intended frequency range for the digital signal that would alias into it) and digitalized; these steps may be chained in a different order. Note that aliasing is sometimes intentional (specifically, whenundersamplingis used) but filtering is still required to discard frequencies not intended to be present in the digital representation.
For each satellite used by the receiver, the receiver must firstacquirethe signal and thentrackit as long as that satellite is in use; both are performed in the digital domain in by far most (if not all) receivers.
Acquiring a signal is the process of determining the frequency and code phase (both relative to receiver time) when it was previously unknown. Code phase must be determined within an accuracy that depends on the receiver design (especially the tracking loop); 0.5 times the duration of code chips (approx. 0.489 μs) is a representative value.
Tracking is the process of continuously adjusting the estimated frequency and phase to match the received signal as close as possible and therefore is aphase locked loop. Note that acquisition is performed to start using a particular satellite, but tracking is performed as long as that satellite is in use.
In this section, one possible procedure is described for L1 C/A acquisition and tracking, but the process is very similar for the other signals. The described procedure is based on computing thecorrelationof the received signal with a locally generated replica of the ranging code and detecting the highest peak or lowest valley. The offset of the highest peak or lowest valley contains information about the code phase relative to receiver time. The duration of the local replica is set by receiver design and is typically shorter than the duration of navigation data bits, which is 20 ms.
Acquisition of a given PRN number can be conceptualized as searching for a signal in a bidimensional search space where the dimensions are (1) code phase, (2) frequency. In addition, a receiver may not know which PRN number to search for, and in that case a third dimension is added to the search space: (3) PRN number.
If the almanac information has previously been acquired, the receiver picks which satellites to listen for by their PRNs. If the almanac information is not in memory, the receiver enters a search mode and cycles through the PRN numbers until a lock is obtained on one of the satellites. To obtain a lock, it is necessary that there be an unobstructed line of sight from the receiver to the satellite. The receiver can then decode the almanac and determine the satellites it should listen for. As it detects each satellite's signal, it identifies it by its distinct C/A code pattern.
The simplest way to acquire the signal (not necessarily the most effective or least computationally expensive) is to compute thedot productof a window of the digitalized signal with a set of locally generated replicas. The locally generated replicas vary in carrier frequency and code phase to cover all the already mentioned search space which is theCartesian productof the frequency search space and the code phase search space. The carrier is a complex number where real and imaginary components are bothsinusoidsas described byEuler's formula. The replica that generates the highest magnitude of dot product is likely the one that best matches the code phase and frequency of the signal; therefore, if that magnitude is above a threshold, the receiver proceeds to track the signal or further refine the estimated parameters before tracking. The threshold is used to minimize false positives (apparently detecting a signal when there is in fact no signal), but some may still occur occasionally.
Using a complex carrier allows the replicas to match the digitalized signal regardless of the signal's carrier phase and to detect that phase (the principle is the same used by theFourier transform). The dot product is a complex number; its magnitude represents the level of similarity between the replica and the signal, as with an ordinarycorrelationof real-valued time series. Theargumentof the dot product is an approximation of the corresponding carrier in the digitalized signal.
As an example, assume that the granularity for the search in code phase is 0.5 chips and in frequency is 500 Hz, then there are1,023/0.5=2,046 code phasesand10,000 Hz/500 Hz=20 frequenciesto try for a total of20×2,046=40,920 local replicas. Note that each frequency bin is centered on its interval and therefore covers 250 Hz in each direction; for example, the first bin has a carrier at −4.750 Hz and covers the interval −5,000 Hz to −4,500 Hz. Code phases are equivalent modulo 1,023 because the ranging code is periodic; for example, phase −0.5 is equivalent to phase 1,022.5.
The following table depicts the local replicas that would be compared against the digitalized signal in this example. "•" means a single local replica while "..." is used for elided local replicas:
As an improvement over the simple correlation method, it is possible to implement the computation of dot products more efficiently with aFourier transform. Instead of performing one dot product for each element in the Cartesian product of code and frequency, a single operation involvingFFTand covering all frequencies is performed for each code phase; each such operation is more computationally expensive, but it may still be faster overall than the previous method due to the efficiency of FFT algorithms, and it recovers carrier frequency with a higher accuracy, because the frequency bins are much closely spaced in aDFT.
Specifically, for all code phases in the search space, the digitalized signal window is multiplied element by element with a local replica of the code (with no carrier), then processed with adiscrete Fourier transform.
Given the previous example to be processed with this method, assume real-valued data (as opposed to complex data, which would have in-phase and quadrature components), a sampling rate of 5 MHz, a signal window of 10 ms, and an intermediate frequency of 2.5 MHz. There will be 5 MHz×10 ms=50,000 samples in the digital signal, and therefore 25,001 frequency components ranging from 0 Hz to 2.5 MHz in steps of 100 Hz (note that the 0 Hz component is real because it is the average of a real-valued signal and the 2.5 MHz component is real as well because it is thecritical frequency). Only the components (or bins) within 5 kHz of the central frequency are examined, which is the range from 2.495 MHz to 2.505 MHz, and it is covered by51 frequency components. There are2,046 code phasesas in the previous case, thus in total51×2,046=104,346 complex frequency componentswill be examined.
Likewise, as an improvement over the simple correlation method, it is possible to perform a single operation covering all code phases for each frequency bin. The operation performed for each code phase bin involves forward FFT, element-wise multiplication in the frequency domain. inverse FFT, and extra processing so that overall, it computes circularcorrelationinstead of circularconvolution. This yields more accuratecode phase determinationthan thesimple correlation methodin contrast with the previous method, which yields more accuratecarrier frequency determinationthan the previous method.
Since the carrier frequency received can vary due toDopplershift, the points where received PRN sequences begin may not differ from O by an exact integral number of milliseconds. Because of this, carrier frequency tracking along with PRN code tracking are used to determine when the received satellite's PRN code begins.[48]Unlike the earlier computation of offset in which trials of all 1,023 offsets could potentially be required, the tracking to maintain lock usually requires shifting of half a pulse width or less. To perform this tracking, the receiver observes two quantities, phase error and received frequency offset. The correlation of the received PRN code with respect to the receiver generated PRN code is computed to determine if the bits of the two signals are misaligned. Comparisons of the received PRN code with receiver generated PRN code shifted half a pulse width early and half a pulse width late are used to estimate adjustment required.[49]The amount of adjustment required for maximum correlation is used in estimating phase error. Received frequency offset from the frequency generated by the receiver provides an estimate of phase rate error. The command for the frequency generator and any further PRN code shifting required are computed as a function of the phase error and the phase rate error in accordance with the control law used. The Doppler velocity is computed as a function of the frequency offset from the carrier nominal frequency. The Doppler velocity is the velocity component along the line of sight of the receiver relative to the satellite.
As the receiver continues to read successive PRN sequences, it will encounter a sudden change in the phase of the 1,023-bit received PRN signal. This indicates the beginning of a data bit of the navigation message.[50]This enables the receiver to begin reading the 20 millisecond bits of the navigation message. The TLM word at the beginning of each subframe of a navigation frame enables the receiver to detect the beginning of a subframe and determine the receiver clock time at which the navigation subframe begins. The HOW word then enables the receiver to determine which specific subframe is being transmitted.[11][12]There can be a delay of up to 30 seconds before the first estimate of position because of the need to read the ephemeris data before computing the intersections of sphere surfaces.
After a subframe has been read and interpreted, the time the next subframe was sent can be calculated through the use of the clock correction data and the HOW. The receiver knows the receiver clock time of when the beginning of the next subframe was received from detection of the Telemetry Word thereby enabling computation of the transit time and thus the pseudorange. The receiver is potentially capable of getting a new pseudorange measurement at the beginning of each subframe or every 6 seconds.
Then the orbital position data, orephemeris, from the navigation message is used to calculate precisely where the satellite was at the start of the message. A more sensitive receiver will potentially acquire the ephemeris data more quickly than a less sensitive receiver, especially in a noisy environment.[51]
GPS Interface Specification
|
https://en.wikipedia.org/wiki/GPS_signals
|
Intelecommunication technology, aBarker codeorBarker sequenceis a finite sequence of digital values with the idealautocorrelationproperty. It is used as a synchronising pattern between the sender and receiver of a stream of bits.
Binary digitshave very little meaning unless the significance of the individual digits is known. The transmission of a pre-arranged synchronising pattern of digits can enable asignalto be regenerated by areceiverwith a low probability of error. In simple terms it is equivalent to tying a label to one digit after which others may be related by counting. This is achieved by transmitting a special pattern of digits which is unambiguously recognised by the receiver. The longer the pattern the more accurately the data can besynchronisedand errors due todistortionomitted. These patterns are called Barker sequences or Barker codes, after the inventorRonald Hugh Barker. The process is described in "Group Synchronisation of Binary Digital Systems" published in 1953.[1]These sequences were initially developed forradar,telemetry, and digital speech encryption in the 1940s and 1950s.
During and after WWII digital technology became a key subject for research e.g. for radar, missile and gun fire control and encryption. In the 1950s scientists were trying various methods around the world to reduce errors in transmissions using code and to synchronise the received data. The problem being transmission noise, time delay and accuracy of received data. In 1948 the mathematicianClaude Shannonpublished an article '"A mathematical Theory of Communication"' which laid out the basic elements ofcommunication. In it he discusses the problems ofnoise.
Shannon realised that “communication signals must be treated in isolation from the meaning of the messages that they transmit” and laid down the theoretical foundations fordigital circuits. “The problem of communication was primarily viewed as a deterministic signal-reconstruction problem: how to transform a received signal, distorted by the physical medium, to reconstruct the original as accurately as possible”[2]or see original.[3]In 1948 electronics was advancing fast but the problem of receiving accurate data had not. This is demonstrated in an article on Frequency Shift Keying published by Wireless World.[4]In 1953 RH Barker published a paper demonstrating how this problem to synchronise the data in transmissions could be overcome. The process is described in “Group Synchronisation of Binary Digital Systems”. When used in data transmissions the receiver can read and if necessary correct the data to be error free byautocorrelationandcross correlationby achieving zero autocorrelation except at the incidence position using specific codes. The Barker sequence process at the time produced great interest, particularly in the United States as his method solved the problem, initiating a huge leap forward intelecommunications. The process has remained at the forefront of radar,data transmissionand telemetry and is now a very well known industry standard, still being researched in many technology fields.
“In a pioneering examination of group synchronization of binary digital systems, Barker reasoned it would be desirable to start with an autocorrelation function having very low sidelobes. The governing code pattern, he insisted, could be unambiguously recognized by thedetector. To assure this premise, Barker contended the selected pattern should be sufficiently unlikely to occur by chance, in a random series of noise generated bits”[5]
ABarker codeorBarker sequenceis a finite sequence ofNvalues of +1 and −1,
with the ideal autocorrelation property, such that the off-peak (non-cyclic)autocorrelationcoefficients
are as small as possible:
for all1≤v<N{\displaystyle 1\leq v<N}.[1]
Only nine Barker sequences[6]are known, all of lengthNat most 13.[7]Barker's 1953 paper asked for sequences with the stronger condition
Only four such sequences are known, shown in bold in the table below.[8]
Here is a table of all known Barker codes, where negations and reversals of the codes have been omitted. A Barker code has a maximum autocorrelation sequence which has sidelobes no larger than 1. It is generally accepted that no other perfect binary phase codes exist.[9][10](It has been proven that there are no further odd-length codes,[11]nor even-length codes ofN< 1022.[12])
Barker codes of lengthNequal to 11 and 13 are used indirect-sequence spread spectrumandpulse compression radarsystems because of their low autocorrelation properties (the sidelobe level of amplitude of the Barker codes is 1/Nthat of the peak signal).[15]A Barker code resembles a discrete version of a continuouschirp, another low-autocorrelation signal used in other pulse compression radars.
The positive and negative amplitudes of the pulses forming the Barker codes imply the use of biphase modulation or binaryphase-shift keying; that is, thechange of phasein thecarrier waveis 180 degrees.
Similar to the Barker codes are thecomplementary sequences, which cancel sidelobes exactly when summed; the even-length Barker code pairs are also complementary pairs. There is a simple constructive method to create arbitrarily long complementary sequences.
For the case of cyclic autocorrelation, other sequences have the same property of having perfect (and uniform) sidelobes, such as prime-lengthLegendre sequences,Zadoff–Chu sequences(used in 3rd- and 4th-generation cellular radio) and2n−1{\displaystyle 2^{n}-1}maximum length sequences(MLS). Arbitrarily long cyclic sequences can be constructed.
In wireless communications, sequences are usually chosen for their spectral properties and for low cross correlation with other sequences likely to interfere. In the 802.11 standard, an 11-chip Barker sequence is used for the 1 and 2 Mbit/s rates. The value of the autocorrelation function for the Barker sequence is 0 or −1 at all offsets except zero, where it is +11. This makes for a more uniform spectrum, and better performance in the receivers.[16]
Applications of Barker codes are found inradar,[17]mobile phone,[18]telemetry,[19]ultrasoundimaging and testing,[20][21]GPS,[22]andWi-Fi.[23]
Many of these technologies useDSSS. This technique incorporates Barker code to improve the received signal quality and improve security.[24]
These codes also used in radio frequency identificationRFID. Some examples where Barker code is used are: pet and livestock tracking, bar code scanners, inventory management, vehicle, parcel, asset and equipment tracking, inventory control, cargo and supply chain logistics.[25]It is also used extensively forIntelligent Transport Systems(ITS) i.e. for vehicle guidance[26]
Barker's algorithm is an alternative to Metropolis–Hastings, which doesn't satisfy the detailed balance condition. Barker's algorithm does converge to the target distribution. Given the current state, x, and the proposed state, x', the acceptance probability is defined as:α(x→x′)=P(x′)P(x)+P(x′){\displaystyle \alpha \left(x\rightarrow x^{\prime }\right)={\frac {P\left(x^{\prime }\right)}{P\left(x\right)+P\left(x^{\prime }\right)}}}The formula doesn't satisfy detailed balance, but makes sure that the balanced condition is met.
|
https://en.wikipedia.org/wiki/Barker_code
|
AGold code, also known asGold sequence, is a type of binarysequence, used intelecommunications(CDMA)[1]and satellite navigation (GPS).[2]Gold codes are named after Robert Gold.[3][4]Gold codes have bounded smallcross-correlationswithin a set, which is useful when multiple devices are broadcasting in the same frequency range. A set of Gold code sequences consists of 2n+ 1 sequences each one with a period of 2n− 1.
A set of Gold codes can be generated with the following steps. Pick twomaximum length sequencesof the same length 2n− 1 such that their absolutecross-correlationis less than or equal to 2(n+2)/2, wherenis the size of thelinear-feedback shift registerused to generate the maximum length sequence (Gold '67). The set of the 2n− 1exclusive-orsof the two sequences in their various phases (i.e. translated into all relative positions) together with the two maximum length sequences form a set of 2n+ 1 Gold code sequences. The highest absolute cross-correlation in this set of codes is 2(n+2)/2+ 1 for evennand 2(n+1)/2+ 1 for oddn.
Theexclusive orof two different Gold codes from the same set is another Gold code in some phase.
Within a set of Gold codes about half of the codes are balanced – the number of ones and zeros differs by only one.[5]
Gold codes are used inGPS. TheGPS C/Aranging codes are Gold codes of period 1,023.
|
https://en.wikipedia.org/wiki/Gold_code
|
AZadoff–Chu (ZC) sequence[1]: 152is acomplex-valuedmathematicalsequencewhich, when applied to asignal, gives rise to a new signal of constantamplitude. Whencyclically shiftedversions of a Zadoff–Chu sequence are imposed upon a signal the resulting set of signals detected at the receiver areuncorrelatedwith one another.
Zadoff–Chu sequences exhibit the useful property that cyclically shifted versions of themselves areorthogonalto one another.
A generated Zadoff–Chu sequence that has not been shifted is known as aroot sequence.
The complex value at each positionnof each root Zadoff–Chu sequence parametrised byuis given by
where
Zadoff–Chu sequences are CAZAC sequences (constant amplitude zero autocorrelation waveform).
Note that the special caseq=0{\displaystyle q=0}results in a Chu sequence,.[1]: 151Settingq≠0{\displaystyle q\neq 0}produces a sequence that is equal to the cyclically shifted version of the Chu sequence byq{\displaystyle q},andmultipliedby a complex, modulus 1 number, where by multiplied we mean that each element is multiplied by the same number.
1. They areperiodicwith periodNZC{\displaystyle N_{\text{ZC}}}.
2. IfNZC{\displaystyle N_{\text{ZC}}}is prime, theDiscrete Fourier Transformof a Zadoff–Chu sequence is another Zadoff–Chu sequence conjugated, scaled and time scaled.
3. The auto correlation of a Zadoff–Chu sequence with a cyclically shifted version of itself is zero, i.e., it is non-zero only at one instant which corresponds to the cyclic shift.
4. Thecross-correlationbetween two prime length Zadoff–Chu sequences, i.e. different values ofu,u=u1,u=u2{\displaystyle u,u=u_{1},u=u_{2}}, is constant1/NZC{\displaystyle 1/{\sqrt {N_{\text{ZC}}}}}, provided thatu1−u2{\displaystyle u_{1}-u_{2}}is relatively prime toNZC{\displaystyle N_{\text{ZC}}}.[2]
Zadoff–Chu sequences are used in the3GPPLong Term Evolution(LTE)air interfacein the Primary Synchronization Signal (PSS), random access preamble (PRACH), uplink control channel (PUCCH), uplink traffic channel (PUSCH) and sounding reference signals (SRS).
By assigningorthogonalZadoff–Chu sequences to each LTEeNodeBand multiplying their transmissions by their respective codes, thecross-correlationof simultaneous eNodeB transmissions is reduced, thus reducing inter-cell interference and uniquely identifying eNodeB transmissions.
Zadoff–Chu sequences are an improvement over theWalsh–Hadamard codesused inUMTSbecause they result in a constant-amplitude output signal, reducing the cost and complexity of theradio's power amplifier.[3]
|
https://en.wikipedia.org/wiki/Zadoff%E2%80%93Chu_sequence
|
Apseudorandomsequence of numbers is one that appears to bestatistically random, despite having been produced by a completelydeterministicand repeatable process.[1]Pseudorandom number generatorsare often used in computer programming, as traditional sources of randomness available to humans (such as rolling dice) rely on physical processes not readily available to computer programs, although developments inhardware random number generatortechnology have challenged this.
The generation of random numbers has many uses, such as forrandom sampling,Monte Carlo methods,board games, orgambling. Inphysics, however, most processes, such as gravitational acceleration, are deterministic, meaning that they always produce the same outcome from the same starting point. Some notable exceptions areradioactive decayandquantum measurement, which are both modeled as being truly random processes in the underlying physics. Since these processes are not practical sources of random numbers, pseudorandom numbers are used, which ideally have the unpredictability of a truly random sequence, despite being generated by a deterministic process.[2]
In many applications, the deterministic process is acomputer algorithmcalled apseudorandom number generator, which must first be provided with a number called arandom seed. Since the same seed will yield the same sequence every time, it is important that the seed be well chosen and kept hidden, especially insecurityapplications, where the pattern's unpredictability is a critical feature.[3]
In some cases where it is important for the sequence to be demonstrably unpredictable, physical sources of random numbers have been used, such as radioactive decay, atmospheric electromagnetic noise harvested from a radio tuned between stations, or intermixed timings ofkeystrokes.[1][4]The time investment needed to obtain these numbers leads to a compromise: using some of these physics readings as a seed for a pseudorandom number generator.
Before modern computing, researchers requiring random numbers would either generate them through various means (dice,cards,roulette wheels,[5]etc.) or use existing random number tables.
The first attempt to provide researchers with a ready supply of random digits was in 1927, when the Cambridge University Press published a table of 41,600 digits developed byL.H.C. Tippett. In 1947, theRAND Corporationgenerated numbers by the electronic simulation of a roulette wheel;[5]the results were eventually published in 1955 asA Million Random Digits with 100,000 Normal Deviates.
Intheoretical computer science, adistributionispseudorandomagainst a class of adversaries if no adversary from the class can distinguish it from the uniform distribution with significant advantage.[6]This notion of pseudorandomness is studied incomputational complexity theoryand has applications tocryptography.
Formally, letSandTbe finite sets and letF= {f:S→T} be a class of functions. AdistributionDoverSis ε-pseudorandomagainstFif for everyfinF, thestatistical distancebetween the distributionsf(X){\displaystyle f(X)}andf(Y){\displaystyle f(Y)}, whereX{\displaystyle X}is sampled fromDandY{\displaystyle Y}is sampled from theuniform distributiononS, is at most ε.
In typical applications, the classFdescribes a model of computation with bounded resources and one is interested in designing distributionsDwith certain properties that are pseudorandom againstF. The distributionDis often specified as the output of apseudorandom generator.[7]
|
https://en.wikipedia.org/wiki/Pseudorandomness
|
Intelecommunications,direct-sequence spread spectrum(DSSS) is aspread-spectrummodulationtechnique primarily used to reduce overall signalinterference. The direct-sequence modulation makes the transmitted signal wider in bandwidth than the information bandwidth.
After the despreading or removal of the direct-sequence modulation in the receiver, the information bandwidth is restored, while the unintentional and intentional interference is substantially reduced.[1]
Swissinventor,Gustav Guanellaproposed a "means for and method of secret signals".[2]With DSSS, the message symbols are modulated by a sequence of complex values known asspreading sequence. Each element of the spreading sequence, a so-calledchip, has a shorter duration than the original message symbols. The modulation of the message symbols scrambles and spreads the signal in the spectrum, and thereby results in a bandwidth of the spreading sequence. The smaller the chip duration, the larger the bandwidth of the resulting DSSS signal; more bandwidth multiplexed to the message signal results in better resistance against narrowband interference.[1][3]
Some practical and effective uses of DSSS include thecode-division multiple access(CDMA) method, theIEEE 802.11bspecification used inWi-Finetworks, and theGlobal Positioning System.[4][5]
Direct-sequence spread-spectrum transmissions multiply the symbol sequence being transmitted with a spreading sequence that has a higher rate than the original message rate. Usually, sequences are chosen such that the resulting spectrum is spectrallywhite. Knowledge of the same sequence is used to reconstruct the original data at the receiving end. This is commonly implemented by the element-wise multiplication with the spreading sequence, followed by summation over a message symbol period. This process,despreading, is mathematically acorrelationof the transmitted spreading sequence with the spreading sequence. In an AWGN channel, the despreaded signal'ssignal-to-noise ratiois increased by the spreading factor, which is the ratio of the spreading-sequence rate to the data rate.
While a transmitted DSSS signal occupies a wider bandwidth than the direct modulation of the original signal would require, its spectrum can be restricted by conventionalpulse-shape filtering.
If an undesired transmitter transmits on the same channel but with a different spreading sequence, the despreading process reduces the power of that signal. This effect is the basis for thecode-division multiple access(CDMA) method of multi-user medium access, which allows multiple transmitters to share the same channel within the limits of thecross-correlationproperties of their spreading sequences.
|
https://en.wikipedia.org/wiki/Direct-sequence_spread_spectrum
|
Electromagnetic compatibility(EMC) is the ability of electrical equipment and systems to function acceptably in theirelectromagnetic environment, by limiting the unintentional generation, propagation and reception of electromagnetic energy which may cause unwanted effects such aselectromagnetic interference(EMI) or even physical damage to operational equipment.[1][2]The goal of EMC is the correct operation of different equipment in a common electromagnetic environment. It is also the name given to the associated branch ofelectrical engineering.
EMC pursues three main classes of issue.Emissionis the generation of electromagnetic energy, whether deliberate or accidental, by some source and its release into the environment. EMC studies the unwanted emissions and the countermeasures which may be taken in order to reduce unwanted emissions. The second class,susceptibility, is the tendency of electrical equipment, referred to as the victim, to malfunction or break down in the presence of unwanted emissions, which are known asRadio frequency interference(RFI).Immunityis the opposite of susceptibility, being the ability of equipment to function correctly in the presence of RFI, with the discipline of "hardening" equipment being known equally as susceptibility or immunity. A third class studied iscoupling, which is the mechanism by which emitted interference reaches the victim.
Interference mitigation and hence electromagnetic compatibility may be achieved by addressing any or all of these issues, i.e., quieting the sources of interference, inhibiting coupling paths and/or hardening the potential victims. In practice, many of the engineering techniques used, such as grounding and shielding, apply to all three issues.
The earliest EMC issue waslightningstrike (lightningelectromagnetic pulse, or LEMP) on ships and buildings.Lightning rodsor lightning conductors began to appear in the mid-18th century. With the advent of widespreadelectricity generationand power supply lines from the late 19th century on, problems also arose with equipmentshort-circuitfailure affecting the power supply, and with local fire and shock hazard when the power line was struck by lightning. Power stations were provided with outputcircuit breakers. Buildings and appliances would soon be provided with inputfuses, and later in the 20th century miniature circuit breakers (MCB) would come into use.
It may be said that radio interference and its correction arose with the first spark-gap experiment ofMarconiin the late 1800s.[3]Asradio communicationsdeveloped in the first half of the 20th century, interference betweenbroadcastradio signals began to occur and an international regulatory framework was set up to ensure interference-free communications.
Switching devices became commonplace through the middle of the 20th century, typically in petrol powered cars and motorcycles but also in domestic appliances such as thermostats and refrigerators. This caused transient interference with domestic radio and (after World War II) TV reception, and in due course laws were passed requiring the suppression of such interference sources.
ESD problems first arose with accidentalelectric sparkdischarges in hazardous environments such as coal mines and when refuelling aircraft or motor cars. Safe working practices had to be developed.
After World War II the military became increasingly concerned with the effects of nuclear electromagnetic pulse (NEMP), lightning strike, and even high-poweredradarbeams, on vehicle and mobile equipment of all kinds, and especially aircraft electrical systems.
When high RF emission levels from other sources became a potential problem (such as with the advent ofmicrowave ovens), certain frequency bands were designated for Industrial, Scientific and Medical (ISM) use, allowing emission levels limited only by thermal safety standards. Later, the International Telecommunication Union adopted a Recommendation providing limits of radiation from ISM devices in order to protect radiocommunications. A variety of issues such as sideband and harmonic emissions, broadband sources, and the ever-increasing popularity of electrical switching devices and their victims, resulted in a steady development of standards and laws.
From the late 1970s, the popularity of modern digital circuitry rapidly grew. As the technology developed, with ever-faster switching speeds (increasing emissions) and lower circuit voltages (increasing susceptibility), EMC increasingly became a source of concern. Many more nations became aware of EMC as a growing problem and issued directives to the manufacturers of digital electronic equipment, which set out the essential manufacturer requirements before their equipment could be marketed or sold. Organizations in individual nations, across Europe and worldwide, were set up to maintain these directives and associated standards. In 1979, the AmericanFCCpublished a regulation that required the electromagnetic emissions of all "digital devices" to be below certain limits.[3]This regulatory environment led to a sharp growth in the EMC industry supplying specialist devices and equipment, analysis and design software, and testing and certification services. Low-voltage digital circuits, especially CMOS transistors, became more susceptible to ESD damage as they were miniaturised and, despite the development of on-chip hardening techniques, a new ESD regulatory regime had to be developed.
From the 1980s on the explosive growth inmobile communicationsand broadcast media channels put huge pressure on the available airspace. Regulatory authorities began squeezing band allocations closer and closer together, relying on increasingly sophisticated EMC control methods, especially in the digital communications domain, to keep cross-channel interference to acceptable levels. Digital systems are inherently less susceptible than analogue systems, and also offer far easier ways (such as software) to implement highly sophisticated protection anderror-correctionmeasures.
In 1985, the USA released the ISM bands for low-power mobile digital communications, leading to the development ofWi-Fiand remotely-operated car door keys. This approach relies on the intermittent nature of ISM interference and use of sophisticated error-correction methods to ensure lossless reception during the quiet gaps between any bursts of interference.
"Electromagnetic interference" (EMI) is defined as the "degradation in the performance of equipment or transmission channel or a system caused by an electromagnetic disturbance" (IEV161-01-06) while "electromagnetic disturbance" is defined as "an electromagnetic phenomenon that can degrade the performance of a device, equipment or system, or adversely affect living or inert matter(IEV 161-01-05). The terms "electromagnetic disturbance" and "electromagnetic interference" designate respectively the cause and the effect,[1]
Electromagnetic compatibility (EMC) is an equipmentcharacteristicorpropertyand is defined as "the ability of equipment or a system to function satisfactorily in its electromagnetic environment without introducing intolerable electromagnetic disturbances to anything in that environment" (IEV 161-01-07).[1]
EMC ensures the correct operation, in the same electromagnetic environment, of different equipment items which use or respond to electromagnetic phenomena, and the avoidance of any interference. Another way of saying this is that EMC is the control of EMI so that unwanted effects are prevented.
Besides understanding the phenomena in themselves, EMC also addresses the countermeasures, such as control regimes, design and measurement, which should be taken in order to prevent emissions from causing any adverse effect.
EMC is often understood as the control of electromagnetic interference (EMI). Electromagnetic interference divides into several categories according to the source and signal characteristics.
The origin of interference, often called "noise" in this context, can be man-made (artificial) or natural.
Continuous, or continuous wave (CW), interference comprises a given range of frequencies. This type is naturally divided into sub-categories according to frequency range, and as a whole is sometimes referred to as "DC to daylight". One common classification is intonarrowbandandbroadband, according to the spread of thefrequency range.
Anelectromagnetic pulse(EMP), sometimes called atransientdisturbance, is a short-duration pulse of energy. This energy is usually broadband by nature, although it often excites a relatively narrow-banddamped sine waveresponse in the victim. Pulse signals divide broadly into isolated and repetitive events.
When a source emits interference, it follows a route to the victim known as the coupling path. There are four basic coupling mechanisms:conductive,capacitive,magneticor inductive, andradiative. Any coupling path can be broken down into one or more of these coupling mechanisms working together.
Conductive couplingoccurs when the coupling path between the source and victim is formed by direct electrical contact with a conducting body.
Capacitive couplingoccurs when a varyingelectrical fieldexists between two adjacent conductors, inducing a change involtageon the receiving conductor.
Inductive couplingor magnetic coupling occurs when a varyingmagnetic fieldexists between two parallel conductors, inducing a change involtagealong the receiving conductor.
Radiative coupling or electromagnetic coupling occurs when source and victim are separated by a large distance. Source and victim act as radio antennas: the source emits or radiates anelectromagnetic wavewhich propagates across the space in between and is picked up or received by the victim.
The damaging effects of electromagnetic interference pose unacceptable risks in many areas of technology, and it is necessary to control such interference and reduce the risks to acceptable levels.
The control of electromagnetic interference (EMI) and assurance of EMC comprises a series of related disciplines:
The risk posed by the threat is usually statistical in nature, so much of the work in threat characterisation and standards setting is based on reducing the probability of disruptive EMI to an acceptable level, rather than its assured elimination.
For a complex or novel piece of equipment, this may require the production of a dedicatedEMC control plansummarizing the application of the above and specifying additional documents required.
Characterisation of the problem requires understanding of:
Breaking a coupling path is equally effective at either the start or the end of the path, therefore many aspects of good EMC design practice apply equally to potential sources and to potential victims. A design which easily couples energy to the outside world will equally easily couple energy in and will be susceptible. A single improvement will often reduce both emissions and susceptibility. Grounding and shielding aim to reduce emissions or divert EMI away from the victim by providing an alternative, low-impedance path. Techniques include:
Other general measures include:
Additional measures to reduce emissions include:
Additional measures to reduce susceptibility include:
Testing is required to confirm that a particular device meets the required standards. It is divided broadly into emissions testing and susceptibility testing. Open-area test sites, or OATS,[4]are the reference sites in most standards. They are especially useful for emissions testing of large equipment systems. However, RF testing of a physical prototype is most often carried out indoors, in a specialized EMC test chamber. Types of the chamber includeanechoic,reverberationand thegigahertz transverse electromagnetic cell(GTEM cell). Sometimescomputational electromagneticssimulations are used to test virtual models. Like all compliance testing, it is important that the test equipment, including the test chamber or site and any software used, be properly calibrated and maintained. Typically, a given run of tests for a particular piece of equipment will require anEMC test planand a follow-uptest report. The full test program may require the production of several such documents.
Emissions are typically measured for radiated field strength and where appropriate for conducted emissions along cables and wiring. Inductive (magnetic) and capacitive (electric) field strengths are near-field effects and are only important if the device under test (DUT) is designed for a location close to other electrical equipment. For conducted emissions, typical transducers include theLISN(line impedance stabilization network) or AMN (artificial mains network) and the RFcurrent clamp. For radiated emission measurement, antennas are used as transducers. Typical antennas specified includedipole,biconical,log-periodic, double ridged guide and conical log-spiral designs. Radiated emissions must be measured in all directions around the DUT. Specialized EMI test receivers or EMI analyzers are used for EMC compliance testing. These incorporate bandwidths and detectors as specified by international EMC standards. An EMI receiver may be based on aspectrum analyserto measure the emission levels of the DUT across a wide band of frequencies (frequency domain), or on a tunable narrower-band device which is swept through the desired frequency range. EMI receivers along with specified transducers can often be used for both conducted and radiated emissions. Pre-selector filters may also be used to reduce the effect of strong out-of-band signals on the front-end of the receiver. Some pulse emissions are more usefully characterized using anoscilloscopeto capture the pulse waveform in the time domain.
Radiated field susceptibility testing typically involves a high-powered source of RF or EM energy and a radiating antenna to direct the energy at the potential victim or device under test (DUT). Conducted voltage and current susceptibility testing typically involves a high-powered signal generator, and acurrent clampor other type oftransformerto inject the test signal. Transient or EMP signals are used to test the immunity of the DUT against powerline disturbances including surges, lightning strikes and switching noise.[5]In motor vehicles, similar tests are performed on battery and signal lines.[6][7]The transient pulse may be generated digitally and passed through a broadband pulse amplifier, or applied directly to the transducer from a specialized pulse generator.Electrostatic dischargetesting is typically performed with apiezo spark generatorcalled an "ESD pistol". Higher energy pulses, such as lightning or nuclear EMP simulations, can require a largecurrent clampor a large antenna which completely surrounds the DUT. Some antennas are so large that they are located outdoors, and care must be taken not to cause an EMP hazard to the surrounding environment.
Several organizations, both national and international, work to promote international co-operation on standardization (harmonization), including publishing various EMC standards. Where possible, a standard developed by one organization may be adopted with little or no change by others. This helps for example to harmonize national standards across Europe.
International standards organizations include:
Among the main national organizations are:
Compliance with national or international standards is usually laid down by laws passed by individual nations. Different nations can require compliance with different standards.
InEuropean law, EU directive 2014/30/EU (previously 2004/108/EC) on EMC defines the rules for the placing on the market/putting into service of electric/electronic equipment within theEuropean Union. The Directive applies to a vast range of equipment including electrical and electronic appliances, systems and installations. Manufacturers ofelectric and electronic devicesare advised to run EMC tests in order to comply with compulsoryCE-labeling. More are given in thelist of EMC directives. Compliance with the applicable harmonised standards whose reference is listed in the OJEU under the EMC Directive gives presumption of conformity with the corresponding essential requirements of the EMC Directive.
In 2019, the USA adopted a program for the protection of critical infrastructure against an electromagnetic pulse, whether caused by ageomagnetic stormor a high-altitude nuclear weapon.[8]
|
https://en.wikipedia.org/wiki/Electromagnetic_compatibility
|
Electromagnetic interference(EMI), also calledradio-frequency interference(RFI) when in theradio frequencyspectrum, is a disturbance generated by an external source that affects an electrical circuit byelectromagnetic induction,electrostatic coupling, or conduction.[1]The disturbance may degrade the performance of the circuit or even stop it from functioning. In the case of a data path, these effects can range from an increase in error rate to a total loss of the data.[2]Both human-made and natural sources generate changing electrical currents and voltages that can cause EMI:ignition systems,cellular networkof mobile phones,lightning,solar flares, andauroras(northern/southern lights).[citation needed]EMI frequently affectsAM radios. It can also affectmobile phones,FM radios, andtelevisions, as well as observations forradio astronomyandatmospheric science.
EMI can be used intentionally forradio jamming, as inelectronic warfare.
Since the earliest days of radio communications, the negative effects of interference from both intentional and unintentional transmissions have been felt and the need to manage the radio frequency spectrum became apparent.[3]
In 1933, a meeting of theInternational Electrotechnical Commission(IEC) in Paris recommended the International Special Committee on Radio Interference (CISPR) be set up to deal with the emerging problem of EMI. CISPR subsequently produced technical publications covering measurement and test techniques and recommended emission and immunity limits. These have evolved over the decades and form the basis of much of the world'sEMCregulations today.[4]
In 1979, legal limits were imposed on electromagnetic emissions from all digital equipment by theFCCin the US in response to the increased number of digital systems that were interfering with wired and radio communications. Test methods and limits were based on CISPR publications, although similar limits were already enforced in parts of Europe.[5]
In the mid 1980s, the European Union member states adopted a number of "new approach" directives with the intention of standardizing technical requirements for products so that they do not become a barrier to trade within the EC. One of these was the EMC Directive (89/336/EC)[6]and it applies to all equipment placed on the market or taken into service. Its scope covers all apparatus "liable to cause electromagnetic disturbance or the performance of which is liable to be affected by such disturbance".[5]
This was the first time there was a legal requirement on immunity, as well as emissions on apparatus intended for the general population. Although there may be additional costs involved for some products to give them a known level of immunity, it increases their perceived quality as they are able to co-exist with apparatus in the active EM environment of modern times and with fewer problems.[5]
Many countries now have similar requirements for products to meet some level ofelectromagnetic compatibility(EMC) regulation.[5]
Electromagnetic interference divides into several categories according to the source and signal characteristics.
The origin of interference, often called "noise" in this context, can be human-made (artificial) or natural.
Continuous, or continuous wave (CW), interference arises where the source continuously emits at a given range of frequencies. This type is naturally divided into sub-categories according to frequency range, and as a whole is sometimes referred to as "DC to daylight". One common classification is into narrowband and broadband, according to the spread of the frequency range.
Anelectromagnetic pulse(EMP), sometimes called atransientdisturbance, arises where the source emits a short-duration pulse of energy. The energy is usually broadband by nature, although it often excites a relatively narrow-banddamped sine waveresponse in the victim.
Sources divide broadly into isolated and repetitive events.
Sources of isolated EMP events include:
Sources of repetitive EMP events, sometimes as regularpulsetrains, include:
Conducted electromagnetic interference is caused by the physical contact of the conductors as opposed to radiated EMI, which is caused by induction (without physical contact of the conductors). Electromagnetic disturbances in the EM field of a conductor will no longer be confined to the surface of the conductor and will radiate away from it. This persists in all conductors and mutual inductance between two radiatedelectromagnetic fieldswill result in EMI.[7]
Some of the technical terms which are employed can be used with differing meanings. Some phenomena may be referred to by various different terms. These terms are used here in a widely accepted way, which is consistent with other articles in the encyclopedia.
The basic arrangement ofnoiseemitter or source,couplingpath and victim,receptoror sink is shown in the figure below. Source and victim are usuallyelectronic hardwaredevices, though the source may be a natural phenomenon such as alightning strike,electrostatic discharge(ESD) or, inone famous case, theBig Bangat the origin of the Universe.
There are four basic coupling mechanisms:conductive,capacitive,magneticor inductive, andradiative. Any coupling path can be broken down into one or more of these coupling mechanisms working together. For example the lower path in the diagram involves inductive, conductive and capacitive modes.
Conductive couplingoccurs when the coupling path between the source and victim is formed by direct electrical contact with a conducting body, for example a transmission line, wire, cable,PCBtrace or metal enclosure. Conducted noise is also characterised by the way it appears on different conductors:
Inductive coupling occurs where the source and victim are separated by a short distance (typically less than awavelength). Strictly, "Inductive coupling" can be of two kinds, electrical induction and magnetic induction. It is common to refer to electrical induction ascapacitive coupling, and to magnetic induction asinductive coupling.
Capacitive couplingoccurs when a varyingelectrical fieldexists between two adjacent conductors typically less than a wavelength apart, inducing a change involtageon the receiving conductor.
Inductive couplingor magnetic coupling occurs when a varyingmagnetic fieldexists between two parallel conductors typically less than a wavelength apart, inducing a change involtagealong the receiving conductor.
Radiative coupling or electromagnetic coupling occurs when source and victim are separated by a large distance, typically more than a wavelength. Source and victim act as radio antennas: the source emits or radiates anelectromagnetic wavewhich propagates across the space in between and is picked up or received by the victim.
Interferencewith the meaning ofelectromagnetic interference, alsoradio-frequency interference(EMIorRFI) is – according toArticle 1.166of theInternational Telecommunication Union's (ITU)Radio Regulations(RR)[8]– defined as "The effect of unwanted energy due to one or a combination ofemissions,radiations, orinductionsupon reception in aradiocommunicationsystem, manifested by any performance degradation, misinterpretation, or loss of information which could be extracted in the absence of such unwanted energy".
This is also a definition used by thefrequency administrationto providefrequency assignmentsand assignment of frequency channels toradio stationsor systems, as well as to analyzeelectromagnetic compatibilitybetweenradiocommunication services.
In accordance with ITU RR (article 1) variations of interference are classified as follows:[9]
Conducted EMI is caused by the physical contact of the conductors as opposed to radiated EMI which is caused byinduction(without physical contact of the conductors).
For lower frequencies, EMI is caused by conduction and, for higher frequencies, by radiation.
EMI through the ground wire is also very common in an electrical facility.
Interference tends to be more troublesome with older radio technologies such as analogueamplitude modulation, which have no way of distinguishing unwanted in-band signals from the intended signal, and the omnidirectional antennas used with broadcast systems. Newer radio systems incorporate several improvements that enhance theselectivity. In digital radio systems, such asWi-Fi,error-correctiontechniques can be used.Spread-spectrumandfrequency-hoppingtechniques can be used with both analogue and digital signalling to improve resistance to interference. A highlydirectionalreceiver, such as aparabolic antennaor adiversity receiver, can be used to select one signal in space to the exclusion of others.
The most extreme example of digitalspread-spectrumsignalling to date is ultra-wideband (UWB), which proposes the use of large sections of theradio spectrumat low amplitudes to transmit high-bandwidth digital data. UWB, if used exclusively, would enable very efficient use of the spectrum, but users of non-UWB technology are not yet prepared to share the spectrum with the new system because of the interference it would cause to their receivers (the regulatory implications of UWB are discussed in theultra-widebandarticle).
In theUnited States, the 1982 Public Law 97-259 allowed theFederal Communications Commission(FCC) to regulate the susceptibility of consumer electronic equipment.[10][11]
Potential sources of RFI and EMI include:[12]various types oftransmitters, doorbell transformers,toaster ovens,electric blankets, ultrasonic pest control devices, electricbug zappers,heating pads, andtouch controlled lamps. MultipleCRTcomputer monitors or televisions sitting too close to one another can sometimes cause a "shimmy" effect in each other, due to the electromagnetic nature of their picture tubes, especially when one of theirde-gaussingcoils is activated.
Electromagnetic interference at 2.4 GHzmay be caused by802.11b,802.11gand802.11nwireless devices,Bluetoothdevices,baby monitorsandcordless telephones,video senders, andmicrowave ovens.
Switchingloads (inductive,capacitive, andresistive), such as electric motors, transformers, heaters, lamps, ballast, power supplies, etc., all cause electromagnetic interference especially at currents above 2A. The usual method used for suppressing EMI is by connecting asnubbernetwork, a resistor in series with acapacitor, across a pair of contacts. While this may offer modest EMI reduction at very low currents, snubbers do not work at currents over 2 A withelectromechanicalcontacts.[13][14]
Another method for suppressing EMI is the use of ferrite core noise suppressors (orferrite beads), which are inexpensive and which clip on to the power lead of the offending device or the compromised device.
Switched-mode power suppliescan be a source of EMI, but have become less of a problem as design techniques have improved, such as integratedpower factor correction.
Most countries have legal requirements that mandateelectromagnetic compatibility: electronic and electrical hardware must still work correctly when subjected to certain amounts of EMI, and should not emit EMI, which could interfere with other equipment (such as radios).
Radio frequency signal quality has declined throughout the 21st century by roughly one decibel per year as the spectrum becomes increasingly crowded.[additional citation(s) needed]This has inflicted aRed Queen's raceon the mobile phone industry as companies have been forced to put up more cellular towers (at new frequencies) that then cause more interference thereby requiring more investment by the providers and frequent upgrades of mobile phones to match.[15]
The International Special Committee for Radio Interference or CISPR (French acronym for "Comité International Spécial des Perturbations Radioélectriques"), which is a committee of the International Electrotechnical Commission (IEC) sets international standards for radiated and conducted electromagnetic interference. These are civilian standards for domestic, commercial, industrial and automotive sectors. These standards form the basis of other national or regional standards, most notably the European Norms (EN) written by CENELEC (European committee for electrotechnical standardisation). US organizations include the Institute of Electrical and Electronics Engineers (IEEE), the American National Standards Institute (ANSI), and the US Military (MILSTD).
Integrated circuits are often a source of EMI, but they must usually couple their energy to larger objects such as heatsinks, circuit board planes and cables to radiate significantly.[16]
Onintegrated circuits, important means of reducing EMI are: the use of bypass ordecoupling capacitorson each active device (connected across the power supply, as close to the device as possible),rise timecontrol of high-speed signals using series resistors,[17]andIC power supply pinfiltering. Shielding is usually a last resort after other techniques have failed, because of the added expense of shielding components such as conductive gaskets.
The efficiency of the radiation depends on the height above theground planeorpower plane(atRF, one is as good as the other) and the length of the conductor in relation to the wavelength of the signal component (fundamental frequency,harmonicortransientsuch as overshoot, undershoot or ringing). At lower frequencies, such as 133MHz, radiation is almost exclusively via I/O cables; RF noise gets onto the power planes and is coupled to the line drivers via the VCC and GND pins. The RF is then coupled to the cable through the line driver ascommon-mode noise. Since the noise is common-mode, shielding has very little effect, even withdifferential pairs. The RF energy iscapacitively coupledfrom the signal pair to the shield and the shield itself does the radiating. One cure for this is to use abraid-breakerorchoketo reduce the common-mode signal.
At higher frequencies, usually above 500 MHz, traces get electrically longer and higher above the plane. Two techniques are used at these frequencies: wave shaping with series resistors and embedding the traces between the two planes. If all these measures still leave too much EMI, shielding such as RF gaskets and copper or conductive tape can be used. Most digital equipment is designed with metal or conductive-coated plastic cases.[citation needed]
Any unshielded semiconductor (e.g. an integrated circuit) will tend to act as a detector for those radio signals commonly found in the domestic environment (e.g. mobile phones).[18]Such a detector can demodulate the high frequency mobile phone carrier (e.g., GSM850 and GSM1900, GSM900 and GSM1800) and produce low-frequency (e.g., 217 Hz) demodulated signals.[19]This demodulation manifests itself as unwanted audible buzz in audio appliances such asmicrophoneamplifier,speakeramplifier, car radio, telephones etc. Adding onboard EMI filters or special layout techniques can help in bypassing EMI or improving RF immunity.[20]Some ICs are designed (e.g., LMV831-LMV834,[21]MAX9724[22]) to have integrated RF filters or a special design that helps reduce any demodulation of high-frequency carrier.
Designers often need to carry out special tests for RF immunity of parts to be used in a system. These tests are often done in ananechoic chamberwith a controlled RF environment where the test vectors produce a RF field similar to that produced in an actual environment.[19]
Interference inradio astronomy, where it is commonly referred to as radio-frequency interference (RFI), is any source of transmission that is within the observed frequency band other than the celestial sources themselves. Because transmitters on and around the Earth can be many times stronger than the astronomical signal of interest, RFI is a major concern for performing radio astronomy.[23]Natural sources of interference, such as lightning and the Sun, are also often referred to as RFI.[citation needed]
Some of the frequency bands that are very important for radio astronomy, such as the21-cm HI lineat 1420 MHz, are protected by regulation.[citation needed]However, modern radio-astronomical observatories such asVLA,LOFAR, andALMAhave a very large bandwidth over which they can observe.[citation needed]Because of the limited spectral space at radio frequencies, these frequency bands cannot be completely allocated to radio astronomy; for example,redshiftedimages of the 21-cm line from thereionizationepoch can overlap with theFM broadcast band(88–108 MHz), and therefore radio telescopes need to deal with RFI in this bandwidth.[23]
Techniques to deal with RFI range from filters in hardware to advanced algorithms in software. One way to deal with strong transmitters is to filter out the frequency of the source completely. This is for example the case for the LOFAR observatory, which filters out the FM radio stations between 90 and 110 MHz. It is important to remove such strong sources of interference as soon as possible, because they might "saturate" the highly sensitive receivers (amplifiersandanalogue-to-digital converters), which means that the received signal is stronger than the receiver can handle. However, filtering out a frequency band implies that these frequencies can never be observed with the instrument.[citation needed]
A common technique to deal with RFI within the observed frequency bandwidth, is to employ RFI detection in software. Such software can find samples in time, frequency or time-frequency space that are contaminated by an interfering source. These samples are subsequently ignored in further analysis of the observed data. This process is often referred to asdata flagging. Because most transmitters have a small bandwidth and are not continuously present such as lightning orcitizens' band(CB) radio devices, most of the data remains available for the astronomical analysis. However, data flagging can not solve issues with continuous broad-band transmitters, such as windmills,digital videoordigital audiotransmitters.[citation needed]
Another way to manage RFI is to establish aradio quiet zone(RQZ). RQZ is a well-defined area surrounding receivers that has special regulations to reduce RFI in favor of radio astronomy observations within the zone. The regulations may include special management of spectrum and power flux or power flux-density limitations. The controls within the zone may cover elements other than radio transmitters or radio devices. These include aircraft controls and control of unintentional radiators such as industrial, scientific and medical devices, vehicles, and power lines. The first RQZ for radio astronomy isUnited States National Radio Quiet Zone(NRQZ), established in 1958.[24]
Prior to the introduction of Wi-Fi, one of the biggest applications of 5 GHz band was theTerminal Doppler Weather Radar.[25][26]The decision to use 5 GHz spectrum for Wi-Fi was finalized at theWorld Radiocommunication Conferencein 2003; however, meteorological authorities were not involved in the process.[27][28]The subsequent lax implementation and misconfiguration of DFS had caused significant disruption in weather radar operations in a number of countries around the world. In Hungary, the weather radar system was declared non-operational for more than a month. Due to the severity of interference, South African weather services ended up abandoning C band operation, switching their radar network toS band.[26][29]
Transmissions on adjacent bands to those used by passiveremote sensing, such asweather satellites, have caused interference, sometimes significant.[30]There is concern that adoption of insufficiently regulated5Gcould produce major interference issues. Significant interference can impairnumerical weather predictionperformance and incur negative economic and public safety impacts.[31][32][33]These concerns led US Secretary of CommerceWilbur Rossand NASA AdministratorJim Bridenstinein February 2019 to urge the FCC to cancel a proposedspectrum auction, which was rejected.[34]
|
https://en.wikipedia.org/wiki/Electromagnetic_interference
|
Frequency-hopping spread spectrum(FHSS) is a method of transmitting radio signals by rapidly changing the carrier frequency among many frequencies occupying a large spectral band. The changes are controlled by a code known to bothtransmitterandreceiver. FHSS is used to avoid interference, to prevent eavesdropping, and to enablecode-division multiple access(CDMA) communications.
The frequency band is divided into smaller sub-bands. Signals rapidly change ("hop") their carrier frequencies among the center frequencies of these sub-bands in a determined order. Interference at a specific frequency will affect the signal only during a short interval.[1]
FHSS offers four main advantages over a fixed-frequency transmission:
Spread-spectrumsignals are highly resistant to deliberatejammingunless the adversary has knowledge of the frequency-hopping pattern. Military radios generate the frequency-hopping pattern under the control of a secretTransmission Security Key(TRANSEC) that the sender and receiver share in advance. This key is generated by devices such as the KY-57 Speech Security Equipment. United States military radios that use frequency hopping include the JTIDS/MIDS family, theHAVE QUICKAeronautical Mobile communications system, and theSINCGARSCombat Net Radio,Link-16.
In the US, since theFederal Communications Commission(FCC) amended rules to allow FHSS systems in the unregulated 2.4 GHz band, many consumer devices in that band have employed various FHSS modes. eFCC CFR 47 part 15.247 covers the regulations in the US for 902–928 MHz, 2400–2483.5 MHz, and 5725–5850 MHz bands, and the requirements for frequency hopping.[2]
Somewalkie-talkiesthat employ FHSS technology have been developed for unlicensed use on the 900 MHz band. FHSS technology is also used in many hobby transmitters and receivers used forradio-controlled modelcars, airplanes, and drones. A type of multiple access is achieved allowing hundreds of transmitter/receiver pairs to be operated simultaneously on the same band, in contrast to previous FM or AM radio-controlled systems that had limited simultaneous channels.
The overall bandwidth required for frequency hopping is much wider than that required to transmit the same information using only onecarrier frequency. But because transmission occurs only on a small portion of this bandwidth at any given time, the instantaneous interference bandwidth is really the same. While providing no extra protection against widebandthermal noise, the frequency-hopping approach reduces the degradation caused by narrowband interference sources.
One of the challenges of frequency-hopping systems is to synchronize the transmitter and receiver. One approach is to have a guarantee that the transmitter will use all the channels in a fixed period of time. The receiver can then find the transmitter by picking a random channel and listening for valid data on that channel. The transmitter's data is identified by a special sequence of data that is unlikely to occur over the segment of data for this channel, and the segment can also have achecksumfor integrity checking and further identification. The transmitter and receiver can use fixed tables of frequency-hopping patterns, so that once synchronized they can maintain communication by following the table.
In the US,FCC part 15on unlicensed spread spectrum systems in the 902–928 MHz and 2.4 GHz bands permits more power than is allowed for non-spread-spectrum systems. Both FHSS and direct-sequence spread-spectrum (DSSS) systems can transmit at 1 watt, a thousandfold increase from the 1 milliwatt limit on non-spread-spectrum systems. The FCC also prescribes a minimum number of frequency channels and a maximum dwell time for each channel.
In 1899,Guglielmo Marconiexperimented with frequency-selective reception in an attempt to minimise interference.[3]
The earliest mentions of frequency hopping in open literature are inUS patent 725,605, awarded toNikola Teslaon March 17, 1903,[4]and in radio pioneerJonathan Zenneck's bookWireless Telegraphy(German, 1908, English translation McGraw Hill, 1915),[5][a]although Zenneck writes thatTelefunkenhad already tried it. Nikola Tesla doesn't mention the phrase "frequency hopping" directly, but certainly alludes to it. EntitledMethod of Signaling, the patent describes a system that would enable radio communicationwithout any danger of the signals or messages being disturbed, intercepted, interfered with in any way.[6]
The German military made limited use of frequency hopping for communication between fixed command points inWorld War Ito prevent eavesdropping by British forces, who did not have the technology to follow the sequence.[7]Jonathan Zenneck's bookWireless Telegraphywas originally published in German in 1908, but was translated into English in 1915 as the enemy started using frequency hopping on the front line.
In 1920, Otto B. Blackwell, De Loss K. Martin, and Gilbert S. Vernam filed a patent application for a "Secrecy Communication System", granted asU.S. Patent 1,598,673in 1926. This patent described a method of transmitting signals on multiple frequencies in a random manner for secrecy, anticipating key features of later frequency hopping systems.[4]
APolishengineer and inventor,Leonard Danilewicz, claimed to have suggested the concept of frequency hopping in 1929 to thePolish General Staff, but it was rejected.[8]
In 1932,U.S. patent 1,869,659was awarded to Willem Broertjes, named "Method of maintaining secrecy in the transmission of wireless telegraphic messages", which describes a system where "messages are transmitted by means of a group of frequencies... known to the sender and receiver alone, and alternated at will during transmission of the messages".
DuringWorld War II, theUS Army Signal Corpswas inventing a communication system calledSIGSALY, which incorporated spread spectrum in a single frequency context. But SIGSALY was a top-secret communications system, so its existence was not known until the 1980s.
In 1942, actressHedy Lamarrand composerGeorge AntheilreceivedU.S. patent 2,292,387for their "Secret Communications System",[9][10]an early version of frequency hopping using apiano-rollto switch among 88 frequencies to make radio-guidedtorpedoesharder for enemies to detect or jam. They then donated the patent to theU.S. Navy.[11]
Frequency-hopping ideas may have been rediscovered in the 1950s during patent searches when private companies were independently developing direct-sequenceCode Division Multiple Access, a non-frequency-hopping form of spread-spectrum.[citation needed]In 1957, engineers at Sylvania Electronic Systems Division adopted a similar idea, using the recently invented transistor instead of Lamarr's and Antheil's clockwork technology.[9][dubious–discuss]In 1962, the US Navy utilized Sylvania Electronic Systems Division's work during theCuban Missile Crisis.[12]
A practical application of frequency hopping was developed byRay Zinn, co-founder of Micrel Corporation. Zinn developed a method allowing radio devices to operate without the need to synchronize a receiver with a transmitter. Using frequency hopping and sweep modes, Zinn's method is primarily applied in low data rate wireless applications such as utility metering, machine and equipment monitoring and metering, and remote control. In 2006 Zinn receivedU.S. patent 6,996,399for his "Wireless device and method using frequency hopping and sweep modes."
Adaptive frequency-hopping spread spectrum(AFH) as used inBluetoothimproves resistance toradio frequency interferenceby avoiding crowded frequencies in the hopping sequence. This sort of adaptive transmission is easier to implement with FHSS than withDSSS.
The key idea behind AFH is to use only the "good" frequencies and avoid the "bad" ones—those experiencingfrequency selective fading, those on which a third party is trying to communicate, or those being actively jammed. Therefore, AFH should be complemented by a mechanism for detecting good and bad channels.
But if the radio frequency interference is itself dynamic, then AFH's strategy of "bad channel removal" may not work well. For example, if there are several colocated frequency-hopping networks (as BluetoothPiconet), they are mutually interfering and AFH's strategy fails to avoid this interference.
The problem of dynamic interference, gradual reduction of available hopping channels and backward compatibility with legacy Bluetooth devices was resolved in version 1.2 of the Bluetooth Standard (2003). Such a situation can often happen in the scenarios that useunlicensed spectrum.
In addition, dynamic radio frequency interference is expected to occur in the scenarios related tocognitive radio, where the networks and the devices should exhibitfrequency-agileoperation.
Chirp modulationcan be seen as a form of frequency-hopping that simply scans through the available frequencies in consecutive order to communicate.
Frequency hopping can be superimposed on other modulations or waveforms to enhance the system performance.
|
https://en.wikipedia.org/wiki/Frequency-hopping_spread_spectrum
|
George Johann Carl Antheil(/ˈæntaɪl/AN-tyle; July 8, 1900 – February 12, 1959) was an Americanavant-gardecomposer, pianist, author, and inventor whose modernist musical compositions explored the sounds – musical, industrial, and mechanical – of the early 20th century. Spending much of the 1920s in Europe, Antheil returned to the United States in the 1930s, and thereafter composed music for films, and eventually, television. As a result of this work, his style became more tonal. A man of diverse interests and talents, Antheil was constantly reinventing himself. He wrote magazine articles, an autobiography, a mystery novel, and newspaper and music columns.
In 1941, Antheil and the actressHedy Lamarrdeveloped a radio guidance system for Alliedtorpedoesthat used a code (stored on a punched paper tape) to synchronize frequency changes, referred to asfrequency hopping, between the transmitter and receiver. It is one of thespread spectrumtechniques that became widely used in modern telecommunications. This work led to their induction into theNational Inventors Hall of Famein 2014.[1]
Antheil was born George Johann Carl Antheil, and grew up in a family of German immigrants inTrenton, New Jersey.[2]His father owned a shoe store.[3][4]Antheil got his education in the Trenton public schools.[5]He was raised bilingually, writing music, prose, and poetry from an early age, and never formally graduated from high school or college,[4]flunking out ofTrenton Central High Schoolin 1918.[6]According to Antheil's autobiography,The Bad Boy of Music(1945), he was "so crazy about music", that his mother sent him to the countryside where no pianos were available. Undeterred, George simply arranged for a local music store to deliver a piano.[7]His somewhat unreliable memoir mythologized his origins as a futurist, and emphasized his upbringing near a noisy machine shop and an ominous prison.[3][8]George's younger brother wasHenry W. Antheil Jr.; he became a diplomatic courier and died on June 14, 1940, whenhis planewas shot down over the Baltic Sea.[9]
Antheil started studying the piano at the age of six. In 1916, he traveled regularly toPhiladelphiato study under Constantine von Sternberg, a former pupil ofFranz Liszt.[4]From Sternberg, he received formal composition training in the European tradition, but his trips to the city also exposed him to conceptual art, includingDadaism.[10]In 1919, he began to work with the more progressiveErnest Blochin New York.[4][10][11]Initially, Bloch had been skeptical and had rejected him, describing Antheil's compositions as "empty" and "pretentious"; however, the teacher was won over by Antheil's enthusiasm and energy, and helped him financially as he attempted to complete an aborted first symphony.[4][12]Antheil's trips to New York also permitted him to meet important figures of the modernist movement, including the musicianLeo Ornstein, journalist and music criticPaul Rosenfeld, painterJohn Marin, photographerAlfred Stieglitz, andMargaret Anderson, editor ofThe Little Review.[10]
At age 19, Antheil was invited to spend the weekend with Anderson and a group of friends; he stayed six months, and the close-knit group, who includedGeorgette Leblanc, former companion ofMaurice Maeterlinck, were to become influential in Antheil's career. Anderson described Antheil as short with an oddly shaped nose, who played "a compelling mechanical music", and used "the piano exclusively as an instrument of percussion, making it sound like a xylophone or a cymballo". Intensely engaged in his music, during this period, Antheil worked on songs, a piano concerto, and a work that came to be known as "The Mechanisms".[13]
Around this time, von Sternberg introduced Antheil to his patron of the next two decades:Mary Louise Curtis Bok, later the founder of theCurtis Institute of Music.[13]Assured by von Sternberg of Antheil's genius and good character, Bok gave him a monthly stipend of $150, and arranged for him to study at thePhiladelphia Settlement Music School. Though she came to disapprove of his behavior and his work, for the next 20 years, she continued to respond favorably to his letters.[12][13]As her financial support enabled Antheil to maintain a degree of independence in his work, many observers believed he should have given her more credit in his autobiography for the length and extent of her contribution to his career.[14]
Antheil continued his piano studies,[12]and the study of modernist compositions, such as those byIgor Stravinskyand members of theLes Sixgroup of French composers. In 1921, he wrote his first in a series of technology-based works, the solo pianoSecond Sonata, "The Airplane". Other works in the group included theSonata Sauvage(1922–23) and subsequentlyThird Sonata, "Death of Machines"(1923),"Mechanisms"(c.1923), both composed in Europe.[15]He also worked on his first symphony, managing to attractLeopold Stokowskito premiere it. Before the performance could take place, Antheil left for Europe to pursue his career. This may have diminished his chances for success in his native country.[16]
On May 30, 1922, at the age of 21, Antheil sailed for Europe to make his name as "a new ultra-modern pianist composer" and a "futurist terrible".[17][18]He had engagedLeo Ornstein's manager, and opened his European career with a concert atWigmore Hall. The concert featured works byClaude Debussyand Stravinsky, as well as his own compositions.[17]
He spent a year in Berlin, planning to work withArtur Schnabel, and gave concerts inBudapest, Vienna, and at theDonaueschingen Festival. As he had desired, he achieved notoriety, but often had to pay the concert expenses out of his own pocket. His financial situation was not helped by Mrs. Bok's reduction of his stipend by half, though she often responded to requests to fund specific aspects of his concerts.[17]He met Boski Markus, a Hungarian and niece of the Austrian playwrightArthur Schnitzler, who became his companion and whom he married in 1925.[19][20]
In the fall of 1922, Antheil took advantage of a chance meeting to introduce himself to his idol Stravinsky in Berlin. They established a warm intimacy and the more established composer encouraged Antheil to move to Paris.[21][22]He went as far as arranging a concert to launch Antheil's career in the French capital, but the younger man failed to show up, preferring to travel to Poland with Markus.[23]The Antheils finally arrived in Paris in June 1923, in time to attend the premiere of Stravinsky's balletLes Noces, but the relationship with Stravinsky did not survive for long. Stravinsky snubbed the younger man, having discovered that Antheil had boasted that "Stravinsky admired his work". The breach devastated Antheil, and was not ultimately repaired until 1941, when Stravinsky sent the family tickets to a concert he was giving in Hollywood.[3][22]
Despite the inauspicious beginning, Antheil found Paris, at the time, a center of musical and artistic innovation, to be a "green tender morning" compared to the "black night" of Berlin.[24]
The couple lived in a one bedroom apartment aboveSylvia Beach's bookshopShakespeare and Company.[20][25]Beach described him "as fellow withbangs, a squished nose and a big mouth with a grin in it. A regular American high school boy."[26]She was very supportive, and introduced Antheil to her circle of friends and customers includingErik Satie,Ezra Pound,James Joyce,Virgil Thomson, andErnest Hemingway. Joyce and Pound were soon talking of an opera collaboration.[27]Pound, in particular, was to become an extravagant supporter and promoter of Antheil and his work, comparing him variously to Stravinsky andJames Cagney, and describing him as breaking down music to its "musical atom". Pound introduced Antheil toJean Cocteau, who in turn helped launch Antheil into the musical salons of Paris; Pound also commissioned Antheil to write three violin sonatas for his mistress,Olga Rudge. In 1924, Pound publishedAntheil and the Treatise on Harmony, as part of his campaign to boost Antheil's reputation. The book may have done Antheil more harm than good, and the composer was to distance himself from it in his memoir.[28][29][30]Natalie Barneyhelped produce some original works, including the First String Quartet in 1925.[31]
Antheil was asked to make his Paris debut at the opening of theBallets suédois, an important Paris social event. He programmed several recent compositions, including the "Airplane Sonata", the "Sonata Sauvage", and "Mechanism". Halfway through his performance, a riot broke out, much to Antheil's delight. According to Antheil, "People were fighting in the aisles, yelling, clapping, hooting! Pandemonium! ... the police entered, and any number of surrealists, society personages, and people of all descriptions were arrested ... Paris hadn't had such a good time since the premiere of Stravinsky'sSacre du Printemps."[32]The riot was filmed and may in fact have been engineered, as theMarcel L'HerbiermovieL'Inhumaineneeded a riot scene set in a concert hall. In the audience were Erik Satie,Darius Milhaud,Man Ray,Pablo Picasso,Jean Cocteau, andFrancis Picabia. Antheil was delighted when Satie and Milhaud praised his music.[29][32]
Reactions to his first performances were cool at best. His technique was loud, brazen, and percussive. Critics wrote that he hit the piano rather than played it, and indeed he often injured himself by doing so. As part of his "bad boy" behavior, Antheil provocatively pulled a revolver from his jacket and laid it on the piano.[23]
Antheil's best-known composition isBallet Mécanique. The "ballet" was originally conceived to be accompanied bythe film of the same nameby experimental filmmakersFernand LégerandDudley Murphy(with cinematography byMan Ray),[33]although the nature of the collaboration is mysterious.[34]The first productions of Antheil's work in 1925 and 1926 did not include the film, which turned out to last around 19 minutes, only half as long as Antheil's score.[33]
Antheil described his "first big work" as "scored for countless numbers of player pianos. All percussive. Like machines. All efficiency. No LOVE. Written without sympathy. Written cold as an army operates. Revolutionary as nothing has been revolutionary."[35]Antheil's original conception was scored for 16 specially synchronized player pianos, two grand pianos, electronic bells, xylophones, bass drums, a siren and three airplane propellers, but difficulties with the synchronization resulted in a rewrite for a single pianola and multiple human pianists.[36]The piece consisted of periods of music and interludes of silence set against the roar of the airplane propellers.[37]Antheil described as "by far my most radical work ... It is the rhythm of machinery, presented as beautifully as an artist knows how."[38]The Léger-Murphy film and Antheil's score were finally performed together at theMuseum of Modern Art, New York, in 1935.[33]
Antheil assiduously promoted the work, and even engineered his supposed "disappearance" while on a visit to Africa, so as to get media attention for a preview concert.[39]The Paris première in June 1926 was sponsored by an American patroness who at the end of the concert was (according to Antheil) "tossed in a blanket by three baronesses and a duke."[2]At the official première at the Théâtre du Champs Elysées, Antheil's music enraged some of the concert-goers, whose objections were drowned out by the cacophonous music,[31]while others vocally supported the work, and the concert ended with a riot in the streets.[40]
On April 10, 1927, Antheil rented New York'sCarnegie Hallto present an entire concert devoted to his works, including the American debut ofBallet Mécaniquein a scaled-down version. He commissioned elaborate backdrops of skyscrapers and machines, and engaged an African American orchestra to premiere hisA Jazz Symphony.[33][41]The concert started well, but according to the concert's promoter and producer, when the wind machine was turned on, "all hell, in a minor way, broke loose." During the gale, audience members clutched their programs and their hats, one "tied a handkerchief to his cane and waved it wildly in the air in a sign of surrender." Much to the amusement of the audience, the untested siren failed to sound on cue, despite frantic cranking and reached its climax only after the end of the performance, as the audience was clapping and leaving the hall.[42]American critics were hostile, calling the concert "a bitter disappointment" and dismissing theBallet Mécaniqueas "boring, artless, and naive" and Antheil's hoped-for riots failed to materialize. The failure of theBallet Mécaniqueaffected him deeply, and he never fully recovered his reputation during his lifetime,[43]though his interest in the mechanical was emulated by other prominent composers such asArthur Honegger,Sergei Prokofiev, andErik Satie.[44]In 1954, Antheil created a modified version of the work for percussion, four pianos, and a recording of an airplane motor.[2]
In the late 1920s, Antheil again moved to Germany, where he worked as assistant musical director of the municipal theatre in Berlin, and wrote music for the ballet and theatre. In 1930, he premiered his first operaTransatlantic.[33][45]This work, which involved American politics, gangsters, a bathtub scene and aChilds Restaurantwas a success at theFrankfurt Opera.[2]In 1933, the rise of theNazi partymade Antheil'savant gardemusic unwelcome in Germany, and in the depths of theDepression, he returned to the US and settled in New York City.
He re-entered American life with enthusiasm, organizing concerts, working on committees withAaron CoplandandWallingford Riegger, and writing piano, ballet, and film scores, as well as an operaHelen RetiresaboutHelen of Troy; the latter proved a flop. His music had moved away from more extreme aspects of modernism, and more tonal,neoromanticaspects were by now discernible in his work.[2][45][46]
In 1936, Antheil travelled to Hollywood, where he became a sought-after film composer, writing more than 30 scores for such directors asCecil B. DeMilleandNicholas Ray,[2][11]includingThe Scoundrel(1935) andThe Plainsman(1936).[30]The Antheils' only child, a son named Peter, was born in 1937.[47]Antheil found the industry hostile to modern music, complaining that it was a "closed proposition", and describing most background scores as "unmitigated tripe". He became increasingly dependent on more independent producers such asBen Hechtto give him work, such asAngels Over Broadway(1940) andSpecter of the Rose(1946).[48]He also wrote the score for the independent filmDementia(1955) andIn a Lonely Place(1950) starringHumphrey Bogart.[49]Antheil was confident in the ability of his music to save a weak film. "If I say so myself, I've saved a couple of sure flops".[49]
Besides writing scores for movies, he continued to compose other music, including for ballet[50]and six symphonies; his later works were in a more romantic style and influenced by Prokofiev and Shostakovich, as well as American music including jazz.[2][30][49][51]Works such as Serenade No. 1, Piano Sonata No. 4,Songs of ExperienceandEight Fragments from Shelley, written in 1948 showed a self-described desire "to disassociate myself from the passé modern schools of the last half-century, and to create a music for myself and those around me which has no fear of developed melody, real development itself, tonality, or other understandable forms."[47]His 1953 operaVolponewas premiered in New York in 1953 to mixed reviews,[49]while a visit to Spain in the 1950s influenced some of his last works, including the film score forThe Pride and the Passion(1957).[47]He also accepted a commission from theCBS Televisionnetwork to compose a theme for theirnewsreeland documentary film seriesThe Twentieth Century(1957–1966), narrated byWalter Cronkite.
Apart from music, Antheil had many other pursuits. In 1930, as Stacey Bishop, he wrote a murder mystery calledDeath in the Darkwith a character based onEzra Pound.[2][30]He was the film music reporter and critic for the magazineModern Musicfrom 1936 to 1940, writing columns considered lively and thoughtful, noting the comings and goings of musicians and composers during an era when the industry was flirting with more "modern" scores for films. He was disappointed, however, and wrote that "Hollywood, after a grand splurge with new composers and new ideas, has settled back into its old grind of producing easy and sure-fire scores."[48]
BeforeWorld War II, he participated in theHollywood Anti-Nazi League, putting on exhibits of artworks banned in Nazi Germany such as those byKäthe Kollwitz.[45]He also published a book of war predictions, entitledThe Shape of the War to Come.[50]
In 1945, he published a memoir calledBad Boy of Music, which became a bestseller.[47]
Antheil wrote a nationally syndicated newspaper relationship advice column, as well as regular columns in magazines such asEsquireandCoronet. He considered himself an expert on femaleendocrinology, and wrote a series of articles about how to determine the availability of women based on glandular effects on their appearance, with titles such as "The Glandbook for the Questing Male".[2][52]Another book of "glandular criminology" was titledEvery Man His Own Detective.
Antheil's interest in this area brought him into contact with the actressHedy Lamarr, who sought his advice about how she might enhance her upper torso. He suggested glandular extracts, but their conversation then moved totorpedoes.[52]
During World War II, Lamarr realized that a single radio-controlled torpedo could severely damage or sink enemy ships causing irreparable damage. However these radio-controlled torpedoes could easily be detected and jammed, by broadcasting interference at the frequency of the control signal, thereby causing the torpedo to go off course.[53]
Antheil and Lamarr developed the idea of usingfrequency hopping: in this case using aplayer pianoroll to randomly change the signal sent between the control center and torpedo at short bursts within a range of 88 frequencies on the spectrum (88 black and white keys are on a piano keyboard). The specific code for the sequence of frequencies would be held identically by the controlling ship and in the torpedo. This basically encrypted the signal, as it was impossible for the enemy to scan and jam all 88 frequencies because this would have required too much power. Antheil would control the frequency-hopping sequence using a player-piano mechanism, which he had earlier used to score hisBallet Mécanique.[54]
On August 11, 1942,U.S. patent 2,292,387was granted to Antheil and "Hedy Kiesler Markey", Lamarr's married name at the time. This early version of frequency hopping, though novel, soon met with opposition from the U.S. Navy and was not adopted.[55]
The idea was not implemented in the US until 1962, when it was used by U.S. military ships during a blockade of Cuba after the patent had expired.[56]The patent was little known until 1997, when theElectronic Frontier Foundationgave Lamarr a belated award for her contributions. In 1998, an Ottawa wireless technology developer,Wi-LAN, acquired a 49% claim to the long expired patent from Lamarr for an undisclosed amount of stock.[57]Lamarr and Antheil's frequency-hopping scheme shares some concepts with modernspread-spectrumcommunication technology, such asBluetooth,COFDMused inWi-Finetwork connections, andCDMAused in some cordless and wireless telephones. Blackwell, Martin, and Vernam's 1920 Secrecy Communication System (U.S. patent 1,598,673) was a predecessor to Kiesler and Antheil's patent, which employed the techniques in the autonomous control of torpedoes.
Antheil died of a heart attack in the New York City borough of Manhattan.[58]His legacy included three accomplished students,Henry Brant,Benjamin Lees, andRuth White. He is buried inRiverview Cemetery, in Trenton, New Jersey.
"Antheil's my man." Mrs. McKisco turned challengingly to Rosemary, "Antheil and Joyce. I don't suppose you ever hear much about those sort of people in Hollywood, but my husband wrote the first criticism ofUlyssesthat ever appeared in America." (F. Scott Fitzgerald,Tender Is the Night)[59]
|
https://en.wikipedia.org/wiki/George_Antheil
|
Have Quick(alsoHAVEQUICK, shortHQ) is anECM-resistantfrequency-hopping systemused to protect militaryaeronautical mobile (OR)radio traffic.
Since the end ofWorld War II, U.S. and Allied military aircraft have used AM radios in theNATO harmonised 225–400 MHz UHF band(part ofNATOB band[1]) for short range air-to-air and ground-to-air communications. During development and the procurement of UHF radios, military planners did not require features to secure communications foraircraftandhelicoptersfrom jamming until the post-Vietnam Warera. Progress in electronics in the 1970s reached a point where anyone with an inexpensive radio frequency scanner or receiver set could intercept military communications. Once the target frequencies were identified, radio frequencyjammingcould easily be employed to degrade or completely disable communications.
The Have Quick program was a response to this problem. Engineers recognized that newer aircraft radios already included all-channelfrequency synthesizersalong with keyboards and displays for data entry. The only other system requirements to achieve the desiredanti-jamfunctionality were an accurate clock (for timed synchronization) and amicroprocessorto addfrequency hoppingto existing radios.
Aircraft and ground radios that employ HAVE QUICK must be initialized with accuratetime of day(TOD; usually from aGPSreceiver), aword of the day(WOD), and a net identifier (providing mode selection and multiple networks to use the same word of the day). A word of the day is atransmission security variablethat consists of six segments of six digits each. The word of the day is loaded into the radio or its control unit to key the HAVE QUICK system to the proper hopping pattern, rate, and dwell time.[2]The word of the day, time of day and net identifier are input to acryptographic pseudorandom number generatorthat controls the frequency changes.
HAVE QUICK is not anencryptionsystem, though many HAVE QUICK radios can be used with encryption; e.g. the KY-58VINSONsystem. HAVE QUICK is not compatible withSINCGARS, the VHF - FM radios used by ground forces, which operate in a different radio band and use a different frequency hopping method; however some newer radios support both.
HAVE QUICK was well adopted, and as of 2007 is used on nearly all U.S. military and NATO aircraft. Improvements include HAVE QUICK II Phase 2, and a "Second generation Anti-Jam Tactical UHF Radio forNATO" calledSATURN.[3]The latter features more complex frequency hopping.
|
https://en.wikipedia.org/wiki/HAVE_QUICK
|
Hedy Lamarr(/ˈhɛdi/; bornHedwig Eva Maria Kiesler; November 9, 1914[a]– January 19, 2000) was an Austrian-born American actress and inventor. After a brief early film career inCzechoslovakia, including the controversial erotic romantic dramaEcstasy(1933), she fled from her first husband,Friedrich Mandl, and secretly moved to Paris. Traveling to London, she metLouis B. Mayer, who offered her a film contract in Hollywood. Lamarr became a film star with her performance in the romantic dramaAlgiers(1938).[2]She achieved further success with the WesternBoom Town(1940) and the dramaWhite Cargo(1942). Lamarr's most successful film was the religious epicSamson and Delilah(1949).[3]She also acted on television before the release of her final film in 1958. She was honored with a star on theHollywood Walk of Famein 1960.
At the beginning ofWorld War II, along withGeorge Antheil, Lamarr co-invented a radio guidance system forAlliedtorpedoesthat usedspread spectrumandfrequency hoppingtechnology to defeat the threat ofradio jammingby theAxis powers. However, the technology was not used in operational systems until after World War II, and then independently of their patent.[4]
Lamarr was born Hedwig Eva Maria Kiesler in 1914 inVienna,[5]the only child of Gertrud "Trude" Kiesler (née Lichtwitz) and Emil Kiesler.
Her father was born to aGalician-Jewishfamily inLembergin theKingdom of Galicia and Lodomeria, part of theAustrian Empire(nowLvivinUkraine) and was, in the 1920s, deputy director ofWiener Bankverein,[6][7]and at the end of his life a director at the unitedCreditanstalt-Bankverein.[8][9]Her mother, a pianist and a native ofBudapest, had come from an upper-class Hungarian-Jewish family. She hadconverted to Catholicismand was described as a "practicing Christian" who raised her daughter as a Christian, although Hedy was not baptized at the time.[8]: 8
As a child, Lamarr showed an interest in acting and was fascinated by theater and film. At the age of 12, she won a beauty contest in Vienna.[10]She also began to learn about technological inventions with her father, who would take her out on walks, explaining how devices functioned.[11][12]
Lamarr was taking acting classes in Vienna when one day, she forged a note from her mother and went toSascha-Filmand was able to have herself hired as ascript girl. While there, she had a role as anextrain the romantic comedyMoney on the Street(1930), and then a small speaking part in the comedyStorm in a Water Glass(1931). ProducerMax Reinhardtthen cast her in a play entitledThe Weaker Sex, which was performed at theTheater in der Josefstadt. Reinhardt was so impressed with her that he brought her with him back toBerlin.[13]
However, she never actually trained with Reinhardt or appeared in any of his Berlin productions. Instead, she met the Russian theatre producerAlexis Granowsky, who cast her in his film directorial debut,The Trunks of Mr. O.F.(1931), starringWalter AbelandPeter Lorre.[14]Granowsky soon moved to Paris, but Lamarr stayed in Berlin and was given the lead role inNo Money Needed(1932), a comedy directed byCarl Boese.[15]Lamarr then starred in the film which made her internationally famous.
In early 1933, at age 18, Lamarr was given the lead inGustav Machatý's filmEcstasy(Ekstasein German,Extasein Czech). She played the neglected young wife of an indifferent older man.
The film became both celebrated and notorious for showing Lamarr's face in the throes of orgasm as well as close-up and brief scenes of nudity. Lamarr claimed she was "duped" by the director and producer, who used high-power telephoto lenses, although the director contested her claims.[16][b][17]
Although she was dismayed and now disillusioned about taking other roles, the film gained world recognition after winning an award at theVenice Film Festival.[18]Throughout Europe, it was regarded as an artistic work. In America, it was considered overly sexual and received negative publicity, especially among women's groups.[16]It was banned there and in Germany.[19]
Lamarr played a number of stage roles, including a starring one inSissy, a play aboutEmpress Elisabeth of Austriaproduced in Vienna. It won accolades from critics. Admirers sent roses to herdressing roomand tried to get backstage to meet her. She sent most of them away, including a man who was more insistent,Friedrich Mandl.[16]He became obsessed with getting to know her.[20]
Mandl was an Austrian military arms merchant[21]and munitions manufacturer who was reputedly the third-richest man in Austria. She fell for his charming and fascinating personality, partly due to his immense financial wealth.[19]Her parents, both ofJewish descent, did not approve due to Mandl's ties to Italian fascist leaderBenito Mussoliniand, later, German FührerAdolf Hitler, but they could not stop the headstrong Lamarr.[16]
On August 10, 1933, Lamarr married Mandl at theKarlskirche. She was 18 years old and he was 33. In her autobiography,Ecstasy and Me, she described Mandl as an extremely controlling husband who strongly objected to her simulated orgasm scene inEcstasyand prevented her from pursuing her acting career. She claimed she was kept a virtual prisoner in their castle home,Schloss Schwarzenau[de].[19]
Mandl had close social and business ties to the Italian government, selling munitions to the country,[8]and had ties to theNaziregime of Germany as well, even though his own father was Jewish, as was Hedy's. Lamarr wrote that the dictators of both countries attended lavish parties at the Mandl home. Lamarr accompanied Mandl to business meetings, where he conferred with scientists and other professionals involved in military technology. These conferences were her introduction to the field of applied science and nurtured her latent talent in science.[22]
Lamarr's marriage to Mandl eventually became unbearable and she decided to separate herself from both her husband and country in 1937. In herautobiography, she wrote that she disguised herself as her maid and fled toParis, but according to other accounts she persuaded Mandl to let her wear all of her jewelry for a dinner party and then disappeared afterward.[23]She wrote about her marriage:
I knew very soon that I could never be an actress while I was his wife. ... He was the absolute monarch in his marriage. ... I was like a doll. I was like a thing, some object of art which had to be guarded—and imprisoned—having no mind, no life of its own.[24]
After arriving in London[25]in 1937, she metLouis B. Mayer, head ofMGM, who was scouting for talent in Europe.[26]She initially turned down the offer he made her (of $125 a week), but then booked herself onto the same New York-bound liner as him, and managed to impress him enough to secure a $500-a-week contract. Mayer persuaded her to change her name to Hedy Lamarr (to distance herself from her real identity, and "theEcstasylady" reputation associated with it),[23]choosing the surname in homage to the beautiful silent film star,Barbara La Marr, on the suggestion of his wife, who admired La Marr. He brought her to Hollywood in 1938 and began promoting her as the "world's most beautiful woman".[27]
Mayer loaned Lamarr to producerWalter Wanger, who was makingAlgiers(1938), an American version of the French film,Pépé le Moko(1937). Lamarr was cast in the lead oppositeCharles Boyer. The film created a "national sensation", says Shearer.[8]: 77She was billed as an unknown but well-publicized Austrian actress, which created anticipation in audiences. Mayer hoped she would become anotherGreta GarboorMarlene Dietrich.[8]: 77According to one viewer, when her face first appeared on the screen, "everyone gasped ... Lamarr's beauty literally took one's breath away."[8]: 2
In future Hollywood films, she was invariablytypecastas the archetypal glamorous seductress of exotic origin. Her second American film was to beI Take This Woman, co-starring withSpencer Tracyunder the direction of regular Dietrich collaboratorJosef von Sternberg. Von Sternberg was fired during the shoot, replaced byFrank Borzage. The film was put on hold, and Lamarr was put intoLady of the Tropics(1939), where she played a mixed-race seductress in Saigon oppositeRobert Taylor. She returned toI Take This Woman, re-shot byW. S. Van Dyke. The resulting film was a flop.
Far more popular wasBoom Town(1940) withClark Gable,Claudette ColbertandSpencer Tracy; it made $5 million.[28]MGM promptly reteamed Lamarr and Gable inComrade X(1940), a comedy film in the vein ofNinotchka(1939), which was another hit.
Lamarr was teamed withJames StewartinCome Live with Me(1941), playing a Viennese refugee. Stewart was also inZiegfeld Girl(1941), where Lamarr,Judy GarlandandLana Turnerplayed aspiring showgirls – a big success.[28]
Lamarr was top-billed inH. M. Pulham, Esq.(1941), although the film's protagonist was the title role played byRobert Young. She made a third film with Tracy,Tortilla Flat(1942). It was successful at the box office, as wasCrossroads(1942) withWilliam Powell.
Lamarr played the exotic Arab seductress[29]Tondelayo inWhite Cargo(1942), top billed overWalter Pidgeon. It was a huge hit.White Cargocontains arguably her most memorable film quote, delivered with provocative invitation: "I am Tondelayo. I maketiffinfor you?" This line typifies many of Lamarr's roles, which emphasized her beauty and sensuality while giving her relatively few lines. The lack of acting challenges bored Lamarr. She reportedly took up inventing to relieve her boredom.[30]
She was reunited with Powell in a comedyThe Heavenly Body(1944), then was borrowed by Warner Bros forThe Conspirators(1944). This was an attempt to repeat the success ofCasablanca(1943), and RKO borrowed her for a melodramaExperiment Perilous(1944).
Back at MGM Lamarr was teamed withRobert Walkerin the romantic comedyHer Highness and the Bellboy(1945), playing a princess who falls in love with a New Yorker. It was very popular, but would be the last film she made under her MGM contract.[31]
Her off-screen life and personality during those years was quite different from her screen image. She spent much of her time feeling lonely and homesick. She might swim at her agent's pool, but shunned the beaches and staring crowds. When asked for an autograph, she wondered why anyone would want it. Writer Howard Sharpe interviewed her and gave his impression:
Hedy has the most incredible personal sophistication. She knows the peculiarly European art of being womanly; she knows what men want in a beautiful woman, what attracts them, and she forces herself to be these things. She has magnetism with warmth, something that neither Dietrich nor Garbo has managed to achieve.[16]
AuthorRichard Rhodesdescribes herassimilationinto American culture:
Of all the European émigrés who escaped Nazi Germany and Nazi Austria, she was one of the very few who succeeded in moving to another culture and becoming a full-fledged star herself. There were so very few who could make the transition linguistically or culturally. She really was a resourceful human being–I think because of her father's strong influence on her as a child.[32]
Lamarr also had a penchant forspeaking about herself in the third person.[33]
Lamarr wanted to join theNational Inventors Council, but was reportedly told by NIC memberCharles F. Ketteringand others that she could better help the war effort by using her celebrity status to sellwar bonds.[34][35]
She participated in a war-bond-selling campaign with a sailor named Eddie Rhodes. Rhodes was in the crowd at each Lamarr appearance, and she would call him up on stage. She would briefly flirt with him before asking the audience if she should give him a kiss. The crowd would say yes, to which Hedy would reply that she would if enough people bought war bonds. After enough bonds were purchased, she would kiss Rhodes and he would head back into the audience. Then they would head off to the next war bond rally.[36]
After leaving MGM in 1945, Lamarr formed a production company withJack Chertokand made the thrillerThe Strange Woman(1946). It went over budget and only made minor profits.[37]
She and Chertok then madeDishonored Lady(1947), another thriller starring Lamarr, which also went over budget – but was not a commercial success. She tried a comedy withRobert Cummings,Let's Live a Little(1948).
Lamarr enjoyed her biggest success playingDelilahagainstVictor Matureas theBiblical strongmaninCecil B. DeMille'sSamson and Delilah, the highest-grossing film of 1950. The film won two Oscars.[19]
Lamarr returned to MGM for afilm noirwithJohn Hodiak,A Lady Without Passport(1950), which flopped. More popular were two pictures she made at Paramount, a Western withRay Milland,Copper Canyon(1950), and aBob Hopespy spoof,My Favorite Spy(1951).
Her career went into decline. She went to Italy to play multiple roles inLoves of Three Queens(1954), which she also produced. However she lacked the experience necessary to make a success of such an epic production, and lost millions of dollars when she was unable to secure distribution of the picture.
She playedJoan of ArcinIrwin Allen's critically panned epic,The Story of Mankind(1957) and did episodes ofZane Grey Theatre("Proud Woman") andShower of Stars("Cloak and Dagger"). Her last film was a thrillerThe Female Animal(1958).
Lamarr was signed to act in the 1966 filmPicture Mommy Dead,[38]but was let go when she collapsed during filming from nervous exhaustion.[39]She was replaced in the role of Jessica Flagmore Shelley byZsa Zsa Gabor.
Although Lamarr had no formal training and was primarily self-taught, she invested her spare time, including on set between takes, in designing and drafting inventions,[40]which included an improvedtraffic stoplightand atabletthat would dissolve in water to create a flavoredcarbonated drink.[30]
During the late1930s, Lamarr attended arms deals with her then-husband, arms dealer Fritz Mandl, "possibly to improve his chances of making a sale".[41]From the meetings, she learned that navies needed "a way to guide a torpedo as it raced through the water." Radio control had been proposed. However, an enemy might be able tojamsuch a torpedo's guidance system and set it off course.[42]
When later discussing this with a new friend, composer and pianistGeorge Antheil, her idea to prevent jamming by frequency hopping met Antheil's previous work in music. In that earlier work, Antheil attempted synchronizing note-hopping in the avant-garde piece written as a score for the filmBallet Mécanique(1923–24) that involved multiple synchronizedplayer pianos. Antheil's idea in the piece was to synchronize the start time of identical player pianos with identical player piano rolls, so the pianos would play in time with one another. Together, they realized that radio frequencies could be changed similarly, using the same kind of mechanism, but miniaturized.[4][41]
Based on the strength of the initial submission of their ideas to theNational Inventors Council(NIC) in late December 1940, in early 1941 the NIC introduced Antheil to Samuel Stuart Mackeown, Professor of Electrical Engineering atCaltech, to consult on the electrical systems.[43][40]Lamarr hired the legal firm ofLyon & Lyonto draft the application for the patent[44][45]which was granted asU.S. patent 2,292,387on August 11, 1942, under her legal name Hedy Kiesler Markey.[46]The invention was proposed to the Navy, who rejected it on the basis that it would be too large to fit in a torpedo,[47]and Lamarr and Antheil, shunned by the Navy, pursued their invention no further. It was suggested that Lamarr invest her time and attention to selling war bonds since she was a celebrity.[48]
Lamarr became anaturalized citizenof the United States at age 38 on April 10, 1953. Herautobiography,Ecstasy and Me, was published in 1966. She said on TV that it was not written by her, and much of it was fictional.[49]Lamarr later sued the publisher, saying that many details were fabricated by itsghost writer, Leo Guild.[50][51]Lamarr, in turn, was sued by Gene Ringgold, who asserted that the book plagiarized material from an article he had written in 1965 forScreen Factsmagazine.[52]
In the late 1950s, along with former husband W. Howard Lee, Lamarr designed and developed the Villa LaMarr ski resort in Aspen, Colorado.[53][54]
In 1966, Lamarr was arrested in Los Angeles forshoplifting. The charges were eventually dropped. In 1991, she was arrested on the same charge inOrlando, Florida, this time for stealing $21.48 worth of laxatives and eye drops.[55][56]She pleaded no contest to avoid a court appearance, and the charges were dropped in return for her promise to refrain from breaking any laws for a year.[57]
The 1970s was a decade of increasing seclusion for Lamarr. She was offered several scripts, television commercials, and stage projects, but none piqued her interest. In 1974, she filed a $10 million lawsuit againstWarner Bros., claiming that the running parody of her name ("Hedley Lamarr") in theMel BrookscomedyBlazing Saddlesinfringed her right to privacy. Brooks said he was flattered. The studio settled out of court for an undisclosed nominal sum and an apology to Lamarr for "almost using her name". Brooks said that Lamarr "never got the joke".[58][59]In 1981, with her eyesight failing, Lamarr retreated from public life and settled inMiami Beach, Florida.[8]
A largeCorel-drawn image of Lamarr wonCorelDRAW's yearly software suite cover design contest in 1996. For several years, beginning in 1997, it was featured on boxes of the software suite. Lamarr sued the company for using her image without her permission. Corel countered that she did not own rights to the image. The parties reached an undisclosed settlement in 1998.[60][61]
For her contribution to the motion picture industry, Lamarr has a star on theHollywood Walk of Fameat 6247Hollywood Boulevard[62][63]adjacent toVine Streetwhere the walk is centered.
Lamarr became estranged from her older son, James Lamarr Loder, when he was 12 years old. Their relationship ended abruptly, and he moved in with another family. They did not speak again for almost 50 years. Lamarr left James Loder out of her will, and he sued for control of the US$3.3 million estate left by Lamarr in 2000.[64]He eventually settled for US$50,000.[65]
In the last decades of her life, the telephone became Lamarr's only means of communication with the outside world, even with her children and close friends. She often talked up to six or seven hours a day on the phone, but she spent hardly any time with anyone in person in her final years.[66][citation needed]
Lamarr died inCasselberry, Florida,[67]on January 19, 2000, of heart disease, aged 85.[8]Her son Anthony Loder spread part of her ashes in Austria'sVienna Woodsin accordance with her last wishes.[68]
In 2014, a memorial to Lamarr was unveiled in Vienna'sCentral Cemetery.[69]The remainder of her ashes were buried there.[70][71]
On January 7, 1939, Hedy Lamarr was selected the "most promising new actress" of 1938 in a poll ofPhiladelphiafilm fans conducted by Elsie Finn, thePhiladelphia Recordfilm critic.[72]
On January 26, 1939, Lamar was chosen the "ideal type" of woman in a poll of both male and female students conducted by thePomona Collegenewspaper.[73]
On May 9, 1939, Lamarr was named the "most beautiful actress" in "a secret poll of 30 Hollywood correspondents" conducted by the American magazineLook.[74]
On August 30, 1940, Lamarr won "top honors for facial features" in a poll of 400 members of the California Models Association.[75]
In December 1943, makeup expertMax Factor, Jr.included Lamarr among the ten glamorous Hollywood actresses with the most appealing voices.[76]
In 1951, British moviegoers voted Lamarr the year's 10th best actress, for her performance inSamson and Delilah.[77]
In 1960, Lamarr was honored with a star on theHollywood Walk of Famefor her contributions to the motion picture industry.[78]
In 1997, Lamarr andGeorge Antheilwere jointly honored with theElectronic Frontier Foundation'sPioneer Award[79]and Lamarr also was the first woman to receive the Invention Convention's BULBIE Gnass Spirit of Achievement Award, known as the "Oscars of inventing".[80][81][82]given to individuals whose creative lifetime achievements in the arts, sciences, business, or invention fields have significantly contributed to society.[83]The following year, Lamarr's native Austria awarded her the Viktor Kaplan Medal of the Austrian Association of Patent Holders and Inventors.[84]
In 2006, theHedy-Lamarr-Wegwas founded in ViennaMeidling(12th District), named after the actress.
In 2013, theIQOQIinstalled aquantum telescopeon the roof of theUniversity of Vienna, which they named after her in 2014.[85]
In 2014, Lamarr was posthumously inducted into theNational Inventors Hall of Famefor frequency-hopping spread spectrum technology.[86]The same year, Anthony Loder's request that the remaining ashes of his mother should be buried in an honorary grave of the city ofViennawas realized. On November 7, her urn was buried at theVienna Central Cemeteryin Group 33 G, Tomb No. 80, not far from the centrally located presidential tomb.[70][71]
On November 9, 2015,Googlehonored her on the 101st anniversary of her birth, and on her 109th on November 9, 2023 with adoodle.[87]
On August 27, 2019, an asteroid was named after her:32730 Lamarr.[88][89]
On August 6, 2023Star Trek: Prodigyshowrunners Dan and Kevin Hageman debuted the first five minutes of footage from season two, showing the new Lamarr-class USS Voyager-A, in tribute to her.[90]
Lamarr was married and divorced six times and had three children:
Following her sixth and final divorce in 1965, Lamarr remained unmarried for the last 35 years of her life.
Throughout her life, Lamarr claimed that her first son, James Lamarr Loder, was not biologically related to her and was adopted during her marriage to Gene Markey.[94][95]However, years later, her son found documentation that he was theout-of-wedlockson of Lamarr and actor John Loder, whom she later married as her third husband.[96]However, a later DNA test proved him not to be biologically related to either.[97]
Source:Hedy Lamarrat theTCM Movie Database
In the 1952 Pulitzer Prize-winning novelThe Caine MutinybyHerman Wouk, Hedy Lamarr is mentioned by name in Chapter 37 when defense attorney Lieutenant Barney Greenwald confronts Lieutenant Tom Keefer at a party after Lieutenant Stephen Maryk's court-martial acquittal in theCainemutiny.[c]–[104]
TheMel Brooks1974 western parodyBlazing Saddlesfeatures a villain, played byHarvey Korman, named "Hedley Lamarr". As a running gag, various characters mistakenly refer to him as "Hedy Lamarr" prompting him to testily reply "That's Hedley."
In the 1982off-BroadwaymusicalLittle Shop of Horrorsand subsequent film adaptation (1986), Audrey II says to Seymour in the song "Feed Me", that he can get Seymour anything he wants including "A date with Hedy Lamarr."[105]
In the 2004 video gameHalf-Life 2,Dr. Kleiner's petheadcrab, Lamarr, is named after Hedy Lamarr.[106]
Her son, Anthony Loder, was featured in the 2004 documentary filmCalling Hedy Lamarr, in which he played excerpts from tapes of her many telephone calls.
In 2008, anoff-Broadwayplay,Frequency Hopping, features the lives of Lamarr and Antheil. The play was written and staged by Elyse Singer, and the script won a prize for best new play about science and technology fromSTAGE.[8][107]
In the 2009 mockumentaryThe Chronoscope,[108]written and directed by Andrew Legge, the fictional Irish scientist Charlotte Keppel is likely modeled after Hedy Lamarr. The film satirizes the extreme politics of the 1930s and tells the story of a fictionalized fascist group that steals a device invented by Keppel. This chronoscope can see the past and is used by the group to create propaganda films of their heroes from the past.
In 2010, Lamarr was selected out of 150 IT people to be featured in a short film launched by theBritish Computer Societyon May 20.[109]
Also during 2010, theNew York Public LibraryexhibitThirty Years of Photography at the New York Public Libraryincluded a photo of a topless Lamarr (c.1930) by Austrian-born American photographerTrude Fleischmann.[110]
In 2011, the story of Lamarr'sfrequency-hopping spread spectruminvention was explored in an episode of theScience ChannelshowDark Matters: Twisted But True, a series that explores the darker side of scientific discovery and experimentation, which premiered on September 7.[111]Her work in improving wireless security was part of the premiere episode of theDiscovery ChannelshowHow We Invented the World.[112]
Also during 2011,Anne Hathawayrevealed that she had learned that the original Catwoman was based on Lamarr, so she studied all of Lamarr's films and incorporated some of her breathing techniques into her portrayal ofCatwomanin the 2012 filmThe Dark Knight Rises.[113]
In 2015, on November 9, the 101st anniversary of Lamarr's birth, Google paid tribute to Hedy Lamarr's work in film and her contributions to scientific advancement with an animatedGoogle Doodle.[114]
In 2016, Lamarr was depicted in an off-Broadway play,HEDY! The Life and Inventions of Hedy Lamarr, a one-woman show written and performed by Heather Massie.[115][116]
Also in 2016, the off-Broadway, one-actor showStand Still and Look Stupid: The Life Story of Hedy Lamarr, starring Emily Ebertz and written by Mike Broemmel, went into production.[117][118]
Also during 2016,Whitney Frost, a character in the TV showAgent Carterwas inspired by Hedy Lamarr andLauren Bacall.[119]
In 2017, actress Celia Massingham portrayed Lamarr onThe CWtelevision seriesLegends of Tomorrowin the sixth episode of the third season, titled "Helen Hunt". The episode is set in 1937 Hollywoodland. The episode aired on November 14, 2017.[120]
Also during 2017, a documentary about Lamarr's career as an actress and later as an inventor,Bombshell: The Hedy Lamarr Story,premiered at the 2017Tribeca Film Festival. The documentary was written and directed by Alexandra Dean and produced bySusan Sarandon;[121][32]it was released in theaters on November 24, 2017, and aired on PBSAmerican Mastersin May 2018.
In 2018, actressAlyssa Sutherlandportrayed Lamarr on the NBC television seriesTimelessin the third episode of the second season, titled "Hollywoodland". The episode aired March 25, 2018.[122]
In 2019, actor and musicianJohnny Deppcomposed a song called "This Is a Song for Miss Hedy Lamarr" withTommy Henriksen. It was included on Depp andJeff Beck's 2022 album18.[123]
Also in 2019,The Only Woman in the Room, a novel based on Hedy Lamarr's life byMarie Benedict, was published bySourcebooks Landmark. The book is aNew York Timesand USA Today bestseller andBarnes & NobleBook Club Pick.[124]In 2019, it received a space in Library Reads's Hall of Fame.[125]
In 2021, Lamarr was mentioned in the first episode of the Marvel'sWhat If...?.[126]The episode aired on August 11, 2021.
In May 2023, a dance production calledHedy Lamarr: An American Musewas made in her honor by Linze Rickles McRae. She was accompanied by her daughter, Azalea McRae, with whom she performed it, alongside her students at her dancing school, Downtown Dance Conservatory in Gadsden, AL.[127]
In July 2024, the principal setting of the second season of the Netflix/Nickelodeon/Paramount television seriesStar Trek Prodigyis the science vessel USS Voyager, NCC-74656-A, a Starship of the Lamarr class, classified in honor of Lamarr's scientific contributions.[128]
|
https://en.wikipedia.org/wiki/Hedy_Lamarr
|
Open spectrum(also known asfree spectrum) is a movement to get theFederal Communications Commissionto provide moreunlicensedradio-frequencyspectrumthat is available for use by all. Proponents of the "commons model" of open spectrum advocate a future where all the spectrum is shared, and in which people useInternetprotocols to communicate with each other, and smart devices, which would find the most effective energy level, frequency, and mechanism.[1]Previous government-imposed limits on who can have stations and who cannot would be removed,[2]and everyone would be givenequal opportunityto use the airwaves for their own radio station, television station, or even broadcast their own website. A notable advocate for Open Spectrum isLawrence Lessig.
National governments currently allocate bands of spectrum (sometimes based on guidelines from theITU) for use by anyone so long as they respect certain technical limits, most notably, a limit on total transmission power. Unlicensed spectrum isdecentralized: there are no license payments or central control for users. However, sharing spectrum between unlicensed equipment requires that mitigation techniques (e.g.: power limitation, duty cycle, dynamic frequency selection) are imposed to ensure that these devices operate without interference.
Traditional users of unlicensed spectrum include cordless telephones, and baby monitors. A collection of new technologies are taking advantage of unlicensed spectrum includingWi-Fi,Ultra Wideband,spread spectrum,software-defined radio,cognitive radio, andmesh networks.[3]
Astronomers use manyradio telescopesto look up at objects such aspulsarsin our ownGalaxyand at distantradio galaxiesup to about half the distance of the observable sphere of ourUniverse. The use of radio frequencies for communication creates pollution from the point of view of astronomers, at best, creating noise or, at worst, totally blinding the astronomical community for certain types of observations of very faint objects. As more and more frequencies are used for communication, astronomical observations are getting more and more difficult.
Negotiations to defend the parts of the spectrum most useful for observing the Universe are mostly carried out by the international astronomical community, as a grassroots community effort, coordinated in the Scientific Committee on Frequency Allocations for Radio Astronomy and Space Science.
|
https://en.wikipedia.org/wiki/Open_spectrum
|
Spread-spectrum time-domain reflectometry(SSTDR) is a measurement technique to identify faults, usually in electrical wires, by observing reflected spread spectrum signals. This type oftime-domain reflectometrycan be used in various high-noise and live environments. Additionally, SSTDR systems have the additional benefit of being able to precisely locate the position of the fault. Specifically, SSTDR is accurate to within a few centimeters for wires carrying 400 Hz aircraft signals as well asMIL-STD-1553data bus signals.[1]An SSTDR system can be run on a live wire because the spread spectrum signals can be isolated from the system noise and activity.
At the most basic level, the system works by sending spread spectrum signals down a wireline and waiting for those signals to be reflected back to the SSTDR system. The reflected signal is then correlated with a copy of the sent signal. Mathematical algorithms are applied to both the shape and timing of the signals to locate either the short or the end of an open circuit.
Spread-spectrum time domain reflectometry is used in detecting intermittent faults in live wires. From buildings and homes to aircraft and naval ships, this technology can discover irregular shorts on live wire running400 Hz, 115 V. For accurate location of a wiring system's fault the SSTDR associates thePN codewith the signal on the line then stores the exact location of the correlation before the arc dissipates. Present SSTDR can collect a complete data set in under 5 ms.[2]
SSTDR technology allows for analysis of a network of wires. One SSTDR sensor can measure up to 4 junctions in a branched wire system.[3]
|
https://en.wikipedia.org/wiki/Spread-spectrum_time-domain_reflectometry
|
Time-hopping(TH) is a communications signal technique which can be used to achieveanti-jamming(AJ) orlow probability of intercept(LPI). It can also refer topulse-position modulation, which in its simplest form employs 2kdiscrete pulses (referring to the unique positions of the pulse within the transmission window) to transmit k bit(s) per pulse.
To achieve LPI, the transmission time is changed randomly by varying the period and duty cycle of the pulse (carrier) using a pseudo-random sequence. The transmitted signal will then have intermittent start and stop times. Although often used to form hybridspread-spectrum(SS) systems, TH is strictly speaking a non-SS technique. Spreading of the spectrum is caused by other factors associated with TH, such as using pulses with low duty cycle having a wide frequency response. An example of hybrid SS is TH-FHSS or hybrid TDMA (time division multiple access).
This technology-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Time-hopping_spread_spectrum
|
In computing,56-bit encryptionrefers to akey sizeof fifty-sixbits, or sevenbytes, forsymmetric encryption. While stronger than40-bit encryption, this still represents a relatively lowlevel of securityin the context of abrute force attack.
The US government traditionally regulated encryption for reasons of national security, law enforcement and foreign policy. Encryption was regulated from 1976 by theArms Export Control Actuntil control was transferred to theDepartment of Commercein 1996.
56-bit refers to the size of a symmetric key used to encrypt data, with the number of unique possible permutations being256{\displaystyle 2^{56}}(72,057,594,037,927,936). 56-bit encryption has its roots inDES, which was the official standard of the USNational Bureau of Standardsfrom 1976, and later also theRC5algorithm. US government regulations required any users of stronger 56-bit symmetric keys to submit to key recovery through algorithms likeCDMFor key escrow,[1]effectively reducing the key strength to 40-bit, and thereby allowing organisations such as theNSAto brute-force this encryption. Furthermore, from 1996 software productsexportedfrom the United States were not permitted to use stronger than 56-bit encryption, requiring different software editions for the US and export markets.[2]In 1999, US allowed 56-bit encryption to be exported without key escrow or any other key recovery requirements.
The advent ofcommerce on the Internetand faster computers raised concerns about the security ofelectronic transactionsinitially with 40-bit, and subsequently also with 56-bit encryption. In February 1997,RSA Data Securityran a brute force competition with a $10,000 prize to demonstrate the weakness of 56-bit encryption; the contest was won four months later.[3]In July 1998, a successful brute-force attack was demonstrated against 56-bit encryption withDeep Crackin just 56 hours.[4]
In 2000, all restrictions on key length were lifted, except for exports to embargoed countries.[5]
56-bit DES encryption is now obsolete, having been replaced as a standard in 2002 by the 128-bit (and stronger)Advanced Encryption Standard. DES continues to be used as a symmetric cipher in combination withKerberosbecause older products do not support newer ciphers likeAES.[6]
|
https://en.wikipedia.org/wiki/56-bit_encryption
|
TheContent Scramble System(CSS) is adigital rights management(DRM) andencryptionsystem employed on many commercially producedDVD-Videodiscs. CSS utilizes aproprietary40-bitstream cipheralgorithm. The system was introduced around 1996 and was first compromised in 1999.[1]
CSS is one of several complementary systems designed torestrict DVD-Videoaccess.
It has been superseded by newer DRM schemes such asContent Protection for Recordable Media(CPRM), or byAdvanced Encryption Standard(AES) in theAdvanced Access Content System(AACS) DRM scheme used byHD DVDandBlu-ray Disc, which have 56-bit and 128-bitkey sizes, respectively, providing a much higherlevel of securitythan the less secure 40-bit key size of CSS.
The content scramble system (CSS) is a collection of proprietary protection mechanisms forDVD-Videodiscs. CSS attempts to restrict access to the content only for licensed applications. According to theDVD Copy Control Association(CCA), which is the consortium that grants licenses, CSS is supposed to protect the intellectual property rights of the content owner.
The details of CSS are only given to licensees for a fee. The license,[2]which binds the licensee to anon-disclosure agreement, would not permit the development ofopen-source softwarefor DVD-Video playback. Instead, there islibdvdcss, areverse engineeredimplementation of CSS. Libdvdcss is a source for documentation, along with the publicly availableDVD-ROM[3]andMMC[4]specifications. There has also been some effort to collect CSS details from various sources.[5]
A DVD-Video can be produced with or without CSS. A publisher may decide to not use CSS protection in order to save license and production costs.
The content scramble system deals with three participants: the disc, the drive and the player. The disc holds the purported copyright information and the encrypted feature. The drive provides the means to read the disc. The player decrypts and presents the audio and visual content of the feature. All participants must conform to the CCA's license agreement.
There are three protection methods:
The first two protection methods have been broken. Circumvention of regional protection is not possible with every drive—even if the drive grants access to the feature, prediction of title keys may fail.[5]However, DVD players exist which do not enforce regional restrictions (after being disabled manually), which makes regional restrictions less effective as a component of CSS.[6]
The DVD-ROM's main-data (§16[3]), consisting of consecutive logical blocks of 2048 bytes, is structured according to the DVD-Video format. The DVD-Video contains (besides others) anMPEG program streamwhich consists of so-called Packs. If CSS is applied to the disc then a subset of all Packs is encrypted with a title-key.
A DVD-ROM contains, besides the main-data, additional data areas. CSS stores there:
CSS also uses six bytes in the frame header for each logical block of user data (§16.3,[3]§6.29.3.1.5[4]):
The drive treats a DVD-Video disc as any DVD-ROM disc. The player reads the disc's user-data and processes them according to the DVD-Video format. However, if the drive detects a disc that has been compiled with CSS, it denies access to logical blocks that are marked as copyrighted (§6.15.3[4]). The player has to execute an authentication handshake first (§4.10.2.2[4]). The authentication handshake is also used to retrieve the disc-key-block and the title-keys.
The drive may also supportRegional Playback Control(RPC) to limit the playback of DVD-Video content to specific regions of the world (§3.3.26[4]). RPC Phase II drives hold an 8-bit region-code and adhere to all requirements of the CSS license agreement (§6.29.3.1.7[4]). It appears that RPC Phase II drives reject title-key requests on region mismatch. However, reading of user-data may still work.[5]
CSS employs astream cipherand mangles thekeystreamwith the plain-text data to produce the cipher text.[7]The stream cipher is based on twolinear-feedback shift register(LFSR) and set up with a 40-bit seed.
Mangling depends on the type of operation. There are three types:
In order to decrypt a DVD-Video, the player reads the disc-key-block and uses its player-key to decrypt the disc-key. Thereafter, the player reads the title-keys and decrypts them with the disc-key. A different title-key can be assigned for theVideo Managerand for eachVideo Title Set. The title-keys are used to decrypt the encrypted Packs.[5]
CSS employs cryptographic keys with a size of only 40 bits. This makes CSS vulnerable to abrute-force attack. At the time CSS was introduced, it was forbidden in the United States for manufacturers toexportcryptographic systems employing keys in excess of 40 bits, a key length that had already been shown to be wholly inadequate in the face of increasing computer processing power (seeData Encryption Standard).
Based on the leakedDeCSSsource-code,Frank A. Stevensonpublished in November 1999 three exploits that rendered the CSS cipher practically ineffective:[7]
The latter exploit recovers a disk-key from its hash-value in less than 18 seconds on a 450 MHz Intel Pentium III.
The CSS design was prepared for the leak of a few player-keys. New discs would not contain an encrypted variant for these player-keys in the disc-key-block. However, Stevenson's exploits made it possible to generate all player-keys.Libdvdcssuses such a list of generated player-keys.
There are cases when no title-keys are available. A drive may deny access on region mismatch but still permit reading of the encrypted DVD-Video. Ethan Hawke presented a plain-text prediction for data repetitions in theMPEG program streamthat enables the recovery of title-keys in real-time directly from the encrypted DVD-Video.[8]
InGeeks Bearing Gifts, authorTed Nelsonstates "DVD encryption was intentionally made light by the DVD encryption committee, based on arguments in a libertarian bookComputer Lib", a claim cited as originating from personal communication with ananonymous source; Nelson is the author ofComputer Lib.[9]
|
https://en.wikipedia.org/wiki/Content_Scramble_System
|
Inalgebraic geometry, aderived schemeis ahomotopy-theoretic generalization of aschemein which classicalcommutative ringsare replaced with derived versions such asdifferential graded algebras, commutativesimplicial rings, orcommutative ring spectra.
From the functor of points point-of-view, a derived scheme is a sheafXon the category of simplicial commutative rings which admits an open affine covering{Spec(Ai)→X}{\displaystyle \{Spec(A_{i})\to X\}}.
From the locallyringed spacepoint-of-view, a derived scheme is a pair(X,O){\displaystyle (X,{\mathcal {O}})}consisting of atopological spaceXand asheafO{\displaystyle {\mathcal {O}}}either of simplicial commutative rings or ofcommutative ring spectra[1]onXsuch that (1) the pair(X,π0O){\displaystyle (X,\pi _{0}{\mathcal {O}})}is aschemeand (2)πkO{\displaystyle \pi _{k}{\mathcal {O}}}is aquasi-coherentπ0O{\displaystyle \pi _{0}{\mathcal {O}}}-module.
Aderived stackis a stacky generalization of a derived scheme.
Over a field of characteristic zero, the theory is closely related to that of a differential graded scheme.[2]By definition, adifferential graded schemeis obtained by gluing affine differential graded schemes, with respect toétale topology.[3]It was introduced byMaxim Kontsevich[4]"as the first approach to derived algebraic geometry."[5]and was developed further by Mikhail Kapranov and Ionut Ciocan-Fontanine.
Just asaffinealgebraic geometry is equivalent (incategorical sense) to the theory ofcommutative rings(commonly calledcommutative algebra), affinederived algebraic geometryover characteristic zero is equivalent to the theory ofcommutative differential graded rings. One of the main example of derived schemes comes from the derived intersection of subschemes of a scheme, giving theKoszul complex. For example, letf1,…,fk∈C[x1,…,xn]=R{\displaystyle f_{1},\ldots ,f_{k}\in \mathbb {C} [x_{1},\ldots ,x_{n}]=R}, then we can get a derived scheme
where
is theétale spectrum.[citation needed]Since we can construct a resolution
the derived ringR/(f1)⊗RL⋯⊗RLR/(fk){\displaystyle R/(f_{1})\otimes _{R}^{\mathbf {L} }\cdots \otimes _{R}^{\mathbf {L} }R/(f_{k})}, aderived tensor product, is the koszul complexKR(f1,…,fk){\displaystyle K_{R}(f_{1},\ldots ,f_{k})}. The truncation of this derived scheme to amplitude[−1,0]{\displaystyle [-1,0]}provides a classical model motivating derived algebraic geometry. Notice that if we have a projective scheme
wheredeg(fi)=di{\displaystyle \deg(f_{i})=d_{i}}we can construct the derived scheme(Pn,E∙,(f1,…,fk)){\displaystyle (\mathbb {P} ^{n},{\mathcal {E}}^{\bullet },(f_{1},\ldots ,f_{k}))}where
with amplitude[−1,0]{\displaystyle [-1,0]}
Let(A∙,d){\displaystyle (A_{\bullet },d)}be a fixed differential graded algebra defined over a field of characteristic0{\displaystyle 0}. Then aA∙{\displaystyle A_{\bullet }}-differential graded algebra(R∙,dR){\displaystyle (R_{\bullet },d_{R})}is calledsemi-freeif the following conditions hold:
It turns out that everyA∙{\displaystyle A_{\bullet }}differential graded algebra admits a surjective quasi-isomorphism from a semi-free(A∙,d){\displaystyle (A_{\bullet },d)}differential graded algebra, called a semi-free resolution. These are unique up to homotopy equivalence in a suitablemodel category. The (relative)cotangent complexof an(A∙,d){\displaystyle (A_{\bullet },d)}-differential graded algebra(B∙,dB){\displaystyle (B_{\bullet },d_{B})}can be constructed using a semi-free resolution(R∙,dR)→(B∙,dB){\displaystyle (R_{\bullet },d_{R})\to (B_{\bullet },d_{B})}: it is defined as
Many examples can be constructed by taking the algebraB{\displaystyle B}representing a variety over a field of characteristic 0, finding a presentation ofR{\displaystyle R}as a quotient of a polynomial algebra and taking the Koszul complex associated to this presentation. The Koszul complex acts as a semi-free resolution of the differential graded algebra(B∙,0){\displaystyle (B_{\bullet },0)}whereB∙{\displaystyle B_{\bullet }}is the graded algebra with the non-trivial graded piece in degree 0.
Thecotangent complexof a hypersurfaceX=V(f)⊂ACn{\displaystyle X=\mathbb {V} (f)\subset \mathbb {A} _{\mathbb {C} }^{n}}can easily be computed: since we have the dgaKR(f){\displaystyle K_{R}(f)}representing thederived enhancementofX{\displaystyle X}, we can compute the cotangent complex as
whereΦ(gds)=g⋅df{\displaystyle \Phi (gds)=g\cdot df}andd{\displaystyle d}is the usual universal derivation. If we take a complete intersection, then the koszul complex
is quasi-isomorphic to the complex
This implies we can construct the cotangent complex of the derived ringR∙{\displaystyle R^{\bullet }}as the tensor product of the cotangent complex above for eachfi{\displaystyle f_{i}}.
Please note that the cotangent complex in the context of derived geometry differs from the cotangent complex of classical schemes. Namely, if there was a singularity in the hypersurface defined byf{\displaystyle f}then the cotangent complex would have infinite amplitude. These observations provide motivation for thehidden smoothnessphilosophy of derived geometry since we are now working with a complex of finite length.
Given a polynomial functionf:An→Am,{\displaystyle f:\mathbb {A} ^{n}\to \mathbb {A} ^{m},}then consider the (homotopy) pullback diagram
where the bottom arrow is the inclusion of a point at the origin. Then, the derived schemeZ{\displaystyle Z}has tangent complex atx∈Z{\displaystyle x\in Z}is given by the morphism
where the complex is of amplitude[−1,0]{\displaystyle [-1,0]}. Notice that the tangent space can be recovered usingH0{\displaystyle H^{0}}and theH−1{\displaystyle H^{-1}}measures how far awayx∈Z{\displaystyle x\in Z}is from being a smooth point.
Given a stack[X/G]{\displaystyle [X/G]}there is a nice description for the tangent complex:
If the morphism is not injective, theH−1{\displaystyle H^{-1}}measures again how singular the space is. In addition, the Euler characteristic of this complex yields the correct (virtual) dimension of the quotient stack.
In particular, if we look at the moduli stack of principalG{\displaystyle G}-bundles, then the tangent complex is justg[+1]{\displaystyle {\mathfrak {g}}[+1]}.
Derived schemes can be used for analyzing topological properties of affine varieties. For example, consider a smooth affine varietyM⊂An{\displaystyle M\subset \mathbb {A} ^{n}}. If we take a regular functionf:M→C{\displaystyle f:M\to \mathbb {C} }and consider the section ofΩM{\displaystyle \Omega _{M}}
Then, we can take the derived pullback diagram
where0{\displaystyle 0}is the zero section, constructing aderived critical locusof the regular functionf{\displaystyle f}.
Consider the affine variety
and the regular function given byf(x,y)=x2+y3{\displaystyle f(x,y)=x^{2}+y^{3}}. Then,
where we treat the last two coordinates asdx,dy{\displaystyle dx,dy}. The derived critical locus is then the derived scheme
Note that since the left term in the derived intersection is a complete intersection, we can compute a complex representing the derived ring as
whereKdx,dy∙(C[x,y,dx,dy]){\displaystyle K_{dx,dy}^{\bullet }(\mathbb {C} [x,y,dx,dy])}is the koszul complex.
Consider a smooth functionf:M→C{\displaystyle f:M\to \mathbb {C} }whereM{\displaystyle M}is smooth. The derived enhancement ofCrit(f){\displaystyle \operatorname {Crit} (f)}, thederived critical locus, is given by the differential graded scheme(M,A∙,Q){\displaystyle (M,{\mathcal {A}}^{\bullet },Q)}where the underlying graded ring are the polyvector fields
and the differentialQ{\displaystyle Q}is defined by contraction bydf{\displaystyle df}.
For example, if
we have the complex
representing the derived enhancement ofCrit(f){\displaystyle \operatorname {Crit} (f)}.
|
https://en.wikipedia.org/wiki/Derived_scheme
|
Pursuing Stacks(French:À la Poursuite des Champs) is an influential 1983 mathematical manuscript byAlexander Grothendieck.[1]It consists of a 12-page letter toDaniel Quillenfollowed by about 600 pages of research notes.
The topic of the work is a generalizedhomotopy theoryusinghigher category theory. The word "stacks" in the title refers to what are nowadays usually called "∞-groupoids", one possible definition of which Grothendieck sketches in his manuscript. (Thestacksofalgebraic geometry, which also go back to Grothendieck, are not the focus of this manuscript.) Among the concepts introduced in the work arederivatorsandtest categories.
Some parts of the manuscript were later developed in:
Pursuing stacks started out as a letter from Grothendieck to Daniel Quillen. In this letter he discusses Quillen's progress[2]on the foundations forhomotopy theoryand remarked on the lack of progress since then. He remarks how some of his friends atBangor university, includingRonald Brown, were studyinghigherfundamental groupoidsΠn(X){\displaystyle \Pi _{n}(X)}for atopological spaceX{\displaystyle X}and how the foundations for such a topic could belaid downand relativized usingtopostheory making way for highergerbes. Moreover, he was critical of using strict groupoids for laying down these foundations since they would not be sufficient for developing the full theory he envisioned.
He laid down his ideas of what such an ∞-groupoid should look like, and gave some axioms sketching out how he envisioned them. Essentially, they are categories with objects, arrows, arrows between arrows, and so on, analogous to the situation for higher homotopies. It's conjectured this could be accomplished by looking at a successive sequence of categories and functors
C0→C1→⋯→Cn→Cn+1→⋯{\displaystyle C_{0}\to C_{1}\to \cdots \to C_{n}\to C_{n+1}\to \cdots }
that are universal with respect to any kind of higher groupoid. This allows for an inductive definition of an ∞-groupoid that depends on the objectsC0{\displaystyle C_{0}}and the inclusion functorsCn→Cn+1{\displaystyle C_{n}\to C_{n+1}}, where the categoriesCn{\displaystyle C_{n}}keep track of the higher homotopical information up to leveln{\displaystyle n}. Such a structure was later called acoheratorsince it keeps track of all higher coherences. This structure has been formally studied by George Malsiniotis[3]making some progress on setting up these foundations and showing thehomotopy hypothesis.
As a matter of fact, the description is formally analogous, and nearly identical, to the description of thehomology groupsof achain complex– and it would seem therefore that that stacks (more specifically, Gr-stacks) are in a sense the closest possible non-commutative generalization of chain complexes, the homology groups of the chain complex becoming the homotopy groups of the “non-commutative chain complex” or stack. - Grothendieck[1]pg 23
This is later explained by the intuition provided by theDold–Kan correspondence: simplicialabelian groupscorrespond to chain complexes of abelian groups, so a higher stack modeled as asimplicial groupshould correspond to a "non-abelian" chain complexF∙{\displaystyle {\mathcal {F}}_{\bullet }}. Moreover, these should have an abelianization given by homology and cohomology, written suggestively asHk(X,F∙){\displaystyle H^{k}(X,{\mathcal {F}}_{\bullet })}orRF∗(F∙){\displaystyle \mathbf {R} F_{*}({\mathcal {F}}_{\bullet })}, since there should be an associatedsix functor formalism[1]pg 24. Moreover, there should be an associated theory of Lefschetz operations, similar to the thesis ofRaynaud.[4]
Because Grothendieck envisioned an alternative formulation of higher stacks using globular groupoids, and observed there should be a corresponding theory usingcubical sets, he came up with the idea of test categories and test functors.[1]pg 42Essentially,test categoriesshould be categoriesM{\displaystyle M}with a class of weak equivalencesW{\displaystyle W}such that there is a geometric realization functor
|⋅|:M→Spaces{\displaystyle |\cdot |:M\to {\text{Spaces}}}
and a weak equivalence
M[W−1]≃Hot{\displaystyle M[W^{-1}]\simeq {\text{Hot}}}
whereHotdenotes thehomotopy category.
|
https://en.wikipedia.org/wiki/Pursuing_Stacks
|
Inmathematics,derivatorsare a proposed framework[1][2]pg 190-195forhomological algebragiving a foundation for bothabelianandnon-abelianhomological algebra and various generalizations of it. They were introduced to address the deficiencies ofderived categories(such as the non-functorialityof the cone construction) and provide at the same time a language forhomotopical algebra.
Derivators were first introduced byAlexander Grothendieckin his long unpublished 1983 manuscriptPursuing Stacks. They were then further developed by him in the huge unpublished 1991 manuscriptLes Dérivateursof almost 2000 pages. Essentially the same concept was introduced (apparently independently) by Alex Heller.[3]
The manuscript has been edited for on-line publication by Georges Maltsiniotis. The theory has been further developed by several other people, including Heller,Franke, Keller and Groth.
One of the motivating reasons for considering derivators is the lack of functoriality with the cone construction withtriangulated categories. Derivators are able to solve this problem, and solve the inclusion of generalhomotopy colimits, by keeping track of all possible diagrams in a category withweak equivalencesand their relations between each other. Heuristically, given the diagram
∙→∙{\displaystyle \bullet \to \bullet }
which is a category with two objects and one non-identity arrow, and a functor
F:(∙→∙)→A{\displaystyle F:(\bullet \to \bullet )\to A}
to a categoryA{\displaystyle A}with a class of weak-equivalencesW{\displaystyle W}(and satisfying the right hypotheses), we should have an associated functor
C(F):∙→A[W−1]{\displaystyle C(F):\bullet \to A[W^{-1}]}
where the target object is unique up to weak equivalence inC[W−1]{\displaystyle {\mathcal {C}}[W^{-1}]}. Derivators are able to encode this kind of information and provide a diagram calculus to use inderived categoriesand homotopy theory.
Formally, aprederivatorD{\displaystyle \mathbb {D} }is a 2-functor
D:Indop→CAT{\displaystyle \mathbb {D} :{\text{Ind}}^{op}\to {\text{CAT}}}
from a suitable 2-category ofindicesto the category of categories. Typically such 2-functors come from considering the categoriesHom_(Iop,A){\displaystyle {\underline {\text{Hom}}}(I^{op},A)}whereA{\displaystyle A}is called thecategory of coefficients. For example,Ind{\displaystyle {\text{Ind}}}could be the category of small categories which are filtered, whose objects can be thought of as the indexing sets for afiltered colimit. Then, given a morphism of diagrams
f:I→J{\displaystyle f:I\to J}
denotef∗{\displaystyle f^{*}}by
f∗:D(J)→D(I){\displaystyle f^{*}:\mathbb {D} (J)\to \mathbb {D} (I)}
This is called theinverse imagefunctor. In the motivating example, this is just precompositition, so given a functorFI∈Hom_(Iop,A){\displaystyle F_{I}\in {\underline {\text{Hom}}}(I^{op},A)}there is an associated functorFJ=FI∘f{\displaystyle F_{J}=F_{I}\circ f}. Note these 2-functors could be taken to be
Hom_(−,A[W−1]){\displaystyle {\underline {\text{Hom}}}(-,A[W^{-1}])}
whereW{\displaystyle W}is a suitable class of weak equivalences in a categoryA{\displaystyle A}.
There are a number of examples of indexing categories which can be used in this construction
Derivators are then the axiomatization of prederivators which come equipped with adjoint functors
wheref!{\displaystyle f_{!}}is left adjoint tof∗{\displaystyle f^{*}}and so on. Heuristically,f∗{\displaystyle f_{*}}should correspond to inverse limits,f!{\displaystyle f_{!}}to colimits.
|
https://en.wikipedia.org/wiki/Derivator
|
In algebra, anoperad algebrais an "algebra" over anoperad. It is a generalization of anassociative algebraover a commutative ringR, with an operad replacingR.
Given an operadO(say, asymmetric sequencein asymmetric monoidal ∞-categoryC), analgebra over an operad, orO-algebrafor short, is, roughly, a left module overOwith multiplications parametrized byO.
IfOis atopological operad, then one can say an algebra over an operad is anO-monoid object inC. IfCis symmetric monoidal, this recovers the usual definition.
LetCbe symmetric monoidal ∞-category with monoidal structure distributive over colimits. Iff:O→O′{\displaystyle f:O\to O'}is a map of operads and, moreover, iffis a homotopy equivalence, then the ∞-category of algebras overOinCis equivalent to the ∞-category of algebras overO'inC.[1]
Thisabstract algebra-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Algebra_over_an_operad
|
Inmathematics, anEn{\displaystyle {\mathcal {E}}_{n}}-algebrain asymmetric monoidalinfinity categoryCconsists of the following data:
subject to the requirements that the multiplication maps are compatible with composition, and thatμ{\displaystyle \mu }is an equivalence ifm=1{\displaystyle m=1}. An equivalent definition is thatAis analgebrainCover the littlen-disksoperad.
Thisalgebra-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/En-ring
|
Higher Topos Theoryis a treatise on the theory of∞-categorieswritten by American mathematicianJacob Lurie. In addition to introducing Lurie's new theory of∞-topoi, the book is widely considered foundational tohigher category theory.[1]Since 2018, Lurie has been transferring the contents ofHigher Topos Theory(along with new material) to Kerodon, an "online resource for homotopy-coherent mathematics"[2]inspired by theStacks Project.
Higher Topos Theorycovers two related topics: ∞-categories and ∞-topoi (which are a special case of the former). The first five of the book's seven chapters comprise a rigorous development of general ∞-category theory in the language of quasicategories, a special class ofsimplicial setwhich acts as a model for ∞-categories. The path of this development largely parallels classicalcategory theory, with the notable exception of the ∞-categoricalGrothendieck construction; this correspondence, which Lurie refers to as "straightening and unstraightening",[3]gains considerable importance in his treatment.
The last two chapters are devoted to ∞-topoi, Lurie's own invention and the ∞-categorical analogue oftopoiin classical category theory. The material of these chapters is original, and is adapted from an earlier preprint of Lurie's.[4]There are also appendices discussing background material oncategories,model categories, andsimplicial categories.
Higher Topos Theoryfollowed an earlier work by Lurie,On Infinity Topoi, uploaded to thearXivin 2003.[4]Algebraic topologistPeter Maywas critical of this preprint, emailing Lurie's then-advisorMike Hopkins"to say that Lurie’s paper had some interesting ideas, but that it felt preliminary and needed more rigor."[1]Lurie released a draft ofHigher Topos Theoryon the arXiv in 2006,[5]and the book was finally published in 2009.
Lurie released a second book on higher category theory,Higher Algebra, as a preprint on his website in 2017.[6]This book assumes the content ofHigher Topos Theoryand uses it to study algebra in the ∞-categorical context.
|
https://en.wikipedia.org/wiki/Higher_Topos_Theory
|
Inmathematics, an∞-topos(infinity-topos) is, roughly, an∞-categorysuch that its objects behave likesheavesof spaces with some choice ofGrothendieck topology; in other words, it gives an intrinsic notion of sheaves without reference to an external space. The prototypical example of an ∞-topos is the ∞-category of sheaves of spaces on sometopological space. But the notion is more flexible; for example, the ∞-category ofétale sheaveson someschemeis not the ∞-category of sheaves on any topological space but it is still an ∞-topos.
Precisely, in Lurie'sHigher Topos Theory, an ∞-topos is defined[1]as an ∞-categoryXsuch that there is a small ∞-categoryCand an (accessible) left exactlocalization functorfrom the ∞-category ofpresheaves of spacesonCtoX. A theorem of Lurie[2]states that an ∞-category is an ∞-topos if and only if it satisfies an ∞-categorical version of Giraud's axioms inordinary topostheory. A "topos" is a category behaving like the category of sheaves of sets on a topological space. In analogy, Lurie's definition and characterization theorem of an ∞-topos says that an ∞-topos is an ∞-category behaving like the category of sheaves of spaces.
It says:
Theorem—LetX{\displaystyle X}be an ∞-category. Then the following are equivalent.
|
https://en.wikipedia.org/wiki/%E2%88%9E-topos
|
Inalgebraic geometry, a branch ofmathematics, theétale spectrumof acommutative ringor anE∞-ring, denoted by Specétor Spét, is an analog of theprime spectrumSpec of a commutative ring that is obtained by replacingZariski topologywithétale topology. The precise definition depends on one's formalism. But the idea of the definition itself is simple. The usual prime spectrum Spec enjoys the relation: for ascheme(S,OS) and a commutative ringA,
where Hom on the left is formorphisms of schemesand Hom on the rightring homomorphisms. This is to say Spec is theright adjointto theglobal sectionfunctor(S,OS)↦Γ(S,OS){\displaystyle (S,{\mathcal {O}}_{S})\mapsto \Gamma (S,{\mathcal {O}}_{S})}. So, roughly, one can (and typically does) simply define the étale spectrum Spét to be the right adjoint to the global section functor on the category of "spaces" with étale topology.[1][2]
Over afield of characteristic zero, K. Behrend constructs the étale spectrum of agraded algebracalled a perfect resolving algebra.[3]He then defines adifferential graded scheme(a type of aderived scheme) as one that is étale-locally such an étale spectrum.
The notion makes sense in the usual algebraic geometry but appears more frequently in the context ofderived algebraic geometry.
Thisalgebraic geometry–related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/%C3%89tale_spectrum
|
Inmathematics, thefuzzy sphereis one of the simplest and most canonical examples ofnon-commutative geometry. Ordinarily, the functions defined on asphereform a commuting algebra. A fuzzy sphere differs from an ordinary sphere because the algebra of functions on it is not commutative. It is generated byspherical harmonicswhose spinlis at most equal to somej. The terms in the product of two spherical harmonics that involve spherical harmonics with spin exceedingjare simply omitted in the product. This truncation replaces an infinite-dimensional commutative algebra by aj2{\displaystyle j^{2}}-dimensional non-commutative algebra.
The simplest way to see this sphere is to realize this truncated algebra of functions as a matrix algebra on some finite-dimensional vector space.
Take the threej-dimensionalsquare matricesJa,a=1,2,3{\displaystyle J_{a},~a=1,2,3}that form a basis for thejdimensional irreducible representation of the Lie algebrasu(2). They satisfy the relations[Ja,Jb]=iϵabcJc{\displaystyle [J_{a},J_{b}]=i\epsilon _{abc}J_{c}}, whereϵabc{\displaystyle \epsilon _{abc}}is thetotally antisymmetric symbolwithϵ123=1{\displaystyle \epsilon _{123}=1}, and generate via the matrix product the algebraMj{\displaystyle M_{j}}ofjdimensional matrices. The value of thesu(2)Casimir operatorin this representation is
whereI{\displaystyle I}is thej-dimensional identity matrix.
Thus, if we define the 'coordinates'xa=kr−1Ja{\displaystyle x_{a}=kr^{-1}J_{a}}whereris the radius of the sphere andkis a parameter, related torandjby4r4=k2(j2−1){\displaystyle 4r^{4}=k^{2}(j^{2}-1)}, then the above equation concerning the Casimir operator can be rewritten as
which is the usual relation for the coordinates on a sphere of radiusrembedded in three dimensional space.
One can define an integral on this space, by
whereFis the matrix corresponding to the functionf.
For example, the integral of unity, which gives the surface of the sphere in the commutative case is here equal to
which converges to the value of the surface of the sphere if one takesjto infinity.
J. Hoppe, Quantum Theory of a Massless Relativistic Surface and a Two dimensional Bound State Problem. PhD thesis, Massachusetts Institute of Technology, 1982.
|
https://en.wikipedia.org/wiki/Fuzzy_sphere
|
Inmathematics, and especiallydifferential geometryandgauge theory, aconnectionon afiber bundleis a device that defines a notion ofparallel transporton the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of alinear connectionon avector bundle, for which the notion of parallel transport must belinear. A linear connection is equivalently specified by acovariant derivative, an operator that differentiatessectionsof the bundle alongtangent directionsin the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, theLevi-Civita connectionon thetangent bundleof apseudo-Riemannian manifold, which gives a standard way to differentiate vector fields.Nonlinear connectionsgeneralize this concept to bundles whose fibers are not necessarily linear.
Linear connections are also calledKoszul connectionsafterJean-Louis Koszul, who gave an algebraic framework for describing them (Koszul 1950).
This article defines the connection on a vector bundle using a common mathematical notation which de-emphasizes coordinates. However, other notations are also regularly used: ingeneral relativity, vector bundle computations are usually written using indexed tensors; ingauge theory, the endomorphisms of the vector space fibers are emphasized. The different notations are equivalent, as discussed in the article onmetric connections(the comments made there apply to all vector bundles).
LetMbe adifferentiable manifold, such asEuclidean space. A vector-valued functionM→Rn{\displaystyle M\to \mathbb {R} ^{n}}can be viewed as asectionof the trivialvector bundleM×Rn→M.{\displaystyle M\times \mathbb {R} ^{n}\to M.}One may consider a section of a general differentiable vector bundle, and it is therefore natural to ask if it is possible to differentiate a section, as a generalization of how one differentiates a function onM.
The model case is to differentiate a functionX:Rn→Rm{\displaystyle X:\mathbb {R} ^{n}\to \mathbb {R} ^{m}}on Euclidean spaceRn{\displaystyle \mathbb {R} ^{n}}. In this setting the derivativedX{\displaystyle dX}at a pointx∈Rn{\displaystyle x\in \mathbb {R} ^{n}}in the directionv∈Rn{\displaystyle v\in \mathbb {R} ^{n}}may be defined by the standard formula
For everyx∈Rn{\displaystyle x\in \mathbb {R} ^{n}}, this defines a new vectordX(v)(x)∈Rm.{\displaystyle dX(v)(x)\in \mathbb {R} ^{m}.}
When passing to a sectionX{\displaystyle X}of a vector bundleE{\displaystyle E}over a manifoldM{\displaystyle M}, one encounters two key issues with this definition. Firstly, since the manifold has no linear structure, the termx+tv{\displaystyle x+tv}makes no sense onM{\displaystyle M}. Instead one takes a pathγ:(−1,1)→M{\displaystyle \gamma :(-1,1)\to M}such thatγ(0)=x,γ′(0)=v{\displaystyle \gamma (0)=x,\gamma '(0)=v}and computes
However this still does not make sense, becauseX(γ(t)){\displaystyle X(\gamma (t))}andX(γ(0)){\displaystyle X(\gamma (0))}are elements of the distinct vector spacesEγ(t){\displaystyle E_{\gamma (t)}}andEx.{\displaystyle E_{x}.}This means that subtraction of these two terms is not naturally defined.
The problem is resolved by introducing the extra structure of aconnectionto the vector bundle. There are at least three perspectives from which connections can be understood. When formulated precisely, all three perspectives are equivalent.
Unless the base is zero-dimensional, there are always infinitely many connections which exist on a given differentiable vector bundle, and so there is always a correspondingchoiceof how to differentiate sections. Depending on context, there may be distinguished choices, for instance those which are determined by solving certainpartial differential equations. In the case of thetangent bundle, anypseudo-Riemannian metric(and in particular anyRiemannian metric) determines a canonical connection, called theLevi-Civita connection.
LetE→M{\displaystyle E\to M}be a smooth realvector bundleover asmooth manifoldM{\displaystyle M}. Denote the space of smoothsectionsofE→M{\displaystyle E\to M}byΓ(E){\displaystyle \Gamma (E)}. Acovariant derivativeonE→M{\displaystyle E\to M}is either of the following equivalent structures:
Beyond using the canonical identification between the vector spaceTx∗M⊗Ex{\displaystyle T_{x}^{\ast }M\otimes E_{x}}and the vector space of linear mapsTxM→Ex,{\displaystyle T_{x}M\to E_{x},}these two definitions are identical and differ only in the language used.
It is typical to denote(∇s)x(v){\displaystyle (\nabla s)_{x}(v)}by∇vs,{\displaystyle \nabla _{v}s,}withx{\displaystyle x}being implicit inv.{\displaystyle v.}With this notation, the product rule in the second version of the definition given above is written
Remark.In the case of a complex vector bundle, the above definition is still meaningful, but is usually taken to be modified by changing "real" and "R{\displaystyle \mathbb {R} }" everywhere they appear to "complex" and "C.{\displaystyle \mathbb {C} .}" This places extra restrictions, as not every real-linear map between complex vector spaces is complex-linear. There is some ambiguity in this distinction, as a complex vector bundle can also be regarded as a real vector bundle.
Given a vector bundleE→M{\displaystyle E\to M}, there are many associated bundles toE{\displaystyle E}which may be constructed, for example the dual vector bundleE∗{\displaystyle E^{*}}, tensor powersE⊗k{\displaystyle E^{\otimes k}}, symmetric and antisymmetric tensor powersSkE,ΛkE{\displaystyle S^{k}E,\Lambda ^{k}E}, and the direct sumsE⊕k{\displaystyle E^{\oplus k}}. A connection onE{\displaystyle E}induces a connection on any one of these associated bundles. The ease of passing between connections on associated bundles is more elegantly captured by the theory ofprincipal bundle connections, but here we present some of the basic induced connections.
Given∇{\displaystyle \nabla }a connection onE{\displaystyle E}, the induceddual connection∇∗{\displaystyle \nabla ^{*}}onE∗{\displaystyle E^{*}}is defined implicitly by
HereX∈Γ(TM){\displaystyle X\in \Gamma (TM)}is a smooth vector field,s∈Γ(E){\displaystyle s\in \Gamma (E)}is a section ofE{\displaystyle E}, andξ∈Γ(E∗){\displaystyle \xi \in \Gamma (E^{*})}a section of the dual bundle, and⟨⋅,⋅⟩{\displaystyle \langle \cdot ,\cdot \rangle }the natural pairing between a vector space and its dual (occurring on each fibre betweenE{\displaystyle E}andE∗{\displaystyle E^{*}}), i.e.,⟨ξ,s⟩:=ξ(s){\displaystyle \langle \xi ,s\rangle :=\xi (s)}. Notice that this definition is essentially enforcing that∇∗{\displaystyle \nabla ^{*}}be the connection onE∗{\displaystyle E^{*}}so that a naturalproduct ruleis satisfied for pairing⟨⋅,⋅⟩{\displaystyle \langle \cdot ,\cdot \rangle }.
Given∇E,∇F{\displaystyle \nabla ^{E},\nabla ^{F}}connections on two vector bundlesE,F→M{\displaystyle E,F\to M}, define thetensor product connectionby the formula
Here we haves∈Γ(E),t∈Γ(F),X∈Γ(TM){\displaystyle s\in \Gamma (E),t\in \Gamma (F),X\in \Gamma (TM)}. Notice again this is the natural way of combining∇E,∇F{\displaystyle \nabla ^{E},\nabla ^{F}}to enforce the product rule for the tensor product connection. By repeated application of the above construction applied to the tensor productE⊗k=(E⊗(k−1))⊗E{\displaystyle E^{\otimes k}=(E^{\otimes (k-1)})\otimes E}, one also obtains thetensor power connectiononE⊗k{\displaystyle E^{\otimes k}}for anyk≥1{\displaystyle k\geq 1}and vector bundleE{\displaystyle E}.
Thedirect sum connectionis defined by
wheres⊕t∈Γ(E⊕F){\displaystyle s\oplus t\in \Gamma (E\oplus F)}.
Since the symmetric power and exterior power of a vector bundle may be viewed naturally as subspaces of the tensor power,SkE,ΛkE⊂E⊗k{\displaystyle S^{k}E,\Lambda ^{k}E\subset E^{\otimes k}}, the definition of the tensor product connection applies in a straightforward manner to this setting. Indeed, since the symmetric and exterior algebras sit inside thetensor algebraas direct summands, and the connection∇{\displaystyle \nabla }respects this natural splitting, one can simply restrict∇{\displaystyle \nabla }to these summands. Explicitly, define thesymmetric product connectionby
and theexterior product connectionby
for alls,t∈Γ(E),X∈Γ(TM){\displaystyle s,t\in \Gamma (E),X\in \Gamma (TM)}. Repeated applications of these products gives inducedsymmetric powerandexterior power connectionsonSkE{\displaystyle S^{k}E}andΛkE{\displaystyle \Lambda ^{k}E}respectively.
Finally, one may define the induced connection∇EndE{\displaystyle \nabla ^{\operatorname {End} {E}}}on the vector bundle of endomorphismsEnd(E)=E∗⊗E{\displaystyle \operatorname {End} (E)=E^{*}\otimes E}, theendomorphism connection. This is simply the tensor product connection of the dual connection∇∗{\displaystyle \nabla ^{*}}onE∗{\displaystyle E^{*}}and∇{\displaystyle \nabla }onE{\displaystyle E}. Ifs∈Γ(E){\displaystyle s\in \Gamma (E)}andu∈Γ(End(E)){\displaystyle u\in \Gamma (\operatorname {End} (E))}, so that the compositionu(s)∈Γ(E){\displaystyle u(s)\in \Gamma (E)}also, then the following product rule holds for the endomorphism connection:
By reversing this equation, it is possible to define the endomorphism connection as the unique connection satisfying
for anyu,s,X{\displaystyle u,s,X}, thus avoiding the need to first define the dual connection and tensor product connection.
Given a vector bundleE{\displaystyle E}of rankr{\displaystyle r}, and any representationρ:GL(r,K)→G{\displaystyle \rho :\mathrm {GL} (r,\mathbb {K} )\to G}into a linear groupG⊂GL(V){\displaystyle G\subset \mathrm {GL} (V)}, there is an induced connection on the associated vector bundleF=E×ρV{\displaystyle F=E\times _{\rho }V}. This theory is most succinctly captured by passing to the principal bundle connection on theframe bundleofE{\displaystyle E}and using the theory of principal bundles. Each of the above examples can be seen as special cases of this construction: the dual bundle corresponds to the inverse transpose (or inverse adjoint) representation, the tensor product to the tensor product representation, the direct sum to the direct sum representation, and so on.
LetE→M{\displaystyle E\to M}be a vector bundle. AnE{\displaystyle E}-valued differential formof degreer{\displaystyle r}is a section of thetensor productbundle:
The space of such forms is denoted by
where the last tensor product denotes the tensor product ofmodulesover theringof smooth functions onM{\displaystyle M}.
AnE{\displaystyle E}-valued 0-form is just a section of the bundleE{\displaystyle E}. That is,
In this notation a connection onE→M{\displaystyle E\to M}is a linear map
A connection may then be viewed as a generalization of theexterior derivativeto vector bundle valued forms. In fact, given a connection∇{\displaystyle \nabla }onE{\displaystyle E}there is a unique way to extend∇{\displaystyle \nabla }to anexterior covariant derivative
This exterior covariant derivative is defined by the following Leibniz rule, which is specified on simple tensors of the formω⊗s{\displaystyle \omega \otimes s}and extended linearly:
whereω∈Ωr(M){\displaystyle \omega \in \Omega ^{r}(M)}so thatdegω=r{\displaystyle \deg \omega =r},s∈Γ(E){\displaystyle s\in \Gamma (E)}is a section, andω∧∇s{\displaystyle \omega \wedge \nabla s}denotes the(r+1){\displaystyle (r+1)}-form with values inE{\displaystyle E}defined by wedgingω{\displaystyle \omega }with the one-form part of∇s{\displaystyle \nabla s}. Notice that forE{\displaystyle E}-valued 0-forms, this recovers the normal Leibniz rule for the connection∇{\displaystyle \nabla }.
Unlike the ordinary exterior derivative, one generally hasd∇2≠0{\displaystyle d_{\nabla }^{2}\neq 0}. In fact,d∇2{\displaystyle d_{\nabla }^{2}}is directly related to the curvature of the connection∇{\displaystyle \nabla }(seebelow).
Every vector bundle over a manifold admits a connection, which can be proved usingpartitions of unity. However, connections are not unique. If∇1{\displaystyle \nabla _{1}}and∇2{\displaystyle \nabla _{2}}are two connections onE→M{\displaystyle E\to M}then their difference is aC∞(M){\displaystyle C^{\infty }(M)}-linear operator. That is,
for all smooth functionsf{\displaystyle f}onM{\displaystyle M}and all smooth sectionss{\displaystyle s}ofE{\displaystyle E}. It follows that the difference∇1−∇2{\displaystyle \nabla _{1}-\nabla _{2}}can be uniquely identified with a one-form onM{\displaystyle M}with values in the endomorphism bundleEnd(E)=E∗⊗E{\displaystyle \operatorname {End} (E)=E^{*}\otimes E}:
Conversely, if∇{\displaystyle \nabla }is a connection onE{\displaystyle E}andA{\displaystyle A}is a one-form onM{\displaystyle M}with values inEnd(E){\displaystyle \operatorname {End} (E)}, then∇+A{\displaystyle \nabla +A}is a connection onE{\displaystyle E}.
In other words, the space of connections onE{\displaystyle E}is anaffine spaceforΩ1(End(E)){\displaystyle \Omega ^{1}(\operatorname {End} (E))}. This affine space is commonly denotedA{\displaystyle {\mathcal {A}}}.
LetE→M{\displaystyle E\to M}be a vector bundle of rankk{\displaystyle k}and letF(E){\displaystyle {\mathcal {F}}(E)}be theframe bundleofE{\displaystyle E}. Then a(principal) connectiononF(E){\displaystyle {\mathcal {F}}(E)}induces a connection onE{\displaystyle E}. First note that sections ofE{\displaystyle E}are in one-to-one correspondence withright-equivariantmapsF(E)→Rk{\displaystyle {\mathcal {F}}(E)\to \mathbb {R} ^{k}}. (This can be seen by considering thepullbackofE{\displaystyle E}overF(E)→M{\displaystyle {\mathcal {F}}(E)\to M}, which is isomorphic to thetrivial bundleF(E)×Rk{\displaystyle {\mathcal {F}}(E)\times \mathbb {R} ^{k}}.) Given a sections{\displaystyle s}ofE{\displaystyle E}let the corresponding equivariant map beψ(s){\displaystyle \psi (s)}. The covariant derivative onE{\displaystyle E}is then given by
whereXH{\displaystyle X^{H}}is thehorizontal liftofX{\displaystyle X}fromM{\displaystyle M}toF(E){\displaystyle {\mathcal {F}}(E)}. (Recall that the horizontal lift is determined by the connection onF(E){\displaystyle {\mathcal {F}}(E)}.)
Conversely, a connection onE{\displaystyle E}determines a connection onF(E){\displaystyle {\mathcal {F}}(E)}, and these two constructions are mutually inverse.
A connection onE{\displaystyle E}is also determined equivalently by alinear Ehresmann connectiononE{\displaystyle E}. This provides one method to construct the associated principal connection.
The induced connections discussed in#Induced connectionscan be constructed as connections on other associated bundles to the frame bundle ofE{\displaystyle E}, using representations other than the standard representation used above. For example ifρ{\displaystyle \rho }denotes the standard representation ofGL(k,R){\displaystyle \operatorname {GL} (k,\mathbb {R} )}onRk{\displaystyle \mathbb {R} ^{k}}, then the associated bundle to the representationρ⊕ρ{\displaystyle \rho \oplus \rho }ofGL(k,R){\displaystyle \operatorname {GL} (k,\mathbb {R} )}onRk⊕Rk{\displaystyle \mathbb {R} ^{k}\oplus \mathbb {R} ^{k}}is the direct sum bundleE⊕E{\displaystyle E\oplus E}, and the induced connection is precisely that which was described above.
LetE→M{\displaystyle E\to M}be a vector bundle of rankk{\displaystyle k}, and letU{\displaystyle U}be an open subset ofM{\displaystyle M}over whichE{\displaystyle E}trivialises. Therefore over the setU{\displaystyle U},E{\displaystyle E}admits a localsmooth frameof sections
Since the framee{\displaystyle \mathbf {e} }defines a basis of the fibreEx{\displaystyle E_{x}}for anyx∈U{\displaystyle x\in U}, one can expand any local sections:U→E|U{\displaystyle s:U\to \left.E\right|_{U}}in the frame as
for a collection of smooth functionss1,…,sk:U→R{\displaystyle s^{1},\dots ,s^{k}:U\to \mathbb {R} }.
Given a connection∇{\displaystyle \nabla }onE{\displaystyle E}, it is possible to express∇{\displaystyle \nabla }overU{\displaystyle U}in terms of the local frame of sections, by using the characteristic product rule for the connection. For any basis sectionei{\displaystyle e_{i}}, the quantity∇(ei)∈Ω1(U)⊗Γ(U,E){\displaystyle \nabla (e_{i})\in \Omega ^{1}(U)\otimes \Gamma (U,E)}may be expanded in the local framee{\displaystyle \mathbf {e} }as
whereAij∈Ω1(U);j=1,…,k{\displaystyle A_{i}^{\ j}\in \Omega ^{1}(U);\,j=1,\dots ,k}are a collection of local one-forms. These forms can be put into a matrix of one-forms defined by
called thelocal connection form of∇{\displaystyle \nabla }overU{\displaystyle U}. The action of∇{\displaystyle \nabla }on any sections:U→E|U{\displaystyle s:U\to \left.E\right|_{U}}can be computed in terms ofA{\displaystyle A}using the product rule as
If the local sections{\displaystyle s}is also written in matrix notation as a column vector using the local framee{\displaystyle \mathbf {e} }as a basis,
then using regular matrix multiplication one can write
whereds{\displaystyle ds}is shorthand for applying theexterior derivatived{\displaystyle d}to each component ofs{\displaystyle s}as a column vector. In this notation, one often writes locally that∇|U=d+A{\displaystyle \left.\nabla \right|_{U}=d+A}. In this sense a connection is locally completely specified by its connection one-form in some trivialisation.
As explained in#Affine properties of the set of connections, any connection differs from another by an endomorphism-valued one-form. From this perspective, the connection one-formA{\displaystyle A}is precisely the endomorphism-valued one-form such that the connection∇|U{\displaystyle \left.\nabla \right|_{U}}onE|U{\displaystyle \left.E\right|_{U}}differs from the trivial connectiond{\displaystyle d}onE|U{\displaystyle \left.E\right|_{U}}, which exists becauseU{\displaystyle U}is a trivialising set forE{\displaystyle E}.
Inpseudo-Riemannian geometry, theLevi-Civita connectionis often written in terms of theChristoffel symbolsΓijk{\displaystyle \Gamma _{ij}^{\ \ k}}instead of the connection one-formA{\displaystyle A}. It is possible to define Christoffel symbols for a connection on any vector bundle, and not just the tangent bundle of a pseudo-Riemannian manifold. To do this, suppose that in addition toU{\displaystyle U}being a trivialising open subset for the vector bundleE→M{\displaystyle E\to M}, thatU{\displaystyle U}is also alocal chartfor the manifoldM{\displaystyle M}, admitting local coordinatesx=(x1,…,xn);xi:U→R{\displaystyle \mathbf {x} =(x^{1},\dots ,x^{n});\quad x^{i}:U\to \mathbb {R} }.
In such a local chart, there is a distinguished local frame for the differential one-forms given by(dx1,…,dxn){\displaystyle (dx^{1},\dots ,dx^{n})}, and the local connection one-formsAij{\displaystyle A_{i}^{j}}can be expanded in this basis as
for a collection of local smooth functionsΓℓij:U→R{\displaystyle \Gamma _{\ell i}^{\ \ j}:U\to \mathbb {R} }, called theChristoffel symbolsof∇{\displaystyle \nabla }overU{\displaystyle U}. In the case whereE=TM{\displaystyle E=TM}and∇{\displaystyle \nabla }is the Levi-Civita connection, these symbols agree precisely with the Christoffel symbols from pseudo-Riemannian geometry.
The expression for how∇{\displaystyle \nabla }acts in local coordinates can be further expanded in terms of the local chartU{\displaystyle U}and the Christoffel symbols, to be given by
Contracting this expression with the local coordinate tangent vector∂∂xℓ{\displaystyle {\frac {\partial }{\partial x^{\ell }}}}leads to
This defines a collection ofn{\displaystyle n}locally defined operators
with the property that
Supposee′{\displaystyle \mathbf {e'} }is another choice of local frame over the same trivialising setU{\displaystyle U}, so that there is a matrixg=(gij){\displaystyle g=(g_{i}^{\ j})}of smooth functions relatinge{\displaystyle \mathbf {e} }ande′{\displaystyle \mathbf {e'} }, defined by
Tracing through the construction of the local connection formA{\displaystyle A}for the framee{\displaystyle \mathbf {e} }, one finds that the connection one-formA′{\displaystyle A'}fore′{\displaystyle \mathbf {e'} }is given by
whereg−1=((g−1)ij){\displaystyle g^{-1}=\left({(g^{-1})}_{i}^{\ j}\right)}denotes the inverse matrix tog{\displaystyle g}. In matrix notation this may be written
wheredg{\displaystyle dg}is the matrix of one-forms given by taking the exterior derivative of the matrixg{\displaystyle g}component-by-component.
In the case whereE=TM{\displaystyle E=TM}is the tangent bundle andg{\displaystyle g}is the Jacobian of a coordinate transformation ofM{\displaystyle M}, the lengthy formulae for the transformation of the Christoffel symbols of the Levi-Civita connection can be recovered from the more succinct transformation laws of the connection form above.
A connection∇{\displaystyle \nabla }on a vector bundleE→M{\displaystyle E\to M}defines a notion ofparallel transportonE{\displaystyle E}along a curve inM{\displaystyle M}. Letγ:[0,1]→M{\displaystyle \gamma :[0,1]\to M}be a smoothpathinM{\displaystyle M}. A sections{\displaystyle s}ofE{\displaystyle E}alongγ{\displaystyle \gamma }is said to beparallelif
for allt∈[0,1]{\displaystyle t\in [0,1]}. Equivalently, one can consider thepullback bundleγ∗E{\displaystyle \gamma ^{*}E}ofE{\displaystyle E}byγ{\displaystyle \gamma }. This is a vector bundle over[0,1]{\displaystyle [0,1]}with fiberEγ(t){\displaystyle E_{\gamma (t)}}overt∈[0,1]{\displaystyle t\in [0,1]}. The connection∇{\displaystyle \nabla }onE{\displaystyle E}pulls back to a connection onγ∗E{\displaystyle \gamma ^{*}E}. A sections{\displaystyle s}ofγ∗E{\displaystyle \gamma ^{*}E}is parallel if and only ifγ∗∇(s)=0{\displaystyle \gamma ^{*}\nabla (s)=0}.
Supposeγ{\displaystyle \gamma }is a path fromx{\displaystyle x}toy{\displaystyle y}inM{\displaystyle M}. The above equation defining parallel sections is a first-orderordinary differential equation(cf.local expressionabove) and so has a unique solution for each possible initial condition. That is, for each vectorv{\displaystyle v}inEx{\displaystyle E_{x}}there exists a unique parallel sections{\displaystyle s}ofγ∗E{\displaystyle \gamma ^{*}E}withs(0)=v{\displaystyle s(0)=v}. Define aparallel transport map
byτγ(v)=s(1){\displaystyle \tau _{\gamma }(v)=s(1)}. It can be shown thatτγ{\displaystyle \tau _{\gamma }}is alinear isomorphism, with inverse given by following the same procedure with the reversed pathγ−{\displaystyle \gamma ^{-}}fromy{\displaystyle y}tox{\displaystyle x}.
Parallel transport can be used to define theholonomy groupof the connection∇{\displaystyle \nabla }based at a pointx{\displaystyle x}inM{\displaystyle M}. This is the subgroup ofGL(Ex){\displaystyle \operatorname {GL} (E_{x})}consisting of all parallel transport maps coming fromloopsbased atx{\displaystyle x}:
The holonomy group of a connection is intimately related to the curvature of the connection (AmbroseSinger 1953).
The connection can be recovered from its parallel transport operators as follows. IfX∈Γ(TM){\displaystyle X\in \Gamma (TM)}is a vector field ands∈Γ(E){\displaystyle s\in \Gamma (E)}a section, at a pointx∈M{\displaystyle x\in M}pick anintegral curveγ:(−ε,ε)→M{\displaystyle \gamma :(-\varepsilon ,\varepsilon )\to M}forX{\displaystyle X}atx{\displaystyle x}. For eacht∈(−ε,ε){\displaystyle t\in (-\varepsilon ,\varepsilon )}we will writeτt:Eγ(t)→Ex{\displaystyle \tau _{t}:E_{\gamma (t)}\to E_{x}}for the parallel transport map traveling alongγ{\displaystyle \gamma }fromt{\displaystyle t}to0{\displaystyle 0}. In particular for everyt∈(−ε,ε){\displaystyle t\in (-\varepsilon ,\varepsilon )}, we haveτts(γ(t))∈Ex{\displaystyle \tau _{t}s(\gamma (t))\in E_{x}}. Thent↦τts(γ(t)){\displaystyle t\mapsto \tau _{t}s(\gamma (t))}defines a curve in the vector spaceEx{\displaystyle E_{x}}, which may be differentiated. The covariant derivative is recovered as
This demonstrates that an equivalent definition of a connection is given by specifying all the parallel transport isomorphismsτγ{\displaystyle \tau _{\gamma }}between fibres ofE{\displaystyle E}and taking the above expression as the definition of∇{\displaystyle \nabla }.
Thecurvatureof a connection∇{\displaystyle \nabla }onE→M{\displaystyle E\to M}is a 2-formF∇{\displaystyle F_{\nabla }}onM{\displaystyle M}with values in the endomorphism bundleEnd(E)=E∗⊗E{\displaystyle \operatorname {End} (E)=E^{*}\otimes E}. That is,
It is defined by the expression
whereX{\displaystyle X}andY{\displaystyle Y}are tangent vector fields onM{\displaystyle M}ands{\displaystyle s}is a section ofE{\displaystyle E}. One must check thatF∇{\displaystyle F_{\nabla }}isC∞(M){\displaystyle C^{\infty }(M)}-linear in bothX{\displaystyle X}andY{\displaystyle Y}and that it does in fact define a bundle endomorphism ofE{\displaystyle E}.
As mentionedabove, the covariant exterior derivatived∇{\displaystyle d_{\nabla }}need not square to zero when acting onE{\displaystyle E}-valued forms. The operatord∇2{\displaystyle d_{\nabla }^{2}}is, however, strictly tensorial (i.e.C∞(M){\displaystyle C^{\infty }(M)}-linear). This implies that it is induced from a 2-form with values inEnd(E){\displaystyle \operatorname {End} (E)}. This 2-form is precisely the curvature form given above. For anE{\displaystyle E}-valued formσ{\displaystyle \sigma }we have
Aflat connectionis one whose curvature form vanishes identically.
The curvature form has a local description calledCartan's structure equation. If∇{\displaystyle \nabla }has local formA{\displaystyle A}on some trivialising open subsetU⊂M{\displaystyle U\subset M}forE{\displaystyle E}, then
onU{\displaystyle U}. To clarify this notation, notice thatA{\displaystyle A}is a endomorphism-valued one-form, and so in local coordinates takes the form of a matrix of one-forms. The operationd{\displaystyle d}applies the exterior derivative component-wise to this matrix, andA∧A{\displaystyle A\wedge A}denotes matrix multiplication, where the components are wedged rather than multiplied.
In local coordinatesx=(x1,…,xn){\displaystyle \mathbf {x} =(x^{1},\dots ,x^{n})}onM{\displaystyle M}overU{\displaystyle U}, if the connection form is writtenA=Aℓdxℓ=(Γℓij)dxℓ{\displaystyle A=A_{\ell }dx^{\ell }=(\Gamma _{\ell i}^{\ \ j})dx^{\ell }}for a collection of local endomorphismsAℓ=(Γℓij){\displaystyle A_{\ell }=(\Gamma _{\ell i}^{\ \ j})}, then one has
Further expanding this in terms of the Christoffel symbolsΓℓij{\displaystyle \Gamma _{\ell i}^{\ \ j}}produces the familiar expression from Riemannian geometry. Namely ifs=siei{\displaystyle s=s^{i}e_{i}}is a section ofE{\displaystyle E}overU{\displaystyle U}, then
HereR=(Rpqij){\displaystyle R=(R_{pqi}^{\ \ \ j})}is the fullcurvature tensorofF∇{\displaystyle F_{\nabla }}, and in Riemannian geometry would be identified with theRiemannian curvature tensor.
It can be checked that if we define[A,A]{\displaystyle [A,A]}to be wedge product of forms butcommutatorof endomorphisms as opposed to composition, thenA∧A=12[A,A]{\displaystyle A\wedge A={\frac {1}{2}}[A,A]}, and with this alternate notation the Cartan structure equation takes the form
This alternate notation is commonly used in the theory of principal bundle connections, where instead we use a connection formω{\displaystyle \omega }, aLie algebra-valued one-form, for which there is no notion of composition (unlike in the case of endomorphisms), but there is a notion of a Lie bracket.
In some references (see for example (MadsenTornehave1997)) the Cartan structure equation may be written with a minus sign:
This different convention uses an order of matrix multiplication that is different from the standard Einstein notation in the wedge product of matrix-valued one-forms.
A version of the second (differential)Bianchi identityfrom Riemannian geometry holds for a connection on any vector bundle. Recall that a connection∇{\displaystyle \nabla }on a vector bundleE→M{\displaystyle E\to M}induces an endomorphism connection onEnd(E){\displaystyle \operatorname {End} (E)}. This endomorphism connection has itself an exterior covariant derivative, which we ambiguously calld∇{\displaystyle d_{\nabla }}. Since the curvature is a globally definedEnd(E){\displaystyle \operatorname {End} (E)}-valued two-form, we may apply the exterior covariant derivative to it. TheBianchi identitysays that
This succinctly captures the complicated tensor formulae of the Bianchi identity in the case of Riemannian manifolds, and one may translate from this equation to the standard Bianchi identities by expanding the connection and curvature in local coordinates.
There is no analogue in general of thefirst(algebraic) Bianchi identity for a general connection, as this exploits the special symmetries of the Levi-Civita connection. Namely, one exploits that the vector bundle indices ofE=TM{\displaystyle E=TM}in the curvature tensorR{\displaystyle R}may be swapped with the cotangent bundle indices coming fromT∗M{\displaystyle T^{*}M}after using the metric to lower or raise indices. For example this allows the torsion-freeness conditionΓℓij=Γiℓj{\displaystyle \Gamma _{\ell i}^{\ \ j}=\Gamma _{i\ell }^{\ \ j}}to be defined for the Levi-Civita connection, but for a general vector bundle theℓ{\displaystyle \ell }-index refers to the local coordinate basis ofT∗M{\displaystyle T^{*}M}, and thei,j{\displaystyle i,j}-indices to the local coordinate frame ofE{\displaystyle E}andE∗{\displaystyle E^{*}}coming from the splittingEnd(E)=E∗⊗E{\displaystyle \mathrm {End} (E)=E^{*}\otimes E}. However in special circumstance, for example when the rank ofE{\displaystyle E}equals the dimension ofM{\displaystyle M}and asolder formhas been chosen, one can use the soldering to interchange the indices and define a notion of torsion for affine connections which are not the Levi-Civita connection.
Given two connections∇1,∇2{\displaystyle \nabla _{1},\nabla _{2}}on a vector bundleE→M{\displaystyle E\to M}, it is natural to ask when they might be considered equivalent. There is a well-defined notion of anautomorphismof a vector bundleE→M{\displaystyle E\to M}. A sectionu∈Γ(End(E)){\displaystyle u\in \Gamma (\operatorname {End} (E))}is an automorphism ifu(x)∈End(Ex){\displaystyle u(x)\in \operatorname {End} (E_{x})}is invertible at every pointx∈M{\displaystyle x\in M}. Such an automorphism is called agauge transformationofE{\displaystyle E}, and the group of all automorphisms is called thegauge group, often denotedG{\displaystyle {\mathcal {G}}}orAut(E){\displaystyle \operatorname {Aut} (E)}. The group of gauge transformations may be neatly characterised as the space of sections of thecapital A adjoint bundleAd(F(E)){\displaystyle \operatorname {Ad} ({\mathcal {F}}(E))}of theframe bundleof the vector bundleE{\displaystyle E}. This is not to be confused with thelowercase aadjoint bundlead(F(E)){\displaystyle \operatorname {ad} ({\mathcal {F}}(E))}, which is naturally identified withEnd(E){\displaystyle \operatorname {End} (E)}itself. The bundleAdF(E){\displaystyle \operatorname {Ad} {\mathcal {F}}(E)}is theassociated bundleto the principal frame bundle by the conjugation representation ofG=GL(r){\displaystyle G=\operatorname {GL} (r)}on itself,g↦ghg−1{\displaystyle g\mapsto ghg^{-1}}, and has fibre the same general linear groupGL(r){\displaystyle \operatorname {GL} (r)}whererank(E)=r{\displaystyle \operatorname {rank} (E)=r}. Notice that despite having the same fibre as the frame bundleF(E){\displaystyle {\mathcal {F}}(E)}and being associated to it,Ad(F(E)){\displaystyle \operatorname {Ad} ({\mathcal {F}}(E))}is not equal to the frame bundle, nor even a principal bundle itself. The gauge group may be equivalently characterised asG=Γ(AdF(E)).{\displaystyle {\mathcal {G}}=\Gamma (\operatorname {Ad} {\mathcal {F}}(E)).}
A gauge transformationu{\displaystyle u}ofE{\displaystyle E}acts on sectionss∈Γ(E){\displaystyle s\in \Gamma (E)}, and therefore acts on connections by conjugation. Explicitly, if∇{\displaystyle \nabla }is a connection onE{\displaystyle E}, then one definesu⋅∇{\displaystyle u\cdot \nabla }by
fors∈Γ(E),X∈Γ(TM){\displaystyle s\in \Gamma (E),X\in \Gamma (TM)}. To check thatu⋅∇{\displaystyle u\cdot \nabla }is a connection, one verifies the product rule
It may be checked that this defines a leftgroup actionofG{\displaystyle {\mathcal {G}}}on the affine space of all connectionsA{\displaystyle {\mathcal {A}}}.
SinceA{\displaystyle {\mathcal {A}}}is an affine space modelled onΩ1(M,End(E)){\displaystyle \Omega ^{1}(M,\operatorname {End} (E))}, there should exist some endomorphism-valued one-formAu∈Ω1(M,End(E)){\displaystyle A_{u}\in \Omega ^{1}(M,\operatorname {End} (E))}such thatu⋅∇=∇+Au{\displaystyle u\cdot \nabla =\nabla +A_{u}}. Using the definition of the endomorphism connection∇End(E){\displaystyle \nabla ^{\operatorname {End} (E)}}induced by∇{\displaystyle \nabla }, it can be seen that
which is to say thatAu=−d∇(u)u−1{\displaystyle A_{u}=-d^{\nabla }(u)u^{-1}}.
Two connections are said to begauge equivalentif they differ by the action of the gauge group, and the quotient spaceB=A/G{\displaystyle {\mathcal {B}}={\mathcal {A}}/{\mathcal {G}}}is themoduli spaceof all connections onE{\displaystyle E}. In general this topological space is neither a smooth manifold or even aHausdorff space, but contains inside it themoduli space of Yang–Mills connectionsonE{\displaystyle E}, which is of significant interest ingauge theoryandphysics.
|
https://en.wikipedia.org/wiki/Koszul_connection
|
Inmathematics, theMoyal product(afterJosé Enrique Moyal; also called thestar productorWeyl–Groenewold product, afterHermann WeylandHilbrand J. Groenewold) is an example of aphase-space star product. It is an associative, non-commutative product,★, on the functions onR2n{\displaystyle \mathbb {R} ^{2n}}, equipped with itsPoisson bracket(with a generalization tosymplectic manifolds, described below). It is a special case of the★-product of the "algebra of symbols" of auniversal enveloping algebra.
The Moyal product is named afterJosé Enrique Moyal, but is also sometimes called theWeyl–Groenewold product as it was introduced byH. J. Groenewoldin his 1946 doctoral dissertation, in a trenchant appreciation[1]of theWeyl correspondence. Moyal actually appears not to know about the product in his celebrated article[2]and was crucially lacking it in his legendary correspondence with Dirac, as illustrated in his biography.[3]The popular naming after Moyal appears to have emerged only in the 1970s, in homage to his flatphase-space quantizationpicture.[4]
The product forsmooth functionsfandgonR2n{\displaystyle \mathbb {R} ^{2n}}takes the formf⋆g=fg+∑n=1∞ℏnCn(f,g),{\displaystyle f\star g=fg+\sum _{n=1}^{\infty }\hbar ^{n}C_{n}(f,g),}where eachCnis a certain bidifferential operatorof ordern, characterized by the following properties (see below for an explicit formula):
Note that, if one wishes to take functions valued in thereal numbers, then an alternative version eliminates theiin the second condition and eliminates the fourth condition.
If one restricts to polynomial functions, the above algebra is isomorphic to theWeyl algebraAn, and the two offer alternative realizations of theWeyl mapof the space of polynomials innvariables (or thesymmetric algebraof a vector space of dimension2n).
To provide an explicit formula, consider a constantPoisson bivectorΠonR2n{\displaystyle \mathbb {R} ^{2n}}:Π=∑i,jΠij∂i∧∂j,{\displaystyle \Pi =\sum _{i,j}\Pi ^{ij}\partial _{i}\wedge \partial _{j},}whereΠijis a real number for eachi,j.
The star product of two functionsfandgcan then be defined as thepseudo-differential operatoracting on both of them,f⋆g=fg+iℏ2∑i,jΠij(∂if)(∂jg)−ℏ28∑i,j,k,mΠijΠkm(∂i∂kf)(∂j∂mg)+…,{\displaystyle f\star g=fg+{\frac {i\hbar }{2}}\sum _{i,j}\Pi ^{ij}(\partial _{i}f)(\partial _{j}g)-{\frac {\hbar ^{2}}{8}}\sum _{i,j,k,m}\Pi ^{ij}\Pi ^{km}(\partial _{i}\partial _{k}f)(\partial _{j}\partial _{m}g)+\ldots ,}whereħis thereduced Planck constant, treated as a formal parameter here.
This is a special case of what is known as theBerezin formula[5]on the algebra of symbols and can be given a closed form[6](which follows from theBaker–Campbell–Hausdorff formula). The closed form can be obtained by using theexponential:f⋆g=m∘eiℏ2Π(f⊗g),{\displaystyle f\star g=m\circ e^{{\frac {i\hbar }{2}}\Pi }(f\otimes g),}wheremis the multiplication map,m(a⊗b) =ab, and the exponential is treated as a power series,eA=∑n=0∞1n!An.{\displaystyle e^{A}=\sum _{n=0}^{\infty }{\frac {1}{n!}}A^{n}.}
That is, the formula forCnisCn=in2nn!m∘Πn.{\displaystyle C_{n}={\frac {i^{n}}{2^{n}n!}}m\circ \Pi ^{n}.}
As indicated, often one eliminates all occurrences ofiabove, and the formulas then restrict naturally to real numbers.
Note that if the functionsfandgare polynomials, the above infinite sums become finite (reducing to the ordinary Weyl-algebra case).
The relationship of the Moyal product to the generalized★-product used in the definition of the "algebra of symbols" of auniversal enveloping algebrafollows from the fact that theWeyl algebrais the universal enveloping algebra of theHeisenberg algebra(modulo that the center equals the unit).
On any symplectic manifold, one can, at least locally, choose coordinates so as to make the symplectic structureconstant, byDarboux's theorem; and, using the associated Poisson bivector, one may consider the above formula. For it to work globally,
as a function on the whole manifold (and not just a local formula), one must equip the symplectic manifold with a torsion-free symplecticconnection. This makes it aFedosov manifold.
More general results forarbitrary Poisson manifolds(where the Darboux theorem does not apply) are given by theKontsevich quantization formula.
A simple explicit example of the construction and utility of the★-product (for the simplest case of a two-dimensional euclideanphase space) is given in the article on theWigner–Weyl transform: two Gaussians compose with this★-product according to a hyperbolic tangent law:[7]exp[−a(q2+p2)]⋆exp[−b(q2+p2)]=11+ℏ2abexp[−a+b1+ℏ2ab(q2+p2)].{\displaystyle \exp \left[-a\left(q^{2}+p^{2}\right)\right]\star \exp \left[-b\left(q^{2}+p^{2}\right)\right]={\frac {1}{1+\hbar ^{2}ab}}\exp \left[-{\frac {a+b}{1+\hbar ^{2}ab}}\left(q^{2}+p^{2}\right)\right].}Equivalently,e−tanh(a)q2+p2ℏ⋆e−tanh(b)q2+p2ℏ=tanh(a+b)tanh(a)+tanh(b)e−tanh(a+b)q2+p2ℏ{\displaystyle e^{-\tanh(a){\frac {q^{2}+p^{2}}{\hbar }}}\star e^{-\tanh(b){\frac {q^{2}+p^{2}}{\hbar }}}={\frac {\tanh(a+b)}{\tanh(a)+\tanh(b)}}e^{-\tanh(a+b){\frac {q^{2}+p^{2}}{\hbar }}}}The classical limit atℏ→0,a/ℏ→α,b/ℏ→β{\displaystyle \hbar \to 0,a/\hbar \to \alpha ,b/\hbar \to \beta }ise−α(q2+p2)e−β(q2+p2)=e−(α+β)(q2+p2){\displaystyle e^{-\alpha (q^{2}+p^{2})}e^{-\beta (q^{2}+p^{2})}=e^{-(\alpha +\beta )(q^{2}+p^{2})}}, as expected.
Every correspondence prescriptionbetween phase space and Hilbert space, however, inducesits ownproper★-product.[8][9]
Similar results are seen in theSegal–Bargmann spaceand in thetheta representationof theHeisenberg group, where the creation and annihilation operatorsa∗=zanda=∂/∂zare understood to act on the complex plane (respectively, theupper half-planefor the Heisenberg group), so that the position and momenta operators are given byq=a+a∗2{\displaystyle q={\frac {a+a^{*}}{2}}}andp=a−a∗2i{\displaystyle p={\frac {a-a^{*}}{2i}}}. This situation is clearly different from the case where the positions are taken to be real-valued, but does offer insights into the overall algebraic structure of the Heisenberg algebra and its envelope, the Weyl algebra.
Inside a phase-space integral, justonestar product of the Moyal type may be dropped,[10]resulting in plain multiplication, as evident by integration by parts,∫dxdpf⋆g=∫dxdpfg,{\displaystyle \int dx\,dp\;f\star g=\int dx\,dp~f~g,}making the cyclicity of the phase-space trace manifest. This is a unique property of the above specific Moyal product, and does not hold for other correspondence rules' star products, such as Husimi's, etc.
|
https://en.wikipedia.org/wiki/Moyal_product
|
Inmathematics,noncommutative topologyis a term used for the relationship betweentopologicalandC*-algebraicconcepts. The term has its origins in theGelfand–Naimark theorem, which implies thedualityof thecategoryoflocally compactHausdorff spacesand the category ofcommutativeC*-algebras. Noncommutative topology is related to analyticnoncommutative geometry.
The premise behind noncommutative topology is that a noncommutative C*-algebra can be treated like the algebra of complex-valuedcontinuous functionson a 'noncommutative space' which does not exist classically. Several topological properties can be formulated as properties for the C*-algebras without making reference to commutativity or the underlying space, and so have an immediate generalization.
Among these are:
Individual elements of a commutative C*-algebra correspond with continuous functions. And so certain types of functions can correspond to certain properties of a C*-algebra. For example,self-adjointelements of a commutative C*-algebra correspond to real-valued continuous functions. Also,projections(i.e. self-adjointidempotents) correspond toindicator functionsofclopen sets.
Categorical constructions lead to some examples. For example, thecoproductof spaces is thedisjoint unionand thus corresponds to thedirect sum of algebras, which is theproductof C*-algebras. Similarly,product topologycorresponds to the coproduct of C*-algebras, thetensor product of algebras. In a more specialized setting,
compactifications of topologies correspond to unitizations of algebras. So theone-point compactificationcorresponds to the minimal unitization of C*-algebras, theStone–Čech compactificationcorresponds to themultiplier algebra, andcorona setscorrespond withcorona algebras.
There are certain examples of properties where multiple generalizations are possible and it is not clear which is preferable. For example,probability measurescan correspond either tostatesor tracial states. Since all states are vacuously
tracial states in the commutative case, it is not clear whether the tracial condition is necessary to be a useful generalization.
One of the major examples of this idea is the generalization oftopological K-theoryto noncommutative C*-algebras in the form ofoperator K-theory.
A further development in this is abivariantversion of K-theory calledKK-theory, which has a composition product
KK(A,B)×KK(B,C)→KK(A,C){\displaystyle KK(A,B)\times KK(B,C)\rightarrow KK(A,C)}
of which theringstructure in ordinary K-theory is a special case. The product gives the structure of acategoryto KK. It has been related tocorrespondencesofalgebraic varieties.[1]
Thistopology-relatedarticle is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Noncommutative_topology
|
Thephase-space formulationis a formulation ofquantum mechanicsthat places thepositionandmomentumvariables on equal footing inphase space. The two key features of the phase-space formulation are that the quantum state is described by aquasiprobability distribution(instead of awave function,state vector, ordensity matrix) and operator multiplication is replaced by astar product.
The theory was fully developed byHilbrand Groenewoldin 1946 in his PhD thesis,[1]and independently byJoe Moyal,[2]each building on earlier ideas byHermann Weyl[3]andEugene Wigner.[4]
In contrast to the phase-space formulation, theSchrödinger pictureuses the positionormomentum representations (see alsoposition and momentum space).
The chief advantage of the phase-space formulation is that it makes quantum mechanics appear as similar toHamiltonian mechanicsas possible by avoiding the operator formalism, thereby "'freeing' the quantization of the 'burden' of theHilbert space".[5]This formulation is statistical in nature and offers logical connections between quantum mechanics and classicalstatistical mechanics, enabling a natural comparison between the two (seeclassical limit). Quantum mechanics in phase space is often favored in certainquantum opticsapplications (seeoptical phase space), or in the study ofdecoherenceand a range of specialized technical problems, though otherwise the formalism is less commonly employed in practical situations.[6]
The conceptual ideas underlying the development of quantum mechanics in phase space have branched into mathematical offshoots such as Kontsevich's deformation-quantization (seeKontsevich quantization formula) andnoncommutative geometry.
The phase-space distributionf(x,p)of a quantum state is a quasiprobability distribution. In the phase-space formulation, the phase-space distribution may be treated as the fundamental, primitive description of the quantum system, without any reference to wave functions or density matrices.[7]
There are several different ways to represent the distribution, all interrelated.[8][9]The most noteworthy is theWigner representation,W(x,p), discovered first.[4]Other representations (in approximately descending order of prevalence in the literature) include theGlauber–Sudarshan P,[10][11]Husimi Q,[12]Kirkwood–Rihaczek, Mehta, Rivier, and Born–Jordan representations.[13][14]These alternatives are most useful when the Hamiltonian takes a particular form, such asnormal orderfor the Glauber–Sudarshan P-representation. Since the Wigner representation is the most common, this article will usually stick to it, unless otherwise specified.
The phase-space distribution possesses properties akin to the probability density in a 2n-dimensional phase space. For example, it isreal-valued, unlike the generally complex-valued wave function. We can understand the probability of lying within a position interval, for example, by integrating the Wigner function over all momenta and over the position interval:
IfÂ(x,p)is an operator representing an observable, it may be mapped to phase space asA(x,p)through theWigner transform. Conversely, this operator may be recovered by theWeyl transform.
The expectation value of the observable with respect to the phase-space distribution is[2][15]
A point of caution, however: despite the similarity in appearance,W(x,p)is not a genuinejoint probability distribution, because regions under it do not represent mutually exclusive states, as required in thethird axiom of probability theory. Moreover, it can, in general, takenegative valueseven for pure states, with the unique exception of (optionallysqueezed)coherent states, in violation of thefirst axiom.
Regions of such negative value are provable to be "small": they cannot extend to compact regions larger than a fewħ, and hence disappear in theclassical limit. They are shielded by theuncertainty principle, which does not allow precise localization within phase-space regions smaller thanħ, and thus renders such "negative probabilities" less paradoxical. If the left side of the equation is to be interpreted as an expectation value in the Hilbert space with respect to an operator, then in the context ofquantum opticsthis equation is known as theoptical equivalence theorem. (For details on the properties and interpretation of the Wigner function, see itsmain article.)
An alternative phase-space approach to quantum mechanics seeks to define a wave function (not just a quasiprobability density) on phase space, typically by means of theSegal–Bargmann transform. To be compatible with the uncertainty principle, the phase-space wave function cannot be an arbitrary function, or else it could be localized into an arbitrarily small region of phase space. Rather, the Segal–Bargmann transform is aholomorphic functionofx+ip{\displaystyle x+ip}. There is a quasiprobability density associated to the phase-space wave function; it is theHusimi Q representationof the position wave function.
The fundamental noncommutative binary operator in the phase-space formulation that replaces the standard operator multiplication is thestar product, represented by the symbol★.[1]Each representation of the phase-space distribution has adifferentcharacteristic star product. For concreteness, we restrict this discussion to the star product relevant to the Wigner–Weyl representation.
For notational convenience, we introduce the notion ofleft and right derivatives. For a pair of functionsfandg, the left and right derivatives are defined as
Thedifferential definitionof the star product is
where the argument of the exponential function can be interpreted as apower series.
Additional differential relations allow this to be written in terms of a change in the arguments offandg:
It is also possible to define the★-product in a convolution integral form,[16]essentially through theFourier transform:
(Thus, e.g.,[7]Gaussians composehyperbolically:
or
etc.)
The energyeigenstatedistributions are known asstargenstates,★-genstates,stargenfunctions, or★-genfunctions, and the associated energies are known asstargenvaluesor★-genvalues. These are solved, analogously to the time-independentSchrödinger equation, by the★-genvalue equation,[17][18]
whereHis the Hamiltonian, a plain phase-space function, most often identical to the classical Hamiltonian.
Thetime evolutionof the phase space distribution is given by a quantum modification ofLiouville flow.[2][9][19]This formula results from applying theWigner transformationto the density matrix version of thequantum Liouville equation,
thevon Neumann equation.
In any representation of the phase space distribution with its associated star product, this is
or, for the Wigner function in particular,
where {{ , }} is theMoyal bracket, the Wigner transform of the quantum commutator, while { , } is the classicalPoisson bracket.[2]
This yields a concise illustration of thecorrespondence principle: this equation manifestly reduces to the classical Liouville equation in the limitħ→ 0. In the quantum extension of the flow, however,the density of points in phase space is not conserved; the probability fluid appears "diffusive" and compressible.[2]The concept of quantum trajectory is therefore a delicate issue here.[20]See the movie for the Morse potential, below, to appreciate the nonlocality of quantum phase flow.
N.B. Given the restrictions placed by the uncertainty principle on localization,Niels Bohrvigorously denied the physical existence of such trajectories on the microscopic scale. By means of formal phase-space trajectories, the time evolution problem of the Wigner function can be rigorously solved using the path-integral method[21]and themethod of quantum characteristics,[22]although there are severe practical obstacles in both cases.
The Hamiltonian for the simple harmonic oscillator in one spatial dimension in the Wigner–Weyl representation is
The★-genvalue equation for thestaticWigner function then reads
Consider, first, the imaginary part of the★-genvalue equation,
This implies that one may write the★-genstates as functions of a single argument:
With this change of variables, it is possible to write the real part of the★-genvalue equation in the form of a modified Laguerre equation (notHermite's equation!), the solution of which involves theLaguerre polynomialsas[18]
introduced by Groenewold,[1]with associated★-genvalues
For the harmonic oscillator, the time evolution of an arbitrary Wigner distribution is simple. An initialW(x,p;t= 0) =F(u)evolves by the above evolution equation driven by the oscillator Hamiltonian given, by simplyrigidly rotating in phase space,[1]
Typically, a "bump" (or coherent state) of energyE≫ħωcan represent a macroscopic quantity and appear like a classical object rotating uniformly in phase space, a plain mechanical oscillator (see the animated figures). Integrating over all phases (starting positions att= 0) of such objects, a continuous "palisade", yields a time-independent configuration similar to the above static★-genstatesF(u), an intuitive visualization of theclassical limitfor large-action systems.[6]
The eigenfunctions can also be characterized by being rotationally symmetric (thus time-invariant) pure states. That is, they are functions of formW(x,p)=f(x2+p2){\displaystyle W(x,p)=f({\sqrt {x^{2}+p^{2}}})}that satisfyW⋆W=(2πℏ)−1W{\displaystyle W\star W=(2\pi \hbar )^{-1}W}.
Suppose a particle is initially in a minimally uncertainGaussian state, with the expectation values of position and momentum both centered at the origin in phase space. The Wigner function for such a state propagating freely is
whereαis a parameter describing the initial width of the Gaussian, andτ=m/α2ħ.
Initially, the position and momenta are uncorrelated. Thus, in 3 dimensions, we expect the position and momentum vectors to be twice as likely to be perpendicular to each other as parallel.
However, the position and momentum become increasingly correlated as the state evolves, because portions of the distribution farther from the origin in position require a larger momentum to be reached: asymptotically,
(This relative"squeezing"reflects the spreading of the freewave packetin coordinate space.)
Indeed, it is possible to show that the kinetic energy of the particle becomes asymptotically radial only, in agreement with the standard
quantum-mechanical notion of the ground-state nonzero angular momentum specifying orientation independence:[24]
TheMorse potentialis used to approximate the vibrational structure of a diatomic molecule.
Tunnelingis a hallmark quantum effect where a quantum particle, not having sufficient energy to fly above, still goes through a barrier. This effect does not exist in classical mechanics.
|
https://en.wikipedia.org/wiki/Phase_space_formulation
|
In abstract algebra, aquasi-free algebrais anassociative algebrathat satisfies the lifting property similar to that of aformally smooth algebraincommutative algebra. The notion was introduced by Cuntz and Quillen for the applications tocyclic homology.[1]A quasi-free algebra generalizes afree algebra, as well as thecoordinate ringof a smooth affinecomplex curve. Because of the latter generalization, a quasi-free algebra can be thought of as signifying smoothness on anoncommutative space.[2]
LetAbe an associative algebra over the complex numbers. ThenAis said to bequasi-freeif the following equivalent conditions are met:[3][4][5]
Let(ΩA,d){\displaystyle (\Omega A,d)}denotes thedifferential envelopeofA; i.e., the universaldifferential-graded algebragenerated byA.[6][7]ThenAis quasi-free if and only ifΩ1A{\displaystyle \Omega ^{1}A}isprojectiveas abimoduleoverA.[3]
There is also a characterization in terms of a connection. Given anA-bimoduleE, aright connectiononEis a linear map
that satisfies∇r(as)=a∇r(s){\displaystyle \nabla _{r}(as)=a\nabla _{r}(s)}and∇r(sa)=∇r(s)a+s⊗da{\displaystyle \nabla _{r}(sa)=\nabla _{r}(s)a+s\otimes da}.[8]A left connection is defined in the similar way. ThenAis quasi-free if and only ifΩ1A{\displaystyle \Omega ^{1}A}admits a right connection.[9]
One of basic properties of a quasi-free algebra is that the algebra is left and righthereditary(i.e., asubmoduleof a projective left or right module is projective or equivalently the left or right global dimension is at most one).[10]This puts a strong restriction for algebras to be quasi-free. For example, a hereditary (commutative)integral domainis precisely aDedekind domain. In particular, apolynomial ringover a field is quasi-free if and only if the number of variables is at most one.
An analog of thetubular neighborhood theorem, called theformal tubular neighborhood theorem, holds for quasi-free algebras.[11]
Thisabstract algebra-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Quasi-free_algebra
|
In mathematics,derived noncommutative algebraic geometry,[1]the derived version ofnoncommutative algebraic geometry, is the geometric study ofderived categoriesand related constructions of triangulated categories using categorical tools. Some basic examples include the bounded derived category of coherent sheaves on a smooth variety,Db(X){\displaystyle D^{b}(X)}, called its derived category, or the derived category of perfect complexes on an algebraic variety, denotedDperf(X){\displaystyle D_{\operatorname {perf} }(X)}. For instance, the derived category of coherent sheavesDb(X){\displaystyle D^{b}(X)}on a smooth projective variety can be used as an invariant of the underlying variety for many cases (ifX{\displaystyle X}has an ample (anti-)canonical sheaf). Unfortunately, studying derived categories as geometric objects of themselves does not have a standardized name.
The derived category ofP1{\displaystyle \mathbb {P} ^{1}}is one of the motivating examples for derived non-commutative schemes due to its easy categorical structure. Recall that theEuler sequenceofP1{\displaystyle \mathbb {P} ^{1}}is the short exact sequence
if we consider the two terms on the right as a complex, then we get the distinguished triangle
SinceCone(ϕ)≅O(−2)[+1]{\displaystyle \operatorname {Cone} (\phi )\cong {\mathcal {O}}(-2)[+1]}we have constructed this sheafO(−2){\displaystyle {\mathcal {O}}(-2)}using only categorical tools. We could repeat this again by tensoring the Euler sequence by the flat sheafO(−1){\displaystyle {\mathcal {O}}(-1)}, and apply the cone construction again. If we take the duals of the sheaves, then we can construct all of the line bundles inCoh(P1){\displaystyle \operatorname {Coh} (\mathbb {P} ^{1})}using only its triangulated structure. It turns out the correct way of studying derived categories from its objects and triangulated structure is with exceptional collections.
The technical tools for encoding this construction are semiorthogonal decompositions and exceptional collections.[2]Asemiorthogonal decompositionof a triangulated categoryT{\displaystyle {\mathcal {T}}}is a collection of full triangulated subcategoriesT1,…,Tn{\displaystyle {\mathcal {T}}_{1},\ldots ,{\mathcal {T}}_{n}}such that the following two properties hold
(1) For objectsTi∈Ob(Ti){\displaystyle T_{i}\in \operatorname {Ob} ({\mathcal {T}}_{i})}we haveHom(Ti,Tj)=0{\displaystyle \operatorname {Hom} (T_{i},T_{j})=0}fori>j{\displaystyle i>j}
(2) The subcategoriesTi{\displaystyle {\mathcal {T}}_{i}}generateT{\displaystyle {\mathcal {T}}}, meaning every objectT∈Ob(T){\displaystyle T\in \operatorname {Ob} ({\mathcal {T}})}can be decomposed in to a sequence ofTi∈Ob(T){\displaystyle T_{i}\in \operatorname {Ob} ({\mathcal {T}})},
such thatCone(Ti→Ti−1)∈Ob(Ti){\displaystyle \operatorname {Cone} (T_{i}\to T_{i-1})\in \operatorname {Ob} ({\mathcal {T}}_{i})}. Notice this is analogous to a filtration of an object in anabelian categorysuch that the cokernels live in a specific subcategory.
We can specialize this a little further by considering exceptional collections of objects, which generate their own subcategories. An objectE{\displaystyle E}in a triangulated category is calledexceptionalif the following property holds
wherek{\displaystyle k}is the underlying field of the vector space of morphisms. A collection of exceptional objectsE1,…,Er{\displaystyle E_{1},\ldots ,E_{r}}is anexceptional collectionof lengthr{\displaystyle r}if for anyi>j{\displaystyle i>j}and anyℓ{\displaystyle \ell }, we have
and is astrong exceptional collectionif in addition, for anyℓ≠0{\displaystyle \ell \neq 0}andanyi,j{\displaystyle i,j}, we have
We can then decompose our triangulated category into the semiorthogonal decomposition
whereT′=⟨E1,…,Er⟩⊥{\displaystyle {\mathcal {T}}'=\langle E_{1},\ldots ,E_{r}\rangle ^{\perp }}, the subcategory of objects inE∈Ob(T){\displaystyle E\in \operatorname {Ob} ({\mathcal {T}})}such thatHom(E,Ei[+ℓ])=0{\displaystyle \operatorname {Hom} (E,E_{i}[+\ell ])=0}. If in additionT′=0{\displaystyle {\mathcal {T}}'=0}then the strong exceptional collection is calledfull.
Beilinson provided the first example of a full strong exceptional collection. In the derived categoryDb(Pn){\displaystyle D^{b}(\mathbb {P} ^{n})}the line bundlesO(−n),O(−n+1),…,O(−1),O{\displaystyle {\mathcal {O}}(-n),{\mathcal {O}}(-n+1),\ldots ,{\mathcal {O}}(-1),{\mathcal {O}}}form a full strong exceptional collection.[2]He proves the theorem in two parts. First showing these objects are an exceptional collection and second by showing the diagonalOΔ{\displaystyle {\mathcal {O}}_{\Delta }}ofPn×Pn{\displaystyle \mathbb {P} ^{n}\times \mathbb {P} ^{n}}has a resolution whose compositions are tensors of the pullback of the exceptional objects.
Technical Lemma
An exceptional collection of sheavesE1,E2,…,Er{\displaystyle E_{1},E_{2},\ldots ,E_{r}}onX{\displaystyle X}is full if there exists a resolution
inDb(X×X){\displaystyle D^{b}(X\times X)}whereFi{\displaystyle F_{i}}are arbitrary coherent sheaves onX{\displaystyle X}.
Another way to reformulate this lemma forX=Pn{\displaystyle X=\mathbb {P} ^{n}}is by looking at the Koszul complex associated to
⨁i=0nO(−Di)→ϕO{\displaystyle \bigoplus _{i=0}^{n}{\mathcal {O}}(-D_{i})\xrightarrow {\phi } {\mathcal {O}}}
whereDi{\displaystyle D_{i}}are hyperplane divisors ofPn{\displaystyle \mathbb {P} ^{n}}. This gives the exact complex
0→O(−∑i=1nDi)→⋯→⨁i≠jO(−Di−Dj)→⨁i=1nO(−Di)→O→0{\displaystyle 0\to {\mathcal {O}}\left(-\sum _{i=1}^{n}D_{i}\right)\to \cdots \to \bigoplus _{i\neq j}{\mathcal {O}}(-D_{i}-D_{j})\to \bigoplus _{i=1}^{n}{\mathcal {O}}(-D_{i})\to {\mathcal {O}}\to 0}
which gives a way to constructO(−n−1){\displaystyle {\mathcal {O}}(-n-1)}using the sheavesO(−n),…,O(−1),O{\displaystyle {\mathcal {O}}(-n),\ldots ,{\mathcal {O}}(-1),{\mathcal {O}}}, since they are the sheaves used in all terms in the above exact sequence, except for
O(−∑i=0nDi)≅O(−n−1){\displaystyle {\mathcal {O}}\left(-\sum _{i=0}^{n}D_{i}\right)\cong {\mathcal {O}}(-n-1)}
which gives a derived equivalence of the rest of the terms of the above complex withO(−n−1){\displaystyle {\mathcal {O}}(-n-1)}. Forn=2{\displaystyle n=2}the Koszul complex above is the exact complex
0→O(−3)→O(−2)⊕O(−2)→O(−1)⊕O(−1)→O→0{\displaystyle 0\to {\mathcal {O}}(-3)\to {\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2)\to {\mathcal {O}}(-1)\oplus {\mathcal {O}}(-1)\to {\mathcal {O}}\to 0}
giving the quasi isomorphism ofO(−3){\displaystyle {\mathcal {O}}(-3)}with the complex
0→O(−2)⊕O(−2)→O(−1)⊕O(−1)→O→0{\displaystyle 0\to {\mathcal {O}}(-2)\oplus {\mathcal {O}}(-2)\to {\mathcal {O}}(-1)\oplus {\mathcal {O}}(-1)\to {\mathcal {O}}\to 0}
IfX{\displaystyle X}is a smooth projective variety with ample (anti-)canonical sheaf and there is an equivalence of derived categoriesF:Db(X)→Db(Y){\displaystyle F:D^{b}(X)\to D^{b}(Y)}, then there is an isomorphism of the underlying varieties.[3]
The proof starts out by analyzing two induced Serre functors onDb(Y){\displaystyle D^{b}(Y)}and finding an isomorphism between them. It particular, it shows there is an objectωY=F(ωX){\displaystyle \omega _{Y}=F(\omega _{X})}which acts like the dualizing sheaf onY{\displaystyle Y}. The isomorphism between these two functors gives an isomorphism of the set of underlying points of the derived categories. Then, what needs to be check is an ismorphismF(ωX⊗k)≅ωY⊗k{\displaystyle F(\omega _{X}^{\otimes k})\cong \omega _{Y}^{\otimes k}}, for anyk∈N{\displaystyle k\in \mathbb {N} }, giving an isomorphism of canonical rings
IfωY{\displaystyle \omega _{Y}}can be shown to be (anti-)ample, then the proj of these rings will give an isomorphismX→Y{\displaystyle X\to Y}. All of the details are contained in Dolgachev's notes.
This theorem fails in the caseX{\displaystyle X}is Calabi-Yau, sinceωX≅OX{\displaystyle \omega _{X}\cong {\mathcal {O}}_{X}}, or is the product of a variety which isCalabi-Yau.Abelian varietiesare a class of examples where a reconstruction theorem couldneverhold. IfX{\displaystyle X}is an abelian variety andX^{\displaystyle {\hat {X}}}is its dual, theFourier–Mukai transformwith kernelP{\displaystyle {\mathcal {P}}}, the Poincare bundle,[4]gives an equivalence
of derived categories. Since an abelian variety is generally not isomorphic to its dual, there are derived equivalent derived categories without isomorphic underlying varieties.[5]There is an alternative theory oftensor triangulated geometrywhere we consider not only a triangulated category, but also a monoidal structure, i.e. a tensor product. This geometry has a full reconstruction theorem using the spectrum of categories.[6]
K3 surfacesare another class of examples where reconstruction fails due to their Calabi-Yau property. There is a criterion for determining whether or not two K3 surfaces are derived equivalent: the derived category of the K3 surfaceDb(X){\displaystyle D^{b}(X)}is derived equivalent to another K3Db(Y){\displaystyle D^{b}(Y)}if and only if there is a Hodge isometryH2(X,Z)→H2(Y,Z){\displaystyle H^{2}(X,\mathbb {Z} )\to H^{2}(Y,\mathbb {Z} )}, that is, an isomorphism ofHodge structure.[3]Moreover, this theorem is reflected in the motivic world as well, where the Chow motives are isomorphic if and only if there is an isometry of Hodge structures.[7]
One nice application of the proof of this theorem is the identification of autoequivalences of the derived category of a smooth projective variety with ample (anti-)canonical sheaf. This is given by
Where an autoequivalenceF{\displaystyle F}is given by an automorphismf:X→X{\displaystyle f:X\to X}, then tensored by a line bundleL∈Pic(X){\displaystyle {\mathcal {L}}\in \operatorname {Pic} (X)}and finally composed with a shift. Note thatAut(X){\displaystyle \operatorname {Aut} (X)}acts onPic(X){\displaystyle \operatorname {Pic} (X)}via the polarization map,g↦g∗(L)⊗L−1{\displaystyle g\mapsto g^{*}(L)\otimes L^{-1}}.[8]
The bounded derived categoryDb(X){\displaystyle D^{b}(X)}was used extensively in SGA6 to construct an intersection theory withK(X){\displaystyle K(X)}andGrγK(X)⊗Q{\displaystyle Gr_{\gamma }K(X)\otimes \mathbb {Q} }. Since these objects are intimately relative with theChow ringofX{\displaystyle X}, itschow motive, Orlov asked the following question: given a fully-faithful functor
is there an induced map on the chow motives
such thatM(X){\displaystyle M(X)}is a summand ofM(Y){\displaystyle M(Y)}?[9]In the case of K3 surfaces, a similar result has been confirmed since derived equivalent K3 surfaces have an isometry of Hodge structures, which gives an isomorphism of motives.
On a smooth variety there is an equivalence between the derived categoryDb(X){\displaystyle D^{b}(X)}and the thick[10][11]full triangulatedDperf(X){\displaystyle D_{\operatorname {perf} }(X)}of perfect complexes. Forseparated,Noetherianschemes of finiteKrull dimension(called theELFcondition)[12]this is not the case, and Orlov defines the derived category of singularities as their difference using a quotient of categories. For an ELF schemeX{\displaystyle X}its derived category of singularities is defined as
for a suitable definition oflocalizationof triangulated categories.
Although localization of categories is defined for a class of morphismsΣ{\displaystyle \Sigma }in the category closed under composition, we can construct such a class from a triangulated subcategory. Given a full triangulated subcategoryN⊂T{\displaystyle {\mathcal {N}}\subset {\mathcal {T}}}the class of morphismsΣ(N){\displaystyle \Sigma ({\mathcal {N}})},s{\displaystyle s}inT{\displaystyle {\mathcal {T}}}wheres{\displaystyle s}fits into a distinguished triangle
X→sY→N→X[+1]{\displaystyle X{\xrightarrow {s}}Y\to N\to X[+1]}
withX,Y∈T{\displaystyle X,Y\in {\mathcal {T}}}andN∈N{\displaystyle N\in {\mathcal {N}}}. It can be checked this forms a multiplicative system using the octahedral axiom for distinguished triangles. Given
with distinguished triangles
whereN,N′∈N{\displaystyle N,N'\in {\mathcal {N}}}, then there are distinguished triangles
Kontsevich proposed a model for Landau–Ginzburg models which was worked out to the following definition:[14]aLandau–Ginzburg modelis a smooth varietyX{\displaystyle X}together with a morphismW:X→A1{\displaystyle W:X\to \mathbb {A} ^{1}}which isflat. There are three associated categories which can be used to analyze the D-branes in a Landau–Ginzburg model using matrix factorizations from commutative algebra.
With this definition, there are three categories which can be associated to any pointw0∈A1{\displaystyle w_{0}\in \mathbb {A} ^{1}}, aZ/2{\displaystyle \mathbb {Z} /2}-graded categoryDGw0(W){\displaystyle DG_{w_{0}}(W)}, an exact categoryPairw0(W){\displaystyle \operatorname {Pair} _{w_{0}}(W)}, and a triangulated categoryDBw0(W){\displaystyle DB_{w_{0}}(W)}, each of which has objects
There is also a shift functor[+1]{\displaystyle [+1]}sendP¯{\displaystyle {\overline {P}}}to
P¯[+1]=(−p0:P0→P1,−p1:P1→P0){\displaystyle {\overline {P}}[+1]=(-p_{0}:P_{0}\to P_{1},-p_{1}:P_{1}\to P_{0})}.
The difference between these categories are their definition of morphisms. The most general of which isDGw0(W){\displaystyle DG_{w_{0}}(W)}whose morphisms are theZ/2{\displaystyle \mathbb {Z} /2}-graded complex
where the grading is given by(i−j)mod2{\displaystyle (i-j){\bmod {2}}}and differential acting on degreed{\displaystyle d}homogeneous elements by
InPairw0(W){\displaystyle \operatorname {Pair} _{w_{0}}(W)}the morphisms are the degree0{\displaystyle 0}morphisms inDGw0(W){\displaystyle DG_{w_{0}}(W)}. Finally,DBw0(W){\displaystyle DB_{w_{0}}(W)}has the morphisms inPairw0(W){\displaystyle \operatorname {Pair} _{w_{0}}(W)}modulo the null-homotopies. Furthermore,DBw0(W){\displaystyle DB_{w_{0}}(W)}can be endowed with a triangulated structure through a graded cone-construction inPairw0(W){\displaystyle \operatorname {Pair} _{w_{0}}(W)}. Givenf¯:P¯→Q¯{\displaystyle {\overline {f}}:{\overline {P}}\to {\overline {Q}}}there is a mapping codeC(f){\displaystyle C(f)}with maps
and
Then, a diagramP¯→Q¯→R¯→P¯[+1]{\displaystyle {\overline {P}}\to {\overline {Q}}\to {\overline {R}}\to {\overline {P}}[+1]}inDBw0(W){\displaystyle DB_{w_{0}}(W)}is a distinguished triangle if it is isomorphic to a cone fromPairw0(W){\displaystyle \operatorname {Pair} _{w_{0}}(W)}.
Using the construction ofDBw0(W){\displaystyle DB_{w_{0}}(W)}we can define the category of D-branes of type B onX{\displaystyle X}with superpotentialW{\displaystyle W}as the product category
This is related to the singularity category as follows: Given a superpotentialW{\displaystyle W}with isolated singularities only at0{\displaystyle 0}, denoteX0=W−1(0){\displaystyle X_{0}=W^{-1}(0)}. Then, there is an exact equivalence of categories
given by a functor induced from cokernel functorCok{\displaystyle \operatorname {Cok} }sending a pairP¯↦Coker(p1){\displaystyle {\overline {P}}\mapsto \operatorname {Coker} (p_{1})}. In particular, sinceX{\displaystyle X}is regular,Bertini's theoremshowsDB(W){\displaystyle DB(W)}is only a finite product of categories.
There is a Fourier-Mukai transformΦZ{\displaystyle \Phi _{Z}}on the derived categories of two related varieties giving an equivalence of their singularity categories. This equivalence is calledKnörrer periodicity. This can be constructed as follows: given a flat morphismf:X→A1{\displaystyle f:X\to \mathbb {A} ^{1}}from a separated regular Noetherian scheme of finite Krull dimension, there is an associated schemeY=X×A2{\displaystyle Y=X\times \mathbb {A} ^{2}}and morphismg:Y→A1{\displaystyle g:Y\to \mathbb {A} ^{1}}such thatg=f+xy{\displaystyle g=f+xy}wherexy{\displaystyle xy}are the coordinates of theA2{\displaystyle \mathbb {A} ^{2}}-factor. Consider the fibersX0=f−1(0){\displaystyle X_{0}=f^{-1}(0)},Y0=g−1(0){\displaystyle Y_{0}=g^{-1}(0)}, and the induced morphismx:Y0→A1{\displaystyle x:Y_{0}\to \mathbb {A} ^{1}}. And the fiberZ=x−1(0){\displaystyle Z=x^{-1}(0)}. Then, there is an injectioni:Z→Y0{\displaystyle i:Z\to Y_{0}}and a projectionq:Z→X0{\displaystyle q:Z\to X_{0}}forming anA1{\displaystyle \mathbb {A} ^{1}}-bundle. The Fourier-Mukai transform
induces an equivalence of categories
calledKnörrer periodicity. There is another form of this periodicity wherexy{\displaystyle xy}is replaced by the polynomialx2+y2{\displaystyle x^{2}+y^{2}}.[15][16]These periodicity theorems are the main computational techniques because it allows for a reduction in the analysis of the singularity categories.
If we take the Landau–Ginzburg model(C2k+1,W){\displaystyle (\mathbb {C} ^{2k+1},W)}whereW=z0n+z12+⋯+z2k2{\displaystyle W=z_{0}^{n}+z_{1}^{2}+\cdots +z_{2k}^{2}}, then the only fiber singular fiber ofW{\displaystyle W}is the origin. Then, the D-brane category of the Landau–Ginzburg model is equivalent to the singularity categoryDsing(Spec(C[z]/(zn))){\displaystyle D_{\text{sing}}(\operatorname {Spec} (\mathbb {C} [z]/(z^{n})))}. Over the algebraA=C[z]/(zn){\displaystyle A=\mathbb {C} [z]/(z^{n})}there are indecomposable objects
whose morphisms can be completely understood. For any pairi,j{\displaystyle i,j}there are morphismsαji:Vi→Vj{\displaystyle \alpha _{j}^{i}:V_{i}\to V_{j}}where
where every other morphism is a composition and linear combination of these morphisms. There are many other cases which can be explicitly computed, using the table of singularities found in Knörrer's original paper.[16]
|
https://en.wikipedia.org/wiki/Derived_noncommutative_algebraic_geometry
|
In mathematics, aQ-categoryoralmost quotient category[1]is acategorythat is a "milder version of a Grothendieck site."[2]A Q-category is acoreflective subcategory.[1][clarification needed]The Q stands for a quotient.
The concept of Q-categories was introduced by Alexander Rosenberg in 1988.[2]The motivation for the notion was its use innoncommutative algebraic geometry; in this formalism,noncommutative spacesare defined assheaveson Q-categories.
A Q-category is defined by the formula[1][further explanation needed]A:(u∗⊣u∗):A¯→u∗←u∗A{\displaystyle \mathbb {A} :(u^{*}\dashv u_{*}):{\bar {A}}{\stackrel {\overset {u^{*}}{\leftarrow }}{\underset {u_{*}}{\to }}}A}whereu∗{\displaystyle u^{*}}is the left adjoint in a pair ofadjoint functorsand is afull and faithful functor.
Thiscategory theory-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Q-category
|
Inmathematics, theSelberg trace formula, introduced bySelberg (1956), is an expression for the character of theunitary representationof aLie groupGon the spaceL2(Γ\G)ofsquare-integrable functions, whereΓis a cofinitediscrete group. The character is given by the trace of certain functions onG.
The simplest case is whenΓiscocompact, when the representation breaks up into discrete summands. Here the trace formula is an extension of theFrobenius formulafor the character of aninduced representationof finite groups. WhenΓis the cocompact subgroupZof the real numbersG=R, the Selberg trace formula is essentially thePoisson summation formula.
The case whenΓ\Gis not compact is harder, because there is acontinuous spectrum, described usingEisenstein series. Selberg worked out the non-compact case whenGis the groupSL(2,R); the extension to higher rank groups is theArthur–Selberg trace formula.
WhenΓis the fundamental group of aRiemann surface, the Selberg trace formula describes the spectrum of differential operators such as theLaplacianin terms of geometric data involving the lengths of geodesics on the Riemann surface. In this case the Selberg trace formula is formally similar to theexplicit formulasrelating the zeros of theRiemann zeta functionto prime numbers, with the zeta zeros corresponding to eigenvalues of the Laplacian, and the primes corresponding to geodesics. Motivated by the analogy, Selberg introduced theSelberg zeta functionof a Riemann surface, whose analytic properties are encoded by the Selberg trace formula.
Cases of particular interest include those for which the space is acompact Riemann surfaceS. The initial publication in 1956 ofAtle Selbergdealt with this case, itsLaplaciandifferential operator and its powers. The traces of powers of a Laplacian can be used to define theSelberg zeta function. The interest of this case was the analogy between the formula obtained, and theexplicit formulaeofprime numbertheory. Here theclosed geodesicsonSplay the role of prime numbers.
At the same time, interest in the traces ofHecke operatorswas linked to theEichler–Selberg trace formula, of Selberg andMartin Eichler, for a Hecke operator acting on a vector space ofcusp formsof a given weight, for a givencongruence subgroupof themodular group. Here the trace of the identity operator is the dimension of the vector space, i.e. the dimension of the space of modular forms of a given type: a quantity traditionally calculated by means of theRiemann–Roch theorem.
The trace formula has applications toarithmetic geometryandnumber theory. For instance, using the trace theorem,Eichler and Shimuracalculated theHasse–Weil L-functionsassociated tomodular curves;Goro Shimura's methods by-passed the analysis involved in the trace formula. The development ofparabolic cohomology(fromEichler cohomology) provided a purely algebraic setting based ongroup cohomology, taking account of thecuspscharacteristic of non-compact Riemann surfaces and modular curves.
The trace formula also has purelydifferential-geometricapplications. For instance, by a result of Buser, thelength spectrumof aRiemann surfaceis an isospectral invariant, essentially by the trace formula.
A compact hyperbolic surfaceXcan be written as the space of orbitsΓ∖H,{\displaystyle \Gamma \backslash \mathbf {H} ,}whereΓis a subgroup ofPSL(2,R), andHis theupper half plane, andΓacts onHbylinear fractional transformations.
The Selberg trace formula for this case is easier than the general case because the surface is compact so there is no continuous spectrum, and the groupΓhas no parabolic or elliptic elements (other than the identity).
Then the spectrum for theLaplace–Beltrami operatoronXis discrete and real, since the Laplace operator is self adjoint with compactresolvent; that is0=μ0<μ1≤μ2≤⋯{\displaystyle 0=\mu _{0}<\mu _{1}\leq \mu _{2}\leq \cdots }where the eigenvaluesμncorrespond toΓ-invariant eigenfunctionsuinC∞(H)of the Laplacian; in other words{u(γz)=u(z),∀γ∈Γy2(uxx+uyy)+μnu=0.{\displaystyle {\begin{cases}u(\gamma z)=u(z),\qquad \forall \gamma \in \Gamma \\y^{2}\left(u_{xx}+u_{yy}\right)+\mu _{n}u=0.\end{cases}}}
Using the variable substitutionμ=s(1−s),s=12+ir{\displaystyle \mu =s(1-s),\qquad s={\tfrac {1}{2}}+ir}the eigenvalues are labeledrn,n≥0.{\displaystyle r_{n},n\geq 0.}
Then theSelberg trace formulais given by∑n=0∞h(rn)=μ(X)4π∫−∞∞rh(r)tanh(πr)dr+∑{T}logN(T0)N(T)12−N(T)−12g(logN(T)).{\displaystyle \sum _{n=0}^{\infty }h(r_{n})={\frac {\mu (X)}{4\pi }}\int _{-\infty }^{\infty }r\,h(r)\tanh(\pi r)\,dr+\sum _{\{T\}}{\frac {\log N(T_{0})}{N(T)^{\frac {1}{2}}-N(T)^{-{\frac {1}{2}}}}}g(\log N(T)).}
The right hand side is a sum over conjugacy classes of the groupΓ, with the first term corresponding to the identity element and the remaining terms forming a sum over the other conjugacy classes{T}(which are all hyperbolic in this case). The functionhhas to satisfy the following:
The functiongis the Fourier transform ofh, that is,h(r)=∫−∞∞g(u)eirudu.{\displaystyle h(r)=\int _{-\infty }^{\infty }g(u)e^{iru}\,du.}
LetGbe a unimodular locally compact group, andΓ{\displaystyle \Gamma }a discrete cocompact subgroup ofGandϕ{\displaystyle \phi }a compactly supported continuous function onG. The trace formula in this setting is the following equality:∑γ∈{Γ}aΓG(γ)∫Gγ∖Gϕ(x−1γx)dx=∑π∈G^aΓG(π)trπ(ϕ){\displaystyle \sum _{\gamma \in \{\Gamma \}}a_{\Gamma }^{G}(\gamma )\int _{G^{\gamma }\setminus G}\phi (x^{-1}\gamma x)\,dx=\sum _{\pi \in {\widehat {G}}}a_{\Gamma }^{G}(\pi )\operatorname {tr} \pi (\phi )}where{Γ}{\displaystyle \{\Gamma \}}is the set of conjugacy classes inΓ{\displaystyle \Gamma },G^{\displaystyle {\widehat {G}}}is theunitary dualofGand:
The left-hand side of the formula is called thegeometric sideand the right-hand side thespectral side. The terms∫Gγ∖Gϕ(x−1γx)dx{\displaystyle \int _{G^{\gamma }\setminus G}\phi (x^{-1}\gamma x)\,dx}areorbital integrals.
Define the following operator on compactly supported functions onΓ∖G{\displaystyle \Gamma \backslash G}:R(ϕ)=∫Gϕ(x)R(x)dx,{\displaystyle R(\phi )=\int _{G}\phi (x)R(x)\,dx,}It extends continuously toL2(Γ∖G){\displaystyle L^{2}(\Gamma \setminus G)}and forf∈L2(Γ∖G){\displaystyle f\in L^{2}(\Gamma \setminus G)}we have:(R(ϕ)f)(x)=∫Gϕ(y)f(xy)dy=∫Γ∖G(∑γ∈Γϕ(x−1γy))f(y)dy{\displaystyle (R(\phi )f)(x)=\int _{G}\phi (y)f(xy)\,dy=\int _{\Gamma \setminus G}\left(\sum _{\gamma \in \Gamma }\phi (x^{-1}\gamma y)\right)f(y)\,dy}after a change of variables. AssumingΓ∖G{\displaystyle \Gamma \setminus G}is compact, the operatorR(ϕ){\displaystyle R(\phi )}istrace-classand the trace formula is the result of computing its trace in two ways as explained below.[1]
The trace ofR(ϕ){\displaystyle R(\phi )}can be expressed as the integral of the kernelK(x,y)=∑γ∈Γϕ(x−1γy){\displaystyle K(x,y)=\sum _{\gamma \in \Gamma }\phi (x^{-1}\gamma y)}along the diagonal, that is:trR(ϕ)=∫Γ∖G∑γ∈Γϕ(x−1γx)dx.{\displaystyle \operatorname {tr} R(\phi )=\int _{\Gamma \setminus G}\sum _{\gamma \in \Gamma }\phi (x^{-1}\gamma x)\,dx.}Let{Γ}{\displaystyle \{\Gamma \}}denote a collection of representatives of conjugacy classes inΓ{\displaystyle \Gamma }, andΓγ{\displaystyle \Gamma ^{\gamma }}andGγ{\displaystyle G^{\gamma }}the respective centralizers ofγ{\displaystyle \gamma }.
Then the above integral can, after manipulation, be writtentrR(ϕ)=∑γ∈{Γ}aΓG(γ)∫Gγ∖Gϕ(x−1γx)dx.{\displaystyle \operatorname {tr} R(\phi )=\sum _{\gamma \in \{\Gamma \}}a_{\Gamma }^{G}(\gamma )\int _{G^{\gamma }\setminus G}\phi (x^{-1}\gamma x)\,dx.}This gives thegeometric sideof the trace formula.
Thespectral sideof the trace formula comes from computing the trace ofR(ϕ){\displaystyle R(\phi )}using the decomposition of the regular representation ofG{\displaystyle G}into its irreducible components. ThustrR(ϕ)=∑π∈G^aΓG(π)trπ(ϕ){\displaystyle \operatorname {tr} R(\phi )=\sum _{\pi \in {\hat {G}}}a_{\Gamma }^{G}(\pi )\operatorname {tr} \pi (\phi )}whereG^{\displaystyle {\hat {G}}}is the set of irreducible unitary representations ofG{\displaystyle G}(recall that the positive integeraΓG(π){\displaystyle a_{\Gamma }^{G}(\pi )}is the multiplicity ofπ{\displaystyle \pi }in the unitary representationR{\displaystyle R}onL2(Γ∖G){\displaystyle L^{2}(\Gamma \setminus G)}).
WhenG{\displaystyle G}is a semisimple Lie group with a maximal compact subgroupK{\displaystyle K}andX=G/K{\displaystyle X=G/K}is the associatedsymmetric spacethe conjugacy classes inΓ{\displaystyle \Gamma }can be described in geometric terms using the compact Riemannian manifold (more generally orbifold)Γ∖X{\displaystyle \Gamma \backslash X}. The orbital integrals and the traces in irreducible summands can then be computed further and in particular one can recover the case of the trace formula for hyperbolic surfaces in this way.
The general theory ofEisenstein serieswas largely motivated by the requirement to separate out thecontinuous spectrum, which is characteristic of the non-compact case.[2]
The trace formula is often given for algebraic groups over the adeles rather than for Lie groups, because this makes the corresponding discrete subgroupΓinto an algebraic group over a field which is technically easier to work with. The case of SL2(C) is discussed inGel'fand, Graev & Pyatetskii-Shapiro (1990)andElstrodt, Grunewald & Mennicke (1998). Gel'fand et al also treat SL2(F) whereFis a locally compact topological field withultrametric norm, so a finite extension of thep-adic numbersQpor of theformal Laurent seriesFq((T)); they also handle the adelic case in characteristic 0, combining all completionsRandQpof therational numbersQ.
Contemporary successors of the theory are theArthur–Selberg trace formulaapplying to the case of general semisimpleG, and the many studies of the trace formula in theLanglands philosophy(dealing with technical issues such asendoscopy). The Selberg trace formula can be derived from the Arthur–Selberg trace formula with some effort.
|
https://en.wikipedia.org/wiki/Selberg_trace_formula
|
Inmathematics, theLanglands programis a set ofconjecturesabout connections betweennumber theoryandgeometry. It was proposed byRobert Langlands(1967,1970). It seeks to relateGalois groupsinalgebraic number theorytoautomorphic formsandrepresentation theoryofalgebraic groupsoverlocal fieldsandadeles. It was described byEdward Frenkelas the "grand unified theoryof mathematics."[1]
The Langlands program is built on existing ideas: thephilosophy of cusp formsformulated a few years earlier byHarish-ChandraandGelfand(1963), the work and Harish-Chandra's approach onsemisimple Lie groups, and in technical terms thetrace formulaofSelbergand others.
What was new in Langlands' work, besides technical depth, was the proposed connection to number theory, together with its rich organisational structure hypothesised (so-calledfunctoriality).
Harish-Chandra's work exploited the principle that what can be done for onesemisimple(or reductive)Lie group, can be done for all. Therefore, once the role of some low-dimensional Lie groups such as GL(2) in the theory of modular forms had been recognised, and with hindsight GL(1) inclass field theory, the way was open to speculation about GL(n) for generaln> 2.
The 'cusp form' idea came out of the cusps onmodular curvesbut also had a meaning visible inspectral theoryas "discrete spectrum", contrasted with the "continuous spectrum" fromEisenstein series. It becomes much more technical for bigger Lie groups, because theparabolic subgroupsare more numerous.
In all these approaches technical methods were available, often inductive in nature and based onLevi decompositionsamongst other matters, but the field remained demanding.[2]
From the perspective of modular forms, examples such asHilbert modular forms,Siegel modular forms, andtheta-serieshad been developed.
The conjectures have evolved since Langlands first stated them. Langlands conjectures apply across many different groups over many different fields for which they can be stated, and each field offers several versions of the conjectures.[3]Some versions[which?]are vague, or depend on objects such asLanglands groups, whose existence is unproven, or on theL-group that has several non-equivalent definitions.
Objects for which Langlands conjectures can be stated:
The conjectures can be stated variously in ways that are closely related but not obviously equivalent.
The starting point of the program wasEmil Artin'sreciprocity law, which generalizesquadratic reciprocity. TheArtin reciprocity lawapplies to aGalois extensionof analgebraic number fieldwhoseGalois groupisabelian; it assignsL-functionsto the one-dimensional representations of this Galois group, and states that theseL-functions are identical to certainDirichletL-seriesor more general series (that is, certain analogues of theRiemann zeta function) constructed fromHecke characters. The precise correspondence between these different kinds ofL-functions constitutes Artin's reciprocity law.
For non-abelian Galois groups and higher-dimensional representations of them,L-functions can be defined in a natural way:ArtinL-functions.
Langlands' insight was to find the proper generalization ofDirichletL-functions, which would allow the formulation of Artin's statement in Langland's more general setting.Heckehad earlier related DirichletL-functions withautomorphic forms(holomorphic functionson the upper half plane of thecomplex number planeC{\displaystyle \mathbb {C} }that satisfy certainfunctional equations). Langlands then generalized these toautomorphic cuspidal representations, which are certain infinite dimensional irreducible representations of thegeneral linear groupGL(n) over theadele ringofQ{\displaystyle \mathbb {Q} }(therational numbers). (This ring tracks all the completions ofQ,{\displaystyle \mathbb {Q} ,}seep-adic numbers.)
Langlands attachedautomorphicL-functionsto these automorphic representations, and conjectured that every ArtinL-function arising from a finite-dimensional representation of the Galois group of anumber fieldis equal to one arising from an automorphic cuspidal representation. This is known as hisreciprocity conjecture.
Roughly speaking, this conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from aLanglands groupto anL-group. This offers numerous variations, in part because the definitions of Langlands group andL-group are not fixed.
Overlocal fieldsthis is expected to give a parameterization ofL-packetsof admissible irreducible representations of areductive groupover the local field. For example, over the real numbers, this correspondence is theLanglands classificationof representations of real reductive groups. Overglobal fields, it should give a parameterization of automorphic forms.
The functoriality conjecture states that a suitable homomorphism ofL-groups is expected to give a correspondence between automorphic forms (in the global case) or representations (in the local case). Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial.
Langlands generalized the idea of functoriality: instead of using the general linear group GL(n), other connectedreductive groupscan be used. Furthermore, given such a groupG, Langlands constructs theLanglands dualgroupLG, and then, for every automorphic cuspidal representation ofGand every finite-dimensional representation ofLG, he defines anL-function. One of his conjectures states that theseL-functions satisfy a certain functional equation generalizing those of other knownL-functions.
He then goes on to formulate a very general "Functoriality Principle". Given two reductive groups and a (well behaved)morphismbetween their correspondingL-groups, this conjecture relates their automorphic representations in a way that is compatible with theirL-functions. This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of aninduced representationconstruction—what in the more traditional theory ofautomorphic formshad been called a 'lifting', known in special cases, and so is covariant (whereas arestricted representationis contravariant). Attempts to specify a direct construction have only produced some conditional results.
All these conjectures can be formulated for more general fields in place ofQ{\displaystyle \mathbb {Q} }:algebraic number fields(the original and most important case),local fields, and function fields (finiteextensionsofFp(t) wherepis aprimeandFp(t) is the field of rational functions over thefinite fieldwithpelements).
The geometric Langlands program, suggested byGérard Laumonfollowing ideas ofVladimir Drinfeld, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relatesl-adic representations of theétale fundamental groupof analgebraic curveto objects of thederived categoryofl-adic sheaves on themoduli stackofvector bundlesover the curve.
A 9-person collaborative project led byDennis Gaitsgoryannounced a proof of the (categorical, unramified) geometric Langlands conjecture leveragingHecke eigensheavesas part of the proof.[4][5][6][7]
The Langlands correspondence for GL(1,K) follows from (and are essentially equivalent to)class field theory.
Langlands proved the Langlands conjectures for groups over the archimedean local fieldsR{\displaystyle \mathbb {R} }(thereal numbers) andC{\displaystyle \mathbb {C} }(thecomplex numbers) by giving theLanglands classificationof their irreducible representations.
Lusztig's classification of the irreducible representations of groups of Lie type over finite fields can be considered an analogue of the Langlands conjectures for finite fields.
Andrew Wiles' proof of modularity of semistable elliptic curves over rationals can be viewed as an instance of the Langlands reciprocity conjecture, since the main idea is to relate the Galois representations arising from elliptic curves to modular forms. Although Wiles' results have been substantially generalized, in many different directions, the full Langlands conjecture forGL(2,Q){\displaystyle {\text{GL}}(2,\mathbb {Q} )}remains unproved.
In 1998,Laurent LafforgueprovedLafforgue's theoremverifying the global Langlands correspondence for the general linear group GL(n,K) for function fieldsK. This work continued earlier investigations by Drinfeld, who previously addressed the case of GL(2,K) in the 1980s.
In 2018,Vincent Lafforgueestablished one half of the global Langlands correspondence (the direction from automorphic forms to Galois representations) for connected reductive groups over global function fields.[8][9][10]
Philip Kutzko(1980) proved thelocal Langlands correspondencefor the general linear group GL(2,K) over local fields.
Gérard Laumon,Michael Rapoport, andUlrich Stuhler(1993) proved the local Langlands correspondence for the general linear group GL(n,K) for positive characteristic local fieldsK. Their proof uses a global argument, realizing smooth admissible representations of interest as the local components of automorphic representations of the group of units of a division algebra over a curve, then using the point-counting formula to study the properties of the global Galois representations associated to these representations.
Michael HarrisandRichard Taylor(2001) proved the local Langlands conjectures for the general linear group GL(n,K) for characteristic 0 local fieldsK.Guy Henniart(2000) gave another proof. Both proofs use a global argument of a similar flavor to the one mentioned in the previous paragraph.Peter Scholze(2013) gave another proof.
In 2008,Ngô Bảo Châuproved the "fundamental lemma", which was originally conjectured by Langlands and Shelstad in 1983 and being required in the proof of some important conjectures in the Langlands program.[11][12]
|
https://en.wikipedia.org/wiki/Langlands_program
|
Inmathematics, theorbit method(also known as theKirillov theory,the method of coadjoint orbitsand by a few similar names) establishes a correspondence between irreducibleunitary representationsof aLie groupand itscoadjoint orbits: orbits of theaction of the groupon the dual space of itsLie algebra. The theory was introduced byKirillov(1961,1962) fornilpotent groupsand later extended byBertram Kostant,Louis Auslander,Lajos Pukánszkyand others to the case ofsolvable groups.Roger Howefound a version of the orbit method that applies top-adic Lie groups.[1]David Voganproposed that the orbit method should serve as a unifying principle in the description of the unitary duals of real reductive Lie groups.[2]
One of the key observations of Kirillov was that coadjoint orbits of a Lie groupGhave natural structure ofsymplectic manifoldswhose symplectic structure is invariant underG. If an orbit is thephase spaceof aG-invariantclassical mechanical systemthen the corresponding quantum mechanical system ought to be described via an irreducible unitary representation ofG. Geometric invariants of the orbit translate into algebraic invariants of the corresponding representation. In this way the orbit method may be viewed as a precise mathematical manifestation of a vague physical principle of quantization. In the case of a nilpotent groupGthe correspondence involves all orbits, but for a generalGadditional restrictions on the orbit are necessary (polarizability, integrality, Pukánszky condition). This point of view has been significantly advanced by Kostant in his theory ofgeometric quantizationof coadjoint orbits.
For aLie groupG{\displaystyle G}, theKirillov orbit methodgives a heuristic method inrepresentation theory. It connects theFourier transformsofcoadjoint orbits, which lie in thedual spaceof theLie algebraofG, to theinfinitesimal charactersof theirreducible representations. The method got its name after theRussianmathematicianAlexandre Kirillov.
At its simplest, it states that a character of a Lie group may be given by theFourier transformof theDirac delta functionsupportedon the coadjoint orbits, weighted by the square-root of theJacobianof theexponential map, denoted byj{\displaystyle j}. It does not apply to all Lie groups, but works for a number of classes ofconnectedLie groups, includingnilpotent, somesemisimplegroups, andcompact groups.
LetGbe aconnected,simply connectednilpotentLie group. Kirillov proved that the equivalence classes ofirreducibleunitary representationsofGare parametrized by thecoadjoint orbitsofG, that is the orbits of the actionGon the dual spaceg∗{\displaystyle {\mathfrak {g}}^{*}}of its Lie algebra. TheKirillov character formulaexpresses theHarish-Chandra characterof the representation as a certain integral over the corresponding orbit.
Complex irreducible representations ofcompact Lie groupshave been completely classified. They are always finite-dimensional, unitarizable (i.e. admit an invariant positive definiteHermitian form) and are parametrized by theirhighest weights, which are precisely the dominant integral weights for the group. IfGis a compactsemisimple Lie groupwith aCartan subalgebrahthen its coadjoint orbits areclosedand each of them intersects the positive Weyl chamberh*+in a single point. An orbit isintegralif this point belongs to the weight lattice ofG.
The highest weight theory can be restated in the form of a bijection between the set of integral coadjoint orbits and the set of equivalence classes of irreducible unitary representations ofG: the highest weight representationL(λ) with highest weightλ∈h*+corresponds to the integral coadjoint orbitG·λ. TheKirillov character formulaamounts to the character formula earlier proved byHarish-Chandra.
|
https://en.wikipedia.org/wiki/Kirillov_orbit_theory
|
Inmathematics, adiscrete series representationis an irreducibleunitary representationof alocally compact topological groupGthat is a subrepresentation of the leftregular representationofGon L²(G). In thePlancherel measure, such representations have positive measure. The name comes from the fact that they are exactly the representations that occur discretely in the decomposition of the regular representation.
IfGisunimodular, an irreducible unitary representation ρ ofGis in the discrete series if and only if one (and hence all)matrix coefficient
withv,wnon-zero vectors issquare-integrableonG, with respect toHaar measure.
WhenGis unimodular, the discrete series representation has a formal dimensiond, with the property that
forv,w,x,yin the representation. WhenGis compact this coincides with the dimension when the Haar measure onGis normalized so thatGhas measure 1.
Harish-Chandra(1965,1966) classified the discrete series representations of connectedsemisimple groupsG. In particular, such a group has discrete series representations if and only if it has the same rank as amaximal compact subgroupK. In other words, amaximal torusTinKmust be aCartan subgroupinG. (This result required that thecenterofGbe finite, ruling out groups such as the simply connected cover of SL(2,R).) It applies in particular tospecial linear groups; of these onlySL(2,R)has a discrete series (for this, see therepresentation theory of SL(2,R)).
Harish-Chandra's classification of the discrete series representations of a semisimple connected Lie group is given as follows. IfLis theweight latticeof the maximal torusT, a sublattice ofitwheretis the Lie algebra ofT, then there is a discrete series representation for every vectorvof
where ρ is theWeyl vectorofG, that is not orthogonal to any root ofG. Every discrete series representation occurs in this way. Two such vectorsvcorrespond to the same discrete series representation if and only if they are conjugate under theWeyl groupWKof the maximal compact subgroupK. If we fix afundamental chamberfor the Weyl group ofK, then the discrete series representation are in 1:1 correspondence with the vectors ofL+ ρ in this Weyl chamber that are not orthogonal to any root ofG. The infinitesimal character of the highest weight representation is given byv(mod the Weyl groupWGofG) under theHarish-Chandra correspondenceidentifying infinitesimal characters ofGwith points of
So for each discrete series representation, there are exactly
discrete series representations with the same infinitesimal character.
Harish-Chandra went on to prove an analogue for these representations of theWeyl character formula. In the case whereGis not compact, the representations have infinite dimension, and the notion ofcharacteris therefore more subtle to define since it is aSchwartz distribution(represented by a locally integrable function), with singularities.
The character is given on the maximal torusTby
WhenGis compact this reduces to the Weyl character formula, withv=λ+ρforλthe highest weight of the irreducible representation (where the product is over roots α having positive inner product with the vectorv).
Harish-Chandra's regularity theoremimplies that the character of a discrete series representation is a locally integrable function on the group.
Pointsvin the cosetL+ ρ orthogonal to roots ofGdo not correspond to discrete series representations, but those not orthogonal to roots ofKare related to certain irreducible representations calledlimit of discrete series representations. There is such a representation for every pair (v,C) wherevis a vector ofL+ ρ orthogonal to some root ofGbut not orthogonal to any root ofKcorresponding to a wall ofC, andCis a Weyl chamber ofGcontainingv. (In the case of discrete series representations there is only one Weyl chamber containingvso it is not necessary to include it explicitly.) Two pairs (v,C) give the same limit of discrete series representation if and only if they are conjugate under the Weyl group ofK. Just as for discrete series representationsvgives the infinitesimal character. There are at most |WG|/|WK| limit of discrete series representations with any given infinitesimal character.
Limit of discrete series representations aretempered representations, which means roughly that they only just fail to be discrete series representations.
Harish-Chandra's original construction of the discrete series was not very explicit. Several authors later found more explicit realizations of the discrete series.
|
https://en.wikipedia.org/wiki/Discrete_series_representation
|
Inmathematics, azonal spherical functionor often justspherical functionis a function on alocally compact groupGwith compact subgroupK(often amaximal compact subgroup) that arises as thematrix coefficientof aK-invariant vector in anirreducible representationofG. The key examples are the matrix coefficients of thespherical principal series, the irreducible representations appearing in the decomposition of theunitary representationofGonL2(G/K). In this case thecommutantofGis generated by the algebra of biinvariant functions onGwith respect toKacting by rightconvolution. It iscommutativeif in additionG/Kis asymmetric space, for example whenGis a connected semisimple Lie group with finite centre andKis a maximal compact subgroup. The matrix coefficients of the spherical principal series describe precisely thespectrumof the correspondingC* algebragenerated by the biinvariant functions ofcompact support, often called aHecke algebra. The spectrum of the commutative Banach *-algebra of biinvariantL1functions is larger; whenGis a semisimple Lie group with maximal compact subgroupK, additional characters come from matrix coefficients of thecomplementary series, obtained by analytic continuation of the spherical principal series.
Zonal spherical functions have been explicitly determined for real semisimple groups byHarish-Chandra. Forspecial linear groups, they were independently discovered byIsrael GelfandandMark Naimark. For complex groups, the theory simplifies significantly, becauseGis thecomplexificationofK, and the formulas are related to analytic continuations of theWeyl character formulaonK. The abstractfunctional analytictheory of zonal spherical functions was first developed byRoger Godement. Apart from their group theoretic interpretation, the zonal spherical functions for a semisimple Lie groupGalso provide a set of simultaneouseigenfunctionsfor the natural action of the centre of theuniversal enveloping algebraofGonL2(G/K), asdifferential operatorson the symmetric spaceG/K. For semisimplep-adicLie groups, the theory of zonal spherical functions and Hecke algebras was first developed by Satake andIan G. Macdonald. The analogues of thePlancherel theoremandFourier inversion formulain this setting generalise the eigenfunction expansions of Mehler, Weyl and Fock forsingular ordinary differential equations: they were obtained in full generality in the 1960s in terms ofHarish-Chandra's c-function.
The name "zonal spherical function" comes from the case whenGis SO(3,R) acting on a 2-sphere andKis the subgroup fixing a point: in this case the zonal spherical functions can be regarded as certain functions on the sphere invariant under rotation about a fixed axis.
LetGbe alocally compactunimodulartopological groupandKacompactsubgroupand letH1=L2(G/K). Thus,H1admits aunitary representationπ ofGby left translation. This is a subrepresentation of the regular representation, since ifH=L2(G) with left and rightregular representationsλ and ρ ofGandPis theorthogonal projection
fromHtoH1thenH1can naturally be identified withPHwith the action ofGgiven by the restriction of λ.
On the other hand, byvon Neumann's commutation theorem[1]
whereS'denotes thecommutantof a set of operatorsS, so that
Thus the commutant of π is generated as avon Neumann algebraby operators
wherefis a continuous function of compact support onG.[a]
HoweverPρ(f)Pis just the restriction of ρ(F) toH1, where
is theK-biinvariant continuous function of compact support obtained by averagingfbyKon both sides.
Thus the commutant of π is generated by the restriction of the operators ρ(F) withFinCc(K\G/K), theK-biinvariant continuous functions of compact support onG.
These functions form a* algebraunderconvolutionwith involution
often called theHecke algebrafor the pair (G,K).
LetA(K\G/K) denote theC* algebragenerated by the operators ρ(F) onH1.
The pair (G,K)
is said to be aGelfand pair[2]if one, and hence all, of the following algebras arecommutative:
SinceA(K\G/K) is a commutativeC* algebra, by theGelfand–Naimark theoremit has the formC0(X),
whereXis the locally compact space of norm continuous *homomorphismsofA(K\G/K) intoC.
A concrete realization of the * homomorphisms inXasK-biinvariantuniformly boundedfunctions onGis obtained as follows.[2][3][4][5][6]
Because of the estimate
the representation π ofCc(K\G/K) inA(K\G/K) extends by continuity
to L1(K\G/K), the* algebraofK-biinvariant integrable functions. The image forms
a dense * subalgebra ofA(K\G/K). The restriction of a * homomorphism χ continuous for the operator norm is
also continuous for the norm ||·||1. Since theBanach space dualof L1is L∞,
it follows that
for some unique uniformly boundedK-biinvariant functionhonG. These functionshare exactly thezonal spherical functionsfor the pair (G,K).
A zonal spherical functionhhas the following properties:[2]
These are easy consequences of the fact that the bounded linear functional χ defined byhis a homomorphism. Properties 2, 3 and 4 or properties 3, 4 and 5 characterize zonal spherical functions. A more general class of zonal spherical functions can be obtained by dropping positive definiteness from the conditions, but for these functions there is no longer any connection
withunitary representations. For semisimple Lie groups, there is a further characterization as eigenfunctions ofinvariant differential operatorsonG/K(see below).
In fact, as a special case of theGelfand–Naimark–Segal construction, there is one-one correspondence between
irreducible representations σ ofGhaving a unit vectorvfixed byKand zonal spherical functionshgiven by
Such irreducible representations are often described as havingclass one. They are precisely the irreducible representations required to decompose theinduced representationπ onH1. Each representation σ extends uniquely by continuity
toA(K\G/K), so that each zonal spherical function satisfies
forfinA(K\G/K). Moreover, since the commutant π(G)' is commutative,
there is a unique probability measure μ on the space of * homomorphismsXsuch that
μ is called thePlancherel measure. Since π(G)' is thecentreof the von Neumann algebra generated byG, it also gives the measure associated with thedirect integraldecomposition ofH1in terms of the irreducible representations σχ.
IfGis aconnectedLie group, then, thanks to the work ofCartan,Malcev,IwasawaandChevalley,Ghas amaximal compact subgroup, unique up to conjugation.[7][8]In this caseKis connected and the quotientG/Kis diffeomorphic to a Euclidean space. WhenGis in additionsemisimple, this can be seen directly using theCartan decompositionassociated to thesymmetric spaceG/K, a generalisation of thepolar decompositionof invertible matrices. Indeed, if τ is the associated period two automorphism ofGwith fixed point subgroupK, then
where
Under theexponential map,Pis diffeomorphic to the -1 eigenspace of τ in theLie algebraofG.
Since τ preservesK, it induces an automorphism of the Hecke algebraCc(K\G/K). On the
other hand, ifFlies inCc(K\G/K), then
so that τ induces an anti-automorphism, because inversion does. Hence, whenGis semisimple,
More generally the same argument gives the following criterion of Gelfand for (G,K) to be a Gelfand pair:[9]
The two most important examples covered by this are when:
The three cases cover the three types ofsymmetric spacesG/K:[5]
LetGbe a compact semisimple connected and simply connected Lie group and τ a period two automorphism of aGwith fixed point subgroupK=Gτ. In this caseKis a connected compact Lie group.[5]In addition letTbe amaximal torusofGinvariant under τ, such thatT∩{\displaystyle \cap }Pis a maximal torus inP, and set[12]
Sis the direct product of a torus and anelementary abelian 2-group.
In 1929Élie Cartanfound a rule to determine the decomposition of L2(G/K) into the direct sum of finite-dimensionalirreducible representationsofG, which was proved rigorously only in 1970 bySigurdur Helgason. Because the commutant ofGon L2(G/K) is commutative, each irreducible representation appears with multiplicity one. ByFrobenius reciprocityfor compact groups, the irreducible representationsVthat occur are precisely those admitting a non-zero vector fixed byK.
From therepresentation theory of compact semisimple groups, irreducible representations ofGare classified by theirhighest weight. This is specified by a homomorphism of the maximal torusTintoT.
TheCartan–Helgason theorem[13][14]states that
The corresponding irreducible representations are calledspherical representations.
The theorem can be proved[5]using theIwasawa decomposition:
whereg{\displaystyle {\mathfrak {g}}},k{\displaystyle {\mathfrak {k}}},a{\displaystyle {\mathfrak {a}}}are the complexifications of theLie algebrasofG,K,A=T∩{\displaystyle \cap }Pand
summed over all eigenspaces forTing{\displaystyle {\mathfrak {g}}}corresponding topositive rootsα not fixed by τ.
LetVbe a spherical representation with highest weight vectorv0andK-fixed vectorvK. Sincev0is an eigenvector of the solvable Lie algebraa⊕n{\displaystyle {\mathfrak {a}}\oplus {\mathfrak {n}}}, thePoincaré–Birkhoff–Witt theoremimplies that theK-module generated byv0is the whole ofV. IfQis the orthogonal projection onto the fixed points ofKinVobtained by averaging overGwith respect toHaar measure, it follows that
for some non-zero constantc. BecausevKis fixed bySandv0is an eigenvector forS, the subgroupSmust actually fixv0, an equivalent form of the triviality condition onS.
Conversely ifv0is fixed byS, then it can be shown[15]that the matrix coefficient
is non-negative onK. Sincef(1) > 0, it follows that (Qv0,v0) > 0 and hence thatQv0is a non-zero vector fixed byK.
IfGis a non-compact semisimple Lie group, its maximal compact subgroupKacts by conjugation on the componentPin theCartan decomposition. IfAis a maximal Abelian subgroup ofGcontained inP, thenAis diffeomorphic to its Lie algebra under theexponential mapand, as afurther generalisationof thepolar decompositionof matrices, every element ofPis conjugate underKto an element ofA, so that[16]
There is also an associatedIwasawa decomposition
whereNis a closed nilpotent subgroup, diffeomorphic to its Lie algebra under the exponential map and normalised byA. ThusS=ANis a closedsolvable subgroupofG, thesemidirect productofNbyA, andG=KS.
If α in Hom(A,T) is acharacterofA, then α extends to a character ofS, by defining it to be trivial onN. There is a correspondingunitaryinduced representationσ ofGon L2(G/S) = L2(K),[17]a so-called(spherical) principal series representation.
This representation can be described explicitly as follows. UnlikeGandK, the solvable Lie groupSis not unimodular. Letdxdenote left invariant Haar measure onSand ΔSthemodular functionofS. Then[5]
The principal series representation σ is realised on L2(K) as[18]
where
is the Iwasawa decomposition ofgwithU(g) inKandX(g) inSand
forkinKandxinS.
The representation σ is irreducible, so that ifvdenotes the constant function 1 onK, fixed byK,
defines a zonal spherical function ofG.
Computing the inner product above leads toHarish-Chandra's formulafor the zonal spherical function
as an integral overK.
Harish-Chandra proved that these zonal spherical functions exhaust the characters of theC* algebragenerated by theCc(K\G/K) acting by right convolution onL2(G/K). He also showed that two different characters α and β give the same zonal spherical function if and only if α = β·s, wheresis in theWeyl groupofA
the quotient of thenormaliserofAinKby itscentraliser, afinite reflection group.
It can also be verified directly[2]that this formula defines a zonal spherical function, without using representation theory. The proof for general semisimple Lie groups that every zonal spherical formula arises in this way requires the detailed study ofG-invariant differential operatorsonG/Kand their simultaneouseigenfunctions(see below).[4][5]In the case of complex semisimple groups, Harish-Chandra andFelix Berezinrealised independently that the formula simplified considerably and could be proved more directly.[5][19][20][21][22]
The remaining positive-definite zonal spherical functions are given
by Harish-Chandra's formula with α in Hom(A,C*) instead of Hom(A,T). Only certain α are permitted and the corresponding irreducible
representations arise as analytic continuations of the spherical principal series. This so-called "complementary series" was first studied byBargmann (1947)forG= SL(2,R) and byHarish-Chandra (1947)andGelfand & Naimark (1947)forG= SL(2,C).
Subsequently in the 1960s, the construction of acomplementary seriesby analytic continuation of the spherical principal series was systematically developed for general semisimple Lie groups by Ray Kunze,Elias SteinandBertram Kostant.[23][24][25]Since these irreducible representations are nottempered, they are not usually required for harmonic analysis onG(orG/K).
Harish-Chandra proved[4][5]that zonal spherical functions can be characterised as those normalised positive definiteK-invariant functions onG/Kthat are eigenfunctions ofD(G/K), the algebra of invariant differential operators onG. This algebra acts onG/Kand commutes with the natural action ofGby left translation. It can be identified with the subalgebra of theuniversal enveloping algebraofGfixed under theadjoint actionofK. As for the commutant ofGon L2(G/K) and the corresponding Hecke algebra, this algebra of operators iscommutative; indeed it is a subalgebra of thealgebra of mesurable operatorsaffiliated with the commutant π(G)', an Abelian von Neumann algebra. As Harish-Chandra proved, it is isomorphic to the algebra ofW(A)-invariant polynomials on the Lie algebra ofA, which itself is apolynomial ringby theChevalley–Shephard–Todd theoremon polynomial invariants offinite reflection groups. The simplest invariant differential operator onG/Kis theLaplacian operator; up to a sign this operator is just the image under π of theCasimir operatorin the centre of the universal enveloping algebra ofG.
Thus a normalised positive definiteK-biinvariant functionfonGis a zonal spherical function if and only if for eachDinD(G/K) there is a constant λDsuch that
i.e.fis a simultaneouseigenfunctionof the operators π(D).
If ψ is a zonal spherical function, then, regarded as a function onG/K, it is an eigenfunction of the Laplacian
there, anelliptic differential operatorwithreal analyticcoefficients. Byanalytic elliptic regularity,
ψ is a real analytic function onG/K, and henceG.
Harish-Chandra used these facts about the structure of the invariant operators to prove that his formula gave all zonal spherical functions for real semisimple Lie groups.[26][27][28]Indeed, the commutativity of the commutant implies that the simultaneous eigenspaces of the algebra of invariant differential operators all have dimension one; and the polynomial structure of this algebra forces the simultaneous eigenvalues to be precisely those already associated with Harish-Chandra's formula.
The groupG= SL(2,C) is thecomplexificationof thecompact Lie groupK= SU(2) and thedouble coverof theLorentz group. The infinite-dimensional representations of the Lorentz group were first studied byDiracin 1945, who considered thediscrete seriesrepresentations, which he termedexpansors. A systematic study was taken up shortly afterwards by Harish-Chandra, Gelfand–Naimark and
Bargmann. The irreducible representations of class one, corresponding to the zonal spherical functions, can be determined easily using the radial
component of theLaplacian operator.[5]
Indeed, any unimodular complex 2×2 matrixgadmits a uniquepolar decompositiong=pvwithvunitary andppositive. In turnp=uau*, withuunitary andaa diagonal matrix with positive entries. Thusg=uawwithw=u*v, so that anyK-biinvariant function onGcorresponds to a function of the diagonal matrix
invariant under the Weyl group. IdentifyingG/Kwith hyperbolic 3-space, the zonal hyperbolic functions ψ correspond to radial functions that are eigenfunctions of the Laplacian. But in terms of the radial coordinater, the Laplacian is given by[29]
Settingf(r) = sinh (r)·ψ(r), it follows thatfis anodd functionofrand an eigenfunction of∂r2{\displaystyle \partial _{r}^{2}}.
Hence
whereℓ{\displaystyle \ell }is real.
There is a similar elementary treatment for thegeneralized Lorentz groupsSO(N,1) inTakahashi (1963)andFaraut & Korányi (1994)(recall that SO0(3,1) = SL(2,C) / ±I).
IfGis a complex semisimple Lie group, it is thecomplexificationof its maximal compact subgroupK. Ifg{\displaystyle {\mathfrak {g}}}andk{\displaystyle {\mathfrak {k}}}are their Lie algebras, then
LetTbe amaximal torusinKwith Lie algebrat{\displaystyle {\mathfrak {t}}}. Then
Let
be theWeyl groupofTinK. Recall characters in Hom(T,T) are calledweightsand can be identified with elements of theweight latticeΛ in
Hom(t{\displaystyle {\mathfrak {t}}},R) =t∗{\displaystyle {\mathfrak {t}}^{*}}. There is a natural ordering on weights and every finite-dimensional irreducible representation (π,V) ofKhas a unique highest weight λ. The weights of theadjoint representationofKonk⊖t{\displaystyle {\mathfrak {k}}\ominus {\mathfrak {t}}}are called roots and ρ is used to denote half the sum of thepositive rootsα,Weyl's character formulaasserts that forz= expXinT
where, for μ int∗{\displaystyle {\mathfrak {t}}^{*}},Aμdenotes the antisymmetrisation
and ε denotes thesign characterof thefinite reflection groupW.
Weyl's denominator formulaexpresses the denominatorAρas a product:
where the product is over the positive roots.
Weyl's dimension formulaasserts that
where theinner productont∗{\displaystyle {\mathfrak {t}}^{*}}is that associated with theKilling formonk{\displaystyle {\mathfrak {k}}}.
Now
TheBerezin–Harish–Chandra formula[5]asserts that forXinit{\displaystyle i{\mathfrak {t}}}
In other words:
One of the simplest proofs[30]of this formula involves theradial componentonAof the Laplacian onG, a proof formally parallel to Helgason's reworking ofFreudenthal's classical proof of theWeyl character formula, using the radial component onTof the Laplacian onK.[31]
In the latter case theclass functionsonKcan be identified withW-invariant functions onT. The
radial component of ΔKonTis just the expression for the restriction of ΔKtoW-invariant functions onT, where
it is given by the formula
where
forXint{\displaystyle {\mathfrak {t}}}. If χ is a character with highest weight λ, it follows that φ =h·χ satisfies
Thus for every weight μ with non-zeroFourier coefficientin φ,
The classical argument of Freudenthal shows that μ + ρ must have the forms(λ + ρ) for somesinW, so the character formula
follows from the antisymmetry of φ.
SimilarlyK-biinvariant functions onGcan be identified withW(A)-invariant functions onA. The
radial component of ΔGonAis just the expression for the restriction of ΔGtoW(A)-invariant functions onA.
It is given by the formula
where
forXinit{\displaystyle i{\mathfrak {t}}}.
The Berezin–Harish–Chandra formula for a zonal spherical function φ can be established by introducing the antisymmetric function
which is an eigenfunction of the Laplacian ΔA. SinceKis generated by copies of subgroups that are homomorphic images of SU(2) corresponding tosimple roots, its complexificationGis generated by the corresponding homomorphic images of SL(2,C). The formula for zonal spherical functions of SL(2,C) implies thatfis aperiodic functiononit{\displaystyle i{\mathfrak {t}}}with respect to somesublattice. Antisymmetry under the Weyl group and the argument of Freudenthal again imply that ψ must have the stated form up to a multiplicative constant, which can be determined using the Weyl dimension formula.
The theory of zonal spherical functions forSL(2,R)originated in the work ofMehlerin 1881 on hyperbolic geometry. He discovered the analogue of the Plancherel theorem, which was rediscovered by Fock in 1943. The corresponding eigenfunction expansion is termed theMehler–Fock transform. It was already put on a firm footing in 1910 byHermann Weyl's important work on thespectral theory of ordinary differential equations. The radial part of the Laplacian in this case leads to ahypergeometric differential equation, the theory of which was treated in detail by Weyl. Weyl's approach was subsequently generalised by Harish-Chandra to study zonal spherical functions and the corresponding Plancherel theorem for more general semisimple Lie groups. Following the work of Dirac on the discrete series representations of SL(2,R), the general theory of unitary irreducible representations of SL(2,R) was developed independently by Bargmann, Harish-Chandra and Gelfand–Naimark. The irreducible representations of class one, or equivalently the theory of zonal spherical functions, form an important special case of this theory.
The groupG=SL(2,R)is adouble coverof the 3-dimensionalLorentz groupSO(2,1), thesymmetry groupof thehyperbolic planewith itsPoincaré metric. It acts byMöbius transformations. The upper half-plane can be identified with the unit disc by theCayley transform. Under this identificationGbecomes identified with the groupSU(1,1), also acting by Möbius transformations. Because the action istransitive, both spaces can be identified withG/K, whereK=SO(2). The metric is invariant underGand the associated Laplacian isG-invariant, coinciding with the image of theCasimir operator. In the upper half-plane model the Laplacian is given by the formula[5][6]
Ifsis a complex number andz=x + i ywithy> 0, the function
is an eigenfunction of Δ:
Since Δ commutes withG, any left translate offsis also an eigenfunction with the same eigenvalue. In particular, averaging overK, the function
is aK-invariant eigenfunction of Δ onG/K. When
with τ real, these functions give all the zonal spherical functions onG. As with Harish-Chandra's more general formula for semisimple Lie groups, φsis a zonal spherical function because it is the matrix coefficient corresponding to a vector fixed byKin theprincipal series. Various arguments are available to prove that there are no others. One of the simplest classicalLie algebraicarguments[5][6][32][33][34]is to note that, since Δ is an elliptic operator with analytic coefficients, by analytic elliptic regularity any eigenfunction is necessarily real analytic. Hence, if the zonal spherical function corresponds to the matrix coefficient for a vectorvand representation σ, the vectorvis ananalytic vectorforGand
forXinit{\displaystyle i{\mathfrak {t}}}. The infinitesimal form of the irreducible unitary representations with a vector fixed byKwere worked out classically by Bargmann.[32][33]They correspond precisely to the principal series of SL(2,R). It follows that the zonal spherical function corresponds to a principal series representation.
Another classical argument[35]proceeds by showing that on radial functions the Laplacian has the form
so that, as a function ofr, the zonal spherical function φ(r) must satisfy theordinary differential equation
for some constant α. The change of variablest= sinhrtransforms this equation into thehypergeometric differential equation. The general solution in terms ofLegendre functionsof complex index is given by[2][36]
where α = ρ(ρ+1). Further restrictions on ρ are imposed by boundedness and positive-definiteness of the zonal spherical function onG.
There is yet another approach, due to Mogens Flensted-Jensen, which derives the properties of the zonal spherical functions on SL(2,R), including the Plancherel formula, from the corresponding results for SL(2,C), which are simple consequences of the Plancherel formula and Fourier inversion formula forR. This "method of descent" works more generally, allowing results for a real semisimple Lie group to be derived by descent from the corresponding results for its complexification.[37][38]
|
https://en.wikipedia.org/wiki/Zonal_spherical_function
|
Inmathematics, specifically in therepresentation theoryofgroupsandalgebras, anirreducible representation(ρ,V){\displaystyle (\rho ,V)}orirrepof an algebraic structureA{\displaystyle A}is a nonzero representation that has no proper nontrivial subrepresentation(ρ|W,W){\displaystyle (\rho |_{W},W)}, withW⊂V{\displaystyle W\subset V}closed under theactionof{ρ(a):a∈A}{\displaystyle \{\rho (a):a\in A\}}.
Every finite-dimensionalunitary representationon aHilbert spaceV{\displaystyle V}is thedirect sumof irreducible representations. Irreducible representations are alwaysindecomposable(i.e. cannot be decomposed further into a direct sum of representations), but the converse may not hold, e.g. the two-dimensional representation of the real numbers acting by upper triangularunipotentmatrices is indecomposable but reducible.
Group representation theory was generalized byRichard Brauerfrom the 1940s to givemodular representation theory, in which the matrix operators act on a vector space over afieldK{\displaystyle K}of arbitrarycharacteristic, rather than a vector space over the field ofreal numbersor over the field ofcomplex numbers. The structure analogous to an irreducible representation in the resulting theory is asimple module.[citation needed]
Letρ{\displaystyle \rho }be a representation i.e. ahomomorphismρ:G→GL(V){\displaystyle \rho :G\to GL(V)}of a groupG{\displaystyle G}whereV{\displaystyle V}is avector spaceover afieldF{\displaystyle F}. If we pick a basisB{\displaystyle B}forV{\displaystyle V},ρ{\displaystyle \rho }can be thought of as a function (a homomorphism) from a group into a set of invertible matrices and in this context is called amatrix representation. However, it simplifies things greatly if we think of the spaceV{\displaystyle V}without a basis.ρ{\displaystyle \rho }isd-dimensionalif the vector spaceV{\displaystyle V}it acts over has dimensiond{\displaystyle d}.
Alinear subspaceW⊂V{\displaystyle W\subset V}is calledG{\displaystyle G}-invariantifρ(g)w∈W{\displaystyle \rho (g)w\in W}for allg∈G{\displaystyle g\in G}and allw∈W{\displaystyle w\in W}. The co-restriction ofρ{\displaystyle \rho }to the general linear group of aG{\displaystyle G}-invariant subspaceW⊂V{\displaystyle W\subset V}is known as asubrepresentation. A representationρ:G→GL(V){\displaystyle \rho :G\to GL(V)}is said to beirreducibleif it has onlytrivialsubrepresentations (all representations can form a subrepresentation with the trivialG{\displaystyle G}-invariant subspaces, e.g. the whole vector spaceV{\displaystyle V}, and{0}). If there is a proper nontrivial invariant subspace,ρ{\displaystyle \rho }is said to bereducible.
Group elements can be represented bymatrices, although the term "represented" has a specific and precise meaning in this context. A representation of a group is a mapping from the group elements to thegeneral linear groupof matrices. As notation, leta,b,c, ...denote elements of a groupGwith group product signified without any symbol, soabis the group product ofaandband is also an element ofG, and let representations be indicated byD. Therepresentation ofais written as
By definition of group representations, the representation of a group product is translated intomatrix multiplicationof the representations:
Ifeis theidentity elementof the group (so thatae=ea=a, etc.), thenD(e)is anidentity matrix, or identically a block matrix of identity matrices, since we must have
and similarly for all other group elements. The last two statements correspond to the requirement thatDis agroup homomorphism.
A representation is reducible if it contains a nontrivial G-invariant subspace, that is to say, all the matricesD(a){\displaystyle D(a)}can be put in upper triangular block form by the same invertible matrixP{\displaystyle P}. In other words, if there is a similarity transformation:
which maps every matrix in the representation into the same pattern upper triangular blocks. Every ordered sequence minor block is a group subrepresentation. That is to say, if the representation is, for example, of dimension 2, then we have:D′(a)=P−1D(a)P=(D(11)(a)D(12)(a)0D(22)(a)),{\displaystyle D'(a)=P^{-1}D(a)P={\begin{pmatrix}D^{(11)}(a)&D^{(12)}(a)\\0&D^{(22)}(a)\end{pmatrix}},}
whereD(11)(a){\displaystyle D^{(11)}(a)}is a nontrivial subrepresentation. If we are able to find a matrixP{\displaystyle P}that makesD(12)(a)=0{\displaystyle D^{(12)}(a)=0}as well, thenD(a){\displaystyle D(a)}is not only reducible but also decomposable.
Notice:Even if a representation is reducible, its matrix representation may still not be the upper triangular block form. It will only have this form if we choose a suitable basis, which can be obtained by applying the matrixP−1{\displaystyle P^{-1}}above to the standard basis.
A representation is decomposable if all the matricesD(a){\displaystyle D(a)}can be put in block-diagonal form by the same invertible matrixP{\displaystyle P}. In other words, if there is asimilarity transformation:[1]
whichdiagonalizesevery matrix in the representation into the same pattern ofdiagonalblocks. Each such block is then a group subrepresentation independent from the others. The representationsD(a)andD′(a)are said to beequivalent representations.[2]The (k-dimensional, say) representation can be decomposed into adirect sum ofk> 1matrices:
soD(a)isdecomposable, and it is customary to label the decomposed matrices by a superscript in brackets, as inD(n)(a)forn= 1, 2, ...,k, although some authors just write the numerical label without parentheses.
The dimension ofD(a)is the sum of the dimensions of the blocks:
If this is not possible, i.e.k= 1, then the representation is indecomposable.[1][3]
Notice: Even if a representation is decomposable, its matrix representation may not be the diagonal block form. It will only have this form if we choose a suitable basis, which can be obtained by applying the matrixP−1{\displaystyle P^{-1}}above to the standard basis.
An irreducible representation is by nature an indecomposable one. However, the converse may fail.
But under some conditions, we do have an indecomposable representation being an irreducible representation.
All groupsG{\displaystyle G}have a zero-dimensional, irreducible trivial representation by mapping all group elements to the identity transformation.
Any one-dimensional representation is irreducible since it has no proper nontrivial invariant subspaces.
The irreducible complex representations of a finite group G can be characterized using results fromcharacter theory. In particular, all complex representations decompose as a direct sum of irreps, and the number of irreps ofG{\displaystyle G}is equal to the number of conjugacy classes ofG{\displaystyle G}.[5]
Inquantum physicsandquantum chemistry, each set ofdegenerate eigenstatesof theHamiltonian operatorcomprises a vector spaceVfor a representation of the symmetry group of the Hamiltonian, a "multiplet", best studied through reduction to its irreducible parts. Identifying the irreducible representations therefore allows one to label the states, predict how they willsplitunder perturbations; or transition to other states inV. Thus, in quantum mechanics, irreducible representations of the symmetry group of the system partially or completely label the energy levels of the system, allowing theselection rulesto be determined.[6]
The irreps ofD(K)andD(J), whereJis the generator of rotations andKthe generator of boosts, can be used to build to spin representations of the Lorentz group, because they are related to the spin matrices of quantum mechanics. This allows them to deriverelativistic wave equations.[7]
|
https://en.wikipedia.org/wiki/Irreducible_representations
|
Ingroup theory, a branch ofabstract algebra, acharacter tableis a two-dimensional table whose rows correspond toirreducible representations, and whose columns correspond toconjugacy classesofgroupelements. The entries consist ofcharacters, thetracesof thematricesrepresenting group elements of the column's class in the given row's group representation. Inchemistry,crystallography, andspectroscopy,character tables of point groupsare used to classifye.g.molecularvibrations according to their symmetry, and to predict whether a transition between two states is forbidden for symmetry reasons. Many university level textbooks onphysical chemistry,quantum chemistry,spectroscopyandinorganic chemistrydevote a chapter to the use of symmetry group character tables.[1][2][3][4][5][6]
The irreduciblecomplexcharacters of afinite groupform acharacter tablewhich encodes much useful information about thegroupGin a concise form. Each row is labelled by anirreducible characterand the entries in the row are the values of that character on any representative of the respectiveconjugacy classofG(because characters areclass functions). The columns are labelled by (representatives of) the conjugacy classes ofG. It is customary to label the first row by the character of thetrivial representation, which is the trivial action ofGon a1-dimensionalvector spacebyρ(g)=1{\displaystyle \rho (g)=1}for allg∈G{\displaystyle g\in G}. Each entry in the first row is therefore 1. Similarly, it is customary to label the first column by theidentity. The entries of the first column are the values of the irreducible characters at the identity, thedegreesof the irreducible characters. Characters of degree 1 are known aslinear characters.
Here is the character table ofC3=<u>, thecyclic groupwith three elements andgeneratoru:
where ω is a primitive cuberoot of unity. The character table for general cyclic groups is (a scalar multiple of) theDFT matrix.
Another example is the character table ofS3{\displaystyle S_{3}}:
where (12) represents the conjugacy class consisting of (12), (13), (23), while (123) represents the conjugacy class consisting of (123), (132). To learn more about character table of symmetric groups, see[2].
The first row of the character table always consists of 1s, and corresponds to thetrivial representation(the 1-dimensional representation consisting of 1×1 matrices containing the entry 1). Further, the character table is always square because (1) irreducible characters are pairwise orthogonal, and (2) no other non-trivial class function is orthogonal to every character. (A class function is one that is constant on conjugacy classes.) This is tied to the important fact that the irreducible representations of a finite groupGare inbijectionwith its conjugacy classes. This bijection also follows by showing that the class sums form abasisfor thecenterof thegroup algebraofG, which has dimension equal to the number of irreducible representations ofG.
The space of complex-valued class functions of a finite groupGhas a naturalinner product:
whereβ(g)¯{\displaystyle {\overline {\beta (g)}}}denotes thecomplex conjugateof the value ofβ{\displaystyle \beta }ong{\displaystyle g}. With respect to this inner product, the irreducible characters form anorthonormal basisfor the space of class functions, and this yields the orthogonality relation for the rows of the character
table:
Forg,h∈G{\displaystyle g,h\in G}the orthogonality relation for columns is as follows:
where the sum is over all of the irreducible charactersχi{\displaystyle \chi _{i}}ofGand the symbol|CG(g)|{\displaystyle \left|C_{G}(g)\right|}denotes theorderof thecentralizerofg{\displaystyle g}.
For an arbitrary characterχi{\displaystyle \chi _{i}}, it is irreducibleif and only if⟨χi,χi⟩=1{\displaystyle \left\langle \chi _{i},\chi _{i}\right\rangle =1}.
The orthogonality relations can aid many computations including:
If the irreducible representationVis non-trivial, then∑gχ(g)=0.{\displaystyle \sum _{g}\chi (g)=0.}
More specifically, consider theregular representationwhich is the permutation obtained from a finite groupGacting on (thefree vector spacespanned by) itself. The characters of this representation areχ(e)=|G|{\displaystyle \chi (e)=\left|G\right|}andχ(g)=0{\displaystyle \chi (g)=0}forg{\displaystyle g}not the identity. Then given an irreducible representationVi{\displaystyle V_{i}},
Then decomposing the regular representations as a sum of irreducible representations ofG, we getVreg=⨁VidimVi{\displaystyle V_{\text{reg}}=\bigoplus V_{i}^{\operatorname {dim} V_{i}}}, from which we conclude
over all irreducible representationsVi{\displaystyle V_{i}}. This sum can help narrow down the dimensions of the irreducible representations in a character table. For example, if the group has order 10 and 4 conjugacy classes (for instance, thedihedral groupof order 10) then the only way to express the order of the group as a sum of four squares is10=12+12+22+22{\displaystyle 10=1^{2}+1^{2}+2^{2}+2^{2}}, so we know the dimensions of all the irreducible representations.
Complex conjugation acts on the character table: since the complex conjugate of a representation is again a representation, the same is true for characters, and thus a character that takes on non-realcomplex values has a conjugate character.
Certain properties of the groupGcan be deduced from its character table:
The character table does not in general determine the groupup toisomorphism: for example, thequaternion groupand thedihedral groupof order 8 have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism. In 1964, this was answered in the negative byE. C. Dade.
The linear representations ofGare themselves a group under thetensor product, since the tensor product of1-dimensionalvector spaces is again1-dimensional. That is, ifρ1:G→V1{\displaystyle \rho _{1}:G\to V_{1}}andρ2:G→V2{\displaystyle \rho _{2}:G\to V_{2}}are linear representations, thenρ1⊗ρ2(g)=(ρ1(g)⊗ρ2(g)){\displaystyle \rho _{1}\otimes \rho _{2}(g)=(\rho _{1}(g)\otimes \rho _{2}(g))}defines a new linear representation. This gives rise to a group of linear characters, called thecharacter groupunder the operation[χ1∗χ2](g)=χ1(g)χ2(g){\displaystyle [\chi _{1}*\chi _{2}](g)=\chi _{1}(g)\chi _{2}(g)}. This group is connected toDirichlet charactersandFourier analysis.
Theouter automorphismgroup acts on the character table by permuting columns (conjugacy classes) and accordingly rows, which gives another symmetry to the table. For example, abelian groups have the outer automorphismg↦g−1{\displaystyle g\mapsto g^{-1}}, which is non-trivial except forelementary abelian 2-groups, and outer because abelian groups are precisely those for which conjugation (inner automorphisms) acts trivially. In the example ofC3{\displaystyle C_{3}}above, this map sendsu↦u2,u2↦u,{\displaystyle u\mapsto u^{2},u^{2}\mapsto u,}and accordingly switchesχ1{\displaystyle \chi _{1}}andχ2{\displaystyle \chi _{2}}(switching their values ofω{\displaystyle \omega }andω2{\displaystyle \omega ^{2}}). Note that this particularautomorphism(negative in abelian groups) agrees with complex conjugation.
Formally, ifϕ:G→G{\displaystyle \phi \colon G\to G}is an automorphism ofGandρ:G→GL{\displaystyle \rho \colon G\to \operatorname {GL} }is a representation, thenρϕ:=g↦ρ(ϕ(g)){\displaystyle \rho ^{\phi }:=g\mapsto \rho (\phi (g))}is a representation. Ifϕ=ϕa{\displaystyle \phi =\phi _{a}}is aninner automorphism(conjugation by some elementa), then it acts trivially on representations, because representations are class functions (conjugation does not change their value). Thus a given class of outer automorphisms, it acts on the characters – because inner automorphisms act trivially, the action of the automorphism groupAut{\displaystyle \mathrm {Aut} }descends to thequotientOut{\displaystyle \mathrm {Out} }.
This relation can be used both ways: given an outer automorphism, one can produce new representations (if the representation is not equal on conjugacy classes that are interchanged by the outer automorphism), and conversely, one can restrict possible outer automorphisms based on the character table.
To find the total number of vibrational modes of a water molecule, the irreducible representation Γirreducibleneeds to calculate from the character table of a water molecule first.
Water (H2O{\displaystyle {\ce {H2O}}}) molecule falls under the point groupC2v{\displaystyle C_{2v}}.[7]Below is the character table ofC2v{\displaystyle C_{2v}}point group, which is also the character table for a water molecule.
In here, the first row describes the possible symmetry operations of this point group and the first column represents the Mulliken symbols. The fifth and sixth columns are functions of the axis variables.
Functions:
When determining the characters for a representation, assign1{\displaystyle 1}if it remains unchanged,0{\displaystyle 0}if it moved, and−1{\displaystyle -1}if it reversed its direction. A simple way to determine the characters for the reducible representationΓreducible{\displaystyle \Gamma _{\text{reducible}}}, is to multiply the "number of unshifted atom(s)" with "contribution per atom" along each of three axis (x,y,z{\displaystyle x,y,z}) when a symmetry operation is carried out.
Unless otherwise stated, for the identity operationE{\displaystyle E}, "contribution per unshifted atom" for each atom is always3{\displaystyle 3}, as none of the atom(s) change their position during this operation. For any reflective symmetry operationσ{\displaystyle \sigma }, "contribution per atom" is always1{\displaystyle 1}, as for any reflection, an atom remains unchanged along with two axis and reverse its direction along with the other axis. For the inverse symmetry operationi{\displaystyle i}, "contribution per unshifted atom" is always−3{\displaystyle -3}, as each of three axis of an atom reverse its direction during this operation. An easiest way to calculate "contribution per unshifted atom" forCn{\displaystyle C_{n}}andSn{\displaystyle S_{n}}symmetry operation is to use below formulas[8]
where,θ=360n{\displaystyle \theta ={\frac {360}{n}}}
A simplified version of above statements is summarized in the table below
per unshifted atom
Character ofΓreducible{\displaystyle \Gamma _{\text{reducible}}}for any symmetry operation={\displaystyle =}Number of unshifted atom(s) during this operation×{\displaystyle \times }Contribution per unshifted atom along each of three axis
From the above discussion, a new character table for a water molecule (C2v{\displaystyle C_{2v}}point group) can be written as
Using the new character table includingΓred{\displaystyle \Gamma _{\text{red}}}, the reducible representation for all motion of theH2O{\displaystyle {\ce {H2O}}}molecule can be reduced using below formula
where,
So,
NA1=14[(9×1×1)+((−1)×1×1)+(3×1×1)+(1×1×1)]=3{\displaystyle N_{A_{1}}={\frac {1}{4}}[(9\times 1\times 1)+((-1)\times 1\times 1)+(3\times 1\times 1)+(1\times 1\times 1)]=3}
NA2=14[(9×1×1+((−1)×1×1)+(3×(−1)×1)+(1×(−1)×1)]=1{\displaystyle N_{A_{2}}={\frac {1}{4}}[(9\times 1\times 1+((-1)\times 1\times 1)+(3\times (-1)\times 1)+(1\times (-1)\times 1)]=1}
NB1=14[(9×1×1)+((−1)×(−1)×1)+(3×1×1)+(1×(−1)×1)]=3{\displaystyle N_{B_{1}}={\frac {1}{4}}[(9\times 1\times 1)+((-1)\times (-1)\times 1)+(3\times 1\times 1)+(1\times (-1)\times 1)]=3}
NB2=14[(9×1×1)+((−1)×(−1)×1)+(3×(−1)×1)+(1×1×1)]=2{\displaystyle N_{B_{2}}={\frac {1}{4}}[(9\times 1\times 1)+((-1)\times (-1)\times 1)+(3\times (-1)\times 1)+(1\times 1\times 1)]=2}
So, the reduced representation for all motions of water molecule will be
Γirreducible=3A1+A2+3B1+2B2{\displaystyle \Gamma _{\text{irreducible}}=3A_{1}+A_{2}+3B_{1}+2B_{2}}
Translational motion will corresponds with the reducible representations in the character table, which havex{\displaystyle x},y{\displaystyle y}andz{\displaystyle z}function
As only the reducible representationsB1{\displaystyle B_{1}},B2{\displaystyle B_{2}}andA1{\displaystyle A_{1}}correspond to thex{\displaystyle x},y{\displaystyle y}andz{\displaystyle z}function,
Γtranslational=A1+B1+B2{\displaystyle \Gamma _{\text{translational}}=A_{1}+B_{1}+B_{2}}
Rotational motion will corresponds with the reducible representations in the character table, which haveRx{\displaystyle R_{x}},Ry{\displaystyle R_{y}}andRz{\displaystyle R_{z}}function
As only the reducible representationsB2{\displaystyle B_{2}},B1{\displaystyle B_{1}}andA2{\displaystyle A_{2}}correspond to thex{\displaystyle x},y{\displaystyle y}andz{\displaystyle z}function,
Γrotational=A2+B1+B2{\displaystyle \Gamma _{\text{rotational}}=A_{2}+B_{1}+B_{2}}
Total vibrational mode,Γvibrational=Γirreducible−Γtranslational−Γrotational{\displaystyle \Gamma _{\text{vibrational}}=\Gamma _{\text{irreducible}}-\Gamma _{\text{translational}}-\Gamma _{\text{rotational}}}
=(3A1+A2+3B1+2B2)−(A1+B1+B2)−(A2+B1+B2){\displaystyle =(3A_{1}+A_{2}+3B_{1}+2B_{2})-(A_{1}+B_{1}+B_{2})-(A_{2}+B_{1}+B_{2})}
=2A1+B1{\displaystyle =2A_{1}+B_{1}}
So, total2+1=3{\displaystyle 2+1=3}vibrational modes are possible for water molecules and two of them are symmetric vibrational modes (as2A1{\displaystyle 2A_{1}}) and the other vibrational mode is antisymmetric (as1B1{\displaystyle 1B_{1}})
There is some rules to be IR active or Raman active for a particular mode.
As the vibrational modes for water moleculeΓvibrational{\displaystyle \Gamma _{\text{vibrational}}}contains bothx{\displaystyle x},y{\displaystyle y}orz{\displaystyle z}and quadratic functions, it has both the IR active vibrational modes and Raman active vibrational modes.
Similar rules will apply for rest of the irreducible representationsΓirreducible,Γtranslational,Γrotational{\displaystyle \Gamma _{\text{irreducible}},\Gamma _{\text{translational}},\Gamma _{\text{rotational}}}
Ethylene is a member of the D2h point group, which has eight Mulliken symbols in the first column. Besides, the ethylene molecule contains six atoms, each with an x, y, and z axis. So, the molecule has a total of 18 axes.
For vibrational modes of the molecule, it is necessary to calculate the irreducible representation Γirreducible. Also, the irreducible representation is related with the reduible representation.
Here is another method to calculate the representation calculation. It is necessary to find the change of x, y and z axes. If the atom changes the place after the operation, there is no contribution to the Γreducible. If the atom keeps the same place after the operation, then check the axis, if the axis keeps same direction, the contribution to the Γreducible.is 1; if the axis reverses to the opposite direction, the contribution to the Γreducible.is -1; if the axis rotates at a certain angleθ, the contribution is cosθ. After calculating all axes of all atoms, there is the value of the reducible representation Γreduciblefor this operation.
In this case, ethylene is the D2hpoint group with eight symmetry operations in the first line, each operation provides the different Γreducible.
E: Identity Symmetry. All atoms remain in their original positions, so they all have the same x, y, and z axes. The 18 axes remain in the same position, each contributing one to the reducible. The reducible number for E is 18.
C2(x), C2(y): As the molecule rotates along the x or y axis, each atom moves and contributes zero to the reducible. The overall Γreduciblefor C2(x) and C2(y) are 0.
C2(z): The molecule rotates along the z axis, with only two carbon atoms remaining in the same position. The x and y axes of each carbon atom reverse to the opposite place, but z axis keeps the same direction, contributing negative one of each atom. The overall Γreducibleis -2.
i: The molecule is inverse through the center. Since all atoms move places, the overall Γreduciblefor i is 0.
σ(xy): The molecule flips across the xy plane. The overall Γreduciblefor σ(xy) is 0, as all atoms move places.
σ(xz): The molecule flips across the xz plane, but two carbon atoms remain in the same place. The x and z axes remain unchanged, each contributing to a single reducible number. However, the y axis reverses and contributes to negative one Γreducible. So, each carbon contributes one Γreducible, the overall Γreducibleis 2.
σ(yz): It is different from other operations. All six atoms maintain their original positions. The y and z axes remain the same, but the x axis reverses, resulting in one Γreduciblefor each atom. The total Γreducibleis 6.
New character table for ethyleneΓred{\displaystyle \Gamma _{\text{red}}}
The next step is to calculate the irreducible presentation based on the reducible presentation. Here is the calculation.
Γirreducible= 3Ag+1B1g+2B2g+3B3g+1Au+3B1u+3B2u+2B3u
Translational motion has x, y and z functions in “linear functions, roatations”. So, Γtrans= 1B1u+1B2u+1B3u
Rotational motion has Rx, Ryand Rzfunctions in “linear functions, roatations”. So, Γrot= 1B1g+1B2g+1B3g
Vibrational motio: Γvib= Γirreducible-Γtrans-Γrot= 3Ag+1B2g+2B3g+1Au+2B1u+2B2u+1B3u
The final step is to determine which vibrations are IR or Raman active. This means that the symmetry operation can be detected using the infrared or Raman spectrum.
First, for IR to work, they must have x, y, and z functions in "linear functions, rotations". In Γvib, only 2B1u+2B2u+1B3uare IR active.
To be Raman active, "quadratic functions" must include x2, y2, z2,xy, xz, yz, x2+y2or x2-y2functions. In Γvib, only 3Ag+1B2g+2B3gare Raman active.
|
https://en.wikipedia.org/wiki/Character_table
|
Inmathematics, more specifically ingroup theory, thecharacterof agroup representationis afunctionon thegroupthat associates to each group element thetraceof the correspondingmatrix. The character carries the essential information about the representation in a more condensed form.Georg Frobeniusinitially developedrepresentation theory of finite groupsentirely based on the characters, and without any explicit matrix realization of representations themselves. This is possible because acomplexrepresentation of afinite groupis determined (up toisomorphism) by its character. The situation with representations over afieldof positivecharacteristic, so-called "modular representations", is more delicate, butRichard Brauerdeveloped a powerful theory of characters in this case as well. Many deep theorems on the structure of finite groups use characters ofmodular representations.
Characters ofirreducible representationsencode many important properties of a group and can thus be used to study its structure. Character theory is an essential tool in theclassification of finite simple groups. Close to half of theproofof theFeit–Thompson theoreminvolves intricate calculations with character values. Easier, but still essential, results that use character theory includeBurnside's theorem(a purely group-theoretic proof of Burnside's theorem has since been found, but that proof came over half a century after Burnside's original proof), and a theorem ofRichard BrauerandMichio Suzukistating that a finitesimple groupcannot have ageneralized quaternion groupas itsSylow2-subgroup.
LetVbe afinite-dimensionalvector spaceover afieldFand letρ:G→ GL(V)be arepresentationof a groupGonV. Thecharacterofρis the functionχρ:G→Fgiven by
whereTris thetrace.
A characterχρis calledirreducibleorsimpleifρis anirreducible representation. Thedegreeof the characterχis thedimensionofρ; in characteristic zero this is equal to the valueχ(1). A character of degree 1 is calledlinear. WhenGis finite andFhas characteristic zero, thekernelof the characterχρis thenormal subgroup:
which is precisely the kernel of the representationρ. However, the character isnota group homomorphism in general.
Let ρ and σ be representations ofG. Then the following identities hold:
whereρ⊕σis thedirect sum,ρ⊗σis thetensor product,ρ∗denotes theconjugate transposeofρ, andAlt2is thealternating productAlt2ρ=ρ∧ρandSym2is thesymmetric square, which is determined byρ⊗ρ=(ρ∧ρ)⊕Sym2ρ.{\displaystyle \rho \otimes \rho =\left(\rho \wedge \rho \right)\oplus {\textrm {Sym}}^{2}\rho .}
The irreduciblecomplexcharacters of a finite group form acharacter tablewhich encodes much useful information about the groupGin a compact form. Each row is labelled by an irreducible representation and the entries in the row are the characters of the representation on the respective conjugacy class ofG. The columns are labelled by (representatives of) the conjugacy classes ofG. It is customary to label the first row by the character of thetrivial representation, which is the trivial action ofGon a 1-dimensional vector space byρ(g)=1{\displaystyle \rho (g)=1}for allg∈G{\displaystyle g\in G}. Each entry in the first row is therefore 1. Similarly, it is customary to label the first column by the identity. Therefore, the first column contains the degree of each irreducible character.
Here is the character table of
thecyclic groupwith three elements and generatoru:
whereωis aprimitivethird root of unity.
The character table is always square, because the number of irreducible representations is equal to the number of conjugacy classes.[2]
The space of complex-valuedclass functionsof a finite groupGhas a naturalinner product:
whereβ(g)is thecomplex conjugateofβ(g). With respect to this inner product, the irreducible characters form anorthonormal basisfor the space of class-functions, and this yields the orthogonality relation for the rows of the character table:
Forg,hinG, applying the same inner product to the columns of the character table yields:
where the sum is over all of the irreducible charactersχiofGand the symbol|CG(g)|denotes the order of thecentralizerofg. Note that sincegandhare conjugate iff they are in the same column of the character table, this implies that the columns of the character table are orthogonal.
The orthogonality relations can aid many computations including:
Certain properties of the groupGcan be deduced from its character table:
The character table does not in general determine the groupup toisomorphism: for example, thequaternion groupQand thedihedral groupof8elements,D4, have the same character table. Brauer asked whether the character table, together with the knowledge of how the powers of elements of its conjugacy classes are distributed, determines a finite group up to isomorphism. In 1964, this was answered in the negative byE. C. Dade.
The linear representations ofGare themselves a group under thetensor product, since the tensor product of 1-dimensional vector spaces is again 1-dimensional. That is, ifρ1:G→V1{\displaystyle \rho _{1}:G\to V_{1}}andρ2:G→V2{\displaystyle \rho _{2}:G\to V_{2}}are linear representations, thenρ1⊗ρ2(g)=(ρ1(g)⊗ρ2(g)){\displaystyle \rho _{1}\otimes \rho _{2}(g)=(\rho _{1}(g)\otimes \rho _{2}(g))}defines a new linear representation. This gives rise to a group of linear characters, called thecharacter groupunder the operation[χ1∗χ2](g)=χ1(g)χ2(g){\displaystyle [\chi _{1}*\chi _{2}](g)=\chi _{1}(g)\chi _{2}(g)}. This group is connected toDirichlet charactersandFourier analysis.
The characters discussed in this section are assumed to be complex-valued. LetHbe a subgroup of the finite groupG. Given a characterχofG, letχHdenote its restriction toH. Letθbe a character ofH.Ferdinand Georg Frobeniusshowed how to construct a character ofGfromθ, using what is now known asFrobenius reciprocity. Since the irreducible characters ofGform an orthonormal basis for the space of complex-valued class functions ofG, there is a unique class functionθGofGwith the property that
for each irreducible characterχofG(the leftmost inner product is for class functions ofGand the rightmost inner product is for class functions ofH). Since the restriction of a character ofGto the subgroupHis again a character ofH, this definition makes it clear thatθGis a non-negativeintegercombination of irreducible characters ofG, so is indeed a character ofG. It is known asthe character ofGinduced fromθ. The defining formula of Frobenius reciprocity can be extended to general complex-valued class functions.
Given a matrix representationρofH, Frobenius later gave an explicit way to construct a matrix representation ofG, known as the representationinduced fromρ, and written analogously asρG. This led to an alternative description of the induced characterθG. This induced character vanishes on all elements ofGwhich are not conjugate to any element ofH. Since the induced character is a class function ofG, it is only now necessary to describe its values on elements ofH. If one writesGas adisjoint unionof rightcosetsofH, say
then, given an elementhofH, we have:
Becauseθis a class function ofH, this value does not depend on the particular choice of coset representatives.
This alternative description of the induced character sometimes allows explicit computation from relatively little information about the embedding ofHinG, and is often useful for calculation of particular character tables. Whenθis the trivial character ofH, the induced character obtained is known as thepermutation characterofG(on the cosets ofH).
The general technique of character induction and later refinements found numerous applications infinite group theoryand elsewhere in mathematics, in the hands of mathematicians such asEmil Artin,Richard Brauer,Walter FeitandMichio Suzuki, as well as Frobenius himself.
The Mackey decomposition was defined and explored byGeorge Mackeyin the context ofLie groups, but is a powerful tool in the character theory and representation theory of finite groups. Its basic form concerns the way a character (or module) induced from a subgroupHof a finite groupGbehaves on restriction back to a (possibly different) subgroupKofG, and makes use of the decomposition ofGinto(H,K)-double cosets.
IfG=⋃t∈THtK{\textstyle G=\bigcup _{t\in T}HtK}is a disjoint union, andθis a complex class function ofH, then Mackey's formula states that
whereθtis the class function oft−1Htdefined byθt(t−1ht) =θ(h)for allhinH. There is a similar formula for the restriction of an induced module to a subgroup, which holds for representations over anyring, and has applications in a wide variety of algebraic andtopologicalcontexts.
Mackey decomposition, in conjunction with Frobenius reciprocity, yields a well-known and useful formula for the inner product of two class functionsθandψinduced from respective subgroupsHandK, whose utility lies in the fact that it only depends on how conjugates ofHandKintersect each other. The formula (with its derivation) is:
(whereTis a full set of(H,K)-double coset representatives, as before). This formula is often used whenθandψare linear characters, in which case all the inner products appearing in the right hand sum are either1or0, depending on whether or not the linear charactersθtandψhave the same restriction tot−1Ht∩K. Ifθandψare both trivial characters, then the inner product simplifies to|T|.
One may interpret the character of a representation as the "twisted"dimension of a vector space.[3]Treating the character as a function of the elements of the groupχ(g), its value at theidentityis the dimension of the space, sinceχ(1) = Tr(ρ(1)) = Tr(IV) = dim(V). Accordingly, one can view the other values of the character as "twisted" dimensions.[clarification needed]
One can find analogs or generalizations of statements about dimensions to statements about characters or representations. A sophisticated example of this occurs in the theory ofmonstrous moonshine: thej-invariantis thegraded dimensionof an infinite-dimensional graded representation of theMonster group, and replacing the dimension with the character gives theMcKay–Thompson seriesfor each element of the Monster group.[3]
IfG{\displaystyle G}is aLie groupandρ{\displaystyle \rho }a finite-dimensional representation ofG{\displaystyle G}, the characterχρ{\displaystyle \chi _{\rho }}ofρ{\displaystyle \rho }is defined precisely as for any group as
Meanwhile, ifg{\displaystyle {\mathfrak {g}}}is aLie algebraandρ{\displaystyle \rho }a finite-dimensional representation ofg{\displaystyle {\mathfrak {g}}}, we can define the characterχρ{\displaystyle \chi _{\rho }}by
The character will satisfyχρ(Adg(X))=χρ(X){\displaystyle \chi _{\rho }(\operatorname {Ad} _{g}(X))=\chi _{\rho }(X)}for allg{\displaystyle g}in the associated Lie groupG{\displaystyle G}and allX∈g{\displaystyle X\in {\mathfrak {g}}}. If we have a Lie group representation and an associated Lie algebra representation, the characterχρ{\displaystyle \chi _{\rho }}of the Lie algebra representation is related to the characterXρ{\displaystyle \mathrm {X} _{\rho }}of the group representation by the formula
Suppose now thatg{\displaystyle {\mathfrak {g}}}is a complexsemisimple Lie algebrawith Cartan subalgebrah{\displaystyle {\mathfrak {h}}}. The value of the characterχρ{\displaystyle \chi _{\rho }}of an irreducible representationρ{\displaystyle \rho }ofg{\displaystyle {\mathfrak {g}}}is determined by its values onh{\displaystyle {\mathfrak {h}}}. The restriction of the character toh{\displaystyle {\mathfrak {h}}}can easily be computed in terms of theweight spaces, as follows:
where the sum is over allweightsλ{\displaystyle \lambda }ofρ{\displaystyle \rho }and wheremλ{\displaystyle m_{\lambda }}is the multiplicity ofλ{\displaystyle \lambda }.[4]
The (restriction toh{\displaystyle {\mathfrak {h}}}of the) character can be computed more explicitly by the Weyl character formula.
|
https://en.wikipedia.org/wiki/Character_theory
|
Inchemistry,molecular symmetrydescribes thesymmetrypresent inmoleculesand the classification of these molecules according to their symmetry. Molecular symmetry is a fundamental concept in chemistry, as it can be used to predict or explain many of a molecule'schemical properties, such as whether or not it has adipole moment, as well as its allowedspectroscopic transitions. To do this it is necessary to usegroup theory. This involves classifying the states of the molecule using theirreducible representationsfrom thecharacter tableof the symmetry group of the molecule. Symmetry is useful in the study ofmolecular orbitals, with applications to theHückel method, toligand field theory, and to theWoodward–Hoffmann rules.[1][2]Many university level textbooks onphysical chemistry,quantum chemistry,spectroscopy[3]andinorganic chemistrydiscuss symmetry.[4][5][6][7][8]Another framework on a larger scale is the use ofcrystal systemsto describecrystallographicsymmetry in bulk materials.
There are many techniques for determining the symmetry of a given molecule, includingX-ray crystallographyand various forms ofspectroscopy.Spectroscopic notationis based on symmetry considerations.
The point group symmetry of a molecule is defined by the presence or absence of 5 types ofsymmetry element.
The five symmetry elements have associated with them five types ofsymmetry operation, which leave the geometry of the molecule indistinguishable from the starting geometry. They are sometimes distinguished from symmetry elements by acaretorcircumflex. Thus,Ĉnis the rotation of a molecule around an axis andÊis the identity operation. A symmetry element can have more than one symmetry operation associated with it. For example, theC4axis of thesquarexenon tetrafluoride(XeF4) molecule is associated with twoĈ4rotations in opposite directions (90° and 270°), aĈ2rotation (180°) andĈ1(0° or 360°). BecauseĈ1is equivalent toÊ,Ŝ1to σ andŜ2toî, all symmetry operations can be classified as either proper or improper rotations.
For linear molecules, either clockwise or counterclockwise rotation about the molecular axis by any angle Φ is a symmetry operation.
The symmetry operations of a molecule (or other object) form agroup. In mathematics, a group is a set with abinary operationthat satisfies the four properties listed below.
In asymmetry group, the group elements are the symmetry operations (not the symmetry elements), and the binary combination consists of applying first one symmetry operation and then the other. An example is the sequence of aC4rotation about the z-axis and a reflection in the xy-plane, denoted σ(xy)C4. By convention the order of operations is from right to left.
A symmetry group obeys the defining properties of any group.
Theorderof a group is the number of elements in the group. For groups of small orders, the group properties can be easily verified by considering its composition table, a table whose rows and columns correspond to elements of the group and whose entries correspond to their products.
The successive application (orcomposition) of one or more symmetry operations of a molecule has an effect equivalent to that of some single symmetry operation of the molecule. For example, aC2rotation followed by a σvreflection is seen to be a σv' symmetry operation: σv*C2= σv'. ("OperationAfollowed byBto formC" is writtenBA=C).[11]Moreover, the set of all symmetry operations (including this composition operation) obeys all the properties of a group, given above. So (S,*) is a group, whereSis the set of all symmetry operations of some molecule, and * denotes the composition (repeated application) of symmetry operations.
This group is called thepoint groupof that molecule, because the set of symmetry operations leave at least one point fixed (though for some symmetries an entire axis or an entire plane remains fixed). In other words, a point group is a group that summarises all symmetry operations that all molecules in that category have.[11]The symmetry of a crystal, by contrast, is described by aspace groupof symmetry operations, which includestranslationsin space.
Assigning each molecule a point group classifies molecules into categories with similar symmetry properties. For example, PCl3, POF3, XeO3, and NH3all share identical symmetry operations.[12]They all can undergo the identity operationE, two differentC3rotation operations, and three different σvplane reflections without altering their identities, so they are placed in one point group,C3v, with order 6.[11]Similarly, water (H2O) and hydrogen sulfide (H2S) also share identical symmetry operations. They both undergo the identity operationE, oneC2rotation, and two σvreflections without altering their identities, so they are both placed in one point group,C2v, with order 4.[13]This classification system helps scientists to study molecules more efficiently, since chemically related molecules in the same point group tend to exhibit similar bonding schemes, molecular bonding diagrams, and spectroscopic properties.[11]Point group symmetry describes the symmetry of a molecule when fixed at its equilibrium configuration in a particular electronic state. It does not allow for tunneling between minima nor for the change in shape that can come about from the centrifugal distortion effects of molecular rotation.
The following table lists many of thepoint groupsapplicable to molecules, labelled using theSchoenflies notation, which is common in chemistry and molecular spectroscopy. The descriptions include common shapes of molecules, which can be explained by theVSEPR model. In each row, the descriptions and examples have no higher symmetries, meaning that the named point group capturesallof the point symmetries.
All of the group operations described above and the symbols for crystallographic point groups themselves were first published byArthur Schoenfliesin 1891 but the groups had been applied by other researchers to the external morphology of crystals much earlier in the 19th century.
In 1914Max von Lauepublished the results of experiments using x-ray diffraction to elucidate the internal structures of crystals producing a limited version of the table of "Laue classes" shown. When adapted for molecular work this table first divides point groups into three kinds: asymmetric, symmetric and spherical tops. These are categories based on the angular momentum of molecules, having respectively 3, 2 and 1 distinct values of angular momentum, becoming more symmetrical down the table. A further sub-division into systems is defined by the rotational groupGin the leftmost column then into rows of Laue classes that take the form of cyclic and dihedral groups in the first two categories and tetrahedral and octahedral classes in the third. Rotational groups occur in the first column and define the non-rotational groups in their class. The second and third columns contain non-rotational groups belonging to the same abstract group as that in the first column A fourth column contains groups that are a direct product of their defining rotational group with space inversion (parity inversion) and so are of twice the order of other members of the class. Groups in this column contain the inversion operation itself as a member. For example, seven groups in the hexagonal system all contain the C6cyclic system, mostly as physical rotational group but in the third column of the table as an abstract group. So, C6and C3hare distinct manifestations of the same group while C6his simply C6xi. Groups D6, C6vand D3hare also example of the same abstract group and D6his the direct product D6xi.
It is difficult to overstate the importance of the Laue class in the applications of point groups to the description of physical properties at the molecular level. Since all the point groups of a Laue class have the same abstract structure, they also have exactly the same irreducible representations and character tables. Any representation of one is automatically a representation of the class and any group in it. One important point is that higher symmetry molecules do not cease to have the lower symmetry of their subgroups. Using the hexagonal example again, picture the series of groups C6of order 6, D6of order 12 and D6hof order 24, each group being of twice the order of the previous one. Once the irreducible representations of a physical example of a cyclic group have been found it is usually a simple process to extend this to the higher order groups. Laue found that x-ray diffraction was unable to distinguish between such groups Tetrahedral and octahedral point groups have a relationship similar to that between cyclic and dihedral groups and the tetrahedral g occurs in all cubic groups.
A set ofmatricesthat multiply together in a way that mimics the multiplication table of the elements of a group is called arepresentationof the group. The simplest method of obtaining a representation of molecular group transformations is to trace the movements of atoms in a molecule when symmetry operations are applied. For example, a water molecule belonging to theC2vpoint group might have an oxygen atom labelled 1 and two hydrogen atoms labelled 2 and 3 as shown in the right hand column vector below. If the hydrogen atoms are imagined to rotate by 180 degrees about an axis passing through the oxygen atom we have the familiarC2operation of this point group. The oxygen atom in position number 1 stays in position but the atoms in positions 2 and 3 are moved to positions 3 and 2 in the resulting column vector. The matrix connecting the two provides a 3 x 3 representation for this operation.
This point group only contains four operations and matrices for the other three operations are obtained similarly, including the identity matrix which just contains 1's on the leading diagonal (top left to bottom right) and 0's elsewhere. Having obtained the representation matrices in this way it is not difficult to show that they multiply out in exactly the same way as the operations themselves.
Although an infinite number of such representations exist, theirreducible representations(or "irreps") of the group are all that are needed as all other representations of the group can be described as adirect sumof the irreducible representations. The first step in finding the irreps making up a given representation is to sum up the values of the leading diagonals for each matrix so, taking the identity matrix first then the matrices in the order above, one obtains (3, 1, 3, 1). These values are the traces or characters of the four matrices. Asymmetric point groups such asC2vonly have 1-dimensional irreps so the character of an irrep is exactly the same is the irrep itself and the following table can be interpreted as irreps or characters.
Looking again at the characters obtained for the 3D representation above (3, 1, 3, 1), we only need simple arithmetic to break this down into irreps. Clearly, E = 3 means there are three irreps and a C2representation sum of 1 means there must be two A and one B irreps so the only combination that adds up to the characters derived is 2A1+ B1
Robert Mullikenwas the first to publish character tables in English and so the notation used to label irreps in the above table is called Mulliken notation. For asymmetric groups it consists of letters A and B with subscripts 1 and 2 as above and subscripts g and u as in the C2hexample below. (Subscript 3 also appears in D2) The irreducible representations are those matrix representations in which the matrices are in their most diagonal form possible and for asymmetric groups this means totally diagonal. One further thing to note about the irrep/character table above is the appearance of polar and axial base vector symbols on the right hand side. This tells us that, for example, cartesian base vector x transforms as irrep B1under the operations of this group. The same collection of product base vectors is used for all asymmetric groups but symmetric and spherical groups use different sets of product base vectors.
Point group C2hhas the operations {E, C2, i, σh} and the 1,5-dibromonapthalene (C10H6Br2) shown in the figure belongs to this symmetry group. It is possible to construct four 18 x 18 matrices representing the transformations of atoms during its symmetry operations in the style of the water molecule example above and reduce it to 18 1D irreps. Notice however that carbon atom number 1 either stays in place or it is exchanged with carbon atom number 5 and these two atoms can be analysed separately from all the other atoms in the molecule. The transformation matrix for these two toms alone during the molecular C2rotation is
with character 0. When this computation is carried out for each of the operations above the characters obtained are (2,0,0.2) because two operations leave the atoms in place and two move them. The irrep table for this group is below. The first column tells us there are two1D irreps, the second column (C2) that there is one A and one B while columns 3 and 4 reveal that one irrep has subscript g the other has to have subscript u. This means that the irreps resulting from the two atoms are Ag+ Bu. In fact, the 18 atoms in this molecule are paired off in exactly the same way as carbon atoms 1 and 5 so that, from a symmetry perspective, the atom consists of 9 pairs of equivalent atoms related through symmetry. It follows that each pair contributes the same irreps as the pair examined above to give a total 18 dimensional irrep result of 9(Ag+ Bu).
Symmetric point groups are divided into systems based on the increasing order of the main rotational axis from three to infinity. Systems are in turn divided into cyclic and dihedral groups and within a system the order of the dihedral group is twice that of the cyclic group. Cyclic groups only have one dimensional representations as shown in the table of irreps and the number of irreps is equal to the order of the group. The irreps shown use standard notation for the rotational group of a class but Mulliken sometimes gave different symbols to other members of the same class even though they belong to the same abstract group and therefore have the same irreps.
Dihedral point groups contain a cyclic group of the same rotational order: so group Dnalways contains group Cnas an index-2 subgroup. It follows that dihedral irreps are superimposed on cyclic irreps because the cyclic group within a dihedral one does not cease to be a cyclic group. A dihedral group also contains a 2-fold rotational axis at right angles to the main cyclic axis and this has two consequences. Firstly, the A and B cyclic irreps are split into pairs of one dimensional irreps identified by subscripts 1 and 2. Secondly, pairs of 1D E−xand E+xcyclic irreps combine to form single Ex2D irreps in the dihedral group because the 2-fold horizontal rotation makes pairs of rotations equivalent. For example, a 60 degree rotation about the main axis becomes equivalent to a (360 − 60) degree rotation because the 2-fold horizontal rotation makes them equivalent. Combinations of this kind are said to form a class. Infinite order dihedral group irreps sometimes use Greek symbol descriptions,Σ,Π,Δthat follow from early linear molecule calculations.
Spherical point group representations
Spherical classes are defined by the tetrahedral, octahedral and icosahedral rotational groups T, O and I. The first two of these, T and O, are related in much the same way as cyclic and dihedral groups are related in symmetric groups. Both tetrahedral and octahedral molecules are often shown with their atoms inscribed in the apices or faces of cubes and might be considered as a single "cubic" system. Every point group in this system contains the simple tetrahedral rotational group as a subgroup. Methane (CH4) is often used as an example and, although often described as a tetrahedral molecule because of the very visible rotational symmetry, it really belongs to the octahedral symmetry class. Considering methane first as a tetrahedral molecule the 12 operations of group T are {E, 3 x c, 4 x b, 4 x b3} where c is a 180 degree rotation along x,y and z axes and b is a 120 degree rotation about the apices of a cube. Character tables under these four headings exhibit the corresponding four irreps A, E+1, E-1and T and it would it is not difficult to convert the transformations of atoms during the symmetry operations to reducible matrices and thence to molecular irreps but this not necessary. Methane has two sets of equivalent atoms that are transformed into each other during operations: a single carbon atom and 4 hydrogen atoms. A single atom can only ever be transformed into itself and therefore always contributes the most symmetrical irrep to the end total irrep count. Additionally, there is a rule of group theory that the most symmetrical irrep must occur once and only once in the irreps of any equivalent atom set so the five dimensions of irreps being sought contain 2A and three others. The only way of filling the remaining three dimensions is to adopt 3D irrep T so the irreps are 2A + T. (E irreps have to be taken in pairs in physical molecular applications).
methane sulfur hexafluoride
Extending this treatment to the octahedral group Tdrequires six 4-fold roto-inversion operations (f) about the main axes and six 2-fold roto-inversions (a), appearing as mirror reflections through opposite edges of the imaginary cube in which methane is placed. So half the operations of this group are rotational and half non-rotational. Rotational group T exists within non-rotational group Td= {E, 3 x c, 8 x b/b3, 6 x f, 6 x a} T irreps A, E+1, E-1and T are promoted to Tdirreps A1, A2, E1and T1and T2. (E irreps collapse into one because the 3-fold irreps become equivalent while A and T expand into two). Reasoning as above we know that the irreps in Tdmust be 2A1+ Txso the last step is to find the 3D subscript. A brief look at the 4 x 4 transformation matrix for the 4-fold rotation operation f shows character Ch(f) = 0 and the subscript x has to be 2 to balance the 1 on the A irrep.
Sulfur hexafluoride can be treated first as a tetrahedral molecule T, then as octahedral O and finally as centred molecule Oi.There are two sets of equivalent atoms consisting of a single sulfur atom and six fluorine atoms. Transformations of the fluorine atoms generate a six dimensional representation that can only reduce into the direct sum of tetrahedral irreps A, E+1, E-1and T because the direct sum must include the most symmetrical irrep once and only once. A direct sum of 5 can only be made up from a 2 and a 3 - no other combination is possible. These irreps are "promoted" to 2A1+ E1+ Txin group O. To get the x observe that the 4-fold rotation in SF6 has character Ch(f) = 2 because two atoms stay in position and a glance at this column of the table suggests A1+ E1+ T1. Finally the inversion operation (i) applied go the fluorine atoms has character Ch(i) = 0 indicating equal numbers ofgandusubscripts (because none of the atoms remains in position). Since the most symmetrical irrep must occur once the only possible result is A1g+ E1g+ T1u. The single sulfur atom always has the most symmetric irrep to the final reduction of the seven dimensional matrices to a direct sum is 2A1g+ E1g+ T1u.
The following reference of character tables uses symbols Zxfor abstract cyclic groups Cxwith A4and S4(alternating and symmetric permutations of 4 objects for T and O. Many authors just use C, T and O in two senses, making it clear which is intended.
Much of our understanding of quantum theory was developed during the early years of the 20th century, leading up to Schrodinger's three dimensional wave equation.
E. Bright Wilsonused character tables in 1934 to predict the symmetry of vibrationalnormal modes.[16]
Hans Betheused characters of point group operations in his study ofligand field theoryin 1929, andEugene Wignerused group theory to explain the selection rules ofatomic spectroscopy.[17]The first character tables were compiled byLászló Tisza(1933), in connection to vibrational spectra.
The complete set of 32 crystallographic point groups was published in 1936 by Rosenthal and Murphy[18]
When Schrodinger's 3D wave equation is applied to a one-electron atom it provides a number of solutions called wave functions that are then used to label the allowed energy levels in that atom. Exact solutions of this kind are usually described by three quantum numbers, n, l and m from which the probable radial and angular distribution of the electron around the atom can be computed. This type of deduction leads to the familiar s, p, d, f, ... description of atomic orbitals based on the l and m quantum numbers. Each solution is a base vector from which more complex structures may be constructed. Descriptions of many-electron atoms use the one-electron model to build models that are sometimes pictured as multiple electrons in the simple structure. Molecular orbitals then take linear combinations of atomic orbitals (LCAOs) to explain the distribution of electrons over multiple atoms within a molecule. Atomic orbital symmetry follows from the angular part of the wave function which increases in complexity in the series s,p,d,f,... so that s orbitals only have radial symmetry while p orbital base vectors have a symmetry identical to that of the Cartesian polar base vectors.
Consider the example of water (H2O), which has theC2vsymmetry described above. The 2pxorbitalof oxygen has, like the x base vector, B1symmetry. It is oriented perpendicular to the plane of the molecule and switches sign with aC2and a σv'(yz) operation, but remains unchanged with the other two operations (obviously, the character for the identity operation is always +1). This orbital's character set is thus {1, −1, 1, −1}, corresponding to the B1irreducible representation. Likewise, the 2pzorbital is seen to have the symmetry of the A1irreducible representation (i.e.: none of the symmetry operations change it), 2pyB2, and the 3dxyorbital A2. These assignments are noted in the rightmost columns of the table.
Eachmolecular orbitalalso has the symmetry of one irreducible representation. For example,ethylene(C2H4) has symmetry group D2h, and its highest occupied molecular orbital (HOMO) is the bonding pi orbital which forms a basis for its irreducible representation B1u.[19]
One can determine the symmetry operations of the point group for a particular molecule by considering the geometrical symmetry of its molecular model. However, when one uses a point group to classify molecular states, the operations in it are not to be interpreted in the same way. Instead the operations are interpreted as rotating and/or reflecting the vibronic (vibration-electronic) coordinates and these operations commute with the vibronic Hamiltonian.[20]They are "symmetry operations" for that vibronic Hamiltonian. The point group is used to classify by symmetry the vibronic eigenstates of a rigid molecule. The symmetry classification of the rotational levels, the eigenstates of the full (rotation-vibration-electronic) Hamiltonian, can be achieved through the use of the appropriate permutation-inversion group (called themolecular symmetry group), as introduced byLonguet-Higgins.[21]
Each normal mode ofmolecular vibrationhas a symmetry which forms a basis for one irreducible representation of the molecular symmetry group.[22]For example, the water molecule has three normal modes of vibration: symmetric stretch in which the two O-H bond lengths vary in phase with each other, asymmetric stretch in which they vary out of phase, and bending in which the bond angle varies. The molecular symmetry of water is C2vwith four irreducible representations A1, A2, B1and B2. The symmetric stretching and the bending modes have symmetry A1, while the asymmetric mode has symmetry B2. The overall symmetry of the three vibrational modes is therefore Γvib= 2A1+ B2.[22][23]
The molecular symmetry ofammonia(NH3) is C3v, withsymmetry operationsE, C3and σv.[9]For N = 4 atoms, the number of vibrational modes for a non-linear molecule is 3N-6 = 6, due to the relative motion of thenitrogenatom and the three hydrogen atoms. All threehydrogenatoms travel symmetrically along the N-H bonds, either in the direction of the nitrogen atom or away from it. This mode is known assymmetric stretch(v₁) and reflects the symmetry in the N-H bond stretching. Of the three vibrational modes, this one has the highestfrequency.[24]
In theBending(ν₂) vibration, the nitrogen atom stays on the axis of symmetry, while the three hydrogen atoms move in different directions from one another, leading to changes in the bond angles. The hydrogen atoms move like an umbrella, so this mode is often referred to as the "umbrella mode".[26]
There is also anAsymmetric Stretchmode (ν₃) in which one hydrogen atom approaches the nitrogen atom while the other two hydrogens move away.
The total number of degrees of freedom for each symmetry species (orirreducible representation) can be determined. Ammonia has four atoms, and each atom is associated with threevector components. The symmetry group C3vfor NH3has the three symmetry species A1, A2and E. The modes of vibration include the vibrational, rotational and translational modes.
Total modes = 3A1+ A2+ 4E. This is a total of 12 modes because each E corresponds to 2 degenerate modes (at the same energy).
Rotational modes= A2+ E (3 modes)
Translational modes = A1+ E
Vibrational modes= Total modes - Rotational modes - Translational modes = 3A1+ A2+ 4E - A2- E - A1- E = 2A1+ 2E (6 modes).
As discussed above in§ The molecular symmetry group, point groups are useful for classifying the vibrational and electronic states ofrigidmolecules (sometimes calledsemi-rigidmolecules) which undergo only small oscillations about a single equilibrium geometry.Longuet-Higginsintroduced the molecular symmetry group (a more general type of symmetry group)[21]suitable not only for classifying the vibrational and electronic states of rigid molecules but also for classifying their rotational and nuclear spin states. Further, such groups can be used to classify the states ofnon-rigid(orfluxional) molecules that tunnel between equivalent geometries[27]and to allow for the distorting effects of molecular rotation. The symmetry operations in the molecular symmetry group are so-called 'feasible' permutations of identical nuclei, or inversion with respect to the center of mass (theparityoperation), or a combination of the two, so that the group is sometimes called a "permutation-inversion group".[21][28]
Examples of molecular nonrigidity abound. For example,ethane(C2H6) has three equivalentstaggered conformations. Tunneling between the conformations occurs at ordinary temperatures byinternal rotation of one methyl grouprelative to the other. This is not a rotation of the entire molecule about theC3axis, although each conformation hasD3dsymmetry, as in the table above. The molecule2-butyne(dimethylacetylene) has the same molecular symmetry group (G36} as ethane but a very much
lower torsional barrier. Similarly,ammonia(NH3) has two equivalent pyramidal (C3v) conformations which are interconverted by the process known asnitrogen inversion.
Additionally, themethanemolecule (CH4) andTrihydrogen cation{H3+) have highly symmetric equilibrium structures withTdandD3hpoint group symmetries respectively; they lack permanent electric dipole moments but they do have very weak pure rotation spectra because of rotational centrifugal distortion.[29][30]
Sometimes it is necessary to consider together electronic states having different point group symmetries at equilibrium. For example, in its ground (N) electronic state the ethylene molecule C2H4hasD2hpoint group symmetry whereas in the excited (V) state it hasD2dsymmetry. To treat these two states together it is necessary to
allow torsion and to use thedouble groupof the molecular symmetry groupG16.[31]
|
https://en.wikipedia.org/wiki/Molecular_symmetry
|
This is alist ofharmonic analysistopics. See alsolist of Fourier analysis topicsandlist of Fourier-related transforms, which are more directed towards the classicalFourier seriesandFourier transformofmathematical analysis,mathematical physicsandengineering.
|
https://en.wikipedia.org/wiki/List_of_harmonic_analysis_topics
|
This is alist ofrepresentation theorytopics, by Wikipedia page. See alsolist of harmonic analysis topics, which is more directed towards themathematical analysisaspects of representation theory.
See also:Glossary of representation theory
|
https://en.wikipedia.org/wiki/List_of_representation_theory_topics
|
Therepresentation theoryofgroupsis a part of mathematics which examines how groups act on given structures.
Here the focus is in particular onoperations of groupsonvector spaces. Nevertheless, groups acting on other groups or onsetsare also considered. For more details, please refer to the section onpermutation representations.
Other than a few marked exceptions, only finite groups will be considered in this article. We will also restrict ourselves to vector spaces overfieldsofcharacteristiczero. Because the theory ofalgebraically closed fieldsof characteristic zero iscomplete, a theory valid for a special algebraically closed field of characteristic zero is also valid for every other algebraically closed field of characteristic zero. Thus, without loss of generality, we can study vector spaces overC.{\displaystyle \mathbb {C} .}
Representation theory is used in many parts of mathematics, as well as in quantum chemistry and physics. Among other things it is used inalgebrato examine the structure of groups. There are also applications inharmonic analysisandnumber theory. For example, representation theory is used in the modern approach to gain new results about automorphic forms.
LetV{\displaystyle V}be aK{\displaystyle K}–vector space andG{\displaystyle G}a finite group. Alinear representationofG{\displaystyle G}is agroup homomorphismρ:G→GL(V)=Aut(V).{\displaystyle \rho :G\to {\text{GL}}(V)={\text{Aut}}(V).}HereGL(V){\displaystyle {\text{GL}}(V)}is notation for ageneral linear group, andAut(V){\displaystyle {\text{Aut}}(V)}for anautomorphism group. This means that a linear representation is a mapρ:G→GL(V){\displaystyle \rho :G\to {\text{GL}}(V)}which satisfiesρ(st)=ρ(s)ρ(t){\displaystyle \rho (st)=\rho (s)\rho (t)}for alls,t∈G.{\displaystyle s,t\in G.}The vector spaceV{\displaystyle V}is called arepresentation spaceofG.{\displaystyle G.}Often the term "representation ofG{\displaystyle G}" is also used for the representation spaceV.{\displaystyle V.}
The representation of a group in amoduleinstead of a vector space is also called a linear representation.
We write(ρ,Vρ){\displaystyle (\rho ,V_{\rho })}for the representationρ:G→GL(Vρ){\displaystyle \rho :G\to {\text{GL}}(V_{\rho })}ofG.{\displaystyle G.}Sometimes we use the notation(ρ,V){\displaystyle (\rho ,V)}if it is clear to which representation the spaceV{\displaystyle V}belongs.
In this article we will restrict ourselves to the study of finite-dimensional representation spaces, except for the last chapter. As in most cases only a finite number of vectors inV{\displaystyle V}is of interest, it is sufficient to study thesubrepresentationgenerated by these vectors. The representation space of this subrepresentation is then finite-dimensional.
Thedegreeof a representation is thedimensionof its representation spaceV.{\displaystyle V.}The notationdim(ρ){\displaystyle \dim(\rho )}is sometimes used to denote the degree of a representationρ.{\displaystyle \rho .}
Thetrivial representationis given byρ(s)=Id{\displaystyle \rho (s)={\text{Id}}}for alls∈G.{\displaystyle s\in G.}
A representation of degree1{\displaystyle 1}of a groupG{\displaystyle G}is a homomorphism into the multiplicativegroupρ:G→GL1(C)=C×=C∖{0}.{\displaystyle \rho :G\to {\text{GL}}_{1}(\mathbb {C} )=\mathbb {C} ^{\times }=\mathbb {C} \setminus \{0\}.}As every element ofG{\displaystyle G}is of finite order, the values ofρ(s){\displaystyle \rho (s)}areroots of unity. For example, letρ:G=Z/4Z→C×{\displaystyle \rho :G=\mathbb {Z} /4\mathbb {Z} \to \mathbb {C} ^{\times }}be a nontrivial linear representation. Sinceρ{\displaystyle \rho }is a group homomorphism, it has to satisfyρ(0)=1.{\displaystyle \rho ({0})=1.}Because1{\displaystyle 1}generatesG,ρ{\displaystyle G,\rho }is determined by its value onρ(1).{\displaystyle \rho (1).}And asρ{\displaystyle \rho }is nontrivial,ρ(1)∈{i,−1,−i}.{\displaystyle \rho ({1})\in \{i,-1,-i\}.}Thus, we achieve the result that the image ofG{\displaystyle G}underρ{\displaystyle \rho }has to be a nontrivial subgroup of the group which consists of the fourth roots of unity. In other words,ρ{\displaystyle \rho }has to be one of the following three maps:
LetG=Z/2Z×Z/2Z{\displaystyle G=\mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z} }and letρ:G→GL2(C){\displaystyle \rho :G\to {\text{GL}}_{2}(\mathbb {C} )}be the group homomorphism defined by:
In this caseρ{\displaystyle \rho }is a linear representation ofG{\displaystyle G}of degree2.{\displaystyle 2.}
LetX{\displaystyle X}be a finite set and letG{\displaystyle G}be a group acting onX.{\displaystyle X.}Denote byAut(X){\displaystyle {\text{Aut}}(X)}the group of all permutations onX{\displaystyle X}with the composition as group multiplication.
A group acting on a finite set is sometimes considered sufficient for the definition of the permutation representation. However, since we want to construct examples for linear representations - where groups act on vector spaces instead of on arbitrary finite sets - we have to proceed in a different way. In order to construct the permutation representation, we need a vector spaceV{\displaystyle V}withdim(V)=|X|.{\displaystyle \dim(V)=|X|.}A basis ofV{\displaystyle V}can be indexed by the elements ofX.{\displaystyle X.}The permutation representation is the group homomorphismρ:G→GL(V){\displaystyle \rho :G\to {\text{GL}}(V)}given byρ(s)ex=es.x{\displaystyle \rho (s)e_{x}=e_{s.x}}for alls∈G,x∈X.{\displaystyle s\in G,x\in X.}All linear mapsρ(s){\displaystyle \rho (s)}are uniquely defined by this property.
Example.LetX={1,2,3}{\displaystyle X=\{1,2,3\}}andG=Sym(3).{\displaystyle G={\text{Sym}}(3).}ThenG{\displaystyle G}acts onX{\displaystyle X}viaAut(X)=G.{\displaystyle {\text{Aut}}(X)=G.}The associated linear representation isρ:G→GL(V)≅GL3(C){\displaystyle \rho :G\to {\text{GL}}(V)\cong {\text{GL}}_{3}(\mathbb {C} )}withρ(σ)ex=eσ(x){\displaystyle \rho (\sigma )e_{x}=e_{\sigma (x)}}forσ∈G,x∈X.{\displaystyle \sigma \in G,x\in X.}
LetG{\displaystyle G}be a group andV{\displaystyle V}be a vector space of dimension|G|{\displaystyle |G|}with a basis(et)t∈G{\displaystyle (e_{t})_{t\in G}}indexed by the elements ofG.{\displaystyle G.}Theleft-regular representationis a special case of thepermutation representationby choosingX=G.{\displaystyle X=G.}This meansρ(s)et=est{\displaystyle \rho (s)e_{t}=e_{st}}for alls,t∈G.{\displaystyle s,t\in G.}Thus, the family(ρ(s)e1)s∈G{\displaystyle (\rho (s)e_{1})_{s\in G}}of images ofe1{\displaystyle e_{1}}are a basis ofV.{\displaystyle V.}The degree of the left-regular representation is equal to the order of the group.
Theright-regular representationis defined on the same vector space with a similar homomorphism:ρ(s)et=ets−1.{\displaystyle \rho (s)e_{t}=e_{ts^{-1}}.}In the same way as before(ρ(s)e1)s∈G{\displaystyle (\rho (s)e_{1})_{s\in G}}is a basis ofV.{\displaystyle V.}Just as in the case of the left-regular representation, the degree of the right-regular representation is equal to the order ofG.{\displaystyle G.}
Both representations areisomorphicviaes↦es−1.{\displaystyle e_{s}\mapsto e_{s^{-1}}.}For this reason they are not always set apart, and often referred to as "the" regular representation.
A closer look provides the following result: A given linear representationρ:G→GL(W){\displaystyle \rho :G\to {\text{GL}}(W)}isisomorphicto the left-regular representation if and only if there exists aw∈W,{\displaystyle w\in W,}such that(ρ(s)w)s∈G{\displaystyle (\rho (s)w)_{s\in G}}is a basis ofW.{\displaystyle W.}
Example.LetG=Z/5Z{\displaystyle G=\mathbb {Z} /5\mathbb {Z} }andV=R5{\displaystyle V=\mathbb {R} ^{5}}with the basis{e0,…,e4}.{\displaystyle \{e_{0},\ldots ,e_{4}\}.}Then the left-regular representationLρ:G→GL(V){\displaystyle L_{\rho }:G\to {\text{GL}}(V)}is defined byLρ(k)el=el+k{\displaystyle L_{\rho }(k)e_{l}=e_{l+k}}fork,l∈Z/5Z.{\displaystyle k,l\in \mathbb {Z} /5\mathbb {Z} .}The right-regular representation is defined analogously byRρ(k)el=el−k{\displaystyle R_{\rho }(k)e_{l}=e_{l-k}}fork,l∈Z/5Z.{\displaystyle k,l\in \mathbb {Z} /5\mathbb {Z} .}
LetG{\displaystyle G}be a finite group, letK{\displaystyle K}be a commutativeringand letK[G]{\displaystyle K[G]}be thegroup algebraofG{\displaystyle G}overK.{\displaystyle K.}This algebra is free and a basis can be indexed by the elements ofG.{\displaystyle G.}Most often the basis is identified withG{\displaystyle G}. Every elementf∈K[G]{\displaystyle f\in K[G]}can then be uniquely expressed as
The multiplication inK[G]{\displaystyle K[G]}extends that inG{\displaystyle G}distributively.
Now letV{\displaystyle V}be aK{\displaystyle K}–moduleand letρ:G→GL(V){\displaystyle \rho :G\to {\text{GL}}(V)}be a linear representation ofG{\displaystyle G}inV.{\displaystyle V.}We definesv=ρ(s)v{\displaystyle sv=\rho (s)v}for alls∈G{\displaystyle s\in G}andv∈V{\displaystyle v\in V}. By linear extensionV{\displaystyle V}is endowed with the structure of a left-K[G]{\displaystyle K[G]}–module. Vice versa we obtain a linear representation ofG{\displaystyle G}starting from aK[G]{\displaystyle K[G]}–moduleV{\displaystyle V}. Additionally, homomorphisms of representations are in bijective correspondence with group algebra homomorphisms. Therefore, these terms may be used interchangeably.[1][2]This is an example of anisomorphism of categories.
SupposeK=C.{\displaystyle K=\mathbb {C} .}In this case the leftC[G]{\displaystyle \mathbb {C} [G]}–module given byC[G]{\displaystyle \mathbb {C} [G]}itself corresponds to the left-regular representation. In the same wayC[G]{\displaystyle \mathbb {C} [G]}as a rightC[G]{\displaystyle \mathbb {C} [G]}–module corresponds to the right-regular representation.
In the following we will define theconvolution algebra: LetG{\displaystyle G}be a group, the setL1(G):={f:G→C}{\displaystyle L^{1}(G):=\{f:G\to \mathbb {C} \}}is aC{\displaystyle \mathbb {C} }–vector space with the operations addition and scalar multiplication then this vector space is isomorphic toC|G|.{\displaystyle \mathbb {C} ^{|G|}.}The convolution of two elementsf,h∈L1(G){\displaystyle f,h\in L^{1}(G)}defined by
makesL1(G){\displaystyle L^{1}(G)}analgebra. The algebraL1(G){\displaystyle L^{1}(G)}is called theconvolution algebra.
The convolution algebra is free and has a basis indexed by the group elements:(δs)s∈G,{\displaystyle (\delta _{s})_{s\in G},}where
Using the properties of the convolution we obtain:δs∗δt=δst.{\displaystyle \delta _{s}*\delta _{t}=\delta _{st}.}
We define a map betweenL1(G){\displaystyle L^{1}(G)}andC[G],{\displaystyle \mathbb {C} [G],}by definingδs↦es{\displaystyle \delta _{s}\mapsto e_{s}}on the basis(δs)s∈G{\displaystyle (\delta _{s})_{s\in G}}and extending it linearly. Obviously the prior map isbijective. A closer inspection of the convolution of two basis elements as shown in the equation above reveals that the multiplication inL1(G){\displaystyle L^{1}(G)}corresponds to that inC[G].{\displaystyle \mathbb {C} [G].}Thus, the convolution algebra and the group algebra are isomorphic as algebras.
Theinvolution
turnsL1(G){\displaystyle L^{1}(G)}into a∗{\displaystyle ^{*}}–algebra. We haveδs∗=δs−1.{\displaystyle \delta _{s}^{*}=\delta _{s^{-1}}.}
A representation(π,Vπ){\displaystyle (\pi ,V_{\pi })}of a groupG{\displaystyle G}extends to a∗{\displaystyle ^{*}}–algebra homomorphismπ:L1(G)→End(Vπ){\displaystyle \pi :L^{1}(G)\to {\text{End}}(V_{\pi })}byπ(δs)=π(s).{\displaystyle \pi (\delta _{s})=\pi (s).}Since multiplicativity is a characteristic property of algebra homomorphisms,π{\displaystyle \pi }satisfiesπ(f∗h)=π(f)π(h).{\displaystyle \pi (f*h)=\pi (f)\pi (h).}Ifπ{\displaystyle \pi }is unitary, we also obtainπ(f)∗=π(f∗).{\displaystyle \pi (f)^{*}=\pi (f^{*}).}For the definition of a unitary representation, please refer to the chapter onproperties. In that chapter we will see that (without loss of generality) every linear representation can be assumed to be unitary.
Using the convolution algebra we can implement aFourier transformationon a groupG.{\displaystyle G.}In the area ofharmonic analysisit is shown that the following definition is consistent with the definition of the Fourier transformation onR.{\displaystyle \mathbb {R} .}
Letρ:G→GL(Vρ){\displaystyle \rho :G\to {\text{GL}}(V_{\rho })}be a representation and letf∈L1(G){\displaystyle f\in L^{1}(G)}be aC{\displaystyle \mathbb {C} }-valued function onG{\displaystyle G}. The Fourier transformf^(ρ)∈End(Vρ){\displaystyle {\hat {f}}(\rho )\in {\text{End}}(V_{\rho })}off{\displaystyle f}is defined as
This transformation satisfiesf∗g^(ρ)=f^(ρ)⋅g^(ρ).{\displaystyle {\widehat {f*g}}(\rho )={\hat {f}}(\rho )\cdot {\hat {g}}(\rho ).}
A map between two representations(ρ,Vρ),(τ,Vτ){\displaystyle (\rho ,V_{\rho }),\,(\tau ,V_{\tau })}of the same groupG{\displaystyle G}is a linear mapT:Vρ→Vτ,{\displaystyle T:V_{\rho }\to V_{\tau },}with the property thatτ(s)∘T=T∘ρ(s){\displaystyle \tau (s)\circ T=T\circ \rho (s)}holds for alls∈G.{\displaystyle s\in G.}In other words, the following diagram commutes for alls∈G{\displaystyle s\in G}:
Such a map is also calledG{\displaystyle G}–linear, or anequivariant map. Thekernel, theimageand thecokernelofT{\displaystyle T}are defined by default. The composition of equivariant maps is again an equivariant map. There is acategory of representationswith equivariant maps as itsmorphisms. They are againG{\displaystyle G}–modules. Thus, they provide representations ofG{\displaystyle G}due to the correlation described in the previous section.
Letρ:G→GL(V){\displaystyle \rho :G\to {\text{GL}}(V)}be a linear representation ofG.{\displaystyle G.}LetW{\displaystyle W}be aG{\displaystyle G}-invariant subspace ofV,{\displaystyle V,}that is,ρ(s)w∈W{\displaystyle \rho (s)w\in W}for alls∈G{\displaystyle s\in G}andw∈W{\displaystyle w\in W}. The restrictionρ(s)|W{\displaystyle \rho (s)|_{W}}is an isomorphism ofW{\displaystyle W}onto itself. Becauseρ(s)|W∘ρ(t)|W=ρ(st)|W{\displaystyle \rho (s)|_{W}\circ \rho (t)|_{W}=\rho (st)|_{W}}holds for alls,t∈G,{\displaystyle s,t\in G,}this construction is a representation ofG{\displaystyle G}inW.{\displaystyle W.}It is calledsubrepresentationofV.{\displaystyle V.}Any representationVhas at least two subrepresentations, namely the one consisting only of 0, and the one consisting ofVitself. The representation is called anirreducible representation, if these two are the only subrepresentations. Some authors also call these representations simple, given that they are precisely thesimple modulesover the group algebraC[G]{\displaystyle \mathbb {C} [G]}.
Schur's lemmaputs a strong constraint on maps between irreducible representations. Ifρ1:G→GL(V1){\displaystyle \rho _{1}:G\to {\text{GL}}(V_{1})}andρ2:G→GL(V2){\displaystyle \rho _{2}:G\to {\text{GL}}(V_{2})}are both irreducible, andF:V1→V2{\displaystyle F:V_{1}\to V_{2}}is a linear map such thatρ2(s)∘F=F∘ρ1(s){\displaystyle \rho _{2}(s)\circ F=F\circ \rho _{1}(s)}for alls∈G.{\displaystyle s\in G.}, there is the following dichotomy:
Two representations(ρ,Vρ),(π,Vπ){\displaystyle (\rho ,V_{\rho }),(\pi ,V_{\pi })}are calledequivalentorisomorphic, if there exists aG{\displaystyle G}–linear vector space isomorphism between the representation spaces. In other words, they are isomorphic if there exists a bijective linear mapT:Vρ→Vπ,{\displaystyle T:V_{\rho }\to V_{\pi },}such thatT∘ρ(s)=π(s)∘T{\displaystyle T\circ \rho (s)=\pi (s)\circ T}for alls∈G.{\displaystyle s\in G.}In particular, equivalent representations have the same degree.
A representation(π,Vπ){\displaystyle (\pi ,V_{\pi })}is calledfaithfulwhenπ{\displaystyle \pi }isinjective. In this caseπ{\displaystyle \pi }induces an isomorphism betweenG{\displaystyle G}and the imageπ(G).{\displaystyle \pi (G).}As the latter is a subgroup ofGL(Vπ),{\displaystyle {\text{GL}}(V_{\pi }),}we can regardG{\displaystyle G}viaπ{\displaystyle \pi }as subgroup ofAut(Vπ).{\displaystyle {\text{Aut}}(V_{\pi }).}
We can restrict the range as well as the domain:
LetH{\displaystyle H}be a subgroup ofG.{\displaystyle G.}Letρ{\displaystyle \rho }be a linear representation ofG.{\displaystyle G.}We denote byResH(ρ){\displaystyle {\text{Res}}_{H}(\rho )}the restriction ofρ{\displaystyle \rho }to the subgroupH.{\displaystyle H.}
If there is no danger of confusion, we might use onlyRes(ρ){\displaystyle {\text{Res}}(\rho )}or in shortResρ.{\displaystyle {\text{Res}}\rho .}
The notationResH(V){\displaystyle {\text{Res}}_{H}(V)}or in shortRes(V){\displaystyle {\text{Res}}(V)}is also used to denote the restriction of the representationV{\displaystyle V}ofG{\displaystyle G}ontoH.{\displaystyle H.}
Letf{\displaystyle f}be a function onG.{\displaystyle G.}We writeResH(f){\displaystyle {\text{Res}}_{H}(f)}or shortlyRes(f){\displaystyle {\text{Res}}(f)}for the restriction to the subgroupH.{\displaystyle H.}
It can be proven that the number of irreducible representations of a groupG{\displaystyle G}(or correspondingly the number of simpleC[G]{\displaystyle \mathbb {C} [G]}–modules) equals the number ofconjugacy classesofG.{\displaystyle G.}
A representation is calledsemisimpleorcompletely reducibleif it can be written as adirect sumof irreducible representations. This is analogous to the corresponding definition for a semisimple algebra.
For the definition of the direct sum of representations please refer to the section ondirect sums of representations.
A representation is calledisotypicif it is a direct sum of pairwise isomorphic irreducible representations.
Let(ρ,Vρ){\displaystyle (\rho ,V_{\rho })}be a given representation of a groupG.{\displaystyle G.}Letτ{\displaystyle \tau }be an irreducible representation ofG.{\displaystyle G.}Theτ{\displaystyle \tau }–isotypeVρ(τ){\displaystyle V_{\rho }(\tau )}ofG{\displaystyle G}is defined as the sum of all irreducible subrepresentations ofV{\displaystyle V}isomorphic toτ.{\displaystyle \tau .}
Every vector space overC{\displaystyle \mathbb {C} }can be provided with aninner product. A representationρ{\displaystyle \rho }of a groupG{\displaystyle G}in a vector space endowed with an inner product is calledunitaryifρ(s){\displaystyle \rho (s)}isunitaryfor everys∈G.{\displaystyle s\in G.}This means that in particular everyρ(s){\displaystyle \rho (s)}isdiagonalizable. For more details see the article onunitary representations.
A representation is unitary with respect to a given inner product if and only if the inner product is invariant with regard to the induced operation ofG,{\displaystyle G,}i.e. if and only if(v|u)=(ρ(s)v|ρ(s)u){\displaystyle (v|u)=(\rho (s)v|\rho (s)u)}holds for allv,u∈Vρ,s∈G.{\displaystyle v,u\in V_{\rho },s\in G.}
A given inner product(⋅|⋅){\displaystyle (\cdot |\cdot )}can be replaced by an invariant inner product by exchanging(v|u){\displaystyle (v|u)}with
Thus, without loss of generality we can assume that every further considered representation is unitary.
Example.LetG=D6={id,μ,μ2,ν,μν,μ2ν}{\displaystyle G=D_{6}=\{{\text{id}},\mu ,\mu ^{2},\nu ,\mu \nu ,\mu ^{2}\nu \}}be thedihedral groupoforder6{\displaystyle 6}generated byμ,ν{\displaystyle \mu ,\nu }which fulfil the propertiesord(ν)=2,ord(μ)=3{\displaystyle {\text{ord}}(\nu )=2,{\text{ord}}(\mu )=3}andνμν=μ2.{\displaystyle \nu \mu \nu =\mu ^{2}.}Letρ:D6→GL3(C){\displaystyle \rho :D_{6}\to {\text{GL}}_{3}(\mathbb {C} )}be a linear representation ofD6{\displaystyle D_{6}}defined on the generators by:
This representation is faithful. The subspaceCe2{\displaystyle \mathbb {C} e_{2}}is aD6{\displaystyle D_{6}}–invariant subspace. Thus, there exists a nontrivial subrepresentationρ|Ce2:D6→C×{\displaystyle \rho |_{\mathbb {C} e_{2}}:D_{6}\to \mathbb {C} ^{\times }}withν↦−1,μ↦1.{\displaystyle \nu \mapsto -1,\mu \mapsto 1.}Therefore, the representation is not irreducible. The mentioned subrepresentation is of degree one and irreducible.
Thecomplementary subspaceofCe2{\displaystyle \mathbb {C} e_{2}}isD6{\displaystyle D_{6}}–invariant as well. Therefore, we obtain the subrepresentationρ|Ce1⊕Ce3{\displaystyle \rho |_{\mathbb {C} e_{1}\oplus \mathbb {C} e_{3}}}with
This subrepresentation is also irreducible. That means, the original representation is completely reducible:
Both subrepresentations are isotypic and are the two only non-zero isotypes ofρ.{\displaystyle \rho .}
The representationρ{\displaystyle \rho }is unitary with regard to the standard inner product onC3,{\displaystyle \mathbb {C} ^{3},}becauseρ(μ){\displaystyle \rho (\mu )}andρ(ν){\displaystyle \rho (\nu )}are unitary.
LetT:C3→C3{\displaystyle T:\mathbb {C} ^{3}\to \mathbb {C} ^{3}}be any vector space isomorphism. Thenη:D6→GL3(C),{\displaystyle \eta :D_{6}\to {\text{GL}}_{3}(\mathbb {C} ),}which is defined by the equationη(s):=T∘ρ(s)∘T−1{\displaystyle \eta (s):=T\circ \rho (s)\circ T^{-1}}for alls∈D6,{\displaystyle s\in D_{6},}is a representation isomorphic toρ.{\displaystyle \rho .}
By restricting the domain of the representation to a subgroup, e.g.H={id,μ,μ2},{\displaystyle H=\{{\text{id}},\mu ,\mu ^{2}\},}we obtain the representationResH(ρ).{\displaystyle {\text{Res}}_{H}(\rho ).}This representation is defined by the imageρ(μ),{\displaystyle \rho (\mu ),}whose explicit form is shown above.
Letρ:G→GL(V){\displaystyle \rho :G\to {\text{GL}}(V)}be a given representation. Thedual representationorcontragredient representationρ∗:G→GL(V∗){\displaystyle \rho ^{*}:G\to {\text{GL}}(V^{*})}is a representation ofG{\displaystyle G}in thedual vector spaceofV.{\displaystyle V.}It is defined by the property
With regard to the natural pairing⟨α,v⟩:=α(v){\displaystyle \langle \alpha ,v\rangle :=\alpha (v)}betweenV∗{\displaystyle V^{*}}andV{\displaystyle V}the definition above provides the equation:
For an example, see the main page on this topic:Dual representation.
Let(ρ1,V1){\displaystyle (\rho _{1},V_{1})}and(ρ2,V2){\displaystyle (\rho _{2},V_{2})}be a representation ofG1{\displaystyle G_{1}}andG2,{\displaystyle G_{2},}respectively. The direct sum of these representations is a linear representation and is defined as
Letρ1,ρ2{\displaystyle \rho _{1},\rho _{2}}be representations of the same groupG.{\displaystyle G.}For the sake of simplicity, the direct sum of these representations is defined as a representation ofG,{\displaystyle G,}i.e. it is given asρ1⊕ρ2:G→GL(V1⊕V2),{\displaystyle \rho _{1}\oplus \rho _{2}:G\to {\text{GL}}(V_{1}\oplus V_{2}),}by viewingG{\displaystyle G}as the diagonal subgroup ofG×G.{\displaystyle G\times G.}
Example.Let (herei{\displaystyle i}andω{\displaystyle \omega }are the imaginary unit and the primitive cube root of unity respectively):
Then
As it is sufficient to consider the image of the generating element, we find that
Letρ1:G1→GL(V1),ρ2:G2→GL(V2){\displaystyle \rho _{1}:G_{1}\to {\text{GL}}(V_{1}),\rho _{2}:G_{2}\to {\text{GL}}(V_{2})}be linear representations. We define the linear representationρ1⊗ρ2:G1×G2→GL(V1⊗V2){\displaystyle \rho _{1}\otimes \rho _{2}:G_{1}\times G_{2}\to {\text{GL}}(V_{1}\otimes V_{2})}into thetensor productofV1{\displaystyle V_{1}}andV2{\displaystyle V_{2}}byρ1⊗ρ2(s1,s2)=ρ1(s1)⊗ρ2(s2),{\displaystyle \rho _{1}\otimes \rho _{2}(s_{1},s_{2})=\rho _{1}(s_{1})\otimes \rho _{2}(s_{2}),}in whichs1∈G1,s2∈G2.{\displaystyle s_{1}\in G_{1},s_{2}\in G_{2}.}This representation is calledouter tensor productof the representationsρ1{\displaystyle \rho _{1}}andρ2.{\displaystyle \rho _{2}.}The existence and uniqueness is a consequence of theproperties of the tensor product.
Example.We reexamine the example provided for thedirect sum:
The outer tensor product
Using the standard basis ofC2⊗C3≅C6{\displaystyle \mathbb {C} ^{2}\otimes \mathbb {C} ^{3}\cong \mathbb {C} ^{6}}we have the following for the generating element:
Remark.Note that thedirect sumand the tensor products have different degrees and hence are different representations.
Letρ1:G→GL(V1),ρ2:G→GL(V2){\displaystyle \rho _{1}:G\to {\text{GL}}(V_{1}),\rho _{2}:G\to {\text{GL}}(V_{2})}be two linear representations of the same group. Lets{\displaystyle s}be an element ofG.{\displaystyle G.}Thenρ(s)∈GL(V1⊗V2){\displaystyle \rho (s)\in {\text{GL}}(V_{1}\otimes V_{2})}is defined byρ(s)(v1⊗v2)=ρ1(s)v1⊗ρ2(s)v2,{\displaystyle \rho (s)(v_{1}\otimes v_{2})=\rho _{1}(s)v_{1}\otimes \rho _{2}(s)v_{2},}forv1∈V1,v2∈V2,{\displaystyle v_{1}\in V_{1},v_{2}\in V_{2},}and we writeρ(s)=ρ1(s)⊗ρ2(s).{\displaystyle \rho (s)=\rho _{1}(s)\otimes \rho _{2}(s).}Then the maps↦ρ(s){\displaystyle s\mapsto \rho (s)}defines a linear representation ofG,{\displaystyle G,}which is also calledtensor productof the given representations.
These two cases have to be strictly distinguished. The first case is a representation of the group product into the tensor product of the corresponding representation spaces. The second case is a representation of the groupG{\displaystyle G}into the tensor product of two representation spaces of this one group. But this last case can be viewed as a special case of the first one by focusing on the diagonal subgroupG×G.{\displaystyle G\times G.}This definition can be iterated a finite number of times.
LetV{\displaystyle V}andW{\displaystyle W}be representations of the groupG.{\displaystyle G.}ThenHom(V,W){\displaystyle {\text{Hom}}(V,W)}is a representation by virtue of the following identity:Hom(V,W)=V∗⊗W{\displaystyle {\text{Hom}}(V,W)=V^{*}\otimes W}. LetB∈Hom(V,W){\displaystyle B\in {\text{Hom}}(V,W)}and letρ{\displaystyle \rho }be the representation onHom(V,W).{\displaystyle {\text{Hom}}(V,W).}LetρV{\displaystyle \rho _{V}}be the representation onV{\displaystyle V}andρW{\displaystyle \rho _{W}}the representation onW.{\displaystyle W.}Then the identity above leads to the following result:
Letρ:G→V⊗V{\displaystyle \rho :G\to V\otimes V}be a linear representation ofG.{\displaystyle G.}Let(ek){\displaystyle (e_{k})}be a basis ofV.{\displaystyle V.}Defineϑ:V⊗V→V⊗V{\displaystyle \vartheta :V\otimes V\to V\otimes V}by extendingϑ(ek⊗ej)=ej⊗ek{\displaystyle \vartheta (e_{k}\otimes e_{j})=e_{j}\otimes e_{k}}linearly. It then holds thatϑ2=1{\displaystyle \vartheta ^{2}=1}and thereforeV⊗V{\displaystyle V\otimes V}splits up intoV⊗V=Sym2(V)⊕Alt2(V),{\displaystyle V\otimes V={\text{Sym}}^{2}(V)\oplus {\text{Alt}}^{2}(V),}in which
These subspaces areG{\displaystyle G}–invariant and by this define subrepresentations which are called thesymmetric squareand thealternating square, respectively. These subrepresentations are also defined inV⊗m,{\displaystyle V^{\otimes m},}although in this case they are denoted wedge product⋀mV{\displaystyle \bigwedge ^{m}V}and symmetric productSymm(V).{\displaystyle {\text{Sym}}^{m}(V).}In case thatm>2,{\displaystyle m>2,}the vector spaceV⊗m{\displaystyle V^{\otimes m}}is in general not equal to the direct sum of these two products.
In order to understand representations more easily, a decomposition of the representation space into the direct sum of simpler subrepresentations would be desirable.
This can be achieved for finite groups as we will see in the following results. More detailed explanations and proofs may be found in[1]and[2].
A subrepresentation and its complement determine a representation uniquely.
The following theorem will be presented in a more general way, as it provides a very beautiful result about representations ofcompact– and therefore also of finite – groups:
Or in the language ofK[G]{\displaystyle K[G]}-modules: Ifchar(K)=0,{\displaystyle {\text{char}}(K)=0,}the group algebraK[G]{\displaystyle K[G]}is semisimple, i.e. it is the direct sum of simple algebras.
Note that this decomposition is not unique. However, the number of how many times a subrepresentation isomorphic to a given irreducible representation is occurring in this decomposition is independent of the choice of decomposition.
The canonical decomposition
To achieve a unique decomposition, one has to combine all the irreducible subrepresentations that are isomorphic to each other. That means, the representation space is decomposed into a direct sum of its isotypes. This decomposition is uniquely determined. It is called thecanonical decomposition.
Let(τj)j∈I{\displaystyle (\tau _{j})_{j\in I}}be the set of all irreducible representations of a groupG{\displaystyle G}up to isomorphism. LetV{\displaystyle V}be a representation ofG{\displaystyle G}and let{V(τj)|j∈I}{\displaystyle \{V(\tau _{j})|j\in I\}}be the set of all isotypes ofV.{\displaystyle V.}Theprojectionpj:V→V(τj){\displaystyle p_{j}:V\to V(\tau _{j})}corresponding to the canonical decomposition is given by
wherenj=dim(τj),{\displaystyle n_{j}=\dim(\tau _{j}),}g=ord(G){\displaystyle g={\text{ord}}(G)}andχτj{\displaystyle \chi _{\tau _{j}}}is the character belonging toτj.{\displaystyle \tau _{j}.}
In the following, we show how to determine the isotype to the trivial representation:
Definition (Projection formula).For every representation(ρ,V){\displaystyle (\rho ,V)}of a groupG{\displaystyle G}we define
In general,ρ(s):V→V{\displaystyle \rho (s):V\to V}is notG{\displaystyle G}-linear. We define
ThenP{\displaystyle P}is aG{\displaystyle G}-linear map, because
This proposition enables us to determine the isotype to the trivial subrepresentation of a given representation explicitly.
How often the trivial representation occurs inV{\displaystyle V}is given byTr(P).{\displaystyle {\text{Tr}}(P).}This result is a consequence of the fact that the eigenvalues of aprojectionare only0{\displaystyle 0}or1{\displaystyle 1}and that the eigenspace corresponding to the eigenvalue1{\displaystyle 1}is the image of the projection. Since the trace of the projection is the sum of all eigenvalues, we obtain the following result
in whichV(1){\displaystyle V(1)}denotes the isotype of the trivial representation.
LetVπ{\displaystyle V_{\pi }}be a nontrivial irreducible representation ofG.{\displaystyle G.}Then the isotype to the trivial representation ofπ{\displaystyle \pi }is the null space. That means the following equation holds
Lete1,...,en{\displaystyle e_{1},...,e_{n}}be anorthonormal basisofVπ.{\displaystyle V_{\pi }.}Then we have:
Therefore, the following is valid for a nontrivial irreducible representationV{\displaystyle V}:
Example.LetG=Per(3){\displaystyle G={\text{Per}}(3)}be the permutation groups in three elements. Letρ:Per(3)→GL5(C){\displaystyle \rho :{\text{Per}}(3)\to {\text{GL}}_{5}(\mathbb {C} )}be a linear representation ofPer(3){\displaystyle {\text{Per}}(3)}defined on the generating elements as follows:
This representation can be decomposed on first look into the left-regular representation ofPer(3),{\displaystyle {\text{Per}}(3),}which is denoted byπ{\displaystyle \pi }in the following, and the representationη:Per(3)→GL2(C){\displaystyle \eta :{\text{Per}}(3)\to {\text{GL}}_{2}(\mathbb {C} )}with
With the help of theirreducibility criteriontaken from the next chapter, we could realize thatη{\displaystyle \eta }is irreducible butπ{\displaystyle \pi }is not. This is because (in terms of the inner product from”Inner product and characters”below) we have(η|η)=1,(π|π)=2.{\displaystyle (\eta |\eta )=1,(\pi |\pi )=2.}
The subspaceC(e1+e2+e3){\displaystyle \mathbb {C} (e_{1}+e_{2}+e_{3})}ofC3{\displaystyle \mathbb {C} ^{3}}is invariant with respect to the left-regular representation. Restricted to this subspace we obtain the trivial representation.
The orthogonal complement ofC(e1+e2+e3){\displaystyle \mathbb {C} (e_{1}+e_{2}+e_{3})}isC(e1−e2)⊕C(e1+e2−2e3).{\displaystyle \mathbb {C} (e_{1}-e_{2})\oplus \mathbb {C} (e_{1}+e_{2}-2e_{3}).}Restricted to this subspace, which is alsoG{\displaystyle G}–invariant as we have seen above, we obtain the representationτ{\displaystyle \tau }given by
Again, we can use the irreducibility criterion of the next chapter to prove thatτ{\displaystyle \tau }is irreducible. Now,η{\displaystyle \eta }andτ{\displaystyle \tau }are isomorphic becauseη(s)=B∘τ(s)∘B−1{\displaystyle \eta (s)=B\circ \tau (s)\circ B^{-1}}for alls∈Per(3),{\displaystyle s\in {\text{Per}}(3),}in whichB:C2→C2{\displaystyle B:\mathbb {C} ^{2}\to \mathbb {C} ^{2}}is given by the matrix
A decomposition of(ρ,C5){\displaystyle (\rho ,\mathbb {C} ^{5})}in irreducible subrepresentations is:ρ=τ⊕η⊕1{\displaystyle \rho =\tau \oplus \eta \oplus 1}where1{\displaystyle 1}denotes the trivial representation and
is the corresponding decomposition of the representation space.
We obtain the canonical decomposition by combining all the isomorphic irreducible subrepresentations:ρ1:=η⊕τ{\displaystyle \rho _{1}:=\eta \oplus \tau }is theτ{\displaystyle \tau }-isotype ofρ{\displaystyle \rho }and consequently the canonical decomposition is given by
The theorems above are in general not valid for infinite groups. This will be demonstrated by the following example: let
Together with the matrix multiplicationG{\displaystyle G}is an infinite group.G{\displaystyle G}acts onC2{\displaystyle \mathbb {C} ^{2}}by matrix-vector multiplication. We consider the representationρ(A)=A{\displaystyle \rho (A)=A}for allA∈G.{\displaystyle A\in G.}The subspaceCe1{\displaystyle \mathbb {C} e_{1}}is aG{\displaystyle G}-invariant subspace. However, there exists noG{\displaystyle G}-invariant complement to this subspace. The assumption that such a complement exists would entail that every matrix isdiagonalizableoverC.{\displaystyle \mathbb {C} .}This is known to be wrong and thus yields a contradiction.
The moral of the story is that if we consider infinite groups, it is possible that a representation - even one that is not irreducible - can not be decomposed into a direct sum of irreducible subrepresentations.
Thecharacterof a representationρ:G→GL(V){\displaystyle \rho :G\to {\text{GL}}(V)}is defined as the map
Even though the character is a map between two groups, it is not in general agroup homomorphism, as the following example shows.
Letρ:Z/2Z×Z/2Z→GL2(C){\displaystyle \rho :\mathbb {Z} /2\mathbb {Z} \times \mathbb {Z} /2\mathbb {Z} \to {\text{GL}}_{2}(\mathbb {C} )}be the representation defined by:
The characterχρ{\displaystyle \chi _{\rho }}is given by
Characters ofpermutation representationsare particularly easy to compute. IfVis theG-representation corresponding to the left action ofG{\displaystyle G}on a finite setX{\displaystyle X}, then
For example,[5]the character of theregular representationR{\displaystyle R}is given by
wheree{\displaystyle e}denotes the neutral element ofG.{\displaystyle G.}
A crucial property of characters is the formula
This formula follows from the fact that thetraceof a productABof two square matrices is the same as the trace ofBA. FunctionsG→C{\displaystyle G\to \mathbb {C} }satisfying such a formula are calledclass functions. Put differently, class functions and in particular characters are constant on eachconjugacy classCs={tst−1|t∈G}.{\displaystyle C_{s}=\{tst^{-1}|t\in G\}.}It also follows from elementary properties of the trace thatχ(s){\displaystyle \chi (s)}is the sum of theeigenvaluesofρ(s){\displaystyle \rho (s)}with multiplicity. If the degree of the representation isn, then the sum isnlong. Ifshas orderm, these eigenvalues are allm-throots of unity. This fact can be used to show thatχ(s−1)=χ(s)¯,∀s∈G{\displaystyle \chi (s^{-1})={\overline {\chi (s)}},\,\,\,\forall \,s\in G}and it also implies|χ(s)|⩽n.{\displaystyle |\chi (s)|\leqslant n.}
Since the trace of the identity matrix is the number of rows,χ(e)=n,{\displaystyle \chi (e)=n,}wheree{\displaystyle e}is the neutral element ofG{\displaystyle G}andnis the dimension of the representation. In general,{s∈G|χ(s)=n}{\displaystyle \{s\in G|\chi (s)=n\}}is anormal subgroupinG.{\displaystyle G.}The following table shows how the charactersχ1,χ2{\displaystyle \chi _{1},\chi _{2}}of two given representationsρ1:G→GL(V1),ρ2:G→GL(V2){\displaystyle \rho _{1}:G\to {\text{GL}}(V_{1}),\rho _{2}:G\to {\text{GL}}(V_{2})}give rise to characters of related representations.
χ1χ2.{\displaystyle \chi _{1}\chi _{2}.}
By construction, there is a direct sum decomposition ofV⊗V=Sym2(V)⊕⋀2V{\displaystyle V\otimes V=Sym^{2}(V)\oplus \bigwedge ^{2}V}. On characters, this corresponds to the fact that the sum of the last two expressions in the table isχ(s)2{\displaystyle \chi (s)^{2}}, the character ofV⊗V{\displaystyle V\otimes V}.
In order to show some particularly interesting results about characters, it is rewarding to consider a more general type of functions on groups:
Definition (Class functions).A functionφ:G→C{\displaystyle \varphi :G\to \mathbb {C} }is called aclass functionif it is constant on conjugacy classes ofG{\displaystyle G}, i.e.
Note that every character is a class function, as the trace of a matrix is preserved under conjugation.
The set of all class functions is aC{\displaystyle \mathbb {C} }–algebra and is denoted byCclass(G){\displaystyle \mathbb {C} _{\text{class}}(G)}. Its dimension is equal to the number of conjugacy classes ofG.{\displaystyle G.}
Proofs of the following results of this chapter may be found in[1],[2]and[3].
Aninner productcan be defined on the set of all class functions on a finite group:
Orthonormal property.Ifχ1,…,χk{\displaystyle \chi _{1},\ldots ,\chi _{k}}are the distinct irreducible characters ofG{\displaystyle G}, they form an orthonormal basis for the vector space of all class functions with respect to the inner product defined above, i.e.
One might verify that the irreducible characters generateCclass(G){\displaystyle \mathbb {C} _{\text{class}}(G)}by showing that there exists no nonzero class function which is orthogonal to all the irreducible characters. Forρ{\displaystyle \rho }a representation andf{\displaystyle f}a class function, denoteρf=∑gf(g)ρ(g).{\displaystyle \rho _{f}=\sum _{g}f(g)\rho (g).}Then forρ{\displaystyle \rho }irreducible, we haveρf=|G|n⟨f,χV∗⟩∈End(V){\displaystyle \rho _{f}={\frac {|G|}{n}}\langle f,\chi _{V}^{*}\rangle \in End(V)}fromSchur's lemma. Supposef{\displaystyle f}is a class function which is orthogonal to all the characters. Then by the above we haveρf=0{\displaystyle \rho _{f}=0}wheneverρ{\displaystyle \rho }is irreducible. But then it follows thatρf=0{\displaystyle \rho _{f}=0}for allρ{\displaystyle \rho }, by decomposability. Takeρ{\displaystyle \rho }to be the regular representation. Applyingρf{\displaystyle \rho _{f}}to some particular basis elementg{\displaystyle g}, we getf(g)=0{\displaystyle f(g)=0}. Since this is true for allg{\displaystyle g}, we havef=0.{\displaystyle f=0.}
It follows from the orthonormal property that the number of non-isomorphic irreducible representations of a groupG{\displaystyle G}is equal to the number ofconjugacy classesofG.{\displaystyle G.}
Furthermore, a class function onG{\displaystyle G}is a character ofG{\displaystyle G}if and only if it can be written as a linear combination of the distinct irreducible charactersχj{\displaystyle \chi _{j}}with non-negative integer coefficients: ifφ{\displaystyle \varphi }is a class function onG{\displaystyle G}such thatφ=c1χ1+⋯+ckχk{\displaystyle \varphi =c_{1}\chi _{1}+\cdots +c_{k}\chi _{k}}wherecj{\displaystyle c_{j}}non-negative integers, thenφ{\displaystyle \varphi }is the character of the direct sumc1τ1⊕⋯⊕ckτk{\displaystyle c_{1}\tau _{1}\oplus \cdots \oplus c_{k}\tau _{k}}of the representationsτj{\displaystyle \tau _{j}}corresponding toχj.{\displaystyle \chi _{j}.}Conversely, it is always possible to write any character as a sum of irreducible characters.
Theinner productdefined above can be extended on the set of allC{\displaystyle \mathbb {C} }-valued functionsL1(G){\displaystyle L^{1}(G)}on a finite group:
Asymmetric bilinear formcan also be defined onL1(G):{\displaystyle L^{1}(G):}
These two forms match on the set of characters. If there is no danger of confusion the index of both forms(⋅|⋅)G{\displaystyle (\cdot |\cdot )_{G}}and⟨⋅|⋅⟩G{\displaystyle \langle \cdot |\cdot \rangle _{G}}will be omitted.
LetV1,V2{\displaystyle V_{1},V_{2}}be twoC[G]{\displaystyle \mathbb {C} [G]}–modules. Note thatC[G]{\displaystyle \mathbb {C} [G]}–modules are simply representations ofG{\displaystyle G}. Since the orthonormal property yields the number of irreducible representations ofG{\displaystyle G}is exactly the number of its conjugacy classes, then there are exactly as many simpleC[G]{\displaystyle \mathbb {C} [G]}–modules (up to isomorphism) as there are conjugacy classes ofG.{\displaystyle G.}
We define⟨V1,V2⟩G:=dim(HomG(V1,V2)),{\displaystyle \langle V_{1},V_{2}\rangle _{G}:=\dim({\text{Hom}}^{G}(V_{1},V_{2})),}in whichHomG(V1,V2){\displaystyle {\text{Hom}}^{G}(V_{1},V_{2})}is the vector space of allG{\displaystyle G}–linear maps. This form is bilinear with respect to the direct sum.
In the following, these bilinear forms will allow us to obtain some important results with respect to the decomposition and irreducibility of representations.
For instance, letχ1{\displaystyle \chi _{1}}andχ2{\displaystyle \chi _{2}}be the characters ofV1{\displaystyle V_{1}}andV2,{\displaystyle V_{2},}respectively. Then⟨χ1,χ2⟩G=(χ1|χ2)G=⟨V1,V2⟩G.{\displaystyle \langle \chi _{1},\chi _{2}\rangle _{G}=(\chi _{1}|\chi _{2})_{G}=\langle V_{1},V_{2}\rangle _{G}.}
It is possible to derive the following theorem from the results above, along with Schur's lemma and the complete reducibility of representations.
With this we obtain a very useful result to analyse representations:
Irreducibility criterion.Letχ{\displaystyle \chi }be the character of the representationV,{\displaystyle V,}then we have(χ|χ)∈N0.{\displaystyle (\chi |\chi )\in \mathbb {N} _{0}.}The case(χ|χ)=1{\displaystyle (\chi |\chi )=1}holds if and only ifV{\displaystyle V}is irreducible.
Therefore, using the first theorem, the characters of irreducible representations ofG{\displaystyle G}form anorthonormal setonCclass(G){\displaystyle \mathbb {C} _{\text{class}}(G)}with respect to this inner product.
In terms of the group algebra, this means thatC[G]≅⊕jEnd(Wj){\displaystyle \mathbb {C} [G]\cong \oplus _{j}{\text{End}}(W_{j})}as algebras.
As a numerical result we get:
in whichR{\displaystyle R}is the regular representation andχWj{\displaystyle \chi _{W_{j}}}andχR{\displaystyle \chi _{R}}are corresponding characters toWj{\displaystyle W_{j}}andR,{\displaystyle R,}respectively. Recall thate{\displaystyle e}denotes the neutral element of the group.
This formula is a "necessary and sufficient" condition for the problem of classifying the irreducible representations of a group up to isomorphism. It provides us with the means to check whether we found all the isomorphism classes of irreducible representations of a group.
Similarly, by using the character of the regular representation evaluated ats≠e,{\displaystyle s\neq e,}we get the equation:
Using the description of representations via the convolution algebra we achieve an equivalent formulation of these equations:
TheFourier inversion formula:
In addition, thePlancherel formulaholds:
In both formulas(ρ,Vρ){\displaystyle (\rho ,V_{\rho })}is a linear representation of a groupG,s∈G{\displaystyle G,s\in G}andf,h∈L1(G).{\displaystyle f,h\in L^{1}(G).}
The corollary above has an additional consequence:
As was shown in the section onproperties of linear representations, we can - by restriction - obtain a representation of a subgroup starting from a representation of a group. Naturally we are interested in the reverse process: Is it possible to obtain the representation of a group starting from a representation of a subgroup? We will see that the induced representation defined below provides us with the necessary concept. Admittedly, this construction is not inverse but rather adjoint to the restriction.
Letρ:G→GL(Vρ){\displaystyle \rho :G\to {\text{GL}}(V_{\rho })}be a linear representation ofG.{\displaystyle G.}LetH{\displaystyle H}be a subgroup andρ|H{\displaystyle \rho |_{H}}the restriction. LetW{\displaystyle W}be a subrepresentation ofρH.{\displaystyle \rho _{H}.}We writeθ:H→GL(W){\displaystyle \theta :H\to {\text{GL}}(W)}to denote this representation. Lets∈G.{\displaystyle s\in G.}The vector spaceρ(s)(W){\displaystyle \rho (s)(W)}depends only on theleft cosetsH{\displaystyle sH}ofs.{\displaystyle s.}LetR{\displaystyle R}be arepresentative systemofG/H,{\displaystyle G/H,}then
is a subrepresentation ofVρ.{\displaystyle V_{\rho }.}
A representationρ{\displaystyle \rho }ofG{\displaystyle G}inVρ{\displaystyle V_{\rho }}is calledinducedby the representationθ{\displaystyle \theta }ofH{\displaystyle H}inW,{\displaystyle W,}if
HereWr=ρ(s)(W){\displaystyle W_{r}=\rho (s)(W)}for alls∈rH{\displaystyle s\in rH}and for allr∈R.{\displaystyle r\in R.}In other words: the representation(ρ,Vρ){\displaystyle (\rho ,V_{\rho })}is induced by(θ,W),{\displaystyle (\theta ,W),}if everyv∈Vρ{\displaystyle v\in V_{\rho }}can be written uniquely as
wherewr∈Wr{\displaystyle w_{r}\in W_{r}}for everyr∈R.{\displaystyle r\in R.}
We denote the representationρ{\displaystyle \rho }ofG{\displaystyle G}which is induced by the representationθ{\displaystyle \theta }ofH{\displaystyle H}asρ=IndHG(θ),{\displaystyle \rho ={\text{Ind}}_{H}^{G}(\theta ),}or in shortρ=Ind(θ),{\displaystyle \rho ={\text{Ind}}(\theta ),}if there is no danger of confusion. The representation space itself is frequently used instead of the representation map, i.e.V=IndHG(W),{\displaystyle V={\text{Ind}}_{H}^{G}(W),}orV=Ind(W),{\displaystyle V={\text{Ind}}(W),}if the representationV{\displaystyle V}is induced byW.{\displaystyle W.}
By using thegroup algebrawe obtain an alternative description of the induced representation:
LetG{\displaystyle G}be a group,V{\displaystyle V}aC[G]{\displaystyle \mathbb {C} [G]}–module andW{\displaystyle W}aC[H]{\displaystyle \mathbb {C} [H]}–submodule ofV{\displaystyle V}corresponding to the subgroupH{\displaystyle H}ofG.{\displaystyle G.}We say thatV{\displaystyle V}is induced byW{\displaystyle W}ifV=C[G]⊗C[H]W,{\displaystyle V=\mathbb {C} [G]\otimes _{\mathbb {C} [H]}W,}in whichG{\displaystyle G}acts on the first factor:s⋅(et⊗w)=est⊗w{\displaystyle s\cdot (e_{t}\otimes w)=e_{st}\otimes w}for alls,t∈G,w∈W.{\displaystyle s,t\in G,w\in W.}
The results introduced in this section will be presented without proof. These may be found in[1]and[2].
This means that if we interpretV′{\displaystyle V'}as aC[G]{\displaystyle \mathbb {C} [G]}–module, we haveHomH(Wθ,V′)≅HomG(Vρ,V′),{\displaystyle {\text{Hom}}^{H}(W_{\theta },V')\cong {\text{Hom}}^{G}(V_{\rho },V'),}whereHomG(Vρ,V′){\displaystyle {\text{Hom}}^{G}(V_{\rho },V')}is the vector space of allC[G]{\displaystyle \mathbb {C} [G]}–homomorphisms ofVρ{\displaystyle V_{\rho }}toV′.{\displaystyle V'.}The same is valid forHomH(Wθ,V′).{\displaystyle {\text{Hom}}^{H}(W_{\theta },V').}
Induction on class functions.In the same way as it was done with representations, we can - byinduction- obtain a class function on the group from a class function on a subgroup. Letφ{\displaystyle \varphi }be a class function onH.{\displaystyle H.}We define a functionφ′{\displaystyle \varphi '}onG{\displaystyle G}by
We sayφ′{\displaystyle \varphi '}isinducedbyφ{\displaystyle \varphi }and writeIndHG(φ)=φ′{\displaystyle {\text{Ind}}_{H}^{G}(\varphi )=\varphi '}orInd(φ)=φ′.{\displaystyle {\text{Ind}}(\varphi )=\varphi '.}
As a preemptive summary, the lesson to take from Frobenius reciprocity is that the mapsRes{\displaystyle {\text{Res}}}andInd{\displaystyle {\text{Ind}}}areadjointto each other.
LetW{\displaystyle W}be an irreducible representation ofH{\displaystyle H}and letV{\displaystyle V}be an irreducible representation ofG,{\displaystyle G,}then the Frobenius reciprocity tells us thatW{\displaystyle W}is contained inRes(V){\displaystyle {\text{Res}}(V)}as often asInd(W){\displaystyle {\text{Ind}}(W)}is contained inV.{\displaystyle V.}
This statement is also valid for theinner product.
George Mackeyestablished a criterion to verify the irreducibility of induced representations. For this we will first need some definitions and some specifications with respect to the notation.
Two representationsV1{\displaystyle V_{1}}andV2{\displaystyle V_{2}}of a groupG{\displaystyle G}are calleddisjoint, if they have no irreducible component in common, i.e. if⟨V1,V2⟩G=0.{\displaystyle \langle V_{1},V_{2}\rangle _{G}=0.}
LetG{\displaystyle G}be a group and letH{\displaystyle H}be a subgroup. We defineHs=sHs−1∩H{\displaystyle H_{s}=sHs^{-1}\cap H}fors∈G.{\displaystyle s\in G.}Let(ρ,W){\displaystyle (\rho ,W)}be a representation of the subgroupH.{\displaystyle H.}This defines by restriction a representationResHs(ρ){\displaystyle {\text{Res}}_{H_{s}}(\rho )}ofHs.{\displaystyle H_{s}.}We writeRess(ρ){\displaystyle {\text{Res}}_{s}(\rho )}forResHs(ρ).{\displaystyle {\text{Res}}_{H_{s}}(\rho ).}We also define another representationρs{\displaystyle \rho ^{s}}ofHs{\displaystyle H_{s}}byρs(t)=ρ(s−1ts).{\displaystyle \rho ^{s}(t)=\rho (s^{-1}ts).}These two representations are not to be confused.
For the case ofH{\displaystyle H}normal, we haveHs=H{\displaystyle H_{s}=H}andRess(ρ)=ρ{\displaystyle {\text{Res}}_{s}(\rho )=\rho }. Thus we obtain the following:
In this section we present some applications of the so far presented theory to normal subgroups and to a special group, the semidirect product of a subgroup with an abelian normal subgroup.
Note that ifA{\displaystyle A}is abelian, then the isotypic modules ofA{\displaystyle A}are irreducible, of degree one, and all homotheties.
We obtain also the following
IfA{\displaystyle A}is an abelian subgroup ofG{\displaystyle G}(not necessarily normal), generallydeg(τ)|(G:A){\displaystyle \deg(\tau )|(G:A)}is not satisfied, but neverthelessdeg(τ)≤(G:A){\displaystyle \deg(\tau )\leq (G:A)}is still valid.
In the following, letG=A⋊H{\displaystyle G=A\rtimes H}be a semidirect product such that the normal semidirect factor,A{\displaystyle A}, is abelian. The irreducible representations of such a groupG,{\displaystyle G,}can be classified by showing that all irreducible representations ofG{\displaystyle G}can be constructed from certain subgroups ofH{\displaystyle H}. This is the so-called method of “little groups” of Wigner and Mackey.
SinceA{\displaystyle A}isabelian, the irreducible characters ofA{\displaystyle A}have degree one and form the groupX=Hom(A,C×).{\displaystyle \mathrm {X} ={\text{Hom}}(A,\mathbb {C} ^{\times }).}The groupG{\displaystyle G}actsonX{\displaystyle \mathrm {X} }by(sχ)(a)=χ(s−1as){\displaystyle (s\chi )(a)=\chi (s^{-1}as)}fors∈G,χ∈X,a∈A.{\displaystyle s\in G,\chi \in \mathrm {X} ,a\in A.}
Let(χj)j∈X/H{\displaystyle (\chi _{j})_{j\in \mathrm {X} /H}}be arepresentative systemof theorbitofH{\displaystyle H}inX.{\displaystyle \mathrm {X} .}For everyj∈X/H{\displaystyle j\in \mathrm {X} /H}letHj={t∈H:tχj=χj}.{\displaystyle H_{j}=\{t\in H:t\chi _{j}=\chi _{j}\}.}This is a subgroup ofH.{\displaystyle H.}LetGj=A⋅Hj{\displaystyle G_{j}=A\cdot H_{j}}be the corresponding subgroup ofG.{\displaystyle G.}We now extend the functionχj{\displaystyle \chi _{j}}ontoGj{\displaystyle G_{j}}byχj(at)=χj(a){\displaystyle \chi _{j}(at)=\chi _{j}(a)}fora∈A,t∈Hj.{\displaystyle a\in A,t\in H_{j}.}Thus,χj{\displaystyle \chi _{j}}is a class function onGj.{\displaystyle G_{j}.}Moreover, sincetχj=χj{\displaystyle t\chi _{j}=\chi _{j}}for allt∈Hj,{\displaystyle t\in H_{j},}it can be shown thatχj{\displaystyle \chi _{j}}is a group homomorphism fromGj{\displaystyle G_{j}}toC×.{\displaystyle \mathbb {C} ^{\times }.}Therefore, we have a representation ofGj{\displaystyle G_{j}}of degree one which is equal to its own character.
Let nowρ{\displaystyle \rho }be an irreducible representation ofHj.{\displaystyle H_{j}.}Then we obtain an irreducible representationρ~{\displaystyle {\tilde {\rho }}}ofGj,{\displaystyle G_{j},}by combiningρ{\displaystyle \rho }with thecanonical projectionGj→Hj.{\displaystyle G_{j}\to H_{j}.}Finally, we construct thetensor productofχj{\displaystyle \chi _{j}}andρ~.{\displaystyle {\tilde {\rho }}.}Thus, we obtain an irreducible representationχj⊗ρ~{\displaystyle \chi _{j}\otimes {\tilde {\rho }}}ofGj.{\displaystyle G_{j}.}
To finally obtain the classification of the irreducible representations ofG{\displaystyle G}we use the representationθj,ρ{\displaystyle \theta _{j,\rho }}ofG,{\displaystyle G,}which is induced by the tensor productχj⊗ρ~.{\displaystyle \chi _{j}\otimes {\tilde {\rho }}.}Thus, we achieve the following result:
Amongst others, the criterion of Mackey and a conclusion based on the Frobenius reciprocity are needed for the proof of the proposition. Further details may be found in[1].
In other words, we classified all irreducible representations ofG=A⋊H.{\displaystyle G=A\rtimes H.}
The representation ring ofG{\displaystyle G}is defined as the abelian group
With the multiplication provided by thetensor product,R(G){\displaystyle R(G)}becomes a ring. The elements ofR(G){\displaystyle R(G)}are calledvirtual representations.
The character defines aring homomorphismin the set of all class functions onG{\displaystyle G}with complex values
in which theχj{\displaystyle \chi _{j}}are the irreducible characters corresponding to theτj.{\displaystyle \tau _{j}.}
Because a representation is determined by its character,χ{\displaystyle \chi }isinjective. The images ofχ{\displaystyle \chi }are calledvirtual characters.
As the irreducible characters form anorthonormal basisofCclass,χ{\displaystyle \mathbb {C} _{\text{class}},\chi }induces an isomorphism
This isomorphism is defined on a basis out ofelementary tensors(τj⊗1)j=1,…,m{\displaystyle (\tau _{j}\otimes 1)_{j=1,\ldots ,m}}byχC(τj⊗1)=χj{\displaystyle \chi _{\mathbb {C} }(\tau _{j}\otimes 1)=\chi _{j}}respectivelyχC(τj⊗z)=zχj,{\displaystyle \chi _{\mathbb {C} }(\tau _{j}\otimes z)=z\chi _{j},}and extendedbilinearly.
We writeR+(G){\displaystyle {\mathcal {R}}^{+}(G)}for the set of all characters ofG{\displaystyle G}andR(G){\displaystyle {\mathcal {R}}(G)}to denote the group generated byR+(G),{\displaystyle {\mathcal {R}}^{+}(G),}i.e. the set of all differences of two characters. It then holds thatR(G)=Zχ1⊕⋯⊕Zχm{\displaystyle {\mathcal {R}}(G)=\mathbb {Z} \chi _{1}\oplus \cdots \oplus \mathbb {Z} \chi _{m}}andR(G)=Im(χ)=χ(R(G)).{\displaystyle {\mathcal {R}}(G)={\text{Im}}(\chi )=\chi (R(G)).}Thus, we haveR(G)≅R(G){\displaystyle R(G)\cong {\mathcal {R}}(G)}and the virtual characters correspond to the virtual representations in an optimal manner.
SinceR(G)=Im(χ){\displaystyle {\mathcal {R}}(G)={\text{Im}}(\chi )}holds,R(G){\displaystyle {\mathcal {R}}(G)}is the set of all virtual characters. As the product of two characters provides another character,R(G){\displaystyle {\mathcal {R}}(G)}is a subring of the ringCclass(G){\displaystyle \mathbb {C} _{\text{class}}(G)}of all class functions onG.{\displaystyle G.}Because theχi{\displaystyle \chi _{i}}form a basis ofCclass(G){\displaystyle \mathbb {C} _{\text{class}}(G)}we obtain, just as in the case ofR(G),{\displaystyle R(G),}an isomorphismC⊗R(G)≅Cclass(G).{\displaystyle \mathbb {C} \otimes {\mathcal {R}}(G)\cong \mathbb {C} _{\text{class}}(G).}
LetH{\displaystyle H}be a subgroup ofG.{\displaystyle G.}The restriction thus defines a ring homomorphismR(G)→R(H),ϕ↦ϕ|H,{\displaystyle {\mathcal {R}}(G)\to {\mathcal {R}}(H),\phi \mapsto \phi |_{H},}which will be denoted byResHG{\displaystyle {\text{Res}}_{H}^{G}}orRes.{\displaystyle {\text{Res}}.}Likewise, the induction on class functions defines a homomorphism of abelian groupsR(H)→R(G),{\displaystyle {\mathcal {R}}(H)\to {\mathcal {R}}(G),}which will be written asIndHG{\displaystyle {\text{Ind}}_{H}^{G}}or in shortInd.{\displaystyle {\text{Ind}}.}
According to theFrobenius reciprocity, these two homomorphisms are adjoint with respect to the bilinear forms⟨⋅,⋅⟩H{\displaystyle \langle \cdot ,\cdot \rangle _{H}}and⟨⋅,⋅⟩G.{\displaystyle \langle \cdot ,\cdot \rangle _{G}.}Furthermore, the formulaInd(φ⋅Res(ψ))=Ind(φ)⋅ψ{\displaystyle {\text{Ind}}(\varphi \cdot {\text{Res}}(\psi ))={\text{Ind}}(\varphi )\cdot \psi }shows that the image ofInd:R(H)→R(G){\displaystyle {\text{Ind}}:{\mathcal {R}}(H)\to {\mathcal {R}}(G)}is anidealof the ringR(G).{\displaystyle {\mathcal {R}}(G).}
By the restriction of representations, the mapRes{\displaystyle {\text{Res}}}can be defined analogously forR(G),{\displaystyle R(G),}and by the induction we obtain the mapInd{\displaystyle {\text{Ind}}}forR(G).{\displaystyle R(G).}Due to the Frobenius reciprocity, we get the result that these maps are adjoint to each other and that the imageIm(Ind)=Ind(R(H)){\displaystyle {\text{Im}}({\text{Ind}})={\text{Ind}}(R(H))}is anidealof the ringR(G).{\displaystyle R(G).}
IfA{\displaystyle A}is a commutative ring, the homomorphismsRes{\displaystyle {\text{Res}}}andInd{\displaystyle {\text{Ind}}}may be extended toA{\displaystyle A}–linear maps:
in whichηj{\displaystyle \eta _{j}}are all the irreducible representations ofH{\displaystyle H}up to isomorphism.
WithA=C{\displaystyle A=\mathbb {C} }we obtain in particular thatInd{\displaystyle {\text{Ind}}}andRes{\displaystyle {\text{Res}}}supply homomorphisms betweenCclass(G){\displaystyle \mathbb {C} _{\text{class}}(G)}andCclass(H).{\displaystyle \mathbb {C} _{\text{class}}(H).}
LetG1{\displaystyle G_{1}}andG2{\displaystyle G_{2}}be two groups with respective representations(ρ1,V1){\displaystyle (\rho _{1},V_{1})}and(ρ2,V2).{\displaystyle (\rho _{2},V_{2}).}Then,ρ1⊗ρ2{\displaystyle \rho _{1}\otimes \rho _{2}}is the representation of thedirect productG1×G2{\displaystyle G_{1}\times G_{2}}as was shown in aprevious section. Another result of that section was that all irreducible representations ofG1×G2{\displaystyle G_{1}\times G_{2}}are exactly the representationsη1⊗η2,{\displaystyle \eta _{1}\otimes \eta _{2},}whereη1{\displaystyle \eta _{1}}andη2{\displaystyle \eta _{2}}are irreducible representations ofG1{\displaystyle G_{1}}andG2,{\displaystyle G_{2},}respectively. This passes over to the representation ring as the identityR(G1×G2)=R(G1)⊗ZR(G2),{\displaystyle R(G_{1}\times G_{2})=R(G_{1})\otimes _{\mathbb {Z} }R(G_{2}),}in whichR(G1)⊗ZR(G2){\displaystyle R(G_{1})\otimes _{\mathbb {Z} }R(G_{2})}is thetensor productof the representation rings asZ{\displaystyle \mathbb {Z} }–modules.
Induction theorems relate the representation ring of a given finite groupGto representation rings of a familyXconsisting of some subsetsHofG. More precisely, for such a collection of subgroups, the induction functor yields a map
Artin's induction theoremis the most elementary theorem in this group of results. It asserts that the following are equivalent:
SinceR(G){\displaystyle {\mathcal {R}}(G)}is finitely generated as a group, the first point can be rephrased as follows:
Serre (1977)gives two proofs of this theorem. For example, sinceGis the union of its cyclic subgroups, every character ofG{\displaystyle G}is a linear combination with rational coefficients of characters induced by characters ofcyclic subgroupsofG.{\displaystyle G.}Since the representations of cyclic groups are well-understood, in particular the irreducible representations are one-dimensional, this gives a certain control over representations ofG.
Under the above circumstances, it is not in general true thatφ{\displaystyle \varphi }is surjective.Brauer's induction theoremasserts thatφ{\displaystyle \varphi }is surjective, provided thatXis the family of allelementary subgroups.
Here a groupHiselementaryif there is some primepsuch thatHis thedirect productof acyclic groupof order prime top{\displaystyle p}and ap{\displaystyle p}–group.
In other words, everycharacterofG{\displaystyle G}is a linear combination with integer coefficients of characters induced by characters of elementary subgroups.
The elementary subgroupsHarising in Brauer's theorem have a richer representation theory than cyclic groups, they at least have the property that any irreducible representation for suchHis induced by a one-dimensional representation of a (necessarily also elementary) subgroupK⊂H{\displaystyle K\subset H}. (This latter property can be shown to hold for anysupersolvable group, which includesnilpotent groupsand, in particular, elementary groups.) This ability to induce representations from degree 1 representations has some further consequences in the representation theory of finite groups.
For proofs and more information about representations over general subfields ofC{\displaystyle \mathbb {C} }please refer to[2].
If a groupG{\displaystyle G}acts on a real vector spaceV0,{\displaystyle V_{0},}the corresponding representation on the complex vector spaceV=V0⊗RC{\displaystyle V=V_{0}\otimes _{\mathbb {R} }\mathbb {C} }is calledreal(V{\displaystyle V}is called thecomplexificationofV0{\displaystyle V_{0}}). The corresponding representation mentioned above is given bys⋅(v0⊗z)=(s⋅v0)⊗z{\displaystyle s\cdot (v_{0}\otimes z)=(s\cdot v_{0})\otimes z}for alls∈G,v0∈V0,z∈C.{\displaystyle s\in G,v_{0}\in V_{0},z\in \mathbb {C} .}
Letρ{\displaystyle \rho }be a real representation. The linear mapρ(s){\displaystyle \rho (s)}isR{\displaystyle \mathbb {R} }-valued for alls∈G.{\displaystyle s\in G.}Thus, we can conclude that the character of a real representation is always real-valued. But not every representation with a real-valued character is real. To make this clear, letG{\displaystyle G}be a finite, non-abelian subgroup of the group
ThenG⊂SU(2){\displaystyle G\subset {\text{SU}}(2)}acts onV=C2.{\displaystyle V=\mathbb {C} ^{2}.}Since the trace of any matrix inSU(2){\displaystyle {\text{SU}}(2)}is real, the character of the representation is real-valued. Supposeρ{\displaystyle \rho }is a real representation, thenρ(G){\displaystyle \rho (G)}would consist only of real-valued matrices. Thus,G⊂SU(2)∩GL2(R)=SO(2)=S1.{\displaystyle G\subset {\text{SU}}(2)\cap {\text{GL}}_{2}(\mathbb {R} )={\text{SO}}(2)=S^{1}.}However the circle group is abelian butG{\displaystyle G}was chosen to be a non-abelian group. Now we only need to prove the existence of a non-abelian, finite subgroup ofSU(2).{\displaystyle {\text{SU}}(2).}To find such a group, observe thatSU(2){\displaystyle {\text{SU}}(2)}can be identified with the units of thequaternions. Now letG={±1,±i,±j,±ij}.{\displaystyle G=\{\pm 1,\pm i,\pm j,\pm ij\}.}The following two-dimensional representation ofG{\displaystyle G}is not real-valued, but has a real-valued character:
Then the image ofρ{\displaystyle \rho }is not real-valued, but nevertheless it is a subset ofSU(2).{\displaystyle {\text{SU}}(2).}Thus, the character of the representation is real.
An irreducible representation ofG{\displaystyle G}on a real vector space can become reducible when extending the field toC.{\displaystyle \mathbb {C} .}For example, the following real representation of the cyclic group is reducible when considered overC{\displaystyle \mathbb {C} }
Therefore, by classifying all the irreducible representations that are real overC,{\displaystyle \mathbb {C} ,}we still haven't classified all the irreducible real representations. But we achieve the following:
LetV0{\displaystyle V_{0}}be a real vector space. LetG{\displaystyle G}act irreducibly onV0{\displaystyle V_{0}}and letV=V0⊗C.{\displaystyle V=V_{0}\otimes \mathbb {C} .}IfV{\displaystyle V}is not irreducible, there are exactly two irreducible factors which are complex conjugate representations ofG.{\displaystyle G.}
Definition.Aquaternionicrepresentation is a (complex) representationV,{\displaystyle V,}which possesses aG{\displaystyle G}–invariant anti-linear homomorphismJ:V→V{\displaystyle J:V\to V}satisfyingJ2=−Id.{\displaystyle J^{2}=-{\text{Id}}.}Thus, askew-symmetric, nondegenerateG{\displaystyle G}–invariant bilinear form defines a quaternionic structure onV.{\displaystyle V.}
Representation of thesymmetric groupsSn{\displaystyle S_{n}}have been intensely studied. Conjugacy classes inSn{\displaystyle S_{n}}(and therefore, by the above, irreducible representations) correspond topartitionsofn. For example,S3{\displaystyle S_{3}}has three irreducible representations, corresponding to the partitions
of 3. For such a partition, aYoung tableauis a graphical device depicting a partition. The irreducible representation corresponding to such a partition (or Young tableau) is called aSpecht module.
Representations of different symmetric groups are related: any representation ofSn×Sm{\displaystyle S_{n}\times S_{m}}yields a representation ofSn+m{\displaystyle S_{n+m}}by induction, and vice versa by restriction. The direct sum of all these representation rings
inherits from these constructions the structure of aHopf algebrawhich, it turns out, is closely related tosymmetric functions.
To a certain extent, the representations of theGLn(Fq){\displaystyle GL_{n}(\mathbf {F} _{q})}, asnvaries, have a similar flavor as for theSn{\displaystyle S_{n}}; the above-mentioned induction process gets replaced by so-calledparabolic induction. However, unlike forSn{\displaystyle S_{n}}, where all representations can be obtained by induction of trivial representations, this is not true forGLn(Fq){\displaystyle GL_{n}(\mathbf {F} _{q})}. Instead, new building blocks, known ascuspidal representations, are needed.
Representations ofGLn(Fq){\displaystyle GL_{n}(\mathbf {F} _{q})}and more generally, representations offinite groups of Lie typehave been thoroughly studied.Bonnafé (2010)describes the representations ofSL2(Fq){\displaystyle SL_{2}(\mathbf {F} _{q})}. A geometric description of irreducible representations of such groups, including the above-mentioned cuspidal representations, is obtained byDeligne-Lusztig theory, which constructs such representation in thel-adic cohomologyofDeligne-Lusztig varieties.
The similarity of the representation theory ofSn{\displaystyle S_{n}}andGLn(Fq){\displaystyle GL_{n}(\mathbf {F} _{q})}goes beyond finite groups. Thephilosophy of cusp formshighlights the kinship of representation theoretic aspects of these types of groups with general linear groups oflocal fieldssuch asQpand of the ring ofadeles, seeBump (2004).
The theory of representations of compact groups may be, to some degree, extended tolocally compact groups. The representation theory unfolds in this context great importance for harmonic analysis and the study of automorphic forms. For proofs, further information and for a more detailed insight which is beyond the scope of this chapter please consult[4]and[5].
Atopological groupis a group together with atopologywith respect to which the group composition and the inversion arecontinuous.
Such a group is calledcompact, if any cover ofG,{\displaystyle G,}which is open in the topology, has a finite subcover. Closed subgroups of a compact group are compact again.
LetG{\displaystyle G}be a compact group and letV{\displaystyle V}be a finite-dimensionalC{\displaystyle \mathbb {C} }–vector space. A linear representation ofG{\displaystyle G}toV{\displaystyle V}is acontinuous group homomorphismρ:G→GL(V),{\displaystyle \rho :G\to {\text{GL}}(V),}i.e.ρ(s)v{\displaystyle \rho (s)v}is a continuous function in the two variabless∈G{\displaystyle s\in G}andv∈V.{\displaystyle v\in V.}
A linear representation ofG{\displaystyle G}into aBanach spaceV{\displaystyle V}is defined to be a continuous group homomorphism ofG{\displaystyle G}into the set of all bijectivebounded linear operatorsonV{\displaystyle V}with a continuous inverse. Sinceπ(g)−1=π(g−1),{\displaystyle \pi (g)^{-1}=\pi (g^{-1}),}we can do without the last requirement. In the following, we will consider in particular representations of compact groups inHilbert spaces.
Just as with finite groups, we can define thegroup algebraand theconvolution algebra. However, the group algebra provides no helpful information in the case of infinite groups, because the continuity condition gets lost during the construction. Instead the convolution algebraL1(G){\displaystyle L^{1}(G)}takes its place.
Most properties of representations of finite groups can be transferred with appropriate changes to compact groups. For this we need a counterpart to the summation over a finite group:
On a compact groupG{\displaystyle G}there exists exactly onemeasuredt,{\displaystyle dt,}such that:
Such a left-translation-invariant, normed measure is calledHaar measureof the groupG.{\displaystyle G.}
SinceG{\displaystyle G}is compact, it is possible to show that this measure is also right-translation-invariant, i.e. it also applies
By the scaling above the Haar measure on a finite group is given bydt(s)=1|G|{\displaystyle dt(s)={\tfrac {1}{|G|}}}for alls∈G.{\displaystyle s\in G.}
All the definitions to representations of finite groups that are mentioned in the section”Properties”, also apply to representations of compact groups. But there are some modifications needed:
To define a subrepresentation we now need a closed subspace. This was not necessary for finite-dimensional representation spaces, because in this case every subspace is already closed. Furthermore, two representationsρ,π{\displaystyle \rho ,\pi }of a compact groupG{\displaystyle G}are called equivalent, if there exists a bijective, continuous, linear operatorT{\displaystyle T}between the representation spaces whose inverse is also continuous and which satisfiesT∘ρ(s)=π(s)∘T{\displaystyle T\circ \rho (s)=\pi (s)\circ T}for alls∈G.{\displaystyle s\in G.}
IfT{\displaystyle T}is unitary, the two representations are calledunitary equivalent.
To obtain aG{\displaystyle G}–invariantinner productfrom a notG{\displaystyle G}–invariant, we now have to use the integral overG{\displaystyle G}instead of the sum. If(⋅|⋅){\displaystyle (\cdot |\cdot )}is an inner product on aHilbert spaceV,{\displaystyle V,}which is not invariant with respect to the representationρ{\displaystyle \rho }ofG,{\displaystyle G,}then
is aG{\displaystyle G}–invariant inner product onV{\displaystyle V}due to the properties of the Haar measuredt.{\displaystyle dt.}Thus, we can assume every representation on a Hilbert space to be unitary.
LetG{\displaystyle G}be a compact group and lets∈G.{\displaystyle s\in G.}LetL2(G){\displaystyle L^{2}(G)}be the Hilbert space of the square integrable functions onG.{\displaystyle G.}We define the operatorLs{\displaystyle L_{s}}on this space byLsΦ(t)=Φ(s−1t),{\displaystyle L_{s}\Phi (t)=\Phi (s^{-1}t),}whereΦ∈L2(G),t∈G.{\displaystyle \Phi \in L^{2}(G),t\in G.}
The maps↦Ls{\displaystyle s\mapsto L_{s}}is a unitary representation ofG.{\displaystyle G.}It is calledleft-regular representation. Theright-regular representationis defined similarly. As the Haar measure ofG{\displaystyle G}is also right-translation-invariant, the operatorRs{\displaystyle R_{s}}onL2(G){\displaystyle L^{2}(G)}is given byRsΦ(t)=Φ(ts).{\displaystyle R_{s}\Phi (t)=\Phi (ts).}The right-regular representation is then the unitary representation given bys↦Rs.{\displaystyle s\mapsto R_{s}.}The two representationss↦Ls{\displaystyle s\mapsto L_{s}}ands↦Rs{\displaystyle s\mapsto R_{s}}are dual to each other.
IfG{\displaystyle G}is infinite, these representations have no finite degree. Theleft- and right-regular representationas defined at the beginning are isomorphic to the left- and right-regular representation as defined above, if the groupG{\displaystyle G}is finite. This is due to the fact that in this caseL2(G)≅L1(G)≅C[G].{\displaystyle L^{2}(G)\cong L^{1}(G)\cong \mathbb {C} [G].}
The different ways of constructing new representations from given ones can be used for compact groups as well, except for the dual representation with which we will deal later. Thedirect sumand thetensor productwith a finite number of summands/factors are defined in exactly the same way as for finite groups. This is also the case for the symmetric and alternating square. However, we need a Haar measure on thedirect productof compact groups in order to extend the theorem saying that the irreducible representations of the product of two groups are (up to isomorphism) exactly the tensor product of the irreducible representations of the factor groups. First, we note that the direct productG1×G2{\displaystyle G_{1}\times G_{2}}of two compact groups is again a compact group when provided with theproduct topology. The Haar measure on the direct product is then given by the product of the Haar measures on the factor groups.
For the dual representation on compact groups we require thetopological dualV′{\displaystyle V'}of the vector spaceV.{\displaystyle V.}This is the vector space of all continuous linear functionals from the vector spaceV{\displaystyle V}into the base field. Letπ{\displaystyle \pi }be a representation of a compact groupG{\displaystyle G}inV.{\displaystyle V.}
The dual representationπ′:G→GL(V′){\displaystyle \pi ':G\to {\text{GL}}(V')}is defined by the property
Thus, we can conclude that the dual representation is given byπ′(s)v′=v′∘π(s−1){\displaystyle \pi '(s)v'=v'\circ \pi (s^{-1})}for allv′∈V′,s∈G.{\displaystyle v'\in V',s\in G.}The mapπ′{\displaystyle \pi '}is again a continuous group homomorphism and thus a representation.
On Hilbert spaces:π{\displaystyle \pi }is irreducible if and only ifπ′{\displaystyle \pi '}is irreducible.
By transferring the results of the sectiondecompositionsto compact groups, we obtain the following theorems:
Every representation of a compact group is isomorphic to adirect Hilbert sumof irreducible representations.
Let(ρ,Vρ){\displaystyle (\rho ,V_{\rho })}be a unitary representation of the compact groupG.{\displaystyle G.}Just as for finite groups we define for an irreducible representation(τ,Vτ){\displaystyle (\tau ,V_{\tau })}the isotype or isotypic component inρ{\displaystyle \rho }to be the subspace
This is the sum of all invariant closed subspacesU,{\displaystyle U,}which areG{\displaystyle G}–isomorphic toVτ.{\displaystyle V_{\tau }.}
Note that the isotypes of not equivalent irreducible representations are pairwise orthogonal.
The corresponding projection to the canonical decompositionpτ:V→V(τ),{\displaystyle p_{\tau }:V\to V(\tau ),}in whichV(τ){\displaystyle V(\tau )}is an isotype ofV,{\displaystyle V,}is for compact groups given by
wherenτ=dim(V(τ)){\displaystyle n_{\tau }=\dim(V(\tau ))}andχτ{\displaystyle \chi _{\tau }}is the character corresponding to the irreducible representationτ.{\displaystyle \tau .}
For every representation(ρ,V){\displaystyle (\rho ,V)}of a compact groupG{\displaystyle G}we define
In generalρ(s):V→V{\displaystyle \rho (s):V\to V}is notG{\displaystyle G}–linear. Let
The mapP{\displaystyle P}is defined asendomorphismonV{\displaystyle V}by having the property
which is valid for the inner product of the Hilbert spaceV.{\displaystyle V.}
ThenP{\displaystyle P}isG{\displaystyle G}–linear, because of
where we used the invariance of the Haar measure.
If the representation is finite-dimensional, it is possible to determine the direct sum of the trivial subrepresentation just as in the case of finite groups.
Generally, representations of compact groups are investigated onHilbert-andBanach spaces. In most cases they are not finite-dimensional. Therefore, it is not useful to refer tocharacterswhen speaking about representations of compact groups. Nevertheless, in most cases it is possible to restrict the study to the case of finite dimensions:
Since irreducible representations of compact groups are finite-dimensional and unitary (see results from thefirst subsection), we can define irreducible characters in the same way as it was done for finite groups.
As long as the constructed representations stay finite-dimensional, the characters of the newly constructed representations may be obtained in the same way as for finite groups.
Schur's lemmais also valid for compact groups:
Let(π,V){\displaystyle (\pi ,V)}be an irreducible unitary representation of a compact groupG.{\displaystyle G.}Then every boundedoperatorT:V→V{\displaystyle T:V\to V}satisfying the propertyT∘π(s)=π(s)∘T{\displaystyle T\circ \pi (s)=\pi (s)\circ T}for alls∈G,{\displaystyle s\in G,}is a scalar multiple of the identity, i.e. there existsλ∈C{\displaystyle \lambda \in \mathbb {C} }such thatT=λId.{\displaystyle T=\lambda {\text{Id}}.}
Definition.The formula
defines an inner product on the set of all square integrable functionsL2(G){\displaystyle L^{2}(G)}of a compact groupG.{\displaystyle G.}Likewise
defines a bilinear form onL2(G){\displaystyle L^{2}(G)}of a compact groupG.{\displaystyle G.}
The bilinear form on the representation spaces is defined exactly as it was for finite groups and analogous to finite groups the following results are therefore valid:
Therefore, using the first theorem, the characters of irreducible representations ofG{\displaystyle G}form anorthonormal setonL2(G){\displaystyle L^{2}(G)}with respect to this inner product.
As we already know that the non-isomorphic irreducible representations are orthonormal, we only need to verify that they generateL2(G).{\displaystyle L^{2}(G).}This may be done, by proving that there exists no non-zero square integrable function onG{\displaystyle G}orthogonal to all the irreducible characters.
Just as in the case of finite groups, the number of the irreducible representations up to isomorphism of a groupG{\displaystyle G}equals the number of conjugacy classes ofG.{\displaystyle G.}However, because a compact group has in general infinitely many conjugacy classes, this does not provide any useful information.
IfH{\displaystyle H}is a closed subgroup of finiteindexin a compact groupG,{\displaystyle G,}the definition of theinduced representationfor finite groups may be adopted.
However, the induced representation can be defined more generally, so that the definition is valid independent of the index of the subgroupH.{\displaystyle H.}
For this purpose let(η,Vη){\displaystyle (\eta ,V_{\eta })}be a unitary representation of the closed subgroupH.{\displaystyle H.}The continuous induced representationIndHG(η)=(I,VI){\displaystyle {\text{Ind}}_{H}^{G}(\eta )=(I,V_{I})}is defined as follows:
LetVI{\displaystyle V_{I}}denote the Hilbert space of all measurable, square integrable functionsΦ:G→Vη{\displaystyle \Phi :G\to V_{\eta }}with the propertyΦ(ls)=η(l)Φ(s){\displaystyle \Phi (ls)=\eta (l)\Phi (s)}for alll∈H,s∈G.{\displaystyle l\in H,s\in G.}The norm is given by
and the representationI{\displaystyle I}is given as the right-translation:I(s)Φ(k)=Φ(ks).{\displaystyle I(s)\Phi (k)=\Phi (ks).}
The induced representation is then again a unitary representation.
SinceG{\displaystyle G}is compact, the induced representation can be decomposed into the direct sum of irreducible representations ofG.{\displaystyle G.}Note that all irreducible representations belonging to the same isotype appear with a multiplicity equal todim(HomG(Vη,VI))=⟨Vη,VI⟩G.{\displaystyle \dim({\text{Hom}}_{G}(V_{\eta },V_{I}))=\langle V_{\eta },V_{I}\rangle _{G}.}
Let(ρ,Vρ){\displaystyle (\rho ,V_{\rho })}be a representation ofG,{\displaystyle G,}then there exists a canonical isomorphism
TheFrobenius reciprocitytransfers, together with the modified definitions of the inner product and of the bilinear form, to compact groups. The theorem now holds for square integrable functions onG{\displaystyle G}instead of class functions, but the subgroupH{\displaystyle H}must be closed.
Another important result in the representation theory of compact groups is the Peter-Weyl Theorem. It is usually presented and proven inharmonic analysis, as it represents one of its central and fundamental statements.
We can reformulate this theorem to obtain a generalization of the Fourier series for functions on compact groups:
The general features of therepresentation theoryof afinite groupG, over thecomplex numbers, were discovered byFerdinand Georg Frobeniusin the years before 1900. Later themodular representation theoryofRichard Brauerwas developed.
|
https://en.wikipedia.org/wiki/Representation_theory_of_finite_groups
|
Inmathematics, specifically inrepresentation theory, asemisimple representation(also called acompletely reducible representation) is alinear representationof agroupor analgebrathat is adirect sumofsimple representations(also calledirreducible representations).[1]It is an example of the general mathematical notion ofsemisimplicity.
Many representations that appear in applications of representation theory are semisimple or can be approximated by semisimple representations. Asemisimple moduleover an algebra over afieldis an example of a semisimple representation.Conversely, a semisimple representation of a groupGover a fieldkis a semisimple module over thegroup algebrak[G].
LetVbe a representation of a groupG; or more generally, letVbe avector spacewith a set of linearendomorphismsacting on it. In general, a vector space acted on by a set of linear endomorphisms is said to besimple(or irreducible) if the only invariant subspaces for those operators are zero and the vector space itself; a semisimple representation then is a direct sum of simple representations in that sense.[1]
The following are equivalent:[2]
The equivalence of the above conditions can beprovedbased on the followinglemma, which is of independent interest:
Lemma[3]—Letp:V→Wbe asurjectiveequivariant mapbetween representations. IfVis semisimple, thenpsplits; i.e., it admits asection.
Proof of the lemma: WriteV=⨁i∈IVi{\displaystyle V=\bigoplus _{i\in I}V_{i}}whereVi{\displaystyle V_{i}}are simple representations.Without loss of generality, we can assumeVi{\displaystyle V_{i}}are subrepresentations; i.e., we can assume the direct sum is internal. Now, consider the family of all possible direct sumsVJ:=⨁i∈JVi⊂V{\displaystyle V_{J}:=\bigoplus _{i\in J}V_{i}\subset V}with various subsetsJ⊂I{\displaystyle J\subset I}. Put thepartial orderingon it by saying the direct sum overKis less than the direct sum overJifK⊂J{\displaystyle K\subset J}. ByZorn's lemma, we can find a maximalJ⊂I{\displaystyle J\subset I}such thatkerp∩VJ=0{\displaystyle \operatorname {ker} p\cap V_{J}=0}. We claim thatV=kerp⊕VJ{\displaystyle V=\operatorname {ker} p\oplus V_{J}}. By definition,kerp∩VJ=0{\displaystyle \operatorname {ker} p\cap V_{J}=0}so we only need to show thatV=kerp+VJ{\displaystyle V=\operatorname {ker} p+V_{J}}. Ifkerp+VJ{\displaystyle \operatorname {ker} p+V_{J}}is a proper subrepresentatiom ofV{\displaystyle V}then there existsk∈I−J{\displaystyle k\in I-J}such thatVk⊄kerp+VJ{\displaystyle V_{k}\not \subset \operatorname {ker} p+V_{J}}. SinceVk{\displaystyle V_{k}}is simple (irreducible),Vk∩(kerp+VJ)=0{\displaystyle V_{k}\cap (\operatorname {ker} p+V_{J})=0}. This contradicts the maximality ofJ{\displaystyle J}, soV=kerp⊕VJ{\displaystyle V=\operatorname {ker} p\oplus V_{J}}as claimed. Hence,W≃V/kerp≃VJ→V{\displaystyle W\simeq V/\operatorname {ker} p\simeq V_{J}\to V}is a section ofp.◻{\displaystyle \square }
Note that we cannot takeJ{\displaystyle J}to the set ofi{\displaystyle i}such thatker(p)∩Vi=0{\displaystyle \ker(p)\cap V_{i}=0}. The reason is that it can happen, and frequently does, thatX{\displaystyle X}is a subspace ofY⊕Z{\displaystyle Y\oplus Z}and yetX∩Y=0=X∩Z{\displaystyle X\cap Y=0=X\cap Z}. For example, takeX{\displaystyle X},Y{\displaystyle Y}andZ{\displaystyle Z}to be three distinct lines through the origin inR2{\displaystyle \mathbb {R} ^{2}}. For an explicit counterexample, letA=Mat2F{\displaystyle A=\operatorname {Mat} _{2}F}be the algebra of 2-by-2matricesand setV=A{\displaystyle V=A}, the regular representation ofA{\displaystyle A}. SetV1={(a0b0)}{\displaystyle V_{1}={\Bigl \{}{\begin{pmatrix}a&0\\b&0\end{pmatrix}}{\Bigr \}}}andV2={(0c0d)}{\displaystyle V_{2}={\Bigl \{}{\begin{pmatrix}0&c\\0&d\end{pmatrix}}{\Bigr \}}}and setW={(ccdd)}{\displaystyle W={\Bigl \{}{\begin{pmatrix}c&c\\d&d\end{pmatrix}}{\Bigr \}}}. ThenV1{\displaystyle V_{1}},V2{\displaystyle V_{2}}andW{\displaystyle W}are allirreducibleA{\displaystyle A}-modulesandV=V1⊕V2{\displaystyle V=V_{1}\oplus V_{2}}. Letp:V→V/W{\displaystyle p:V\to V/W}be the natural surjection. Thenkerp=W≠0{\displaystyle \operatorname {ker} p=W\neq 0}andV1∩kerp=0=V2∩kerp{\displaystyle V_{1}\cap \operatorname {ker} p=0=V_{2}\cap \operatorname {ker} p}. In this case,W≃V1≃V2{\displaystyle W\simeq V_{1}\simeq V_{2}}butV≠kerp⊕V1⊕V2{\displaystyle V\neq \operatorname {ker} p\oplus V_{1}\oplus V_{2}}because this sum is not direct.
Proof of equivalences[4]1.⇒3.{\displaystyle 1.\Rightarrow 3.}: Takepto be the natural surjectionV→V/W{\displaystyle V\to V/W}. SinceVis semisimple,psplits and so, through a section,V/W{\displaystyle V/W}isisomorphicto a subrepresentation that is complementary toW.
3.⇒2.{\displaystyle 3.\Rightarrow 2.}: We shall first observe that every nonzero subrepresentationWhas a simple subrepresentation. ShrinkingWto a (nonzero)cyclic subrepresentationwe can assume it is finitely generated. Then it has amaximal subrepresentationU. By the condition 3.,V=U⊕U′{\displaystyle V=U\oplus U'}for someU′{\displaystyle U'}. By modular law, it impliesW=U⊕(W∩U′){\displaystyle W=U\oplus (W\cap U')}. Then(W∩U′)≃W/U{\displaystyle (W\cap U')\simeq W/U}is a simple subrepresentation
ofW("simple" because of maximality). This establishes the observation. Now, takeW{\displaystyle W}to be the sum of all simple subrepresentations, which, by 3., admits a complementary representationW′{\displaystyle W'}. IfW′≠0{\displaystyle W'\neq 0}, then, by the early observation,W′{\displaystyle W'}contains a simple subrepresentation and soW∩W′≠0{\displaystyle W\cap W'\neq 0}, a nonsense. Hence,W′=0{\displaystyle W'=0}.
2.⇒1.{\displaystyle 2.\Rightarrow 1.}:[5]The implication is a direct generalization of a basic fact in linear algebra that a basis can be extracted from a spanning set of a vector space. That is we can prove the following slightly more precise statement:
As in the proof of the lemma, we can find a maximal direct sumW{\displaystyle W}that consists of someVi{\displaystyle V_{i}}'s. Now, for eachiinI, by simplicity, eitherVi⊂W{\displaystyle V_{i}\subset W}orVi∩W=0{\displaystyle V_{i}\cap W=0}. In the second case, the direct sumW⊕Vi{\displaystyle W\oplus V_{i}}is a contradiction to the maximality ofW. Hence,Vi⊂W{\displaystyle V_{i}\subset W}.◻{\displaystyle \square }
A finite-dimensionalunitary representation(i.e., a representation factoring through aunitary group) is a basic example of a semisimple representation. Such a representation is semisimple since ifWis a subrepresentation, then the orthogonal complement toWis a complementary representation[6]because ifv∈W⊥{\displaystyle v\in W^{\bot }}andg∈G{\displaystyle g\in G}, then⟨π(g)v,w⟩=⟨v,π(g−1)w⟩=0{\displaystyle \langle \pi (g)v,w\rangle =\langle v,\pi (g^{-1})w\rangle =0}for anywinWsinceWisG-invariant, and soπ(g)v∈W⊥{\displaystyle \pi (g)v\in W^{\bot }}.
For example, given a continuous finite-dimensionalcomplexrepresentationπ:G→GL(V){\displaystyle \pi :G\to GL(V)}of afinite groupor acompact groupG, by the averaging argument, one can define aninner product⟨,⟩{\displaystyle \langle \,,\rangle }onVthat isG-invariant: i.e.,⟨π(g)v,π(g)w⟩=⟨v,w⟩{\displaystyle \langle \pi (g)v,\pi (g)w\rangle =\langle v,w\rangle }, which is to sayπ(g){\displaystyle \pi (g)}is a unitary operator and soπ{\displaystyle \pi }is a unitary representation.[6]Hence, every finite-dimensional continuous complex representation ofGis semisimple.[7]For a finite group, this is a special case ofMaschke's theorem, which says a finite-dimensional representation of a finite groupGover a fieldkwithcharacteristicnot dividing theorderofGis semisimple.[8][9]
ByWeyl's theorem on complete reducibility, every finite-dimensional representation of asemisimple Lie algebraover a field of characteristic zero is semisimple.[10]
Given a linear endomorphismTof a vector spaceV,Vis semisimple as a representation ofT(i.e.,Tis asemisimple operator)if and only ifthe minimal polynomial ofTis separable; i.e., a product of distinct irreducible polynomials.[11]
Given a finite-dimensional representationV, theJordan–Hölder theoremsays there is a filtration by subrepresentations:V=V0⊃V1⊃⋯⊃Vn=0{\displaystyle V=V_{0}\supset V_{1}\supset \cdots \supset V_{n}=0}such that each successive quotientVi/Vi+1{\displaystyle V_{i}/V_{i+1}}is a simple representation. Then the associated vector spacegrV:=⨁i=0n−1Vi/Vi+1{\displaystyle \operatorname {gr} V:=\bigoplus _{i=0}^{n-1}V_{i}/V_{i+1}}is a semisimple representation called anassociated semisimple representation, which,up toan isomorphism, is uniquely determined byV.[12]
A representation of aunipotent groupis generally not semisimple. TakeG{\displaystyle G}to be the group consisting ofrealmatrices[1a01]{\displaystyle {\begin{bmatrix}1&a\\0&1\end{bmatrix}}}; it acts onV=R2{\displaystyle V=\mathbb {R} ^{2}}in a natural way and makesVa representation ofG. IfWis a subrepresentation ofVthat has dimension 1, then a simple calculation shows that it must be spanned by the vector[10]{\displaystyle {\begin{bmatrix}1\\0\end{bmatrix}}}. That is, there are exactly threeG-subrepresentations ofV; in particular,Vis not semisimple (as a unique one-dimensional subrepresentation does not admit a complementary representation).[13]
The decomposition of a semisimple representation into simple ones, called a semisimple decomposition, need not be unique; for example, for a trivial representation, simple representations are one-dimensional vector spaces and thus a semisimple decomposition amounts to a choice of a basis of the representation vector space.[14]Theisotypic decomposition, on the other hand, is an example of a unique decomposition.[15]
However, for a finite-dimensional semisimple representationVover analgebraically closed field, the numbers of simple representations up to isomorphism appearing in the decomposition ofV(1) are unique and (2) completely determine the representation up to isomorphism;[16]this is a consequence ofSchur's lemmain the following way. Suppose a finite-dimensional semisimple representationVover an algebraically closed field is given: by definition, it is a direct sum of simple representations. By grouping together simple representations in the decomposition that are isomorphic to each other, up to an isomorphism, one finds a decomposition (not necessarily unique):[16]
whereVi{\displaystyle V_{i}}are simple representations, mutually non-isomorphic to one another, andmi{\displaystyle m_{i}}are positiveintegers. By Schur's lemma,
whereHomequiv{\displaystyle \operatorname {Hom} _{\text{equiv}}}refers to theequivariant linear maps. Also, eachmi{\displaystyle m_{i}}is unchanged ifVi{\displaystyle V_{i}}is replaced by another simple representation isomorphic toVi{\displaystyle V_{i}}. Thus, the integersmi{\displaystyle m_{i}}are independent of chosen decompositions; they are themultiplicitiesof simple representationsVi{\displaystyle V_{i}}, up to isomorphism, inV.[17]
In general, given a finite-dimensional representationπ:G→GL(V){\displaystyle \pi :G\to GL(V)}of a groupGover a fieldk, the compositionχV:G→πGL(V)→trk{\displaystyle \chi _{V}:G\,{\overset {\pi }{\to }}\,GL(V)\,{\overset {\operatorname {tr} }{\to }}\,k}is called thecharacterof(π,V){\displaystyle (\pi ,V)}.[18]When(π,V){\displaystyle (\pi ,V)}is semisimple with the decompositionV≃⨁iVi⊕mi{\displaystyle V\simeq \bigoplus _{i}V_{i}^{\oplus m_{i}}}as above, the tracetr(π(g)){\displaystyle \operatorname {tr} (\pi (g))}is the sum of the traces ofπ(g):Vi→Vi{\displaystyle \pi (g):V_{i}\to V_{i}}with multiplicities and thus, as functions onG,
whereχVi{\displaystyle \chi _{V_{i}}}are the characters ofVi{\displaystyle V_{i}}. WhenGis a finite group or more generally a compact group andV{\displaystyle V}is a unitary representation with the inner product given by the averaging argument, theSchur orthogonality relationssay:[19]the irreducible characters (characters of simple representations) ofGare an orthonormal subset of the space of complex-valued functions onGand thusmi=⟨χV,χVi⟩{\displaystyle m_{i}=\langle \chi _{V},\chi _{V_{i}}\rangle }.
There is a decomposition of a semisimple representation that is unique, calledtheisotypic decomposition of the representation. By definition, given a simple representationS, theisotypic componentof typeSof a representationVis the sum of all subrepresentations ofVthat are isomorphic toS;[15]note the component is also isomorphic to the direct sum of some choice of subrepresentations isomorphic toS(so the component is unique, while the summands are not necessary so).
Then the isotypic decomposition of a semisimple representationVis the (unique) direct sum decomposition:[15][20]
whereG^{\displaystyle {\widehat {G}}}is the set of isomorphism classes of simple representations ofGandVλ{\displaystyle V^{\lambda }}is the isotypic component ofVof typeSfor someS∈λ{\displaystyle S\in \lambda }.
Theisotypic componentofweightλ{\displaystyle \lambda }of aLie algebra moduleis the sum of all submodules which areisomorphicto the highest weight module with weightλ{\displaystyle \lambda }.
This defines the isotypic component of weightλ{\displaystyle \lambda }ofV{\displaystyle V}:λ(V):=⨁i=1dλVi≃Cdλ⊗Mλ{\displaystyle \lambda (V):=\bigoplus _{i=1}^{d_{\lambda }}V_{i}\simeq \mathbb {C} ^{d_{\lambda }}\otimes M_{\lambda }}wheredλ{\displaystyle d_{\lambda }}is maximal.
LetV{\displaystyle V}be the space of homogeneous degree-three polynomials over the complex numbers in variablesx1,x2,x3{\displaystyle x_{1},x_{2},x_{3}}. ThenS3{\displaystyle S_{3}}acts onV{\displaystyle V}by permutation of the three variables. This is a finite-dimensional complex representation of a finite group, and so is semisimple. Therefore, this 10-dimensional representation can be broken up into three isotypic components, each corresponding to one of the three irreducible representations ofS3{\displaystyle S_{3}}. In particular,V{\displaystyle V}contains three copies of the trivial representation, one copy of the sign representation, and three copies of the two-dimensional irreducible representationW{\displaystyle W}ofS3{\displaystyle S_{3}}. For example, the span ofx12x2−x22x1+x12x3−x22x3{\displaystyle x_{1}^{2}x_{2}-x_{2}^{2}x_{1}+x_{1}^{2}x_{3}-x_{2}^{2}x_{3}}andx22x3−x32x2+x22x1−x32x1{\displaystyle x_{2}^{2}x_{3}-x_{3}^{2}x_{2}+x_{2}^{2}x_{1}-x_{3}^{2}x_{1}}is isomorphic toW{\displaystyle W}. This can more easily be seen by writing this two-dimensional subspace as
Another copy ofW{\displaystyle W}can be written in a similar form:
So can the third:
ThenW1⊕W2⊕W3{\displaystyle W_{1}\oplus W_{2}\oplus W_{3}}is the isotypic component of typeW{\displaystyle W}inV{\displaystyle V}.
InFourier analysis, one decomposes a (nice) function as thelimitof the Fourier series of the function. In much the same way, a representation itself may not be semisimple but it may be the completion (in a suitable sense) of a semisimple representation. The most basic case of this is thePeter–Weyl theorem, which decomposes the left (or right)regular representationof a compact group into the Hilbert-space completion of the direct sum of all simple unitary representations. As acorollary,[21]there is a natural decomposition forW=L2(G){\displaystyle W=L^{2}(G)}= the Hilbert space of (classes of) square-integrable functions on a compact groupG:
where⨁^{\displaystyle {\widehat {\bigoplus }}}means the completion of the direct sum and the direct sum runs over all isomorphism classes of simple finite-dimensional unitary representations(π,V){\displaystyle (\pi ,V)}ofG.[note 1]Note here that every simple unitary representation (up to an isomorphism) appears in the sum with the multiplicity the dimension of the representation.
When the groupGis a finite group, the vector spaceW=C[G]{\displaystyle W=\mathbb {C} [G]}is simply the group algebra ofGand also the completion is vacuous. Thus, the theorem simply says that
That is, each simple representation ofGappears in the regular representation with multiplicity the dimension of the representation.[22]This is one of standard facts in the representation theory of a finite group (and is much easier to prove).
When the groupGis thecircle groupS1{\displaystyle S^{1}}, the theorem exactly amounts to the classical Fourier analysis.[23]
Inquantum mechanicsandparticle physics, theangular momentumof an object can be described bycomplex representations of the rotation group SO(3), all of which are semisimple.[24]Due toconnection between SO(3) and SU(2), the non-relativisticspinof anelementary particleis described bycomplex representations of SU(2)and the relativistic spin is described bycomplex representations of SL2(C), all of which are semisimple.[24]Inangular momentum coupling,Clebsch–Gordan coefficientsarise from the multiplicities of irreducible representations occurring in the semisimple decomposition of a tensor product of irreducible representations.[25]
|
https://en.wikipedia.org/wiki/Semisimple_representation
|
Achild prodigyis, technically, a child under the age of 10 who produces meaningful work in some domain at the level of an adult expert.[1][2][3]The term is also applied more broadly to describe young people who are extraordinarily talented in some field.[4]
The termwunderkind(from GermanWunderkind; literally "wonder child") is sometimes used as a synonym for child prodigy, particularly in media accounts.Wunderkindalso is used to recognise those who achieve success and acclaim early in their adult careers.[5]
Generally, prodigies in all domains are suggested to have relatively elevatedIQ, extraordinary memory, and exceptional attention to detail. Significantly, while math and physics prodigies may have higher IQs, this may be an impediment to art prodigies.[6]
K. Anders Ericssonemphasised the contribution of deliberate practice over their innate talent to prodigies' exceptional performance in chess.[7]The deliberate practice is energy-consuming and requires attention to correct mistakes. As prodigies start formal chess training early with intense dedication to deliberate practice, they may accumulate enough deliberate practice for their exceptional performance. Therefore, this framework provide an arguably reasonable justification for chess prodigies. However, similar amounts of practice also make children differ in their achievements because of other factors such as the quality of deliberate practice, and their interests in chess.
Chess prodigies may have higherIQsthan normal children. This positive link between chess skills of prodigies and intelligence is particularly significant on the “performance intelligence”, regarding fluid reasoning, spatial processing, attentiveness to details, and visual-motor integration, while least significant on the “verbal intelligence”, regarding the ability to understand and reason using concepts framed in words.[8]However, this positive link is absent among adult experts. Remarkably, in the sample of chess prodigies, the more intelligent children played chess worse. This is considered as the result of less practice time of more intelligent chess skills.
Practice-plasticity-processes (PPP) model was proposed to explain the existence of chess prodigies by integrating the practice extreme and innate talent extreme theories. Besides deliberate practice,neuroplasticityis identified as another critical component for developing chess heuristics (e.g., simple search techniques and abstract rules like “occupy the centre”),chunks(e.g., group of pieces locating in specific squares), and templates (e.g., familiarised complex patterns of chunks), which are essential for chess skills. The more plastic the brain is, the easier it is for them to acquire chunks, templates, and heuristics for better performance. On the other hand, inherited individual differences in the brain are circumscribed children to learn these skills.[9]
Music prodigies usually express their talents in exceptional performance or composition.
The Multifactorial Gene-Environment Interaction Model incorporates the roles of adequate practice, certain personality traits, elevated IQ, and exceptional working memory in the explanation of music prodigies.[10]A study comparing current and former prodigies with normal people and musicians who showed their talents or were trained later in life to test this model. It found prodigies neither have exceptional performance in terms of IQ, working memory, nor specific personality. This study also emphasises the significance of frequent practice early in life, when the brain is moreplastic. Besides the quality of practice, and the parental investment, the experience offlowduring the practice is important for efficient and adequate practice for music prodigies. Practice demands high levels of concentration, which is hard for children in general, but flow can provide inherent pleasures of the practice to ensure this focused work.[11]
PET scansperformed on several mathematics prodigies have suggested that they think in terms of[clarification needed]long-term working memory (LTWM).[12]Thismemory, specific to a field of expertise,[clarification needed]is capable of holding relevant information for extended periods, usually hours. For example, experienced waiters have been found to hold the orders of up to twenty customers in their heads while they serve them, but perform only as well as an average person in number-sequence recognition. The PET scans also answer questions about which specific areas of the brain associate themselves with manipulating numbers.[12]
One subject[who?]never excelled as a child in mathematics, but he taught himself algorithms and tricks for calculatory speed, becoming capable of extremely complex mental math. His brain, compared to six other controls, was studied using the PET scan, revealing separate areas of his brain that he manipulated to solve complex problems. Some of the areas that he and presumably prodigies use are brain sectors dealing in visual and spatial memory, as well as visualmental imagery. Other areas of the brain showed use by the subject, including a sector of the brain generally related to childlike "finger counting", probably used in his mind to relate numbers to thevisual cortex.[12]
This finding is consistent with the introspective report of this[which?]calculating prodigy, which states that he used visual images to encode and retrieve numerical information in LTWM. Compared toshort-term memorystrategies, used by normal people on complex mathematical problems, encoding and retrievalepisodic memorystrategies would be more efficient. The prodigy may switch between these two strategies, which reduce the storage retrieval times of long-term memory and circumvent the limited capacities of short-term memory. In turn, they can encode and retrieve specific information (e.g., the intermediate answers during the calculation) in the long-term working memory more accurately and effectively.[13]
Similar strategies were found among prodigies masteringmental abacus calculation. The positions of beads on the physicalabacusact as visual proxies of each digit for prodigies to solve complex computations. This one-to-one corresponding structure allows them to rapidly encode and retrieve digits in the long-term working memory during the calculation.[14]ThefMRIscans showed stronger activation of brain areas related to visual processing for Chinese children being trained with abacus mental compared to control groups. This may indicate a greater demand for visuospatial information processing and visual-motor imagination in abacus mental calculation. Additionally, the right middle frontal gyrus activation is suggested to be the neuroanatomical link between prodigies' abacus mental calculation and the visuospatial working memory. This activation serves a mediation effect on the correlation between abacus-based mental calculation andvisuospatial working memory. A training-inducedneuroplasticityregarding working memory performance for children is proposed.[15]A study examining German calculating prodigies also proposed a similar reason for exceptional calculation abilities. Excellent working memory capacities and neuroplastic changes brought by extensive practice would be essential to enhance this domain-specific skill.[16]
"My mother said that I should finish high school and go to college first."
Noting that thecerebellumacts to streamline the speed and efficiency of all thought processes, Vandervert[18]explained the abilities of prodigies in terms of the collaboration ofworking memoryand the cognitive functions of the cerebellum. Citing extensive imaging evidence, Vandervert first proposed this approach in two publications which appeared in 2003. In addition to imaging evidence, Vandervert's approach is supported by the substantial award-winning studies of the cerebellum by Masao Ito.[19]
Vandervert[20]provided extensive argument that, in the prodigy, the transition from visual-spatial working memory to other forms of thought (language, art, mathematics) is accelerated by the unique emotional disposition of the prodigy and the cognitive functions of the cerebellum. According to Vandervert, in the emotion-driven prodigy (commonly observed as a "rage to master") the cerebellum accelerates the streamlining of the efficiencies of working memory in its manipulation and decomposition/re-composition of visual-spatial content intolanguage acquisitionand into linguistic, mathematical, and artistic precocity.[21]
Essentially, Vandervert has argued that when a child is confronted with a challenging new situation, visual-spatial working memory and speech-related and other notational system-related working memory are decomposed and re-composed (fractionated) by the cerebellum and thenblendedin the cerebral cortex in an attempt to deal with the new situation.[22]In child prodigies, Vandervert believes this blending process isaccelerateddue to their unique emotional sensitivities which result in high levels of repetitious focus on, in most cases, particularrule-governedknowledge domains. He has also argued that child prodigies first began to appear about 10,000 years ago when rule-governed knowledge had accumulated to a significant point, perhaps at the agricultural-religious settlements ofGöbekli TepeorCyprus.[23]
Some researchers believe that prodigious talent tends to arise as a result of the innate talent of the child, and the energetic and emotional investment that the child ventures. Others believe that the environment plays the dominant role, many times in obvious ways. For example,László Polgárset out to raise his children to be chess players, and all three of his daughters went on to become world-class players (two of whom aregrandmasters), emphasising the potency a child's environment can have in determining the pursuits toward which a child's energy will be directed, and showing that an incredible amount of skill can be developed through suitable training.[24]
Co-incidence theory explains the development of prodigies with a continuum of the discussion of nature and nurture. This theory states that the integrative of various factors in the development and expression of human potential, including:[25]
Prodigiousness in childhood is not always maintained into adulthood. Some researchers have found that gifted children fall behind due to lack of effort. Jim Taylor, professor at the University of San Francisco, theorizes that this is because gifted children experience success at an early age with little to no effort and may not develop a sense of ownership of success. Therefore, these children might not develop a connection between effort and outcome. Some children might also believe that they can succeed without effort in the future as well. Dr. Anders Ericcson, professor at Florida State University, researches expert performance in sports, music, mathematics, and other activities. His findings demonstrate that prodigiousness in childhood is not a strong indicator of later success. Rather, the number of hours devoted to the activity was a better indicator.[26]
Rosemary Callard-Szulgit and other educators have written extensively about the problem of perfectionism in bright children, calling it their "number one social-emotional trait". Gifted children often associate even slight imperfection with failure, so that they become fearful of effort, even in their personal lives, and in extreme cases end up virtually immobilized.[27]
Prodigies have been found with the over-representation of relatives with autism on their family pedigrees. Autism traits on theAutism-spectrum quotient(AQ) were reported in both first-degree relatives of child prodigies and of autism, which was higher than normal prevalence.[28]
Some autistic traits can be found among prodigies. Firstly, the social function of arithmetic prodigies may be weaker because of larger activation in certain brain areas enhancing their arithmetic performance, which is also essential for social and emotional functions (i.e., precuneus, lingual and fusiform gyrus). Theseneuroplasticchanges in neural networks may modulate their social performances in terms of emotional face processing and emotional evaluation of complex social interactions. Nevertheless, this emotional or social modulation must not score at psychopathological levels.[16]Additionally, the attentiveness to details, a typical characteristic of AQ, is enhanced among prodigies compared to normal people, even those withAsperger syndrome.[6]
|
https://en.wikipedia.org/wiki/Child_prodigy
|
TheDoomsday rule,Doomsday algorithmorDoomsday methodis analgorithmofdetermination of the day of the weekfor a given date. It provides aperpetual calendarbecause theGregorian calendarmoves in cycles of 400 years. The algorithm formental calculationwas devised byJohn Conwayin 1973,[1][2]drawing inspiration fromLewis Carroll'sperpetual calendar algorithm.[3][4][5]It takes advantage of each year having a certain day of the week upon which certain easy-to-remember dates, called thedoomsdays, fall; for example, the last day of February, April 4 (4/4), June 6 (6/6), August 8 (8/8), October 10 (10/10), and December 12 (12/12) all occur on the same day of the week in the year.
Applying the Doomsday algorithm involves three steps: determination of the anchor day for the century, calculation of the anchor day for the year from the one for the century, and selection of the closest date out of those that always fall on the doomsday, e.g., 4/4 and 6/6, and count of the number of days (modulo 7) between that date and the date in question to arrive at the day of the week. The technique applies to both theGregorian calendarand theJulian calendar, although their doomsdays are usually different days of the week.
The algorithm is simple enough that it can be computed mentally. Conway could usually give the correct answer in under two seconds. To improve his speed, he practiced his calendrical calculations on his computer, which was programmed to quiz him with random dates every time he logged on.[6]
Doomsday for the current year in the Gregorian calendar (2025) is Friday. Simple methods forfinding the doomsday of a yearexist.
One can find the day of the week of a given calendar date by using a nearby doomsday as a reference point. To help with this, the following is a list of easy-to-remember dates for each month that always land on the doomsday.
The last day of February is always a doomsday. For January, January 3 is a doomsday during common years and January 4 a doomsday during leap years, which can be remembered as "the 3rd during 3 years in 4, and the 4th in the 4th year". For March, one can remember eitherPi Dayor "March 0", the latter referring to the day before March 1, i.e. the last day of February.
For the months April through December, the even numbered months are covered by the double dates 4/4, 6/6, 8/8, 10/10, and 12/12, all of which fall on the doomsday. The odd numbered months can be remembered with the mnemonic "I work from9 to 5at the7-11", i.e., 9/5, 7/11, and also 5/9 and 11/7, are all doomsdays (this is true for both the Day/Month and Month/Day conventions).[8]
Several well-known dates, such asIndependence Day in United States,Boxing Day,HalloweenandValentine's Dayin common years, also fall on doomsdays every year.
Since the doomsday for a particular year is directly related to weekdays of dates in the period from March through February of the next year, common years and leap years have to be distinguished for January and February of the same year.
To find which day of the weekChristmas Dayof 2021 is, proceed as follows: in the year 2021, doomsday is on Sunday. Since December 12 is a doomsday, December 25, being thirteen days afterwards (two weeks less a day), fell on a Saturday. Christmas Day is always the day of the week before doomsday. In addition, July 4 (U.S. Independence Day) is always on the same day of the week as a doomsday, as areHalloween(October 31),Pi Day(March 14), and December 26 (Boxing Day).
Since this algorithm involves treating days of the week like numbers modulo 7,John Conwaysuggested thinking of the days of the week as "Noneday" or "Sansday" (for Sunday), "Oneday", "Twosday", "Treblesday", "Foursday", "Fiveday", and "Six-a-day" in order to recall the number-weekday relation without needing to count them out in one's head.[12]
There are some languages, such asSlavic languages,Chinese,Estonian,Greek,Portuguese,GalicianandHebrew, that base some of thenames of the week days in their positional order. The Slavic, Chinese, and Estonian agree with the table above; the other languages mentioned count from Sunday as day one.
First take the anchor day for the century. For the purposes of the doomsday rule, a century starts with '00 and ends with '99. The following table shows the anchor day of centuries 1600–1699, 1700–1799, 1800–1899, 1900–1999, 2000–2099, 2100–2199 and 2200–2299.
For the Gregorian calendar:
For the Julian calendar:
Note:c=⌊year100⌋{\displaystyle c={\biggl \lfloor }{{\text{year}} \over 100}{\biggr \rfloor }}.
Next, find the year's anchor day. To accomplish that according to Conway:[13]
For the twentieth-century year 1966, for example:
As described in bullet 4, above, this is equivalent to:
So doomsday in 1966 fell on Monday.
Similarly, doomsday in 2005 is on a Monday:
The doomsday's anchor day calculation is effectively calculating the number of days between any given date in the base year and the same date in the current year, then taking the remainder modulo 7. When both dates come after the leap day (if any), the difference is just365y+y/4(rounded down). But 365 equals 52 × 7 + 1, so after taking the remainder we get just
This gives a simpler formula if one is comfortable dividing large values ofyby both 4 and 7. For example, we can compute
which gives the same answer as in the example above.
Where 12 comes in is that the pattern of(y+⌊y4⌋)mod7{\displaystyle {\bigl (}y+{\bigl \lfloor }{\tfrac {y}{4}}{\bigr \rfloor }{\bigr )}{\bmod {7}}}almostrepeats every 12 years. After 12 years, we get(12+124)mod7=15mod7=1{\displaystyle {\bigl (}12+{\tfrac {12}{4}}{\bigr )}{\bmod {7}}=15{\bmod {7}}=1}. If we replaceybyymod 12, we are throwing this extra day away; but adding back in⌊y12⌋{\displaystyle {\bigl \lfloor }{\tfrac {y}{12}}{\bigr \rfloor }}compensates for this error, giving the final formula.
For calculating the Gregorian anchor day of a century: three “common centuries” (each having 24 leap years) are followed by a “leap century” (having 25 leap years). A common century moves the doomsday forward by
days (equivalent to two days back). A leap century moves the doomsday forward by 6 days (equivalent to one day back).
Soccenturies move the doomsday forward by
but this is equivalent to
Four centuries move the doomsday forward by
so four centuries form a cycle that leaves the doomsday unchanged (and hence the “mod 4” in the century formula).
A simpler method for finding the year's anchor day was discovered in 2010 by Chamberlain Fong and Michael K. Walters.[14]Called the "odd + 11" method, it is equivalent[14]to computing
It is well suited to mental calculation, because it requires no division by 4 (or 12), and the procedure is easy to remember because of its repeated use of the "odd + 11" rule. Furthermore, addition by 11 is very easy to perform mentally inbase-10 arithmetic.
Extending this to get the anchor day, the procedure is often described as accumulating a running totalTin six steps, as follows:
Applying this method to the year 2005, for example, the steps as outlined would be:
The explicit formula for the odd+11 method is:
Although this expression looks daunting and complicated, it is actually simple[14]because of acommon subexpressiony+ 11(ymod 2)/2that only needs to be calculated once.
Anytime adding 11 is needed, subtracting 17 yields equivalent results. While subtracting 17 may seem more difficult to mentally perform than adding 11, there are cases where subtracting 17 is easier, especially when the number is a two-digit number that ends in 7 (such as 17, 27, 37, ..., 77, 87, and 97).
Another method for calculating the Doomsday was proposed by H. Nakai in 2023.[15]
As above, let the year numbernbe expressed asn=100c+y{\displaystyle n=100c+y}, wherec{\displaystyle c}andy{\displaystyle y}represent the century and the last two digits of the year, respectively. Ifc2{\displaystyle c_{2}}andy2{\displaystyle y_{2}}denote the remainders whenc{\displaystyle c}andy{\displaystyle y}are divided by 4, respectively, then the number representing the day of the week for the Doomsday is given by the remainder5(c2+y2−1)+10ymod(7){\displaystyle 5(c_{2}+y_{2}-1)+10y\quad {\text{mod}}\;(7)}.
(August 7, 1966) The remainder on dividing(c,y)=(19,66){\displaystyle (c,y)=(19,66)}by 4 is(c2,y2)=(3,2){\displaystyle (c_{2},y_{2})=(3,2)}, which gives5⋅(3+2−1)=20≡6{\displaystyle 5\cdot (3+2-1)=20\equiv 6}; 10 timesy{\displaystyle y}is10⋅66=660≡2{\displaystyle 10\cdot 66=660\equiv {2}}, so Doomsday for 1966 is6+2=8≡1{\displaystyle 6+2=8\equiv 1}, that is, Monday. The difference between 7 and the Doomsday in August (namely 8) is7−8=−1≡6{\displaystyle 7-8=-1\equiv 6}, so the answer is1+6=7≡0{\displaystyle 1+6=7\equiv 0}, Sunday.[16]
Doomsday is related to thedominical letterof the year as follows.
Look up the table below for the dominical letter (DL).
For the year 2025, the dominical letter is A + 3 = E.
For computer use, the following formulas for the anchor day of a year are convenient.
For the Gregorian calendar:
For example, the doomsday 2009 is Saturday under the Gregorian calendar (the currently accepted calendar), since
As another example, the doomsday 1946 is Thursday, since
For the Julian calendar:
The formulas apply also for theproleptic Gregorian calendarand theproleptic Julian calendar. They use thefloor functionandastronomical year numberingfor years BC.
For comparison, seethe calculation of a Julian day number.
Since in the Gregorian calendar there are 146,097 days, or exactly 20,871 seven-day weeks, in 400 years, the anchor day repeats every four centuries. For example, the anchor day of 1700–1799 is the same as the anchor day of 2100–2199, i.e. Sunday.
The full 400-year cycle of doomsdays is given in the adjacent table. The centuries are for the Gregorian andproleptic Gregorian calendar, unless marked with a J for Julian. The Gregorian leap years are highlighted.
Negative years useastronomical year numbering. Year 25BC is −24, shown in the column of −100J (proleptic Julian) or −100 (proleptic Gregorian), at the row 76.
A leap year with Monday as doomsday means that Sunday is one of 97 days skipped in the 400-year sequence. Thus the total number of years with Sunday as doomsday is 71 minus the number of leap years with Monday as doomsday, etc. Since Monday as doomsday is skipped across February 29, 2000, and the pattern of leap days is symmetric about that leap day, the frequencies of doomsdays per weekday (adding common and leap years) are symmetric about Monday. The frequencies of doomsdays of leap years per weekday are symmetric about the doomsday of 2000, Tuesday.
The frequency of a particular date being on a particular weekday can easily be derived from the above (for a date from January 1 – February 28, relate it to the doomsday of the previous year).
For example, February 28 is one day after doomsday of the previous year, so it is 58 times each on Tuesday, Thursday and Sunday, etc. February 29 is doomsday of a leap year, so it is 15 times each on Monday and Wednesday, etc.
Regarding the frequency of doomsdays in a Julian 28-year cycle, there are 1 leap year and 3 common years for every weekday, the latter 6, 17 and 23 years after the former (so with intervals of 6, 11, 6, and 5 years; not evenly distributed because after 12 years the day is skipped in the sequence of doomsdays).[citation needed]The same cycle applies for any given date from March 1 falling on a particular weekday.
For any given date up to February 28 falling on a particular weekday, the 3 common years are 5, 11, and 22 years after the leap year, so with intervals of 5, 6, 11, and 6 years. Thus the cycle is the same, but with the 5-year interval after instead of before the leap year.
Thus, for any date except February 29, the intervals between common years falling on a particular weekday are 6, 11, 11. See e.g. at the bottom of the pageCommon year starting on Mondaythe years in the range 1906–2091.
For February 29 falling on a particular weekday, there is just one in every 28 years, and it is of course a leap year.
TheGregorian calendaris currently accurately lining up with astronomical events such assolstices. In 1582 this modification of theJulian calendarwas first instituted. In order to correct for calendar drift, 10 days were skipped, so doomsday moved back 10 days (i.e. 3 weekdays): Thursday, October 4 (Julian, doomsday is Wednesday) was followed by Friday, October 15 (Gregorian, doomsday is Sunday). The table includes Julian calendar years, but the algorithm is for the Gregorian and proleptic Gregorian calendar only.
Note that the Gregorian calendar was not adopted simultaneously in all countries, so for many centuries, different regions used different dates for the same day.
Suppose we want to know the day of the week of September 18, 1985. We begin with the century's anchor day, Wednesday. To this, adda,b, andcabove:
This yieldsa+b+c= 8. Counting 8 days from Wednesday, we reach Thursday, which is the doomsday in 1985. (Using numbers: In modulo 7 arithmetic, 8 is congruent to 1. Because the century's anchor day is Wednesday (index 3), and 3 + 1 = 4, doomsday in 1985 was Thursday (index 4).) We now compare September 18 to a nearby doomsday, September 5. We see that the 18th is 13 past a doomsday, i.e. one day less than two weeks. Hence, the 18th was a Wednesday (the day preceding Thursday). (Using numbers: In modulo 7 arithmetic, 13 is congruent to 6 or, more succinctly, −1. Thus, we take one away from the doomsday, Thursday, to find that September 18, 1985, was a Wednesday.)
Suppose that we want to find the day of week that theAmerican Civil Warbroke out atFort Sumter, which was April 12, 1861. The anchor day for the century was 94 days after Tuesday, or, in other words, Friday (calculated as18 × 5 + ⌊18/4⌋; or just look at the chart, above, which lists the century's anchor days). The digits 61 gave a displacement of six days so doomsday was Thursday. Therefore, April 4 was Thursday so April 12, eight days later, was a Friday.
|
https://en.wikipedia.org/wiki/Doomsday_rule
|
Thedetermination of the day of the weekfor any date may be performed with a variety ofalgorithms. In addition,perpetual calendarsrequire no calculation by the user, and are essentially lookup tables.
A typical application is to calculate theday of the weekon which someone was born or a specific event occurred.
In numerical calculation, the days of the week are represented as weekday numbers. If Monday is the first day of the week, the days may be coded 1 to 7, for Monday through Sunday, as is practiced inISO 8601. The day designated with 7 may also be counted as0, by applying thearithmetic modulo7, which calculates the remainder of a number after division by 7. Thus, the number 7 is treated as 0, the number 8 as 1, the number 9 as 2, the number 18 as 4, and so on. If Sunday is counted as day 1, then 7 days later (i.e.day 8) is also a Sunday, and day 18 is the same as day 4, which is a Wednesday since this falls three days after Sunday (i.e.18 mod 7 = 4).[a]
The basic approach of nearly all of the methods to calculate the day of the week begins by starting from an "anchor date": a known pair (such as 1 January 1800 as a Wednesday), determining the number of days between the known day and the day that you are trying to determine, and using arithmetic modulo 7 to find a new numerical day of the week.
One standard approach is to look up (or calculate, using a known rule) the value of the first day of the week of a given century, look up (or calculate, using a method of congruence) an adjustment for the month, calculate the number of leap years since the start of the century, and then add these together along with the number of years since the start of the century, and the day number of the month. Eventually, one ends up with a day-count to which one applies modulo 7 to determine the day of the week of the date.[4]
Some methods do all the additions first and then cast out sevens, whereas others cast them out at each step, as inLewis Carroll's method. Either way is quite viable: the former is easier for calculators and computer programs, the latter for mental calculation (it is quite possible to do all the calculations in one's head with a little practice). None of the methods given here perform range checks, so unreasonable dates will produce erroneous results.
Every seventh day in a month has the same name as the previous:
"Corresponding months" are those months within the calendar year that start on the same day of the week. For example, September and December correspond, because 1 September falls on the same day as 1 December (as there are precisely thirteen 7-day weeks between the two dates). Months can only correspond if the number of days between their first days is divisible by 7, or in other words, if their first days are a whole number of weeks apart. For example, February of acommon yearcorresponds to March because February has 28 days, a number divisible by 7, 28 days being exactly four weeks.
In aleap year, January and February correspond to different months than in a common year, since adding 29 February means each subsequent month starts a day later.
January corresponds to October in common years and April and July in leap years. February corresponds to March and November in common years and August in leap years. March always corresponds to November, April always corresponds to July, and September always corresponds to December. August does not correspond to any other month in a common year. October doesn't correspond to any other month in a leap year. May and June never correspond to any other month.
In the months table below, corresponding months have the same number, a fact which follows directly from the definition.
There are seven possible days that a year can start on, and leap years will alter the day of the week after 29 February. This means that there are 14 configurations that a year can have. All the configurations can be referenced by adominical letter, but as 29 February has no letter allocated to it, a leap year has two dominical letters, one for January and February and the other (one step back in the alphabetical sequence) for March to December.
2021 is a common year starting on Friday, which means that it corresponds to the 2010 calendar year. The first two months of 2021 correspond to the first two months of 2016. 2022 is a common year starting on Saturday, which means that it corresponds to the 2011 calendar year. The last ten months of 2022 correspond to the last ten months of 2016. 2023 is a common year starting on Sunday, which means that it corresponds to the 2017 calendar year. 2024 is a leap year starting on Monday, which means that it corresponds to the 1996 calendar year. The first two months of 2024 correspond to the first two months of 2018. The last ten months of 2024 correspond to the last ten months of 2019.
Each leap year repeats once every 28 years, and every common year repeats once every 6 years and twice every 11 years. For instance, the last occurrence of a leap year starting on Wednesday was 2020 and the next occurrence will be 2048. Likewise, the next common years starting on Friday will be 2027, 2038, and then 2049. Both of these statements are true unless a leap year is skipped, but that will not happen until 2100.
For details see the table below.
Notes:
"Year 000" is, in normal chronology, the year 1 BC (which precedes AD 1). Inastronomical year numberingthe year 0 comes between 1 BC and AD 1. In theproleptic Julian calendar, (that is, the Julian calendar as it would have been if it had been operated correctly from the start), 1 BC starts on Thursday. In theproleptic Gregorian calendar, (so called because it wasn't devised until 1582), 1 BC starts on Saturday.
For Julian dates before 1300 and after 1999 the year in the table which differs by an exact multiple of 700 years should be used. For Gregorian dates after 2299, the year in the table which differs by an exact multiple of 400 years should be used. The values "r0" through "r6" indicate the remainder when the Hundreds value is divided by 7 and 4 respectively, indicating how the series extend in either direction. Both Julian and Gregorian values are shown 1500-1999 for convenience. Bold figures (e.g.,04) denote leap year. If a year ends in 00 and its hundreds are in bold it is a leap year. Thus 19 indicates that 1900 is not a Gregorian leap year, (but19in the Julian column indicates that itisa Julian leap year, as are all Julianx00 years).20indicates that 2000 is a leap year. UseJanandFebonly in leap years.
For determination of the day of the week (1 January 2000, Saturday)
The formula is w = (d + m + y + c) mod 7.
Note that the date (and hence the day of the week) in the Revised Julian and Gregorian calendars is the same from 14 October 1923 to 28 February AD 2800 inclusive and that for large years it may be possible to subtract 6300 or a multiple thereof before starting so as to reach a year which is within or closer to the table.
To look up the weekday of any date for any year using the table, subtract 100 from the year, divide the difference by 100, multiply the resulting quotient (omitting fractions) by seven and divide the product by nine. Note the quotient (omitting fractions). Enter the table with the Julian year, and just before the final division add 50 and subtract the quotient noted above.
Example: What is the day of the week of 27 January 8315?
8315−6300=2015, 2015−100=1915, 1915/100=19 remainder 15, 19×7=133, 133/9=14 remainder 7. 2015 is 700 years ahead of 1315, so 1315 is used. From table: for hundreds (13): 6. For remaining digits (15): 4. For month (January): 0. For date (27): 27.6 + 4 + 0 + 27 + 50 − 14 = 73. 73/7=10 remainder 3. Day of week = Tuesday.
To find theDominical Letter, calculate the day of the week for either 1 January or 1 October. If it is Sunday, the Dominical Letter is A, if Saturday B, and similarly backwards through the week and forwards through the alphabet to Monday, which is G.
Leap years have two Sunday Letters, so for January and February calculate the day of the week for 1 January and for March to December calculate the day of the week for 1 October.
Leap years are all years which divide exactly by four with the following exceptions:
In the Gregorian calendar– all years which divide exactly by 100 (other than those which divide exactly by 400).
In the Revised Julian calendar– all years which divide exactly by 100 (other than those which give remainder 200 or 600 when divided by 900).
This is an artefact of recreational mathematics. Seedoomsday rulefor an explanation.
Use this table for finding the day of the week without any calculations.
All the C days are doomsdays
Examples:
December is in rowFand 26 is in columnE, so the letter for the date is C located in rowFand columnE. 93 (year mod 100) is in rowD(year row) and the letter C in the year row is located in columnG. 18 ([year/100] in the Gregorian century column) is in rowC(century row) and the letter in the century row and columnGis B, so the day of the week is Tuesday.
October 13 is a F day. The letter F in the year row (07) is located in columnG. The letter in the century row (13) and columnGis E, so the day of the week is Friday.
January 1 corresponds to G, G in the year row (00) corresponds to F in the century row (20), and F corresponds to Saturday.
A pithy formula for the method:"Date letter (G), letter (G) is in year row (00) for the letter (F) in century row (20), and for the day, the letter (F) become weekday (Saturday)".
The Sunday Letter method
Each day of the year (other than 29 February) has a letter allocated to it in the recurring sequence ABCDEFG. The series begins with A on 1 January and continues to A again on 31 December. The Sunday letter is the one which stands against all the Sundays in the year. Since 29 February has no letter, this means that the Sunday Letter for March to December is one step back in the sequence compared to that for January and February. The letter for any date will be found where the row containing the month (in black) at the left of the "Latin square" meets the column containing the date above the "Latin square". The Sunday letter will be found where the column containing the century (below the "Latin square") meets the row containing the year's last two digits to the right of the "Latin square". For a leap year, the Sunday letter thus found is the one which applies to March to December.
So, for example, to find the weekday of 16 June 2020:
Column "20" meets row "20" at "D". Row "June" meets column "16" at "F". As F is two letters on from D, so the weekday is two days on from Sunday, i.e. Tuesday.
TheRata Diemethod works by adding up the number of daysdthat has passed since a date of known day of the weekD. The day of-the-week is then given by(D+d) mod 7, conforming to whatever convention was used to encodeD.
For example, the date of 13 August 2009 is 733632 days from 1 January AD 1. Taking the number mod 7 yields 4, hence a Thursday.
Carl Friedrich Gaussdescribed a method for calculating the day of the week for 1 January in any given year in a handwritten note in a collection of astronomical tables.[5]He never published it. It was finally included in his collected works in 1927.[6]Compared to Rata Die, the result helps simplify the counting of years.
Gauss's method was applicable to the Gregorian calendar. He numbered the weekdays from 0 to 6 starting with Sunday. He defined the following operation.
The above procedure can be condensed into a single expression for the Gregorian case:(D + m + 5((A−1)%4) + 4((A−1)%100) + 6((A−1)%400))%7
For year number 2000,A− 1 = 1999,Y− 1 = 99andC= 19, the weekday of 1 January is
The weekdays for30 April 1777 and 23 February 1855are
and
The algorithm for the day-of-week of 1 Jan can be proven using modulo arithmetic. The main point is that because365 % 7 = 1, each year adds 1 day to the progression. The rest is adjustment for leap year. The century-based versions have36525 % 7 = 6.
The table of month offsets show a divergence in February due to the leap year. A common technique (later used by Zeller) is to shift the month to start with March, so that the leap day is at the tail of the counting. In addition, as later shown by Zeller, the table can be replaced with an arithmetic expression.
This formula was also converted into graphical and tabular methods for calculating any day of the week by Kraitchik and Schwerdtfeger.[6][7]
The following formula is an example of a version without a lookup table. The year is assumed to begin in March, meaning dates in January and February should be treated as being part of the preceding year. The formula for the Gregorian calendar is[8]
where
In Zeller's algorithm, the months are numbered from 3 for March to 14 for February. The year is assumed to begin in March; this means, for example, that January 1995 is to be treated as month 13 of 1994.[9]The formula for the Gregorian calendar isw≡(⌊13(m+1)5⌋+⌊y4⌋+⌊c4⌋+d+y−2c)mod7{\displaystyle w\equiv \left(\left\lfloor {\frac {13(m+1)}{5}}\right\rfloor +\left\lfloor {\frac {y}{4}}\right\rfloor +\left\lfloor {\frac {c}{4}}\right\rfloor +d+y-2c\right){\bmod {7}}}where
The only difference is one between Zeller's algorithm (Z) and the Disparate Gaussian algorithm (G), that isZ−G= 1 = Sunday.
Wang's algorithm[10]for human calculation of the Gregorian calendar is (the formula should be subtracted by 1 if m is 1 or 2 if the year is a leap year)w=(d−d0(m)+y0−y1+⌊y0/4−y1/2⌋−2(cmod4))mod7,{\displaystyle w=\left(d-d_{0}(m)+y_{0}-y_{1}+\left\lfloor y_{0}/4-y_{1}/2\right\rfloor -2\left(c{\bmod {4}}\right)\right){\bmod {7}},}where
An algorithm for the Julian calendar can be derived from the algorithm abovew=(d−d0(m)+y0−y1+⌊y0/4−y1/2⌋−c)mod7,{\displaystyle w=\left(d-d_{0}(m)+y_{0}-y_{1}+\left\lfloor y_{0}/4-y_{1}/2\right\rfloor -c\right){\bmod {7}},}whered0(m){\displaystyle d_{0}(m)}is a doomsday.
In a partly tabular method by Schwerdtfeger, the year is split into the century and the two digit year within the century. The approach depends on the month. Form≥ 3,
sogis between 0 and 99. Form= 1,2,
The formula for the day of the week is[6]
where the positive modulus is chosen.[6]
The value ofeis obtained from the following table:
The value offis obtained from the following table, which depends on the calendar. For the Gregorian calendar,[6]
For the Julian calendar,[6]
Charles Lutwidge Dodgson (Lewis Carroll) devised a method resembling a puzzle, yet partly tabular in using the same index numbers for the months as in the "Complete table: Julian and Gregorian calendars" above. He lists the same three adjustments for the first three months of non-leap years, one 7 higher for the last, and gives cryptic instructions for finding the rest; his adjustments for centuries are to be determined using formulas similar to those for the centuries table. Although explicit in asserting that his method also works forOld Styledates, his example reproduced below to determine that "1676, February 23" is a Wednesday only works on a Julian calendar which starts the year on January 1, instead of March 25 as on the "Old Style"Julian calendar.
Algorithm:[11]
Take the given date in 4 portions, viz. the number of centuries, the number of years over, the month, the day of the month.
Compute the following 4 items, adding each, when found, to the total of the previous items. When an item or total exceeds 7, divide by 7, and keep the remainder only.
Century-item: For 'Old Style' (which ended 2 September 1752) subtract from 18. For 'New Style' (which began 14 September 1752) divide by 4, take overplus [surplus] from 3, multiply remainder by 2.
Year-item: Add together the number of dozens, the overplus, and the number of 4s in the overplus.
Month-item: If it begins or ends with a vowel, subtract the number, denoting its place in the year, from 10. This, plus its number of days, gives the item for the following month. The item for January is "0"; for February or March, "3"; for December, "12".
Day-item: The total, thus reached, must be corrected, by deducting "1" (first adding 7, if the total be "0"), if the date be January or February in a leap year, remembering that every year, divisible by 4, is a Leap Year, excepting only the century-years, in 'New Style', when the number of centuries is not so divisible (e.g. 1800).
The final result gives the day of the week, "0" meaning Sunday, "1" Monday, and so on.
Examples:[11]
17, divided by 4, leaves "1" over; 1 from 3 gives "2"; twice 2 is "4".
83 is 6 dozen and 11, giving 17; plus 2 gives 19, i.e. (dividing by 7) "5". Total 9, i.e. "2"
The item for August is "8 from 10", i.e. "2"; so, for September, it is "2 plus 31", i.e. "5" Total 7, i.e. "0", which goes out.
18 gives "4". Answer, "Thursday".
16 from 18 gives "2"
76 is 6 dozen and 4, giving 10; plus 1 gives 11, i.e. "4".
Total "6"
The item for February is "3". Total 9, i.e. "2"
23 gives "2". Total "4"
Correction for Leap Year gives "3". Answer, "Wednesday".
Dates before 1752 would in England be givenOld Stylewith 25 March as thefirst day of the new year. Carroll's method however assumes 1 January as the first day of the year, thus he fails to arrive at the correct answer, namely "Friday".
Had he noticed that1676, February 23(with 25 March as New Year's Day) is actually1677, February 23(with 1 January as New Year's Day), he would have accounted for differing year numbers—just likeGeorge Washington's birthday differs—between the two calendars. Then his method yields:
16 from 18 gives "2" 77 is 6 dozen and 5, giving 11; plus 1 gives 12, i.e. "5". Total "7" The item for February is "3". Total 10, i.e. "3" 23 gives "2". Total "5". Answer, "Friday".
It is noteworthy that those who have republished Carroll's method have failed to point out his error, most notablyMartin Gardner.[12]
In 1752, the British Empire abandoned its use of theOld StyleJulian calendarupon adopting theGregorian calendar, which has become today's standard in most countries of the world. For more background, seeOld Style and New Style dates.
In theC languageexpressions below,y,manddare, respectively, integer variables representing the year (e.g., 1988), month (1–12) and day of the month (1-31).
In 1990, Michael Keith and Tom Craver published the foregoing expression that seeks to minimize the number of keystrokes needed to enter a self-contained function for converting a Gregorian date into a numerical day of the week.[13]It returns0= Sunday,1= Monday, etc. This expression uses a less cumbersome month component than does Zeller's algorithm.
Shortly afterwards, Hans Lachman streamlined their algorithm for ease of use on low-end devices. As designed originally for four-function calculators, his method needs fewer keypad entries by limiting its range either to A.D. 1905–2099, or to historical Julian dates. It was later modified to convert any Gregorian date, even on anabacus. OnMotorola 68000-based devices, there is similarly less need of eitherprocessor registersoropcodes, depending on the intended design objective.[14]
The tabular forerunner to Tøndering's algorithm is embodied in the followingK&R Cfunction.[15]With minor changes, it was adapted for otherhigh level programming languagessuch asAPL2.[16]Posted by Tomohiko Sakamoto on the comp.lang.cUsenet newsgroupin 1992, it is accurate for any Gregorian date.[17][18]
It returns0= Sunday,1= Monday, etc.
Sakamoto also simultaneously posted a more obfuscated version:
This version encodes the month offsets in the string and as a result requires a computer that uses standardASCIIto run the algorithm correctly, reducing itsportability. In addition, both algorithms omitinttype declarations, which is allowed in the originalK&R Cbut not allowed inANSI C.
(Tøndering's algorithm is, again, similar in structure to Zeller's congruence and Keith's short code, except that the month-related component is31*m/12. Sakamoto's is somewhere between the Disparate Gaussian and the Schwerdtfeger's algorithm, apparently unaware of the expression form.)
0: Sunday 1: Monday .. 6: Saturday
|
https://en.wikipedia.org/wiki/Calculating_the_day_of_the_week
|
Hypercalculiais "a specificdevelopmentalcondition in which the ability to performmathematicalcalculations is significantly superior to general learning ability and to school attainment in maths."[1]A 2002neuroimagingstudy of a child with hypercalculia suggested greater brain volume in the righttemporal lobe. SerialSPECTscans revealed hyperperfusion over rightparietal areasduring performance of arithmetic tasks.[2]
Children at any age may be stronger in language or in mathematics, but very rarely in both. Autistic children are no different. A rare example of a child with multiple savant tendencies is a case study of a thirteen-year-old girl. Pacheva, Panoy, Gillberg, and Neville discovered this individual has not only hypercalculia abilities, but also showcaseshyperlexia, and hypermnesia capabilities.[3]
A study published in 2014 examined the reading and math achievement profiles and their changes over time within a sample of children between the ages 6–9 diagnosed with anautism spectrum disorder. What they found was that there are four distinct achievement profiles: higher-achieving (39%), hyperlexia (9%), hypercalculia (20%) and lower-achieving (32%).[4]A previous study conducted in 2009 estimated the rate of hypercalculia at 16.2% in ASD adolescents.[5][6]
According to Wei, Christiano, Yu, Wagner, and Spiker, research of the ASD achievement profile, hypercalculia, is sometimes overlooked in academic settings. Sometimes this oversight is a result of more resources being spent on understanding the capabilities of children who exhibit hyperlexia. Children with an ASD have shown various results during testing for hypercalculia. Some of these varied results indicate: below average performance of mathematical and problem solving tasks, average proficiency, and high-achievers topping the 99th percentile on 'standardized math achievement measures.'[4]
There is an ongoing debate concerning the cause of hypercalculia along with other savant perceptions. Some researchers theorize that obsessive tendencies may trigger greater attention to certain areas of their lives.[2]
Individuals with autism will sometimes focus a lot of their time, energy, and attention on schedules or routines, calendar calculations, numbers or counting, and/or music.[7]
Other researchers speculate that people with savant tendencies may use different brain areas while they are processing subjects of their higher abilities. Among other debate arguments are hypotheses with regards to neural processes and working memory storage capabilities.[2]
Wallace sometimes refers to these individuals as "mathematical savants" or "arithmetic savants." In his experience, individuals with this ability tend to prefer a chunking or segmentation method of sorts. Their proclivities tend to push them towards breaking bigger things down to smaller things like numbers or equations. This data led Wallace to research, "prime number savants." Prime number savants can calculate which numbers are prime by breaking up the number over and over numerous times until they are at its lowest form.[clarification needed]The next step is figuring out if that number can be evenly divided.[8]
There are five different types of disorders that have been labeled on the autism spectrum. According to theDiagnostic and Statistical Manual of Mental Disorders, Fourth Edition (DSM-IV), the five different types of disorders on the autism spectrum are listed as: Autistic disorder, Asperger disorder, childhood disintegrative disorder, Rett disorder, and pervasive developmental disorder - not otherwise specified (PPD-NOS).[9]
A 2013 study examined the behavioral patterns of children on the autism spectrum who demonstrated measurable intellectual abilities. Their behavior was compared to that of neurotypical children with similar intellectual capacities. The study found that children on the autism spectrum were more likely to internalize personal difficulties. Further analysis suggested that this tendency may be associated with deficits in social and language development. Children exhibiting savant-like traits—such as hypercalculia, hyperlexia, and enhanced semantic memory—also showed a propensity to internalize issues, and were reported to experience higher levels of anxiety, low self-esteem, perfectionism, and challenges in social interactions.
These social difficulties were linked to patterns of social withdrawal and reluctance to engage or share with peers. Many of the children observed in the study were diagnosed with either PDD-NOS or Asperger syndrome. The findings indicated minimal behavioral differences between children categorized as high-performing intellectually and those with lower intellectual performance within the spectrum.[10]
Towards the end of the twentieth century, recognition of autistic children, including autistic children with savant abilities, has increased awareness in the educational system.[11]
There are just a few main names for savant children. The first category of savants was first discovered in London in 1887 by Dr.J. Langdon Down. Down coined the term 'idiot savant.' This term is given to individuals who have an IQ score below 25. These individuals show below average intelligence in most areas, but still show gifted expertise among such areas as music, arithmetic, reading, writing, or art to name a few. Idiot savant is no longer an acceptable name of categorization. It is not used very much anymore and was mainly discontinued after the first century of its discovery. Almost all individuals diagnosed with savant aptitudes test with an IQ of 40 or above.[12]
The second name often used for these children is 'autistic savant.' Just like Down's term, autistic savant is not always appropriate for all savant cases. Only half of individuals with savant syndrome are autistic. The other half of the savant population suffer from other central nervous system deficiencies caused by injuries or other disorders.[12]
Savant syndrome is the more overarching and accurate name to identify children with these higher-cognition skills.[12]
Awareness of savant syndrome has increased in recent years; however, the relatively low prevalence of the condition continues to pose challenges in developing and providing specialized educational resources to meet the specific needs of affected individuals. Advances in diagnostic tools have improved the identification of children with savant characteristics, allowing for better understanding of their cognitive profiles and educational requirements.[11]
Students with savant syndrome may demonstrate exceptional abilities in specific domains, such as mathematics, music, or memory, and some may participate in programs for gifted students. Despite these talents, they may also exhibit difficulties with communication and interpreting social cues, which can result in behaviors that appear inappropriate or socially unskilled in classroom settings.
Educational planning for such students often involves consideration of both their strengths and areas of difficulty, which vary significantly from one individual to another. For example, some children with mathematical savant skills may perform complex calculations with exceptional speed and accuracy—akin to a "human calculator"—yet struggle to apply these abilities in practical, everyday contexts. This disconnect between extraordinary cognitive skills and functional application is a characteristic noted in some cases of savant syndrome.[11]
|
https://en.wikipedia.org/wiki/Hypercalculia
|
Theabacus systemofmental calculationis a system where users mentally visualize anabacusto carry out arithmeticalcalculations.[1]No physical abacus is used; only the answers are written down. Calculations can be made at great speed in this way. For example, in the Flash Anzan event at theAll Japan Soroban Championship, champion Takeo Sasano was able to add fifteen three-digit numbers in just 1.7 seconds.[2]
This system is being propagated in China,[3]Singapore, South Korea, Thailand, Malaysia, and Japan. Mental calculation is said to improve mental capability, increases speed of response, memory power, and concentration power.
Many veteran and prolific abacus users in China, Japan, South Korea, and others who use the abacus daily, naturally tend to not use the abacus any more, but perform calculations by visualizing the abacus. This was verified when theright brainof visualisers showed heightenedEEGactivity when calculating, compared with others using an actual abacus to perform calculations.
The abacus can be used routinely to perform addition, subtraction, multiplication, and division; it can also be used to extract square and cube[4]roots.
This article about anumberis astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Mental_abacus
|
The titlemnemonistrefers to an individual with the ability to remember and recall unusually long lists of data, such as unfamiliar names, lists of numbers, entries in books, etc. Some mnemonists also memorize texts such as long poems, speeches, or even entire books of fiction or non-fiction. The term is derived from the termmnemonic, which refers to a strategy to support remembering (such as themethod of lociormajor system), but not all mnemonists report using mnemonics. Mnemonists may have superior innate ability to recall or remember,[1]in addition to (or instead of) relying on techniques.
While the innateness of mnemonists' skills is debated, the methods that mnemonists use to memorize are well-documented. Many mnemonists have been studied inpsychologylabs over the last century, and most have been found to use mnemonic devices. Currently, all memory champions at theWorld Memory Championshipshave said that they use mnemonic strategies, such as themethod of loci, to perform their memory feats.
Skilled memory theory was proposed byK. Anders Ericssonand Bill Chase to explain the effectiveness of mnemonic devices in memory expertise. Generally,short-term memoryhas a capacity ofseven items;[2]however, in order to memorize long strings of unrelated information, this constraint must be overcome. Skilled memory theory involves three steps: meaningful encoding, retrieval structure, and speed-up.[3]
In encoding, information is encoded in terms of knowledge structures through meaningful associations. This may initially involve breaking down long lists into more manageablechunksthat fall within the capacity of short term memory. Verbal reports of memory experts show a consistent grouping of three or four. A digit sequence 1-9-4-5, for example, can then be remembered as "the year World War Two ended". Luria reported thatSolomon Shereshevskyusedsynesthesiato associate numbers and words as visual images or colors to encode the information presented to him, but Luria did not clearly distinguish between synesthesia and mnemonic techniques like the method of loci and number shapes.[4][5]
Other subjects studied have used previous knowledge such as racing times or historical information[6]to encode new information. This is supported by studies that have shown that previous knowledge about a subject will increase one's ability to remember it. Chess experts, for example, can memorize more pieces of a chess game in progress than a novice chess player.[3]However, while there is some correlation between memory expertise and generalintelligence, as measured by eitherIQor thegeneral intelligence factor, the two are by no means identical. Many memory experts have been shown to be average to above-average by these two measures, but not exceptional.[7]
The next step is to create a retrieval structure by which the associations can be recalled. It serves the function of storing retrieval cues without having to use short term memory. It is used to preserve the order of items to be remembered. Verbal reports of memory experts show two prominent methods of retrieving information: hierarchical nodes and the method of loci. Retrieval structures are hierarchically organized and can be thought of as nodes that are activated when information is retrieved. Verbal reports have shown that memory experts have different retrieval structures. One expert clustered digits into groups, groups into supergroups, and supergroups into clusters of supergroups. However, by far the most common method of retrieval structure is the method of loci.[8]
The method of loci is "the use of an orderly arrangement of locations into which one could place the images of things or people that are to be remembered."[9]The encoding process happens in three steps. First, an architectural area, such as the houses on a street, must be memorized. Second, each item to be remembered must be associated with a separate image. Finally, this set of images can be distributed in a "locus", or place within the architectural area in a pre-determined order. Then, as one tries to recall the information, the mnemonists simply has to "walk" down the street, see each symbol, and recall the associated information. An example of mnemonists who used this method isSolomon Shereshevsky; he would useGorky Street, his own street. When he read, each word would form a graphic image. He would then place this image in a place along the street; later, when he needed to recall the information, he would simply "stroll" down the street again to recall the necessary information.[5]Neuroimagingstudies have shown results that support the method of loci as the retrieval method in world-class memory performers. AnfMRIrecorded brain activity in memory experts and a control group as they were memorizing selected data. Previous studies have shown that teaching a control group the method of loci leads to changes in brain activation during memorization. Consistent with their use of the method of loci, memory experts had higher activity in the medialparietal cortex, retrospenial cortex, and right posteriorhippocampus; these brain areas have been linked tospatial memory and navigation. These differences were observable even when the memory experts were trying to memorize stimuli, such as snowflakes, where they showed no superior ability to the control group.[10]
The final step in skilled memory theory is acceleration. With practice, time necessary for encoding and retrieval operations can be dramatically reduced. As a result, storage of information can then be performed within a few seconds. Indeed, one confounding factor in the study of memory is that the subjects often improve from day-to-day as they are tested over and over.
The innateness of expert performance in the memory field has been studied thoroughly by many scientists; it is a matter which has still not been definitively resolved.
Much evidence exists which points towards memory expertise as a learned skill which can only be learned through hours of deliberate practice. Anecdotally, the performers in top memory competitions like theWorld Memory Championshipsand theExtreme Memory Tournamentall deny any ability of a photographic memory; rather, these experts have averaged 10 years practicing their encoding strategies.[8]Another piece of evidence which points away from an innate superiority of memory is the specificity of memory expertise in memorists. For example, though memory experts have an exceptional ability to remember digits, their ability to remember unrelated items which are more difficult to encode, such as symbols or snowflakes, is the same as that of an average person. The same holds true for memory experts in other fields: studies ofmental calculatorsand chess experts show the same specificity for superior memory.[3][8][10]In some cases, other types of memory, such asvisual memoryfor faces, may even be impaired.[11]Another piece of evidence of memory expertise as a learned ability is the fact that dedicated individuals can make exceptional memory gains when exposed to mnemonics and given a chance to practice. One subject, SF, a college student of average intelligence, was able to attain world-class memory performance after hundreds of hours of practice over two years. His memory, in fact, improved over 70 standard deviations, while his digit span, ormemory spanfor digits, grew to 80 digits, which was higher than the digit span for all memory experts previously recorded.[3]Similarly, adults of average intelligence taught encoding strategies also show large gains in memory performance. Finally, neuroimaging studies performed on memory experts and compared to a control group have found no systematic anatomical differences in the brain between memory experts and a control group.[10]While it is true that there are activation differences between the brains of memory experts and a control group, they are due to the use of spatial techniques to form retrieval structures, not any structural differences.
Much of the evidence for innate superiority of memory is anecdotal and is therefore rejected by scientists who have moved toward accepting only reproducible studies as evidence for elite performance. There have been exceptions, however, that do not fit skilled memory theory as proposed by Chase and Ericsson. Synesthetes, for example, show a memory advantage for material that induces their synesthesia over a control group. This advantage tends to be in retention of new information rather than learning. However, synesthetes are likely to have some brain differences which give them an innate advantage when it comes to memory.[12]
Another group which may have some innate memory advantage areautistic savants. Unfortunately, many savants who have performed memory feats, such asKim Peek, have not been rigorously studied; they do claim not to need to use encoding strategies. A recent imaging study of savants found that there are activation differences between savants and typically developing individuals; these cannot be explained by the method of loci as mnemonic savants do not tend to use encoding strategies for their memory. Savants activated the right inferioroccipital areasof their brain, whereas control participants activated the left parietal region which is generally associated with attentional processes.[13]
Memory sportcontains a more comprehensive list of well-known memory athletes. The complete, up-to-date memory world rankings can be found at theInternational Association of Memorywebsite.[18]
|
https://en.wikipedia.org/wiki/Mnemonist
|
Thesoroban(算盤, そろばん, counting tray)is anabacusdeveloped inJapan. It is derived from theancient Chinesesuanpan, imported to Japan in the 14th century.[1][nb 1]Like the suanpan, the soroban is still used today, despite the proliferation of practical and affordable pocketelectronic calculators.
The soroban is composed of an odd number of columns or rods, each having beads: one separate bead having a value of five, calledgo-dama(五玉, ごだま, "five-bead")and four beads each having a value of one, calledichi-dama(一玉, いちだま, "one-bead"). Each set of beads of each rod is divided by a bar known as a reckoning bar. The number and size of beads in each rod make a standard-sized 13-rod soroban much less bulky than a standard-sized suanpan of similar expressive power.
The number of rods in a soroban is always odd and never fewer than seven. Basic models usually have thirteen rods, but the number of rods on practical or standard models often increases to 21, 23, 27 or even 31, thus allowing calculation of more digits or representations of several different numbers at the same time. Each rod represents a digit, and a larger number of rods allows the representation of more digits, either in singular form or during operations.
The beads and rods are made of a variety of different materials. Most soroban made in Japan are made of wood and have wood, metal,rattan, orbamboorods for the beads to slide on. The beads themselves are usuallybiconal(shaped like a double-cone). They are normally made of wood, although the beads of some soroban, especially those made outside Japan, can bemarble, stone, or even plastic. The cost of a soroban is commensurate with the materials used in its construction.
One unique feature that sets the soroban apart from its Chinese cousin is a dot marking every third rod in a soroban. These areunit rodsand any one of them is designated to denote the last digit of the whole number part of the calculation answer. Any number that is represented on rods to the right of this designated rod is part of the decimal part of the answer, unless the number is part of a division or multiplication calculation. Unit rods to the left of the designated one also aid in place value by denoting the groups in the number (such as thousands, millions, etc.). Suanpan usually do not have this feature.
The soroban uses abi-quinary coded decimalsystem, where each of the rods can represent a single digit from 0 to 9. By moving beads towards the reckoning bar, they are put in the "on" position; i.e., they assume value. For the "five bead" this means it is moved downwards, while "one beads" are moved upwards. In this manner, all digits from 0 to 9 can be represented by different configurations of beads, as shown below:
These digits can subsequently be used to represent multiple-digit numbers. This is done in the same way as in Western, decimal notation: the rightmost digit represents units, the one to the left of it represents tens, etc. The number8036, for instance, is represented by the following configuration:
The soroban user is free to choose which rod is used for the units; typically this will be one of the rods marked with a dot (see the 6 in the example above). Any digits to the right of the units represent decimals: tenths, hundredths, etc. In order to change8036into80.36, for instance, the user places the digits in such a way that the 0 falls on a rod marked with a dot:
The methods ofadditionandsubtractionon a soroban are basically the same as the equivalent operations on a suanpan, with basic addition and subtraction making use of acomplementary numberto add or subtract ten in carrying over.
There are many methods to perform bothmultiplicationanddivisionon a soroban, especially Chinese methods that came with the importation of the suanpan. The authority in Japan on the soroban, theJapan Abacus Committee, has recommended so-called standard methods for both multiplication and division which require only the use of themultiplication table. These methods were chosen for efficiency and speed in calculation.
Because the soroban developed through a reduction in the number of beads from seven, to six, and then to the present five, these methods can be used on the suanpan as well as on soroban produced before the 1930s, which have five "one" beads and one "five" bead.
The "five" beads methods for the olden soroban before the 1930s can be foundhere.
The Japanese abacus has been taught in school for over 500 years, deeply rooted in the value of learning the fundamentals as a form of art.[3]However, the introduction of the West during the Meiji period and then again after World War II has gradually altered the Japanese education system. Now, the strive is for speed and turning out deliverables rather than understanding the subtle intricacies of the concepts behind the product. Calculators have since replaced sorobans, and elementary schools are no longer required to teach students how to use the soroban, though some do so by choice. The growing popularity of calculators within the context of Japanese modernization has driven the study of soroban from public schools to private after school classrooms. Where once it was an institutionally required subject in school for children grades 2 to 6, current laws have made keeping this art form and perspective on math practiced amongst the younger generations more lenient.[4]Today, it shifted from a given to a game where one can take The Japanese Chamber of Commerce and Industry's examination in order to obtain a certificate and license.[5]
There are six levels of mastery, starting from sixth-grade (very skilled) all the way up to first-grade (for those who have completely mastered the use of the soroban). Those obtaining at least a third-grade certificate/license are qualified to work in public corporations.
The soroban is still taught in some primary schools as a way to visualize and grapple with mathematical concepts. The practice of soroban includes the teacher reciting a string of numbers (addition, subtraction, multiplication, and division) in a song-like manner where at the end, the answer is given by the teacher. This helps train the ability to follow the tempo given by the teacher while remaining calm and accurate. In this way, it reflects on a fundamental aspect of Japanese culture of practicing meditative repetition in every aspect of life.[3]Primary school students often bring two soroban to class, one with the modern configuration and the other one having the older configuration of one heavenly bead and five earth beads.
Shortly after the beginning of one's soroban studies, drills to enhancemental calculation, known asanzan(暗算, "blind calculation")in Japanese, are incorporated. Students are asked to solve problems mentally by visualizing the soroban and working out the solution by moving the beads theoretically in one's mind. The mastery of anzan is one reason why, despite the access to handheld calculators, some parents still send their children to private tutors to learn the soroban.
The soroban is also the basis for two kinds of abaci developed for the use of blind people. One is the toggle-type abacus wherein flip switches are used instead of beads. The second is the Cranmer abacus which has circular beads, longer rods, and a leather backcover so the beads do not slide around when in use.
The soroban's physical resemblance is derived from thesuanpanbut the number of beads is identical to theRoman abacus, which had four beads below and one at the top.
Most historians on the soroban agree that it has its roots on the suanpan's importation to Japan via the Korean peninsula around the 14th century.[1][nb 1]When the suanpan first became native to Japan as the soroban (with its beads modified for ease of use), it had two heavenly beads and five earth beads. But the soroban was not widely used until the 17th century, although it was in use by Japanese merchants since its introduction.[6]Once the soroban became popularly known, several Japanese mathematicians, includingSeki Kōwa, studied it extensively. These studies became evident on the improvements on the soroban itself and the operations used on it.
In the construction of the soroban itself, the number of beads had begun to decrease. In around 1850, one heavenly bead was removed from the suanpan configuration of two heavenly beads and five earth beads. This new Japanese configuration existed concurrently with the suanpan until the start of theMeiji era, after which the suanpan fell completely out of use. In 1891, Irie Garyū further removed one earth bead, forming the modern configuration of one heavenly bead and four earth beads.[7]This configuration was later reintroduced in 1930 and became popular in the 1940s.
Also, when the suanpan was imported to Japan, it came along with its division table. The method of using the table was calledkyūkihō(九帰法, "nine returning method")in Japanese, while the table itself was called thehassan(八算, "eight calculation"). Thedivision tableused along with the suanpan was more popular because of the original hexadecimal configuration ofJapanese currency[citation needed]. But because using the division table was complicated and it should be remembered along with the multiplication table, it soon fell out in 1935 (soon after the soroban's present form was reintroduced in 1930), with a so-called standard method replacing the use of the division table. This standard method of division, recommended today by the Japan Abacus Committee, is in fact an old method which usedcounting rods, first suggested by mathematician Momokawa Chubei in 1645,[8]and therefore had to compete with the division table during the latter's heyday.
On November 11, 1946, a contest was held in Tokyo between the Japanese soroban, used byKiyoshi Matsuzaki, and an electric calculator, operated by US Army Private Thomas Nathan Wood. The basis for scoring in the contest was speed and accuracy of results in all four basic arithmetic operations and a problem which combines all four. The soroban won 4 to 1, with the electric calculator prevailing in multiplication.[9]
About the event, theNippon Timesnewspaper reported that "Civilization ... tottered" that day,[10]while theStars and Stripesnewspaper described the soroban's "decisive" victory as an event in which "themachine agetook a step backward....".[11]
The breakdown of results is as follows:
|
https://en.wikipedia.org/wiki/Soroban
|
Inmathematics, aDiophantine equationis anequation, typically apolynomial equationin two or moreunknownswithintegercoefficients, for which onlyintegersolutions are of interest. Alinear Diophantine equationequates the sum of two or more unknowns, with coefficients, to a constant. Anexponential Diophantine equationis one in which unknowns can appear inexponents.
Diophantine problemshave fewer equations than unknowns and involve finding integers that solve all equations simultaneously. Because suchsystems of equationsdefinealgebraic curves,algebraic surfaces, or, more generally,algebraic sets, their study is a part ofalgebraic geometrythat is calledDiophantine geometry.
The wordDiophantinerefers to theHellenistic mathematicianof the 3rd century,DiophantusofAlexandria, who made a study of such equations and was one of the first mathematicians to introducesymbolismintoalgebra. The mathematical study of Diophantine problems that Diophantus initiated is now calledDiophantine analysis.
While individual equations present a kind of puzzle and have been considered throughout history, the formulation of general theories of Diophantine equations, beyond the case of linear andquadraticequations, was an achievement of the twentieth century.
In the following Diophantine equations,w, x, y, andzare the unknowns and the other letters are given constants:
The simplest linear Diophantine equation takes the formax+by=c,{\displaystyle ax+by=c,}wherea,bandcare given integers. The solutions are described by the following theorem:
Proof:Ifdis this greatest common divisor,Bézout's identityasserts the existence of integerseandfsuch thatae+bf=d. Ifcis a multiple ofd, thenc=dhfor some integerh, and(eh, fh)is a solution. On the other hand, for every pair of integersxandy, the greatest common divisordofaandbdividesax+by. Thus, if the equation has a solution, thencmust be a multiple ofd. Ifa=udandb=vd, then for every solution(x, y), we havea(x+kv)+b(y−ku)=ax+by+k(av−bu)=ax+by+k(udv−vdu)=ax+by,{\displaystyle {\begin{aligned}a(x+kv)+b(y-ku)&=ax+by+k(av-bu)\\&=ax+by+k(udv-vdu)\\&=ax+by,\end{aligned}}}showing that(x+kv, y−ku)is another solution. Finally, given two solutions such thatax1+by1=ax2+by2=c,{\displaystyle ax_{1}+by_{1}=ax_{2}+by_{2}=c,}one deduces thatu(x2−x1)+v(y2−y1)=0.{\displaystyle u(x_{2}-x_{1})+v(y_{2}-y_{1})=0.}Asuandvarecoprime,Euclid's lemmashows thatvdividesx2−x1, and thus that there exists an integerksuch that bothx2−x1=kv,y2−y1=−ku.{\displaystyle x_{2}-x_{1}=kv,\quad y_{2}-y_{1}=-ku.}Therefore,x2=x1+kv,y2=y1−ku,{\displaystyle x_{2}=x_{1}+kv,\quad y_{2}=y_{1}-ku,}which completes the proof.
TheChinese remainder theoremdescribes an important class of linear Diophantine systems of equations: letn1,…,nk{\displaystyle n_{1},\dots ,n_{k}}bekpairwise coprimeintegers greater than one,a1,…,ak{\displaystyle a_{1},\dots ,a_{k}}bekarbitrary integers, andNbe the productn1⋯nk.{\displaystyle n_{1}\cdots n_{k}.}The Chinese remainder theorem asserts that the following linear Diophantine system has exactly one solution(x,x1,…,xk){\displaystyle (x,x_{1},\dots ,x_{k})}such that0 ≤x<N, and that the other solutions are obtained by adding toxa multiple ofN:x=a1+n1x1⋮x=ak+nkxk{\displaystyle {\begin{aligned}x&=a_{1}+n_{1}\,x_{1}\\&\;\;\vdots \\x&=a_{k}+n_{k}\,x_{k}\end{aligned}}}
More generally, every system of linear Diophantine equations may be solved by computing theSmith normal formof its matrix, in a way that is similar to the use of thereduced row echelon formto solve asystem of linear equationsover a field. Usingmatrix notationevery system of linear Diophantine equations may be writtenAX=C,{\displaystyle AX=C,}whereAis anm×nmatrix of integers,Xis ann× 1column matrixof unknowns andCis anm× 1column matrix of integers.
The computation of the Smith normal form ofAprovides twounimodular matrices(that is matrices that are invertible over the integers and have ±1 as determinant)UandVof respective dimensionsm×mandn×n, such that the matrixB=[bi,j]=UAV{\displaystyle B=[b_{i,j}]=UAV}is such thatbi,iis not zero forinot greater than some integerk, and all the other entries are zero. The system to be solved may thus be rewritten asB(V−1X)=UC.{\displaystyle B(V^{-1}X)=UC.}Callingyithe entries ofV−1Xanddithose ofD=UC, this leads to the systembi,iyi=di,1≤i≤k0yi=di,k<i≤n.{\displaystyle {\begin{aligned}&b_{i,i}y_{i}=d_{i},\quad 1\leq i\leq k\\&0y_{i}=d_{i},\quad k<i\leq n.\end{aligned}}}
This system is equivalent to the given one in the following sense: A column matrix of integersxis a solution of the given system if and only ifx=Vyfor some column matrix of integersysuch thatBy=D.
It follows that the system has a solution if and only ifbi,idividesdifori≤kanddi= 0fori>k. If this condition is fulfilled, the solutions of the given system areV[d1b1,1⋮dkbk,khk+1⋮hn],{\displaystyle V\,{\begin{bmatrix}{\frac {d_{1}}{b_{1,1}}}\\\vdots \\{\frac {d_{k}}{b_{k,k}}}\\h_{k+1}\\\vdots \\h_{n}\end{bmatrix}}\,,}wherehk+1, …,hnare arbitrary integers.
Hermite normal formmay also be used for solving systems of linear Diophantine equations. However, Hermite normal form does not directly provide the solutions; to get the solutions from the Hermite normal form, one has to successively solve several linear equations. Nevertheless, Richard Zippel wrote that the Smith normal form "is somewhat more than is actually needed to solve linear diophantine equations. Instead of reducing the equation to diagonal form, we only need to make it triangular, which is called the Hermite normal form. The Hermite normal form is substantially easier to compute than the Smith normal form."[6]
Integer linear programmingamounts to finding some integer solutions (optimal in some sense) of linear systems that include alsoinequations. Thus systems of linear Diophantine equations are basic in this context, and textbooks on integer programming usually have a treatment of systems of linear Diophantine equations.[7]
A homogeneous Diophantine equation is a Diophantine equation that is defined by ahomogeneous polynomial. A typical such equation is the equation ofFermat's Last Theorem
As a homogeneous polynomial innindeterminates defines ahypersurfacein theprojective spaceof dimensionn− 1, solving a homogeneous Diophantine equation is the same as finding therational pointsof a projective hypersurface.
Solving a homogeneous Diophantine equation is generally a very difficult problem, even in the simplest non-trivial case of three indeterminates (in the case of two indeterminates the problem is equivalent with testing if arational numberis thedth power of another rational number). A witness of the difficulty of the problem is Fermat's Last Theorem (ford> 2, there is no integer solution of the above equation), which needed more than three centuries of mathematicians' efforts before being solved.
For degrees higher than three, most known results are theorems asserting that there are no solutions (for example Fermat's Last Theorem) or that the number of solutions is finite (for exampleFalting's theorem).
For the degree three, there are general solving methods, which work on almost all equations that are encountered in practice, but no algorithm is known that works for every cubic equation.[8]
Homogeneous Diophantine equations of degree two are easier to solve. The standard solving method proceeds in two steps. One has first to find one solution, or to prove that there is no solution. When a solution has been found, all solutions are then deduced.
For proving that there is no solution, one may reduce the equationmodulop. For example, the Diophantine equation
does not have any other solution than the trivial solution(0, 0, 0). In fact, by dividingx, y, andzby theirgreatest common divisor, one may suppose that they arecoprime. The squares modulo 4 are congruent to 0 and 1. Thus the left-hand side of the equation is congruent to 0, 1, or 2, and the right-hand side is congruent to 0 or 3. Thus the equality may be obtained only ifx, y, andzare all even, and are thus not coprime. Thus the only solution is the trivial solution(0, 0, 0). This shows that there is norational pointon acircleof radius3{\displaystyle {\sqrt {3}}}, centered at the origin.
More generally, theHasse principleallows deciding whether a homogeneous Diophantine equation of degree two has an integer solution, and computing a solution if there exist.
If a non-trivial integer solution is known, one may produce all other solutions in the following way.
Let
be a homogeneous Diophantine equation, whereQ(x1,…,xn){\displaystyle Q(x_{1},\ldots ,x_{n})}is aquadratic form(that is, a homogeneous polynomial of degree 2), with integer coefficients. Thetrivial solutionis the solution where allxi{\displaystyle x_{i}}are zero. If(a1,…,an){\displaystyle (a_{1},\ldots ,a_{n})}is a non-trivial integer solution of this equation, then(a1,…,an){\displaystyle \left(a_{1},\ldots ,a_{n}\right)}are thehomogeneous coordinatesof arational pointof the hypersurface defined byQ. Conversely, if(p1q,…,pnq){\textstyle \left({\frac {p_{1}}{q}},\ldots ,{\frac {p_{n}}{q}}\right)}are homogeneous coordinates of a rational point of this hypersurface, whereq,p1,…,pn{\displaystyle q,p_{1},\ldots ,p_{n}}are integers, then(p1,…,pn){\displaystyle \left(p_{1},\ldots ,p_{n}\right)}is an integer solution of the Diophantine equation. Moreover, the integer solutions that define a given rational point are all sequences of the form
wherekis any integer, anddis the greatest common divisor of thepi.{\displaystyle p_{i}.}
It follows that solving the Diophantine equationQ(x1,…,xn)=0{\displaystyle Q(x_{1},\ldots ,x_{n})=0}is completely reduced to finding the rational points of the corresponding projective hypersurface.
Let nowA=(a1,…,an){\displaystyle A=\left(a_{1},\ldots ,a_{n}\right)}be an integer solution of the equationQ(x1,…,xn)=0.{\displaystyle Q(x_{1},\ldots ,x_{n})=0.}AsQis a polynomial of degree two, a line passing throughAcrosses the hypersurface at a single other point, which is rational if and only if the line is rational (that is, if the line is defined by rational parameters). This allows parameterizing the hypersurface by the lines passing throughA, and the rational points are those that are obtained from rational lines, that is, those that correspond to rational values of the parameters.
More precisely, one may proceed as follows.
By permuting the indices, one may suppose, without loss of generality thatan≠0.{\displaystyle a_{n}\neq 0.}Then one may pass to the affine case by considering theaffine hypersurfacedefined by
which has the rational point
If this rational point is asingular point, that is if allpartial derivativesare zero atR, all lines passing throughRare contained in the hypersurface, and one has acone. The change of variables
does not change the rational points, and transformsqinto a homogeneous polynomial inn− 1variables. In this case, the problem may thus be solved by applying the method to an equation with fewer variables.
If the polynomialqis a product of linear polynomials (possibly with non-rational coefficients), then it defines twohyperplanes. The intersection of these hyperplanes is a rationalflat, and contains rational singular points. This case is thus a special instance of the preceding case.
In the general case, consider theparametric equationof a line passing throughR:
Substituting this inq, one gets a polynomial of degree two inx1, that is zero forx1=r1. It is thus divisible byx1−r1. The quotient is linear inx1, and may be solved for expressingx1as a quotient of two polynomials of degree at most two int2,…,tn−1,{\displaystyle t_{2},\ldots ,t_{n-1},}with integer coefficients:
Substituting this in the expressions forx2,…,xn−1,{\displaystyle x_{2},\ldots ,x_{n-1},}one gets, fori= 1, …,n− 1,
wheref1,…,fn{\displaystyle f_{1},\ldots ,f_{n}}are polynomials of degree at most two with integer coefficients.
Then, one can return to the homogeneous case. Let, fori= 1, …,n,
be thehomogenizationoffi.{\displaystyle f_{i}.}These quadratic polynomials with integer coefficients form a parameterization of the projective hypersurface defined byQ:
A point of the projective hypersurface defined byQis rational if and only if it may be obtained from rational values oft1,…,tn−1.{\displaystyle t_{1},\ldots ,t_{n-1}.}AsF1,…,Fn{\displaystyle F_{1},\ldots ,F_{n}}are homogeneous polynomials, the point is not changed if alltiare multiplied by the same rational number. Thus, one may suppose thatt1,…,tn−1{\displaystyle t_{1},\ldots ,t_{n-1}}arecoprime integers. It follows that the integer solutions of the Diophantine equation are exactly the sequences(x1,…,xn){\displaystyle (x_{1},\ldots ,x_{n})}where, fori= 1, ...,n,
wherekis an integer,t1,…,tn−1{\displaystyle t_{1},\ldots ,t_{n-1}}are coprime integers, anddis the greatest common divisor of thenintegersFi(t1,…,tn−1).{\displaystyle F_{i}(t_{1},\ldots ,t_{n-1}).}
One could hope that the coprimality of theti, could imply thatd= 1. Unfortunately this is not the case, as shown in the next section.
The equation
is probably the first homogeneous Diophantine equation of degree two that has been studied. Its solutions are thePythagorean triples. This is also the homogeneous equation of theunit circle. In this section, we show how the above method allows retrievingEuclid's formulafor generating Pythagorean triples.
For retrieving exactly Euclid's formula, we start from the solution(−1, 0, 1), corresponding to the point(−1, 0)of the unit circle. A line passing through this point may be parameterized by its slope:
Putting this in the circle equation
one gets
Dividing byx+ 1, results in
which is easy to solve inx:
It follows
Homogenizing as described above one gets all solutions as
wherekis any integer,sandtare coprime integers, anddis the greatest common divisor of the three numerators. In fact,d= 2ifsandtare both odd, andd= 1if one is odd and the other is even.
Theprimitive triplesare the solutions wherek= 1ands>t> 0.
This description of the solutions differs slightly from Euclid's formula because Euclid's formula considers only the solutions such thatx, y, andzare all positive, and does not distinguish between two triples that differ by the exchange ofxandy,
The questions asked in Diophantine analysis include:
These traditional problems often lay unsolved for centuries, and mathematicians gradually came to understand their depth (in some cases), rather than treat them as puzzles.
The given information is that a father's age is 1 less than twice that of his son, and that the digitsABmaking up the father's age are reversed in the son's age (i.e.BA). This leads to the equation10A+B= 2(10B+A) − 1, thus19B− 8A= 1. Inspection gives the resultA= 7,B= 3, and thusABequals 73 years andBAequals 37 years. One may easily show that there is not any other solution withAandBpositive integers less than 10.
Many well known puzzles in the field ofrecreational mathematicslead to diophantine equations. Examples include thecannonball problem,Archimedes's cattle problemandthe monkey and the coconuts.
In 1637,Pierre de Fermatscribbled on the margin of his copy ofArithmetica: "It is impossible to separate a cube into two cubes, or a fourth power into two fourth powers, or in general, any power higher than the second into two like powers." Stated in more modern language, "The equationan+bn=cnhas no solutions for anynhigher than 2." Following this, he wrote: "I have discovered a truly marvelous proof of this proposition, which this margin is too narrow to contain." Such a proof eluded mathematicians for centuries, however, and as such his statement became famous asFermat's Last Theorem. It was not until 1995 that it was proven by the British mathematicianAndrew Wiles.
In 1657, Fermat attempted to solve the Diophantine equation61x2+ 1 =y2(solved byBrahmaguptaover 1000 years earlier). The equation was eventually solved byEulerin the early 18th century, who also solved a number of other Diophantine equations. The smallest solution of this equation in positive integers isx= 226153980,y= 1766319049(seeChakravala method).
In 1900,David Hilbertproposed the solvability of all Diophantine equations asthe tenthof hisfundamental problems. In 1970,Yuri Matiyasevichsolved it negatively, building on work ofJulia Robinson,Martin Davis, andHilary Putnamto prove that a generalalgorithmfor solving all Diophantine equationscannot exist.
Diophantine geometry, is the application of techniques fromalgebraic geometrywhich considers equations that also have a geometric meaning. The central idea of Diophantine geometry is that of arational point, namely a solution to a polynomial equation or asystem of polynomial equations, which is a vector in a prescribedfieldK, whenKisnotalgebraically closed.
The oldest general method for solving a Diophantine equation—or for proving that there is no solution— is the method ofinfinite descent, which was introduced byPierre de Fermat. Another general method is theHasse principlethat usesmodular arithmeticmodulo all prime numbers for finding the solutions. Despite many improvements these methods cannot solve most Diophantine equations.
The difficulty of solving Diophantine equations is illustrated byHilbert's tenth problem, which was set in 1900 byDavid Hilbert; it was to find an algorithm to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution.Matiyasevich's theoremimplies that such an algorithm cannot exist.
During the 20th century, a new approach has been deeply explored, consisting of usingalgebraic geometry. In fact, a Diophantine equation can be viewed as the equation of anhypersurface, and the solutions of the equation are the points of the hypersurface that have integer coordinates.
This approach led eventually to theproof by Andrew Wilesin 1994 ofFermat's Last Theorem, stated without proof around 1637. This is another illustration of the difficulty of solving Diophantine equations.
An example of an infinite Diophantine equation is:n=a2+2b2+3c2+4d2+5e2+⋯,{\displaystyle n=a^{2}+2b^{2}+3c^{2}+4d^{2}+5e^{2}+\cdots ,}which can be expressed as "How many ways can a given integernbe written as the sum of a square plus twice a square plus thrice a square and so on?" The number of ways this can be done for eachnforms an integer sequence. Infinite Diophantine equations are related totheta functionsand infinite dimensional lattices. This equation always has a solution for any positiven.[9]Compare this to:n=a2+4b2+9c2+16d2+25e2+⋯,{\displaystyle n=a^{2}+4b^{2}+9c^{2}+16d^{2}+25e^{2}+\cdots ,}which does not always have a solution for positiven.
If a Diophantine equation has as an additional variable or variables occurring asexponents, it is an exponential Diophantine equation. Examples include:
A general theory for such equations is not available; particular cases such asCatalan's conjectureandFermat's Last Theoremhave been tackled. However, the majority are solved via ad-hoc methods such asStørmer's theoremor eventrial and error.
|
https://en.wikipedia.org/wiki/Diophantine_equation
|
Mathematical puzzlesmake up an integral part ofrecreational mathematics. They have specific rules, but they do not usually involve competition between two or more players. Instead, to solve such apuzzle, the solver must find a solution that satisfies the given conditions. Mathematical puzzles require mathematics to solve them.Logic puzzlesare a common type of mathematical puzzle.
Conway's Game of Lifeandfractals, as two examples, may also be considered mathematical puzzles even though the solver interacts with them only at the beginning by providing a set of initial conditions. After these conditions are set, the rules of the puzzle determine all subsequent changes and moves. Many of the puzzles are well known because they were discussed byMartin Gardnerin his "Mathematical Games" column in Scientific American. Mathematical puzzles are sometimes used to motivate students in teaching elementary schoolmath problemsolving techniques.[1]Creative thinking– or "thinking outside the box" – often helps to find the solution.
The fields ofknot theoryandtopology, especially their non-intuitive conclusions, are often seen as a part of recreational mathematics.
|
https://en.wikipedia.org/wiki/Mathematical_puzzle
|
Apuzzleis agame,problem, ortoythat tests a person's ingenuity orknowledge. In a puzzle, the solver is expected to put pieces together (or take them apart) in a logical way, in order to find the solution of the puzzle. There are different genres of puzzles, such ascrossword puzzles, word-search puzzles, number puzzles, relational puzzles, and logic puzzles. The academic study of puzzles is calledenigmatology.
Puzzles are often created to be a form of entertainment but they can also arise from seriousmathematicalorlogicalproblems. In such cases, their solution may be a significant contribution to mathematical research.[1]
TheOxford English Dictionarydates the wordpuzzle(as averb) to the 16th century. Its earliest use documented in theOEDwas in a book titledThe Voyage ofRobert Dudley...to the West Indies, 1594–95, narrated by Capt. Wyatt, by himself, and by Abram Kendall, master(published circa 1595). The word later came to be used as anoun, first as anabstract nounmeaning 'the state or condition of being puzzled', and later developing the meaning of 'a perplexing problem'. TheOED's earliest clear citation in the sense of 'a toy that tests the player's ingenuity' is from SirWalter Scott's 1814 novelWaverley, referring to a toy known as a "reel in a bottle".[2]
The etymology of the verbpuzzleis described byOEDas "unknown"; unproven hypotheses regarding its origin include an Old English verbpuslianmeaning 'pick out', and a derivation of the verbpose.[3]
Puzzles can be categorized as:
Solutions of puzzles often require the recognition ofpatternsand the adherence to a particular kind of order. People with a high level ofinductive reasoning aptitudemay be better at solving such puzzles compared to others. But puzzles based uponinquiryanddiscoverymay be solved more easily by those with gooddeduction skills. Deductive reasoning improves with practice. Mathematical puzzles often involve BODMAS.BODMASis an acronym which stands for Bracket, Of, Division, Multiplication, Addition and Subtraction. In certain regions, PEMDAS (Parentheses, Exponents, Multiplication, Division, Addition and Subtraction) is the synonym of BODMAS. It explains the order of operations to solve an expression. Some mathematical puzzles require top to bottom convention to avoid theambiguityin the order of operations. It is an elegantly simple idea that relies, assudokudoes, on the requirement that numbers appear only once starting from top to bottom as coming along.[4]
Puzzle makers are people who make puzzles. In general terms of occupation, apuzzlerorpuzzlistis someone who composes and/or solves puzzles.
Some notable creators of puzzles are:
The nine linked-rings puzzle, an advanced puzzle device that requires mathematical calculation to solve, was invented in China during theWarring States period(475-221 BCE).[5]Jigsaw puzzleswere invented around 1760, whenJohn Spilsbury, a British engraver andcartographer, mounted a map on a sheet of wood, which he then sawed around the outline of each individual country on the map. He then used the resulting pieces as an aid for the teaching of geography.[6]
After becoming popular among the public, this kind of teaching aid remained the primary use of jigsaw puzzles until about 1820.[7]
The largest puzzle (40,320 pieces) is made by a German game companyRavensburger.[8]The smallest puzzle ever made was created at LaserZentrum Hannover. It is only five square millimeters, the size of a sand grain.
The puzzles that were first documented areriddles. In Europe, Greek mythology produced riddles like theriddle of the Sphinx. Many riddles were produced during the Middle Ages, as well.[9]
By the early 20th century, magazines and newspapers found that they could increase their readership by publishingpuzzle contests, beginning withcrosswordsand in modern dayssudoku.
There are organizations and events that cater to puzzle enthusiasts, such as:
|
https://en.wikipedia.org/wiki/Puzzle
|
Sideways Arithmetic From Wayside Schoolis achildren'snovelbyLouis Sacharin theWayside Schoolseries. The book contains mathematical and logic puzzles for the reader to solve, presented as whatThe New Yorkercalled "absurdist math problems."[1]The problems are interspersed with characteristically quirky stories about the students at Wayside School.
Sideways Arithmetic from Wayside Schoolbegins with a foreword from Sachar in character as Louis the yard teacher, explaining the "sideways" nature of the problems within. He says that when he showed the students at Wayside School a regular math textbook, they laughed, thinking it was a book of jokes.
The first chapter introduces Sue, a new student in Mrs. Jewls's class. She is bewildered to discover that the arithmetic lessons involve adding words instead of numbers usingverbal arithmetic, e.g., "elf + elf = fool." The book presents an explanation for children of how these problems are solved, and then gives them several to do on their own. In chapter 2, Sue protests that math isn't supposed to be done that way, and gives the class a few traditional math problems like "seven + four = eleven." These are also presented as verbal arithmetic puzzles that are, as Mrs. Jewls states, impossible; the reader is tasked with figuring out why. In the next chapter, Mrs. Jewls tells Sue that if she doesn't understand how to do math in her class, she should switch schools. But when Sue inadvertently gets a question correct, Mrs. Jewls lets her stay. Chapter 4 contains more verbal arithmetic problems, this time with multiplication.
Beginning with chapter 5, the book switches to logic and optimization problems. In this chapter, students have to determine what happened at recess through logical elimination. In Chapter 6, Mrs. Jewls is having trouble filling outreport cardsbecause she lost the correct answers to a series of quizzes; the reader must logically deduce those answers based on the scores each student got. Chapter 7 presents an algebraic optimization problem: lunch lady Miss Mush's meals become more and more disgusting the more of them she prepares, and the reader must determine, among other things, how many meals she should cook so that the most students are willing to eat. Chapter 8 involves "false logic" puzzles, with statements presented as questions on true-or-false quizzes. In the final chapter, Sue finally makes a new friend, Joy, who has stayed after school trying to solve her impossible true-or-false test involving theliar's paradox). They go home together.
Charles Ashbacher, writing in theJournal of Recreational Mathematics, calledSideways Arithmetican "excellent supplementary book for elementary school mathematics", and suggested that the verbal arithmetic problems would be particularly useful in teaching.[2]The Guardianpraised the book and its sequel, writing: "Sachar never wastes a moment, a word or a clue."[3]
|
https://en.wikipedia.org/wiki/Sideways_Arithmetic_From_Wayside_School
|
Acryptogramis a type of puzzle that consists of a short piece ofencryptedtext.[1]Generally thecipherused to encrypt the text is simple enough that the cryptogram can be solved by hand.Substitution cipherswhere each letter is replaced by a different letter, number, or symbol are frequently used. To solve the puzzle, one must recover the original lettering. Though once used in more serious applications, they are now mainly printed for entertainment in newspapers and magazines.
Other types ofclassical ciphersare sometimes used to create cryptograms. An example is thebook cipher, where a book or article is used to encrypt a message.
The ciphers used in cryptograms were created not for entertainment purposes, but for real encryption of military orpersonal secrets.[2]
The first use of the cryptogram for entertainment purposes occurred during theMiddle Agesby monks who had spare time for intellectual games. A manuscript found atBambergstates that Irish visitors to the court ofMerfyn Frych ap Gwriad(died 844), king ofGwyneddinWales, were given a cryptogram which could only be solved by transposing the letters from Latin into Greek.[3]Around the thirteenth century, the English monkRoger Baconwrote a book in which he listed seven cipher methods, and stated that "a man is crazy who writes a secret in any other way than one which will conceal it from the vulgar." In the 19th centuryEdgar Allan Poehelped to popularize cryptograms with many newspaper and magazine articles.[4]
Well-known examples of cryptograms in contemporary culture are the syndicated newspaper puzzles Cryptoquip and Cryptoquote, fromKing Features.[5]Celebrity Cipher, distributed byAndrew McMeel, is another cipher game in contemporary culture, challenging the player to decrypt quotes from famous personalities.[6]
A cryptoquip is a specific type of cryptogram that usually comes with a clue or a pun. The solution often involves a humorous or witty phrase.[7]
In a public challenge, writer J.M. Appel announced on September 28, 2014, that the table of contents page of his short story collection,Scouting for the Reaper, doubled as a cryptogram, and he pledged an award for the first to solve it.[8]
Cryptograms based on substitution ciphers can often be solved byfrequency analysisand by recognizing letter patterns in words, such as one-letter words, which, in English, can only be "i" or "a" (and sometimes "o"). Double letters, apostrophes, and the fact that no letter can substitute for itself in the cipher also offer clues to the solution. Occasionally, cryptogram puzzle makers will start the solver off with a few letters.
A printed code key form (the alphabet with a blank under each letter to fill in the substituted letter) is usually not provided but can be drawn to use as a solving aid if needed. Skilled puzzle solvers should require neither a code key form nor starter clue letters.
While the cryptogram has remained popular, over time other puzzles similar to it have emerged. One of these is the Cryptoquote, which is a famous quote encrypted in the same way as a cryptogram. A more recent version, with a biblical twist, is CodedWord. This puzzle makes the solution available only online, where it provides a short exegesis on the biblical text. A third is the Cryptoquiz. The top of this puzzle has a category (unencrypted), such as "Flowers". Below this is a list of encrypted words which are related to the stated category. The person must then solve for the entire list to finish the puzzle. Yet another type involves using numbers as they relate to texting to solve the puzzle.
TheZodiac Killersent four cryptograms to police while he was still active. Despite much research, only two of these have been translated, which was of no help in identifying the serial killer.[9]
|
https://en.wikipedia.org/wiki/Cryptogram
|
Early numeracyis a branch ofnumeracythat aims to enhance numeracy learning for younger learners, particularly those at-risk in the area ofmathematics. Usually the mathematical learning begins with simply learning the first digits, 1 through 10. This is done because it acts as an entry way to the expansion of counting. One can keep track of the digits using any of thefingers.[1]
Thismathematics-related article is astub. You can help Wikipedia byexpanding it.
|
https://en.wikipedia.org/wiki/Early_numeracy
|
Elementary mathematics, also known asprimaryorsecondary schoolmathematics, is the study of mathematics topics that are commonly taught at the primary or secondary school levels around the world. It includes a wide range of mathematical concepts and skills, includingnumber sense,algebra,geometry,measurement, anddata analysis. These concepts and skills form the foundation for more advanced mathematical study and are essential for success in many fields and everyday life. The study of elementary mathematics is a crucial part of a student's education and lays the foundation for future academic and career success.
Number sense is an understanding of numbers and operations. In the 'Number Sense and Numeration' strand students develop an understanding of numbers by being taught various ways of representing numbers, as well as the relationships among numbers.[2]
Properties of thenatural numberssuch asdivisibilityand the distribution ofprime numbers, are studied in basicnumber theory, another part of elementary mathematics.
Elementary Focus:
'Measurement skills and concepts' or 'Spatial Sense' are directly related to the world in which students live. Many of the concepts that students are taught in this strand are also used in other subjects such as science, social studies, and physical education[3]In the measurement strand students learn about the measurable attributes of objects,in addition to the basic metric system.
Elementary Focus:
The measurement strand consists of multiple forms of measurement, as Marian Small states: "Measurement is the process of assigning a qualitative or quantitative description of size to an object based on a particular attribute."[4]
A formula is an entity constructed using the symbols and formation rules of a givenlogical language.[5]For example, determining thevolumeof asphererequires a significant amount ofintegral calculusor its geometrical analogue, themethod of exhaustion;[6]but, having done this once in terms of someparameter(theradiusfor example), mathematicians have produced a formula to describe the volume.
An equation is aformulaof the formA=B, whereAandBareexpressionsthat may contain one or severalvariablescalledunknowns, and "=" denotes theequalitybinary relation. Although written in the form ofproposition, an equation is not astatementthat is either true or false, but a problem consisting of finding the values, calledsolutions, that, when substituted for the unknowns, yield equal values of the expressionsAandB. For example, 2 is the uniquesolutionof theequationx+ 2 = 4, in which theunknownisx.[7]
Data is asetofvaluesofqualitativeorquantitativevariables; restated, pieces of data are individual pieces ofinformation. Data incomputing(ordata processing) is represented in astructurethat is oftentabular(represented byrowsandcolumns), atree(asetofnodeswithparent-childrenrelationship), or agraph(a set ofconnectednodes). Data is typically the result ofmeasurementsand can bevisualizedusinggraphsorimages.
Data as anabstractconceptcan be viewed as the lowest level ofabstraction, from whichinformationand thenknowledgeare derived.
Two-dimensional geometry is a branch ofmathematicsconcerned with questions of shape, size, and relative position of two-dimensional figures. Basic topics in elementary mathematics include polygons, circles, perimeter and areas.
Apolygonis a shape that is bounded by a finite chain of straightline segmentsclosing in a loop to form aclosed chainorcircuit. These segments are called itsedgesorsides, and the points where two edges meet are the polygon'svertices(singular: vertex) orcorners. The interior of the polygon is sometimes called itsbody. Ann-gonis a polygon withnsides. A polygon is a 2-dimensional example of the more generalpolytopein any number of dimensions.
Acircleis a simpleshapeoftwo-dimensional geometrythat is the set of allpointsin aplanethat are at a given distance from a given point, thecenter.The distance between any of the points and the center is called theradius. It can also be defined as the locus of a point equidistant from a fixed point.
Aperimeteris a path that surrounds atwo-dimensionalshape. The term may be used either for the path or its length - it can be thought of as the length of the outline of a shape. The perimeter of acircleorellipseis called itscircumference.
Areais thequantitythat expresses the extent of atwo-dimensionalfigure orshape. There are several well-knownformulasfor the areas of simple shapes such astriangles,rectangles, andcircles.
Two quantities are proportional if a change in one is always accompanied by a change in the other, and if the changes are always related by use of a constant multiplier. The constant is called thecoefficientof proportionality orproportionality constant.
Analytic geometryis the study ofgeometryusing acoordinate system. This contrasts withsynthetic geometry.
Usually theCartesian coordinate systemis applied to manipulateequationsforplanes,straight lines, andsquares, often in two and sometimes in three dimensions. Geometrically, one studies theEuclidean plane(2 dimensions) andEuclidean space(3 dimensions). As taught in school books, analytic geometry can be explained more simply: it is concerned with defining and representing geometrical shapes in a numerical way and extracting numerical information from shapes' numerical definitions and representations.
Transformations are ways of shifting and scaling functions using different algebraic formulas.
Anegative numberis areal numberthat isless thanzero. Such numbers are often used to represent the amount of a loss or absence. For example, adebtthat is owed may be thought of as a negative asset, or a decrease in some quantity may be thought of as a negative increase. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius andFahrenheitscales for temperature.
Exponentiation is amathematicaloperation, written asbn, involving two numbers, thebaseband theexponent(orpower)n. Whennis anatural number(i.e., a positiveinteger), exponentiation corresponds to repeatedmultiplicationof the base: that is,bnis theproductof multiplyingnbases:
Roots are the opposite of exponents. Thenth rootof anumberx(writtenxn{\displaystyle {\sqrt[{n}]{x}}}) is a numberrwhich when raised to the powernyieldsx. That is,
wherenis thedegreeof the root. A root of degree 2 is called asquare rootand a root of degree 3, acube root. Roots of higher degree are referred to by using ordinal numbers, as infourth root,twentieth root, etc.
For example:
Compass-and-straightedge, also known as ruler-and-compass construction, is the construction of lengths,angles, and other geometric figures using only anidealizedrulerandcompass.
The idealized ruler, known as astraightedge, is assumed to be infinite in length, and has no markings on it and only one edge. The compass is assumed to collapse when lifted from the page, so may not be directly used to transfer distances. (This is an unimportant restriction since, using a multi-step procedure, a distance can be transferred even with a collapsing compass, seecompass equivalence theorem.) More formally, the only permissible constructions are those granted by thefirst three postulatesofEuclid.
Two figures or objects are congruent if they have the sameshapeand size, or if one has the same shape and size as the mirror image of the other.[8]More formally, two sets ofpointsare calledcongruentif, and only if, one can be transformed into the other by anisometry, i.e., a combination ofrigid motions, namely atranslation, arotation, and areflection. This means that either object can be repositioned and reflected (but not resized) so as to coincide precisely with the other object. So two distinct plane figures on a piece of paper are congruent if we can cut them out and then match them up completely. Turning the paper over is permitted.
Two geometrical objects are calledsimilarif they both have the sameshape, or one has the same shape as the mirror image of the other. More precisely, one can be obtained from the other by uniformlyscaling(enlarging or shrinking), possibly with additionaltranslation,rotationandreflection. This means that either object can be rescaled, repositioned, and reflected, so as to coincide precisely with the other object. If two objects are similar, each iscongruentto the result of a uniform scaling of the other.
Solid geometrywas the traditional name for thegeometryof three-dimensionalEuclidean space.Stereometrydeals with themeasurementsofvolumesof varioussolid figures(three-dimensionalfigures) includingpyramids,cylinders,cones,truncated cones,spheres, andprisms.
Rational numberis anynumberthat can be expressed as thequotientor fractionp/qof twointegers, with thedenominatorqnot equal to zero.[9]Sinceqmay be equal to 1, every integer is a rational number. Thesetof all rational numbers is usually denoted by a boldfaceQ(orblackboard boldQ{\displaystyle \mathbb {Q} }).
Apatternis a discernible regularity in the world or in a manmade design. As such, the elements of a pattern repeat in a predictable manner. Ageometric patternis a kind of pattern formed of geometric shapes and typically repeating like aaallpaper.
Arelationon asetAis a collection ofordered pairsof elements ofA. In other words, it is asubsetof theCartesian productA2=A×A. Common relations include divisibility between two numbers and inequalities.
Afunction[10]is arelationbetween asetof inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real numberxto its squarex2. The output of a functionfcorresponding to an inputxis denoted byf(x) (read "fofx"). In this example, if the input is −3, then the output is 9, and we may writef(−3) = 9. The input variable(s) are sometimes referred to as the argument(s) of the function.
Theslope of a lineis a number that describes both thedirectionand thesteepnessof the line.[11]Slope is often denoted by the letterm.[12]
Trigonometryis a branch ofmathematicsthat studies relationships involving lengths andanglesoftriangles. The field emerged during the 3rd century BC from applications ofgeometryto astronomical studies.[13]
In the United States, there has been considerable concern about the low level of elementary mathematics skills on the part of many students, as compared to students in other developed countries.[14]TheNo Child Left Behindprogram was one attempt to address this deficiency, requiring that all American students be tested in elementary mathematics.[15]
|
https://en.wikipedia.org/wiki/Elementary_mathematics
|
Inmathematics educationat theprimary schoollevel,chunking(sometimes also called thepartial quotients method) is an elementary approach for solving simpledivisionquestions by repeatedsubtraction. It is also known as thehangman methodwith the addition of a line separating the divisor, dividend, and partial quotients.[1]It has a counterpart in thegrid methodfor multiplication as well.
In general, chunking is more flexible than the traditional method in that the calculation of quotient is less dependent on the place values. As a result, it is often considered to be a more intuitive, but a less systematic approach to divisions – where the efficiency is highly dependent upon one'snumeracyskills.
To calculate thewhole numberquotientof dividing a large number by a small number, the student repeatedly takes away "chunks" of the large number, where each "chunk" is an easy multiple (for example 100×, 10×, 5× 2×, etc.) of the small number, until the large number has been reduced to zero – or theremainderis less than the small number itself. At the same time the student is generating a list of the multiples of the small number (i.e., partial quotients) that have so far been taken away, which when added up together would then become the whole number quotient itself.
For example, to calculate 132÷8, one might successively subtract 80, 40 and 8 to leave 4:
Because 10 + 5 + 1 = 16, 132÷8 is 16 with 4 remaining.
In the UK, this approach for elementary division sums has come into widespread classroom use in primary schools since the late 1990s, when theNational Numeracy Strategyin its "numeracy hour" brought in a new emphasis on more free-form oral and mental strategies for calculations, rather than the rote learning of standard methods.[2]
Compared to theshort divisionandlong divisionmethods that are traditionally taught, chunking may seem strange, unsystematic, and arbitrary. However, it is argued that chunking, rather than moving straight to short division, gives a better introduction to division, in part because the focus is always holistic, focusing throughout on the whole calculation and its meaning, rather than just rules for generating successive digits. The more freeform nature of chunking also means that it requires more genuine understanding – rather than just the ability to follow a ritualised procedure – to be successful.[3]
An alternative way of performing chunking involves the use of the standard long division tableau – except that the partial quotients are stacked up on the top of each other above the long division sign, and that all numbers are spelled out in full. By allowing one to subtract more chunks than what one currently has, it is also possible to expand chunking into a fully bidirectional method as well.
|
https://en.wikipedia.org/wiki/Chunking_(division)
|
Inmathematical logic, thePeano axioms(/piˈɑːnoʊ/,[1][peˈaːno]), also known as theDedekind–Peano axiomsor thePeano postulates, areaxiomsfor thenatural numberspresented by the 19th-century Italian mathematicianGiuseppe Peano. These axioms have been used nearly unchanged in a number ofmetamathematicalinvestigations, including research into fundamental questions of whethernumber theoryisconsistentandcomplete.
Theaxiomatizationofarithmeticprovided by Peano axioms is commonly calledPeano arithmetic.
The importance of formalizingarithmeticwas not well appreciated until the work ofHermann Grassmann, who showed in the 1860s that many facts in arithmetic could be derived from more basic facts about thesuccessor operationandinduction.[2][3]In 1881,Charles Sanders Peirceprovided anaxiomatizationof natural-number arithmetic.[4][5]In 1888,Richard Dedekindproposed another axiomatization of natural-number arithmetic, and in 1889, Peano published a simplified version of them as a collection of axioms in his bookThe principles of arithmetic presented by a new method(Latin:Arithmetices principia, nova methodo exposita).
The nine Peano axioms contain three types of statements. The first axiom asserts the existence of at least one member of the set of natural numbers. The next four are general statements aboutequality; in modern treatments these are often not taken as part of the Peano axioms, but rather as axioms of the "underlying logic".[6]The next three axioms arefirst-orderstatements about natural numbers expressing the fundamental properties of the successor operation. The ninth, final, axiom is asecond-orderstatement of the principle of mathematical induction over the natural numbers, which makes this formulation close tosecond-order arithmetic. A weaker first-order system is obtained by explicitly adding the addition and multiplication operation symbols and replacing thesecond-order inductionaxiom with a first-orderaxiom schema. The termPeano arithmeticis sometimes used for specifically naming this restricted system.
When Peano formulated his axioms, the language ofmathematical logicwas in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation forset membership(∈, which comes from Peano's ε). Peano maintained a clear distinction between mathematical and logical symbols, which was not yet common in mathematics; such a separation had first been introduced in theBegriffsschriftbyGottlob Frege, published in 1879.[7]Peano was unaware of Frege's work and independently recreated his logical apparatus based on the work ofBooleandSchröder.[8]
The Peano axioms define the arithmetical properties ofnatural numbers, usually represented as asetNorN.{\displaystyle \mathbb {N} .}Thenon-logical symbolsfor the axioms consist of a constant symbol 0 and a unary function symbolS.
The first axiom states that the constant 0 is a natural number:
Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number,[9]while the axioms inFormulario mathematicoinclude zero.[10]
The next four axioms describe theequalityrelation. Since they are logically valid in first-order logic with equality, they are not considered to be part of "the Peano axioms" in modern treatments.[8]
The remaining axioms define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor"functionS.
Axioms 1, 6, 7, 8 define aunary representationof the intuitive notion of natural numbers: the number 1 can be defined asS(0), 2 asS(S(0)), etc. However, considering the notion of natural numbers as being defined by these axioms, axioms 1, 6, 7, 8 do not imply that the successor function generates all the natural numbers different from 0.
The intuitive notion that each natural number can be obtained by applyingsuccessorsufficiently many times to zero requires an additional axiom, which is sometimes called theaxiom of induction.
The induction axiom is sometimes stated in the following form:
In Peano's original formulation, the induction axiom is asecond-order axiom. It is now common to replace this second-order principle with a weakerfirst-orderinduction scheme. There are important differences between the second-order and first-order formulations, as discussed in the section§ Peano arithmetic as first-order theorybelow.
If we use the second-order induction axiom, it is possible to defineaddition,multiplication, andtotal (linear) orderingonNdirectly using the axioms. However,with first-order induction, this is not possible[citation needed]and addition and multiplication are often added as axioms. The respective functions and relations are constructed inset theoryorsecond-order logic, and can be shown to be unique using the Peano axioms.
Additionis a function thatmapstwo natural numbers (two elements ofN) to another one. It is definedrecursivelyas:
For example:
To prove commutativity of addition, first prove0+b=b{\displaystyle 0+b=b}andS(a)+b=S(a+b){\displaystyle S(a)+b=S(a+b)}, each by induction onb{\displaystyle b}. Using both results, then provea+b=b+a{\displaystyle a+b=b+a}by induction onb{\displaystyle b}.
Thestructure(N, +)is acommutativemonoidwith identity element 0.(N, +)is also acancellativemagma, and thusembeddablein agroup. The smallest group embeddingNis theintegers.[citation needed]
Similarly,multiplicationis a function mapping two natural numbers to another one. Given addition, it is defined recursively as:
It is easy to see thatS(0){\displaystyle S(0)}is the multiplicativeright identity:
To show thatS(0){\displaystyle S(0)}is also the multiplicative left identity requires the induction axiom due to the way multiplication is defined:
Therefore, by the induction axiomS(0){\displaystyle S(0)}is the multiplicative left identity of all natural numbers. Moreover, it can be shown[14]that multiplication is commutative anddistributes overaddition:
Thus,(N,+,0,⋅,S(0)){\displaystyle (\mathbb {N} ,+,0,\cdot ,S(0))}is a commutativesemiring.
The usualtotal orderrelation ≤ on natural numbers can be defined as follows, assuming 0 is a natural number:
This relation is stable under addition and multiplication: fora,b,c∈N{\displaystyle a,b,c\in \mathbb {N} }, ifa≤b, then:
Thus, the structure(N, +, ·, 1, 0, ≤)is anordered semiring; because there is no natural number between 0 and 1, it is a discrete ordered semiring.
The axiom of induction is sometimes stated in the following form that uses a stronger hypothesis, making use of the order relation "≤":
This form of the induction axiom, calledstrong induction, is a consequence of the standard formulation, but is often better suited for reasoning about the ≤ order. For example, to show that the naturals arewell-ordered—everynonemptysubsetofNhas aleast element—one can reason as follows. Let a nonemptyX⊆Nbe given and assumeXhas no least element.
Thus, by the strong induction principle, for everyn∈N,n∉X. Thus,X∩N= ∅, whichcontradictsXbeing a nonempty subset ofN. ThusXhas a least element.
Amodelof the Peano axioms is a triple(N, 0,S), whereNis a (necessarily infinite) set,0 ∈NandS:N→Nsatisfies the axioms above.Dedekindproved in his 1888 book,The Nature and Meaning of Numbers(German:Was sind und was sollen die Zahlen?, i.e., "What are the numbers and what are they good for?") that any two models of the Peano axioms (including the second-order induction axiom) areisomorphic. In particular, given two models(NA, 0A,SA)and(NB, 0B,SB)of the Peano axioms, there is a uniquehomomorphismf:NA→NBsatisfying
and it is abijection. This means that the second-order Peano axioms arecategorical. (This is not the case with any first-order reformulation of the Peano axioms, below.)
The Peano axioms can be derived fromset theoreticconstructions of thenatural numbersand axioms of set theory such asZF.[15]The standard construction of the naturals, due toJohn von Neumann, starts from a definition of 0 as the empty set, ∅, and an operatorson sets defined as:
The set of natural numbersNis defined as the intersection of all setsclosedundersthat contain the empty set. Each natural number is equal (as a set) to the set of natural numbers less than it:
and so on. The setNtogether with 0 and thesuccessor functions:N→Nsatisfies the Peano axioms.
Peano arithmetic isequiconsistentwith several weak systems of set theory.[16]One such system is ZFC with theaxiom of infinityreplaced by its negation. Another such system consists ofgeneral set theory(extensionality, existence of theempty set, and theaxiom of adjunction), augmented by an axiom schema stating that a property that holds for the empty set and holds of an adjunction whenever it holds of the adjunct must hold for all sets.
The Peano axioms can also be understood usingcategory theory. LetCbe acategorywithterminal object1C, and define the category ofpointed unary systems, US1(C) as follows:
ThenCis said to satisfy the Dedekind–Peano axioms if US1(C) has an initial object; this initial object is known as anatural number objectinC. If(N, 0,S)is this initial object, and(X, 0X,SX)is any other object, then the unique mapu: (N, 0,S) → (X, 0X,SX)is such that
This is precisely the recursive definition of 0XandSX.
When the Peano axioms were first proposed,Bertrand Russelland others agreed that these axioms implicitly defined what we mean by a "natural number".[17]Henri Poincaréwas more cautious, saying they only defined natural numbers if they wereconsistent; if there is a proof that starts from just these axioms and derives a contradiction such as 0 = 1, then the axioms are inconsistent, and don't define anything.[18]In 1900,David Hilbertposed the problem of proving their consistency using onlyfinitisticmethods as thesecondof histwenty-three problems.[19]In 1931,Kurt Gödelproved hissecond incompleteness theorem, which shows that such a consistency proof cannot be formalized within Peano arithmetic itself, if Peano arithmetic is consistent.[20]
Although it is widely claimed that Gödel's theorem rules out the possibility of a finitistic consistency proof for Peano arithmetic, this depends on exactly what one means by a finitistic proof. Gödel himself pointed out the possibility of giving a finitistic consistency proof of Peano arithmetic or stronger systems by using finitistic methods that are not formalizable in Peano arithmetic, and in 1958, Gödel published a method for proving the consistency of arithmetic usingtype theory.[21]In 1936,Gerhard Gentzengavea proof of the consistencyof Peano's axioms, usingtransfinite inductionup to anordinalcalledε0.[22]Gentzen explained: "The aim of the present paper is to prove the consistency of elementary number theory or, rather, to reduce the question of consistency to certain fundamental principles". Gentzen's proof is arguably finitistic, since the transfinite ordinal ε0can be encoded in terms of finite objects (for example, as aTuring machinedescribing a suitable order on the integers, or more abstractly as consisting of the finitetrees, suitably linearly ordered). Whether or not Gentzen's proof meets the requirements Hilbert envisioned is unclear: there is no generally accepted definition of exactly what is meant by a finitistic proof, and Hilbert himself never gave a precise definition.
The vast majority of contemporary mathematicians believe that Peano's axioms are consistent, relying either on intuition or the acceptance of a consistency proof such asGentzen's proof. A small number of philosophers and mathematicians, some of whom also advocateultrafinitism, reject Peano's axioms because accepting the axioms amounts to accepting the infinite collection of natural numbers. In particular, addition (including the successor function) and multiplication are assumed to betotal. Curiously, there areself-verifying theoriesthat are similar to PA but have subtraction and division instead of addition and multiplication, which are axiomatized in such a way to avoid proving sentences that correspond to the totality of addition and multiplication, but which are still able to prove all trueΠ1{\displaystyle \Pi _{1}}theorems of PA, and yet can be extended to a consistent theory that proves its own consistency (stated as the non-existence of a Hilbert-style proof of "0=1").[23]
All of the Peano axioms except the ninth axiom (the induction axiom) are statements infirst-order logic.[24]The arithmetical operations of addition and multiplication and the order relation can also be defined using first-order axioms. The axiom of induction above issecond-order, since itquantifiesover predicates (equivalently, sets of natural numbers rather than natural numbers). As an alternative one can consider a first-orderaxiom schemaof induction. Such a schema includes one axiom per predicate definable in the first-order language of Peano arithmetic, making it weaker than the second-order axiom.[25]The reason that it is weaker is that the number of predicates in first-order language is countable, whereas the number of sets of natural numbers is uncountable. Thus, there exist sets that cannot be described in first-order language (in fact, most sets have this property).
First-order axiomatizations of Peano arithmetic have another technical limitation. In second-order logic, it is possible to define the addition and multiplication operations from thesuccessor operation, but this cannot be done in the more restrictive setting of first-order logic. Therefore, the addition and multiplication operations are directly included in thesignatureof Peano arithmetic, and axioms are included that relate the three operations to each other.
The following list of axioms (along with the usual axioms of equality), which contains six of the seven axioms ofRobinson arithmetic, is sufficient for this purpose:[26]
In addition to this list of numerical axioms, Peano arithmetic contains the induction schema, which consists of arecursively enumerableand even decidable set ofaxioms. For each formulaφ(x,y1, ...,yk)in the language of Peano arithmetic, thefirst-order induction axiomforφis the sentence
wherey¯{\displaystyle {\bar {y}}}is an abbreviation fory1,...,yk. The first-order induction schema includes every instance of the first-order induction axiom; that is, it includes the induction axiom for every formulaφ.
The above axiomatization of Peano arithmetic uses a signature that only has symbols for zero as well as the successor, addition, and multiplications operations. There are many other different, but equivalent, axiomatizations. One such alternative[27]uses an order relation symbol instead of the successor operation and the language ofdiscretely ordered semirings(axioms 1-7 for semirings, 8-10 on order, 11-13 regarding compatibility, and 14-15 for discreteness):
The theory defined by these axioms is known asPA−. It is also incomplete and one of its important properties is that any structureM{\displaystyle M}satisfying this theory has an initial segment (ordered by≤{\displaystyle \leq }) isomorphic toN{\displaystyle \mathbb {N} }. Elements in that segment are calledstandardelements, while other elements are callednonstandardelements.
Finally, Peano arithmeticPAis obtained by adding the first-order induction schema.
According toGödel's incompleteness theorems, the theory ofPA(if consistent) is incomplete. Consequently, there are sentences offirst-order logic(FOL) that are true in the standard model ofPAbut are not a consequence of the FOL axiomatization. Essential incompleteness already arises for theories with weaker axioms, such asRobinson arithmetic.
Closely related to the above incompleteness result (viaGödel's completeness theoremfor FOL) it follows that there is noalgorithmfor deciding whether a given FOL sentence is a consequence of a first-order axiomatization of Peano arithmetic or not. Hence,PAis an example of anundecidable theory. Undecidability arises already for the existential sentences ofPA, due to the negative answer toHilbert's tenth problem, whose proof implies that allcomputably enumerablesets arediophantine sets, and thus definable by existentially quantified formulas (with free variables) ofPA. Formulas ofPAwith higherquantifier rank(more quantifier alternations) than existential formulas are more expressive, and define sets in the higher levels of thearithmetical hierarchy.
Although the usualnatural numberssatisfy the axioms ofPA, there are other models as well (called "non-standard models"); thecompactness theoremimplies that the existence of nonstandard elements cannot be excluded in first-order logic.[28]The upwardLöwenheim–Skolem theoremshows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism.[29]This illustrates one way the first-order system PA is weaker than the second-order Peano axioms.
When interpreted as a proof within a first-orderset theory, such asZFC, Dedekind's categoricity proof for PA shows that each model of set theory has a unique model of the Peano axioms, up to isomorphism, that embeds as an initial segment of all other models of PA contained within that model of set theory. In the standard model of set theory, this smallest model of PA is the standard model of PA; however, in a nonstandard model of set theory, it may be a nonstandard model of PA. This situation cannot be avoided with any first-order formalization of set theory.
It is natural to ask whether a countable nonstandard model can be explicitly constructed. The answer is affirmative asSkolemin 1933 provided an explicit construction of such anonstandard model. On the other hand,Tennenbaum's theorem, proved in 1959, shows that there is no countable nonstandard model of PA in which either the addition or multiplication operation iscomputable.[30]This result shows it is difficult to be completely explicit in describing the addition and multiplication operations of a countable nonstandard model of PA. There is only one possibleorder typeof a countable nonstandard model. Lettingωbe the order type of the natural numbers,ζbe the order type of the integers, andηbe the order type of the rationals, the order type of any countable nonstandard model of PA isω+ζ·η, which can be visualized as a copy of the natural numbers followed by a dense linear ordering of copies of the integers.
Acutin a nonstandard modelMis a nonempty subsetCofMso thatCis downward closed (x<yandy∈C⇒x∈C) andCis closed under successor. Aproper cutis a cut that is a proper subset ofM. Each nonstandard model has many proper cuts, including one that corresponds to the standard natural numbers. However, the induction scheme in Peano arithmetic prevents any proper cut from being definable. The overspill lemma, first proved by Abraham Robinson, formalizes this fact.
Overspill lemma[31]—LetMbe a nonstandard model of PA and letCbe a proper cut ofM. Suppose thata¯{\displaystyle {\bar {a}}}is a tuple of elements ofMandφ(x,a¯){\displaystyle \varphi (x,{\bar {a}})}is a formula in the language of arithmetic so that
Then there is acinMthat is greater than every element ofCsuch that
This article incorporates material fromPAonPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.
|
https://en.wikipedia.org/wiki/Peano_axioms
|
Inmathematics,division by zero,divisionwhere the divisor (denominator) iszero, is a unique and problematic special case. Usingfractionnotation, the general example can be written asa0{\displaystyle {\tfrac {a}{0}}}, wherea{\displaystyle a}is the dividend (numerator).
The usual definition of thequotientinelementary arithmeticis the number which yields the dividend whenmultipliedby the divisor. That is,c=ab{\displaystyle c={\tfrac {a}{b}}}is equivalent toc⋅b=a.{\displaystyle c\cdot b=a.}By this definition, the quotientq=a0{\displaystyle q={\tfrac {a}{0}}}is nonsensical, as the productq⋅0{\displaystyle q\cdot 0}is always0{\displaystyle 0}rather than some other numbera.{\displaystyle a.}Following the ordinary rules ofelementary algebrawhile allowing division by zero can create amathematical fallacy, a subtle mistake leading to absurd results. To prevent this, the arithmetic ofreal numbersand more general numerical structures calledfieldsleaves division by zeroundefined, and situations where division by zero might occur must be treated with care. Since any number multiplied by zero is zero, the expression00{\displaystyle {\tfrac {0}{0}}}is also undefined.
Calculusstudies the behavior offunctionsin thelimitas their input tends to some value. When areal functioncan be expressed as a fraction whose denominator tends to zero, the output of the function becomes arbitrarily large, and is said to "tend to infinity", a type ofmathematical singularity. For example, thereciprocal function,f(x)=1x,{\displaystyle f(x)={\tfrac {1}{x}},}tends to infinity asx{\displaystyle x}tends to0.{\displaystyle 0.}When both the numerator and the denominator tend to zero at the same input, the expression is said to take anindeterminate form, as the resulting limit depends on the specific functions forming the fraction and cannot be determined from their separate limits.
As an alternative to the common convention of working with fields such as the real numbers and leaving division by zero undefined, it is possible to define the result of division by zero in other ways, resulting in different number systems. For example, the quotienta0{\displaystyle {\tfrac {a}{0}}}can be defined to equal zero; it can be defined to equal a new explicitpoint at infinity, sometimes denoted by theinfinity symbol∞{\displaystyle \infty };or it can be defined to result in signed infinity, with positive or negative sign depending on the sign of the dividend. In these number systems division by zero is no longer a special exception per se, but the point or points at infinity involve their own new types of exceptional behavior.
Incomputing, an error may result from an attempt to divide by zero. Depending on the context and the type of number involved, dividing by zero may evaluate topositive or negative infinity, return a specialnot-a-numbervalue, orcrashthe program, among other possibilities.
ThedivisionN/D=Q{\displaystyle N/D=Q}can be conceptually interpreted in several ways.[1]
Inquotitive division, the dividendN{\displaystyle N}is imagined to be split up into parts of sizeD{\displaystyle D}(the divisor), and the quotientQ{\displaystyle Q}is the number of resulting parts. For example, imagine ten slices of bread are to be made into sandwiches, each requiring two slices of bread. A total of five sandwiches can be made(102=5{\displaystyle {\tfrac {10}{2}}=5}).Now imagine instead that zero slices of bread are required per sandwich (perhaps alettuce wrap). Arbitrarily many such sandwiches can be made from ten slices of bread, as the bread is irrelevant.[2]
The quotitive concept of division lends itself to calculation by repeatedsubtraction: dividing entails counting how many times the divisor can be subtracted before the dividend runs out. Because no finite number of subtractions of zero will ever exhaust a non-zero dividend, calculating division by zero in this waynever terminates.[3]Such an interminable division-by-zeroalgorithmis physically exhibited by somemechanical calculators.[4]
Inpartitive division, the dividendN{\displaystyle N}is imagined to be split intoD{\displaystyle D}parts, and the quotientQ{\displaystyle Q}is the resulting size of each part. For example, imagine ten cookies are to be divided among two friends. Each friend will receive five cookies(102=5{\displaystyle {\tfrac {10}{2}}=5}).Now imagine instead that the ten cookies are to be divided among zero friends. How many cookies will each friend receive? Since there are no friends, this is an absurdity.[5]
In another interpretation, the quotientQ{\displaystyle Q}represents theratioN:D.{\displaystyle N:D.}[6]For example, a cake recipe might call for ten cups of flour and two cups of sugar, a ratio of10:2{\displaystyle 10:2}or, proportionally,5:1.{\displaystyle 5:1.}To scale this recipe to larger or smaller quantities of cake, a ratio of flour to sugar proportional to5:1{\displaystyle 5:1}could be maintained, for instance one cup of flour and one-fifth cup of sugar, or fifty cups of flour and ten cups of sugar.[7]Now imagine a sugar-free cake recipe calls for ten cups of flour and zero cups of sugar. The ratio10:0,{\displaystyle 10:0,}or proportionally1:0,{\displaystyle 1:0,}is perfectly sensible:[8]it just means that the cake has no sugar. However, the question "How many parts flour for each part sugar?" still has no meaningful numerical answer.
A geometrical appearance of the division-as-ratio interpretation is theslopeof astraight linein theCartesian plane.[9]The slope is defined to be the "rise" (change in vertical coordinate) divided by the "run" (change in horizontal coordinate) along the line. When this is written using the symmetrical ratio notation, a horizontal line has slope0:1{\displaystyle 0:1}and a vertical line has slope1:0.{\displaystyle 1:0.}However, if the slope is taken to be a singlereal numberthen a horizontal line has slope01=0{\displaystyle {\tfrac {0}{1}}=0}while a vertical line has an undefined slope, since in real-number arithmetic the quotient10{\displaystyle {\tfrac {1}{0}}}is undefined.[10]The real-valued slopeyx{\displaystyle {\tfrac {y}{x}}}of a line through the origin is the vertical coordinate of theintersectionbetween the line and a vertical line at horizontal coordinate1,{\displaystyle 1,}dashed black in the figure. The vertical red and dashed black lines areparallel, so they have no intersection in the plane. Sometimes they are said to intersect at apoint at infinity, and the ratio1:0{\displaystyle 1:0}is represented by a new number∞{\displaystyle \infty };[11]see§ Projectively extended real linebelow. Vertical lines are sometimes said to have an "infinitely steep" slope.
Division is the inverse ofmultiplication, meaning that multiplying and then dividing by the same non-zero quantity, or vice versa, leaves an original quantity unchanged; for example(5×3)/3={\displaystyle (5\times 3)/3={}}(5/3)×3=5{\displaystyle (5/3)\times 3=5}.[12]Thus a division problem such as63=?{\displaystyle {\tfrac {6}{3}}={?}}can be solved by rewriting it as an equivalent equation involving multiplication,?×3=6,{\displaystyle {?}\times 3=6,}where?{\displaystyle {?}}represents the same unknown quantity, and then finding the value for which the statement is true; in this case the unknown quantity is2,{\displaystyle 2,}because2×3=6,{\displaystyle 2\times 3=6,}so therefore63=2.{\displaystyle {\tfrac {6}{3}}=2.}[13]
An analogous problem involving division by zero,60=?,{\displaystyle {\tfrac {6}{0}}={?},}requires determining an unknown quantity satisfying?×0=6.{\displaystyle {?}\times 0=6.}However, any number multiplied by zero is zero rather than six, so there exists no number which can substitute for?{\displaystyle {?}}to make a true statement.[14]
When the problem is changed to00=?,{\displaystyle {\tfrac {0}{0}}={?},}the equivalent multiplicative statement is?×0=0{\displaystyle {?}\times 0=0};in this caseanyvalue can be substituted for the unknown quantity to yield a true statement, so there is no single number which can be assigned as the quotient00.{\displaystyle {\tfrac {0}{0}}.}
Because of these difficulties, quotients where the divisor is zero are traditionally taken to beundefined, and division by zero is not allowed.[15][16]
A compelling reason for not allowing division by zero is that allowing it leads tofallacies.
When working with numbers, it is easy to identify an illegal division by zero. For example:
The fallacy here arises from the assumption that it is legitimate to cancel0like any other number, whereas, in fact, doing so is a form of division by0.
Usingalgebra, it is possible to disguise a division by zero[17]to obtain aninvalid proof. For example:[18]
This is essentially the same fallacious computation as the previous numerical version, but the division by zero was obfuscated because we wrote0asx− 1.
TheBrāhmasphuṭasiddhāntaofBrahmagupta(c. 598–668) is the earliest text to treatzeroas a number in its own right and to define operations involving zero.[17]According to Brahmagupta,
A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator. Zero divided by zero is zero.
In 830,Mahāvīraunsuccessfully tried to correct the mistake Brahmagupta made in his bookGanita Sara Samgraha: "A number remains unchanged when divided by zero."[17]
Bhāskara II'sLīlāvatī(12th century) proposed that division by zero results in an infinite quantity,[19]
A quantity divided by zero becomes a fraction the denominator of which is zero. This fraction is termed an infinite quantity. In this quantity consisting of that which has zero for its divisor, there is no alteration, though many may be inserted or extracted; as no change takes place in the infinite and immutable God when worlds are created or destroyed, though numerous orders of beings are absorbed or put forth.
Historically, one of the earliest recorded references to the mathematical impossibility of assigning a value toa0{\textstyle {\tfrac {a}{0}}}is contained inAnglo-IrishphilosopherGeorge Berkeley's criticism ofinfinitesimal calculusin 1734 inThe Analyst("ghosts of departed quantities").[20]
Calculusstudies the behavior offunctionsusing the concept of alimit, the value to which a function's output tends as its input tends to some specific value. The notationlimx→cf(x)=L{\textstyle \lim _{x\to c}f(x)=L}means that the value of the functionf{\displaystyle f}can be made arbitrarily close toL{\displaystyle L}by choosingx{\displaystyle x}sufficiently close toc.{\displaystyle c.}
In the case where the limit of thereal functionf{\displaystyle f}increases without bound asx{\displaystyle x}tends toc,{\displaystyle c,}the function is not defined atx,{\displaystyle x,}a type ofmathematical singularity. Instead, the function is said to "tend to infinity", denotedlimx→cf(x)=∞,{\textstyle \lim _{x\to c}f(x)=\infty ,}and itsgraphhas the linex=c{\displaystyle x=c}as a verticalasymptote. While such a function is not formally defined forx=c,{\displaystyle x=c,}and theinfinity symbol∞{\displaystyle \infty }in this case does not represent any specificreal number, such limits are informally said to "equal infinity". If the value of the function decreases without bound, the function is said to "tend to negative infinity",−∞.{\displaystyle -\infty .}In some cases a function tends to two different values whenx{\displaystyle x}tends toc{\displaystyle c}from above(x→c+{\displaystyle x\to c^{+}})and below(x→c−{\displaystyle x\to c^{-}}); such a function has two distinctone-sided limits.[21]
A basic example of an infinite singularity is thereciprocal function,f(x)=1/x,{\displaystyle f(x)=1/x,}which tends to positive or negative infinity asx{\displaystyle x}tends to0{\displaystyle 0}:
limx→0+1x=+∞,limx→0−1x=−∞.{\displaystyle \lim _{x\to 0^{+}}{\frac {1}{x}}=+\infty ,\qquad \lim _{x\to 0^{-}}{\frac {1}{x}}=-\infty .}
In most cases, the limit of a quotient of functions is equal to the quotient of the limits of each function separately,
limx→cf(x)g(x)=limx→cf(x)limx→cg(x).{\displaystyle \lim _{x\to c}{\frac {f(x)}{g(x)}}={\frac {\displaystyle \lim _{x\to c}f(x)}{\displaystyle \lim _{x\to c}g(x)}}.}
However, when a function is constructed by dividing two functions whose separate limits are both equal to0,{\displaystyle 0,}then the limit of the result cannot be determined from the separate limits, so is said to take anindeterminate form, informally written00.{\displaystyle {\tfrac {0}{0}}.}(Another indeterminate form,∞∞,{\displaystyle {\tfrac {\infty }{\infty }},}results from dividing two functions whose limits both tend to infinity.) Such a limit may equal any real value, may tend to infinity, or may not converge at all, depending on the particular functions. For example, in
limx→1x2−1x−1,{\displaystyle \lim _{x\to 1}{\dfrac {x^{2}-1}{x-1}},}
the separate limits of the numerator and denominator are0{\displaystyle 0}, so we have the indeterminate form00{\displaystyle {\tfrac {0}{0}}}, but simplifying the quotient first shows that the limit exists:
limx→1x2−1x−1=limx→1(x−1)(x+1)x−1=limx→1(x+1)=2.{\displaystyle \lim _{x\to 1}{\frac {x^{2}-1}{x-1}}=\lim _{x\to 1}{\frac {(x-1)(x+1)}{x-1}}=\lim _{x\to 1}(x+1)=2.}
Theaffinely extended real numbersare obtained from thereal numbersR{\displaystyle \mathbb {R} }by adding two new numbers+∞{\displaystyle +\infty }and−∞,{\displaystyle -\infty ,}read as "positive infinity" and "negative infinity" respectively, and representingpoints at infinity. With the addition of±∞,{\displaystyle \pm \infty ,}the concept of a "limit at infinity" can be made to work like a finite limit. When dealing with both positive and negative extended real numbers, the expression1/0{\displaystyle 1/0}is usually left undefined. However, in contexts where only non-negative values are considered, it is often convenient to define1/0=+∞{\displaystyle 1/0=+\infty }.
The setR∪{∞}{\displaystyle \mathbb {R} \cup \{\infty \}}is theprojectively extended real line, which is aone-point compactificationof the real line. Here∞{\displaystyle \infty }means an unsigned infinity orpoint at infinity, an infinite quantity that is neither positive nor negative. This quantity satisfies−∞=∞{\displaystyle -\infty =\infty }, which is necessary in this context. In this structure,a0=∞{\displaystyle {\frac {a}{0}}=\infty }can be defined for nonzeroa, anda∞=0{\displaystyle {\frac {a}{\infty }}=0}whenais not∞{\displaystyle \infty }. It is the natural way to view the range of thetangent functionand cotangent functions oftrigonometry:tan(x)approaches the single point at infinity asxapproaches either+π/2or−π/2from either direction.
This definition leads to many interesting results. However, the resulting algebraic structure is not afield, and should not be expected to behave like one. For example,∞+∞{\displaystyle \infty +\infty }is undefined in this extension of the real line.
The subject ofcomplex analysisapplies the concepts of calculus in thecomplex numbers. Of major importance in this subject is theextended complex numbersC∪{∞},{\displaystyle \mathbb {C} \cup \{\infty \},}the set of complex numbers with a single additional number appended, usually denoted by theinfinity symbol∞{\displaystyle \infty }and representing apoint at infinity, which is defined to be contained in everyexterior domain, making those itstopologicalneighborhoods.
This can intuitively be thought of as wrapping up the infinite edges of the complex plane and pinning them together at the single point∞,{\displaystyle \infty ,}aone-point compactification, making the extended complex numbers topologically equivalent to asphere. This equivalence can be extended to a metrical equivalence by mapping each complex number to a point on the sphere via inversestereographic projection, with the resultingspherical distanceapplied as a new definition of distance between complex numbers; and in general the geometry of the sphere can be studied using complex arithmetic, and conversely complex arithmetic can be interpreted in terms of spherical geometry. As a consequence, the set of extended complex numbers is often called theRiemann sphere. The set is usually denoted by the symbol for the complex numbers decorated by an asterisk, overline, tilde, or circumflex, for exampleC^=C∪{∞}.{\displaystyle {\hat {\mathbb {C} }}=\mathbb {C} \cup \{\infty \}.}
In the extended complex numbers, for any nonzero complex numberz,{\displaystyle z,}ordinary complex arithmetic is extended by the additional rulesz0=∞,{\displaystyle {\tfrac {z}{0}}=\infty ,}z∞=0,{\displaystyle {\tfrac {z}{\infty }}=0,}∞+0=∞,{\displaystyle \infty +0=\infty ,}∞+z=∞,{\displaystyle \infty +z=\infty ,}∞⋅z=∞.{\displaystyle \infty \cdot z=\infty .}However,00{\displaystyle {\tfrac {0}{0}}},∞∞{\displaystyle {\tfrac {\infty }{\infty }}}, and0⋅∞{\displaystyle 0\cdot \infty }are left undefined.
The four basic operations – addition, subtraction, multiplication and division – as applied to whole numbers (positive integers), with some restrictions, in elementary arithmetic are used as a framework to support the extension of the realm of numbers to which they apply. For instance, to make it possible to subtract any whole number from another, the realm of numbers must be expanded to the entire set ofintegersin order to incorporate the negative integers. Similarly, to support division of any integer by any other, the realm of numbers must expand to therational numbers. During this gradual expansion of the number system, care is taken to ensure that the "extended operations", when applied to the older numbers, do not produce different results. Loosely speaking, since division by zero has no meaning (isundefined) in the whole number setting, this remains true as the setting expands to therealor evencomplex numbers.[22]
As the realm of numbers to which these operations can be applied expands there are also changes in how the operations are viewed. For instance, in the realm of integers, subtraction is no longer considered a basic operation since it can be replaced by addition of signed numbers.[23]Similarly, when the realm of numbers expands to include the rational numbers, division is replaced by multiplication by certain rational numbers. In keeping with this change of viewpoint, the question, "Why can't we divide by zero?", becomes "Why can't a rational number have a zero denominator?". Answering this revised question precisely requires close examination of the definition of rational numbers.
In the modern approach to constructing the field of real numbers, the rational numbers appear as an intermediate step in the development that is founded onset theory. First, the natural numbers (including zero) are established on an axiomatic basis such asPeano's axiom systemand then this is expanded to thering of integers. The next step is to define the rational numbers keeping in mind that this must be done using only the sets and operations that have already been established, namely, addition, multiplication and the integers. Starting with the set ofordered pairsof integers,{(a,b)}withb≠ 0, define abinary relationon this set by(a,b) ≃ (c,d)if and only ifad=bc. This relation is shown to be anequivalence relationand itsequivalence classesare then defined to be the rational numbers. It is in the formal proof that this relation is an equivalence relation that the requirement that the second coordinate is not zero is needed (for verifyingtransitivity).[24][25][26]
Although division by zero cannot be sensibly defined with real numbers and integers, it is possible to consistently define it, or similar operations, in other mathematical structures.
In thehyperreal numbers, division by zero is still impossible, but division by non-zeroinfinitesimalsis possible.[27]The same holds true in thesurreal numbers.[28]
Indistribution theoryone can extend the function1x{\textstyle {\frac {1}{x}}}to a distribution on the whole space of real numbers (in effect by usingCauchy principal values). It does not, however, make sense to ask for a "value" of this distribution atx= 0; a sophisticated answer refers to thesingular supportof the distribution.
Inmatrixalgebra, square or rectangular blocks of numbers are manipulated as though they were numbers themselves: matrices can beaddedandmultiplied, and in some cases, a version of division also exists. Dividing by a matrix means, more precisely, multiplying by itsinverse. Not all matrices have inverses.[29]For example, amatrix containing only zerosis not invertible.
One can define a pseudo-division, by settinga/b=ab+, in whichb+represents thepseudoinverseofb. It can be proven that ifb−1exists, thenb+=b−1. Ifbequals 0, then b+= 0.
Inabstract algebra, the integers, the rational numbers, the real numbers, and the complex numbers can be abstracted to more general algebraic structures, such as acommutative ring, which is a mathematical structure where addition, subtraction, and multiplication behave as they do in the more familiar number systems, but division may not be defined. Adjoining a multiplicative inverses to a commutative ring is calledlocalization. However, the localization of every commutative ring at zero is thetrivial ring, where0=1{\displaystyle 0=1}, so nontrivial commutative rings do not have inverses at zero, and thus division by zero is undefined for nontrivial commutative rings.
Nevertheless, any number system that forms acommutative ringcan be extended to a structure called awheelin which division by zero is always possible.[30]However, the resulting mathematical structure is no longer a commutative ring, as multiplication no longer distributes over addition. Furthermore, in a wheel, division of an element by itself no longer results in the multiplicative identity element1{\displaystyle 1}, and if the original system was anintegral domain, the multiplication in the wheel no longer results in acancellative semigroup.
The concepts applied to standard arithmetic are similar to those in more general algebraic structures, such asringsandfields. In a field, every nonzero element is invertible under multiplication; as above, division poses problems only when attempting to divide by zero. This is likewise true in askew field(which for this reason is called adivision ring). However, in other rings, division by nonzero elements may also pose problems. For example, the ringZ/6Zof integers mod 6. The meaning of the expression22{\textstyle {\frac {2}{2}}}should be the solutionxof the equation2x=2{\displaystyle 2x=2}. But in the ringZ/6Z, 2 is azero divisor. This equation has two distinct solutions,x= 1andx= 4, so the expression22{\textstyle {\frac {2}{2}}}isundefined.
In field theory, the expressionab{\textstyle {\frac {a}{b}}}is only shorthand for the formal expressionab−1, whereb−1is the multiplicative inverse ofb. Since the field axioms only guarantee the existence of such inverses for nonzero elements, this expression has no meaning whenbis zero. Modern texts, that define fields as a special type of ring, include the axiom0 ≠ 1for fields (or its equivalent) so that thezero ringis excluded from being a field. In the zero ring, division by zero is possible, which shows that the other field axioms are not sufficient to exclude division by zero in a field.
In computing, most numerical calculations are done withfloating-point arithmetic, which since the 1980s has been standardized by theIEEE 754specification. In IEEE floating-point arithmetic, numbers are represented using a sign (positive or negative), a fixed-precisionsignificandand an integerexponent. Numbers whose exponent is too large to represent instead "overflow" to positive or negativeinfinity(+∞ or −∞), while numbers whose exponent is too small to represent instead "underflow" topositive or negative zero(+0 or −0). ANaN(not a number) value represents undefined results.
In IEEE arithmetic, division of 0/0 or ∞/∞ results in NaN, but otherwise division always produces a well-defined result. Dividing any non-zero number by positive zero (+0) results in an infinity of the same sign as the dividend. Dividing any non-zero number bynegative zero(−0) results in an infinity of the opposite sign as the dividend. This definition preserves the sign of the result in case ofarithmetic underflow.[31]
For example, using single-precision IEEE arithmetic, ifx= −2−149, thenx/2 underflows to −0, and dividing 1 by this result produces 1/(x/2) = −∞. The exact result −2150is too large to represent as a single-precision number, so an infinity of the same sign is used instead to indicate overflow.
Integerdivision by zero is usually handled differently from floating point since there is no integer representation for the result.CPUsdiffer in behavior: for instancex86processors trigger ahardware exception, whilePowerPCprocessors silently generate an incorrect result for the division and continue, andARMprocessors can either cause a hardware exception or return zero.[32]Because of this inconsistency between platforms, theCandC++programming languagesconsider the result of dividing by zeroundefined behavior.[33]In typicalhigher-level programming languages, such asPython,[34]anexceptionis raised for attempted division by zero, which can be handled in another part of the program.
Manyproof assistants, such asRocq(previously known asCoq) andLean, define 1/0 = 0. This is due to the requirement that all functions aretotal. Such a definition does not create contradictions, as further manipulations (such ascancelling out) still require that the divisor is non-zero.[35][36]
|
https://en.wikipedia.org/wiki/Division_by_zero
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.