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https://en.wikipedia.org/wiki/Posting%20style | When a message is replied to in e-mail, Internet forums, or Usenet, the original can often be included, or "quoted", in a variety of different posting styles.
The main options are interleaved posting (also called inline replying, in which the different parts of the reply follow the relevant parts of the original post), bottom-posting (in which the reply follows the quote) or top-posting (in which the reply precedes the quoted original message). For each of those options, there is also the issue of whether trimming of the original text is allowed, required, or preferred.
For a long time the traditional style was to post the answer below as much of the quoted original as was necessary to understand the reply (bottom or inline). Many years later, when email became widespread in business communication, it became a widespread practice to reply above the entire original and leave it (supposedly untouched) below the reply.
While each online community differs on which styles are appropriate or acceptable, within some communities the use of the "wrong" method risks being seen as a breach of netiquette, and can provoke vehement response from community regulars.
Quoting previous messages
In an e-mail reply, it is sometimes appropriate to include a full or partial copy of the original message that is being replied to; due to the asynchronous nature of Internet communication, people often engage in many conversations at the same time, and email responses may be received long after the original message was sent. For these reasons, the original poster may not be aware of what message a post is intended to be a response to, and providing context is helpful. Many email reading programs (mail user agents) encourage this behaviour by automatically including a copy of the original message in the reply editing window.
Quoted text from previous messages is usually distinguished in some way from the new (reply) text. Often, the two parts are given different indentation. In the exampl |
https://en.wikipedia.org/wiki/Dividing%20a%20circle%20into%20areas | In geometry, the problem of dividing a circle into areas by means of an inscribed polygon with n sides in such a way as to maximise the number of areas created by the edges and diagonals, sometimes called Moser's circle problem, has a solution by an inductive method. The greatest possible number of regions, , giving the sequence 1, 2, 4, 8, 16, 31, 57, 99, 163, 256, ... (). Though the first five terms match the geometric progression , it deviates at , showing the risk of generalising from only a few observations.
Lemma
If there are n points on the circle and one more point is added, n lines can be drawn from the new point to previously existing points. Two cases are possible. In the first case (a), the new line passes through a point where two or more old lines (between previously existing points) cross. In the second case (b), the new line crosses each of the old lines in a different point. It will be useful to know the following fact.
Lemma. The new point A can be chosen so that case b occurs for each of the new lines.
Proof. For the case a, three points must be on one line: the new point A, the old point O to which the line is drawn, and the point I where two of the old lines intersect. There are n old points O, and hence finitely many points I where two of the old lines intersect. For each O and I, the line OI crosses the circle in one point other than O. Since the circle has infinitely many points, it has a point A which will be on none of the lines OI. Then, for this point A and all of the old points O, case b will be true.
This lemma means that, if there are k lines crossing AO, then each of them crosses AO at a different point and k + 1 new areas are created by the line AO.
Solution
Inductive method
The lemma establishes an important property for solving the problem. By employing an inductive proof, one can arrive at a formula for f(n) in terms of f(n − 1).
In the figure the dark lines are connecting points 1 through 4 dividing the circle into 8 |
https://en.wikipedia.org/wiki/Weak%20consistency | The name weak consistency can be used in two senses. In the first sense, strict and more popular, weak consistency is one of the consistency models used in the domain of concurrent programming (e.g. in distributed shared memory, distributed transactions etc.).
A protocol is said to support weak consistency if:
All accesses to synchronization variables are seen by all processes (or nodes, processors) in the same order (sequentially) - these are synchronization operations. Accesses to critical sections are seen sequentially.
All other accesses may be seen in different order on different processes (or nodes, processors).
The set of both read and write operations in between different synchronization operations is the same in each process.
Therefore, there can be no access to a synchronization variable if there are pending write operations. And there can not be any new read/write operation started if the system is performing any synchronization operation.
In the second, more general, sense weak consistency may be applied to any consistency model weaker than sequential consistency.
A stricter condition is strong consistency, where parallel processes can observe only one consistent state.
References
The original paper on weak ordering: M. Dubois, C. Scheurich and F. A. Briggs, Memory Access Buffering in Multiprocessors, in Proceedings of 13th Annual International Symposium on Computer Architecture 14, 2 (June 1986), 434-442.
Sarita V. Adve, Mark D. Hill, Weak ordering - a new definition, in Proceedings of the 17th Annual International Symposium on Computer Architecture.
Consistency models |
https://en.wikipedia.org/wiki/Memory%20coherence | Memory coherence is an issue that affects the design of computer systems in which two or more processors or cores share a common area of memory.
In a uniprocessor system (where there exists only one core), there is only one processing element doing all the work and therefore only one processing element that can read or write from/to a given memory location. As a result, when a value is changed, all subsequent read operations of the corresponding memory location will see the updated value, even if it is cached.
Conversely, in multiprocessor (or multicore) systems, there are two or more processing elements working at the same time, and so it is possible that they simultaneously access the same memory location. Provided none of them changes the data in this location, they can share it indefinitely and cache it as they please. But as soon as one updates the location, the others might work on an out-of-date copy that, e.g., resides in their local cache. Consequently, some scheme is required to notify all the processing elements of changes to shared values; such a scheme is known as a memory coherence protocol, and if such a protocol is employed the system is said to have a coherent memory.
The exact nature and meaning of the memory coherency is determined by the consistency model that the coherence protocol implements. In order to write correct concurrent programs, programmers must be aware of the exact consistency model that is employed by their systems.
When implemented in hardware, the coherency protocol can, for example, be directory-based or snooping-based (also called sniffing). Specific protocols include the MSI protocol and its derivatives MESI, MOSI and MOESI.
See also
Cache coherence
Distributed shared memory
Race condition
References
Computer memory
Parallel computing |
https://en.wikipedia.org/wiki/Distributed%20shared%20memory | In computer science, distributed shared memory (DSM) is a form of memory architecture where physically separated memories can be addressed as a single shared address space. The term "shared" does not mean that there is a single centralized memory, but that the address space is shared—i.e., the same physical address on two processors refers to the same location in memory. Distributed global address space (DGAS), is a similar term for a wide class of software and hardware implementations, in which each node of a cluster has access to shared memory in addition to each node's private (i.e., not shared) memory.
Overview
A distributed-memory system, often called a multicomputer, consists of multiple independent processing nodes with local memory modules which is connected by a general interconnection network. Software DSM systems can be implemented in an operating system, or as a programming library and can be thought of as extensions of the underlying virtual memory architecture. When implemented in the operating system, such systems are transparent to the developer; which means that the underlying distributed memory is completely hidden from the users. In contrast, software DSM systems implemented at the library or language level are not transparent and developers usually have to program them differently. However, these systems offer a more portable approach to DSM system implementations. A DSM system implements the shared-memory model on a physically distributed memory system.
DSM can be achieved via software as well as hardware. Hardware examples include cache coherence circuits and network interface controllers. There are three ways of implementing DSM:
Page-based approach using virtual memory
Shared-variable approach using routines to access shared variables
Object-based approach, ideally accessing shared data through object-oriented discipline
Advantages
Scales well with a large number of nodes
Message passing is hidden
Can handle complex and large data |
https://en.wikipedia.org/wiki/Acclimatization | Acclimatization or acclimatisation (also called acclimation or acclimatation) is the process in which an individual organism adjusts to a change in its environment (such as a change in altitude, temperature, humidity, photoperiod, or pH), allowing it to maintain fitness across a range of environmental conditions. Acclimatization occurs in a short period of time (hours to weeks), and within the organism's lifetime (compared to adaptation, which is evolution, taking place over many generations). This may be a discrete occurrence (for example, when mountaineers acclimate to high altitude over hours or days) or may instead represent part of a periodic cycle, such as a mammal shedding heavy winter fur in favor of a lighter summer coat. Organisms can adjust their morphological, behavioral, physical, and/or biochemical traits in response to changes in their environment. While the capacity to acclimate to novel environments has been well documented in thousands of species, researchers still know very little about how and why organisms acclimate the way that they do.
Names
The nouns acclimatization and acclimation (and the corresponding verbs acclimatize and acclimate) are widely regarded as synonymous, both in general vocabulary and in medical vocabulary. The synonym acclimatation is less commonly encountered, and fewer dictionaries enter it.
Methods
Biochemical
In order to maintain performance across a range of environmental conditions, there are several strategies organisms use to acclimate. In response to changes in temperature, organisms can change the biochemistry of cell membranes making them more fluid in cold temperatures and less fluid in warm temperatures by increasing the number of membrane proteins. In response to certain stressors, some organisms express so-called heat shock proteins that act as molecular chaperones and reduce denaturation by guiding the folding and refolding of proteins. It has been shown that organisms which are acclimated to high or low t |
https://en.wikipedia.org/wiki/Neil%20Sloane |
Neil James Alexander Sloane FLSW (born October 10, 1939) is a British-American mathematician. His major contributions are in the fields of combinatorics, error-correcting codes, and sphere packing. Sloane is best known for being the creator and maintainer of the On-Line Encyclopedia of Integer Sequences (OEIS).
Biography
Sloane was born in Beaumaris, Anglesey, Wales, in 1939, moving to Cowes, Isle of Wight, England in 1946. The family emigrated to Australia, arriving at the start of 1949. Sloane then moved from Melbourne to the United States in 1961.
He studied at Cornell University under Nick DeClaris, Frank Rosenblatt, Frederick Jelinek and Wolfgang Heinrich Johannes Fuchs, receiving his Ph.D. in 1967. His doctoral dissertation was titled Lengths of Cycle Times in Random Neural Networks. Sloane joined AT&T Bell Labs in 1968 and retired from AT&T Labs in 2012. He became an AT&T Fellow in 1998. He is also a Fellow of the Learned Society of Wales, an IEEE Fellow, a Fellow of the American Mathematical Society, and a member of the National Academy of Engineering.
He is a winner of a Lester R. Ford Award in 1978 and the Chauvenet Prize in 1979. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. In 2005 Sloane received the IEEE Richard W. Hamming Medal.
In 2008 he received the Mathematical Association of America David P. Robbins Prize, and in 2013 the George Pólya Award.
In 2014, to celebrate his 75th birthday, Sloane shared some of his favorite integer sequences. Besides mathematics, he loves rock climbing and has authored two rock-climbing guides to New Jersey.
He regularly appears in videos for Brady Haran's YouTube channel Numberphile.
Selected publications
Neil James Alexander Sloane, A Handbook of Integer Sequences, Academic Press, NY, 1973.
Florence Jessie MacWilliams and Neil James Alexander Sloane, The Theory of Error-Correcting Codes, Elsevier/North-Holland, Amsterdam, 1977.
M. Harwit and Neil James Alexander |
https://en.wikipedia.org/wiki/Phenol%20formaldehyde%20resin | Phenol formaldehyde resins (PF) (phenolic resins or phenoplasts) are synthetic polymers obtained by the reaction of phenol or substituted phenol with formaldehyde. Used as the basis for Bakelite, PFs were the first commercial synthetic resins (plastics). They have been widely used for the production of molded products including billiard balls, laboratory countertops, and as coatings and adhesives. They were at one time the primary material used for the production of circuit boards but have been largely replaced with epoxy resins and fiberglass cloth, as with fire-resistant FR-4 circuit board materials.
There are two main production methods. One reacts phenol and formaldehyde directly to produce a thermosetting network polymer, while the other restricts the formaldehyde to produce a prepolymer known as novolac which can be moulded and then cured with the addition of more formaldehyde and heat. There are many variations in both production and input materials that are used to produce a wide variety of resins for special purposes.
Formation and structure
Phenol-formaldehyde resins, as a group, are formed by a step-growth polymerization reaction that can be either acid- or base-catalysed. Since formaldehyde exists predominantly in solution as a dynamic equilibrium of methylene glycol oligomers, the concentration of the reactive form of formaldehyde depends on temperature and pH.
Phenol reacts with formaldehyde at the ortho and para sites (sites 2, 4 and 6) allowing up to 3 units of formaldehyde to attach to the ring. The initial reaction in all cases involves the formation of a hydroxymethyl phenol:
HOC6H5 + CH2O → HOC6H4CH2OH
The hydroxymethyl group is capable of reacting with either another free ortho or para site, or with another hydroxymethyl group. The first reaction gives a methylene bridge, and the second forms an ether bridge:
HOC6H4CH2OH + HOC6H5 → (HOC6H4)2CH2 + H2O
2 HOC6H4CH2OH → (HOC6H4CH2)2O + H2O
The diphenol (HOC6H4)2CH2 (somet |
https://en.wikipedia.org/wiki/GlassBridge%20Enterprises | GlassBridge Enterprises, Inc., formerly Imation Corporation, is an American holding company. Through its subsidiary, Glassbridge focuses primarily on investment and asset management. Prior to the name change, Glassbridge had three core elements: traditional storage (magnetic tape and optical products), secure and scalable storage (data backup, data archive and data security for small and medium businesses) and what the company calls "audio and video information" products.
History
Imation was started in 1996, when 3M spun off its data storage and imaging business. The company underwent a divestment of non-core businesses, and invested in four core product technology areas: secure storage, scalable storage, wireless/connectivity, and magnetic tape. As part of 3M, the company was involved in the development of many technological improvements in data storage, such as the introduction of the first American-made magnetic tape in 1947, the first quarter-inch tape cartridge for data storage (QIC) in 1971, and the floppy disk in 1984.
In February 2012, Imation announced a product set to secure mobile data, identities, and workspaces, based on the idea that employees used portable storage devices to transport corporate data.
The security news followed five acquisitions the company made in 2011 in scalable storage and data security: ENCRYPTX; MXI Security from Memory Experts International; the assets of ProStor; the secure data storage hardware business of IronKey; and intellectual property from Nine Technologies. Imation received an exclusive license from IronKey for its secure storage management software and service and a license to use the IronKey brand for secure storage products.
In October 2011, Imation products for small and medium businesses centered on its DataGuard and InfiniVault multi-tier data protection and data archive appliances.
The company sold consumer electronics, headphones and accessories under the Imation, Memorex, TDK Life on Record, and XtremeMa |
https://en.wikipedia.org/wiki/Schedule%20%28project%20management%29 | In project management, a schedule is a listing of a project's milestones, activities, and deliverables. Usually dependencies and resources are defined for each task, then start and finish dates are estimated from the resource allocation, budget, task duration, and scheduled events. A schedule is commonly used in the project planning and project portfolio management parts of project management. Elements on a schedule may be closely related to the work breakdown structure (WBS) terminal elements, the Statement of work, or a Contract Data Requirements List.
Overview
In many industries, such as engineering and construction, the development and maintenance of the project schedule is the responsibility of a full-time scheduler or team of schedulers, depending on the size and the scope of the project. The techniques of scheduling are well developed but inconsistently applied throughout industry. Standardization and promotion of scheduling best practices are being pursued by the Association for the Advancement of Cost Engineering (AACE), the Project Management Institute (PMI), and the US Government for acquisition and accounting purposes.
Project management is not limited to industry; the average person can use it to organize their own life. Some examples are:
Homeowner renovation project
Keeping track of all the family activities
Coaching a team
Planning a vacation
Planning a wedding
Some project management software programs provide templates, lists, and example schedules to help their users with creating their schedule.
Methods
The project schedule is a calendar that links the tasks to be done with the resources that will do them. It is the core of the project plan used to show the organization how the work will be done, commit people to the project, determine resource needs, and used as a kind of checklist to make sure that every task necessary is performed. Before a project schedule can be created, the schedule maker should have a work breakdown stru |
https://en.wikipedia.org/wiki/Ecological%20crisis | An ecological or environmental crises occurs when changes to the environment of a species or population destabilizes its continued survival. Some of the important causes include:
Degradation of an abiotic ecological factor (for example, increase of temperature, less significant rainfalls)
Increased pressures from predation
Rise in the number of individuals (i.e. overpopulation)
The evolutionary theory of punctuated equilibrium sees infrequent ecological crises as a potential driver of rapid evolution.
Because of the impact of humans on the natural environment in the recent geological period, the term ecological crisis is often applied to environmental issues caused by human civilizations such as: the climate crisis, biodiversity loss and plastic pollution which have emerged as major global challenges during the first few decades of the 21st century.
Examples
Crises caused by abiotic factors
Climate change is starting to have major impacts on ecosystems. With global temperature rising, there is a decrease in snow-fall, and sea levels are rising. Ecosystems will change or evolve to cope with the increase in temperature. Consequently, many species are being driven out of their habitats.
Polar bears are being threatened. They need ice for hunting seals, their primary prey. However, the ice caps are melting, making their hunting periods shorter each year. As a result, the polar bears are not developing enough fat for the winter; therefore, they are not able to reproduce at a healthy rate.
Fresh water and wetland ecosystems are dealing with extreme effects of the increase of temperature. The climate change could be devastating to salmon and trout and to other aquatic life. The increase in temperature will disrupt the current life patterns of the salmon and trout. The cold-water fish will eventually leave their natural geographical range to live in cooler waters by migrating to higher elevations.
While many species have been able to adapt to the new conditions |
https://en.wikipedia.org/wiki/Gr%C3%BCnwald%E2%80%93Letnikov%20derivative | In mathematics, the Grünwald–Letnikov derivative is a basic extension of the derivative in fractional calculus that allows one to take the derivative a non-integer number of times. It was introduced by Anton Karl Grünwald (1838–1920) from Prague, in 1867, and by Aleksey Vasilievich Letnikov (1837–1888) in Moscow in 1868.
Constructing the Grünwald–Letnikov derivative
The formula
for the derivative can be applied recursively to get higher-order derivatives. For example, the second-order derivative would be:
Assuming that the h 's converge synchronously, this simplifies to:
which can be justified rigorously by the mean value theorem. In general, we have (see binomial coefficient):
Removing the restriction that n be a positive integer, it is reasonable to define:
This defines the Grünwald–Letnikov derivative.
To simplify notation, we set:
So the Grünwald–Letnikov derivative may be succinctly written as:
An alternative definition
In the preceding section, the general first principles equation for integer order derivatives was derived. It can be shown that the equation may also be written as
or removing the restriction that n must be a positive integer:
This equation is called the reverse Grünwald–Letnikov derivative. If the substitution h → −h is made, the resulting equation is called the direct Grünwald–Letnikov derivative:
References
Further reading
The Fractional Calculus, by Oldham, K.; and Spanier, J. Hardcover: 234 pages. Publisher: Academic Press, 1974.
Fractional calculus |
https://en.wikipedia.org/wiki/E-gold | e-gold was a digital gold currency operated by Gold & Silver Reserve Inc. (G&SR) that allowed users to make payments, which it called "spends", in grams of gold, silver, and other precious metals. e-gold was launched in 1996 and grew to five million accounts by 2009, when transfers were suspended due to legal issues. At its peak in 2006, e-gold processed more than US$2 billion worth of spends per year, backed by over 85 million worth of gold, about . e-gold Ltd. was incorporated in Nevis, Saint Kitts and Nevis, and its operations were based in Florida.
Beginnings
e-gold was founded by Douglas Jackson, an oncologist, and Barry Downey, an attorney, in 1996. The pair originally backed e-gold accounts with gold coins stored in a safe deposit box in Melbourne, Florida. When e-gold was at its peak, the company stored its gold and platinum in bank vaults in London and Dubai.
By 1998, G&SR was an affiliate member of NACHA and a full member of NACHA's Internet council. The company experienced exponential growth starting in 2000. In a July 13, 1999 article in the Financial Times, Tim Jackson (no relation to Douglas) described e-gold as "the only electronic currency that has achieved critical mass on the web". In 2001, the company said that e-gold had more than 200,000 accounts and more than $14 million of digital gold currency in circulation. By 2004, there were over a million accounts.
By the early 2000s, the capability of immediate settlement, as implemented by e-gold, was recognized as key to the emergence of systems for peer-to-peer transfers of digital rights such as "smart contracts".
e-gold was a founding member of the Financial Coalition Against Child Pornography. In 2005 and 2006, the company took effective action to combat child exploitation.
Criminal abuse
e-gold was a target of financial malware and phishing scams by criminal syndicates and was used for illegal activities. In December 2005, the Federal Bureau of Investigation (FBI) raided G&SR's offices, se |
https://en.wikipedia.org/wiki/Vivisection | Vivisection () is surgery conducted for experimental purposes on a living organism, typically animals with a central nervous system, to view living internal structure. The word is, more broadly, used as a pejorative catch-all term for experimentation on live animals but the term is rarely used by practising scientists. Human vivisection, such as live organ harvesting, has been perpetrated as a form of torture.
Animal vivisection
Research requiring vivisection techniques that cannot be met through other means is often subject to an external ethics review in conception and implementation, and in many jurisdictions use of anesthesia is legally mandated for any surgery likely to cause pain to any vertebrate.
In the United States, the Animal Welfare Act explicitly requires that any procedure that may cause pain use "tranquilizers, analgesics, and anesthetics" with exceptions when "scientifically necessary". The act does not define "scientific necessity" or regulate specific scientific procedures, but approval or rejection of individual techniques in each federally funded lab is determined on a case-by-case basis by the Institutional Animal Care and Use Committee, which contains at least one veterinarian, one scientist, one non-scientist, and one other individual from outside the university.
In the United Kingdom, any experiment involving vivisection must be licensed by the Home Secretary. The Animals (Scientific Procedures) Act 1986 "expressly directs that, in determining whether to grant a licence for an experimental project, 'the Secretary of State shall weigh the likely adverse effects on the animals concerned against the benefit likely to accrue.
In Australia, the Code of Practice "requires that all experiments must be approved by an Animal Experimentation Ethics Committee" that includes a "person with an interest in animal welfare who is not employed by the institution conducting the experiment, and an additional independent person not involved in animal exper |
https://en.wikipedia.org/wiki/GeForce%203%20series | The GeForce 3 series (NV20) is the third generation of Nvidia's GeForce graphics processing units (GPUs). Introduced in February 2001, it advanced the GeForce architecture by adding programmable pixel and vertex shaders, multisample anti-aliasing and improved the overall efficiency of the rendering process.
The GeForce 3 was unveiled during the 2001 Macworld Conference & Expo/Tokyo 2001 in Makuhari Messe and powered realtime demos of Pixar's Junior Lamp and id Software's Doom 3. Apple would later announce launch rights for its new line of computers.
The GeForce 3 family comprises 3 consumer models: the GeForce 3, the GeForce 3 Ti200, and the GeForce 3 Ti500. A separate professional version, with a feature-set tailored for computer aided design, was sold as the Quadro DCC. A derivative of the GeForce 3, known as the NV2A, is used in the Microsoft Xbox game console.
Architecture
Introduced three months after Nvidia acquired the assets of 3dfx and marketed as the nFinite FX Engine, the GeForce 3 was the first Microsoft Direct3D 8.0 compliant 3D-card. Its programmable shader architecture enabled applications to execute custom visual effects programs in Microsoft Shader language 1.1. It is believed that the fixed-function T&L hardware from GeForce 2 was still included on the chip for use with Direct3D 7.0 applications, as the single vertex shader was not fast enough to emulate it yet. With respect to pure pixel and texel throughput, the GeForce 3 has four pixel pipelines which each can sample two textures per clock. This is the same configuration as GeForce 2, excluding the slower GeForce 2 MX line.
To take better advantage of available memory performance, the GeForce 3 has a memory subsystem dubbed Lightspeed Memory Architecture (LMA). This is composed of several mechanisms that reduce overdraw, conserve memory bandwidth by compressing the z-buffer (depth buffer) and better manage interaction with the DRAM.
Other architectural changes include EMBM support (first |
https://en.wikipedia.org/wiki/Nmap | Nmap (Network Mapper) is a network scanner created by Gordon Lyon (also known by his pseudonym Fyodor Vaskovich). Nmap is used to discover hosts and services on a computer network by sending packets and analyzing the responses.
Nmap provides a number of features for probing computer networks, including host discovery and service and operating system detection. These features are extensible by scripts that provide more advanced service detection, vulnerability detection, and other features. Nmap can adapt to network conditions including latency and congestion during a scan.
Nmap started as a Linux utility and was ported to other systems including Windows, macOS, and BSD. It is most popular on Linux, followed by Windows.
Features
Nmap features include:
Fast scan (nmap -F [target]) – Performing a basic port scan for fast result.
Host discovery – Identifying hosts on a network. For example, listing the hosts that respond to TCP and/or ICMP requests or have a particular port open.
Port scanning – Enumerating the open ports on target hosts.
Version detection – Interrogating network services on remote devices to determine application name and version number.
Ping Scan – Check host by sending ping requests.
TCP/IP stack fingerprinting – Determining the operating system and hardware characteristics of network devices based on observations of network activity of said devices.
Scriptable interaction with the target – using Nmap Scripting Engine (NSE) and Lua programming language.
Nmap can provide further information on targets, including reverse DNS names, device types, and MAC addresses.
Typical uses of Nmap:
Auditing the security of a device or firewall by identifying the network connections which can be made to, or through it.
Identifying open ports on a target host in preparation for auditing.
Network inventory, network mapping, maintenance and asset management.
Auditing the security of a network by identifying new servers.
Generating traffic to hosts on a |
https://en.wikipedia.org/wiki/Platform-specific%20model | A platform-specific model is a model of a software or business system that is linked to a specific technological platform (e.g. a specific programming language, operating system, document file format or database). Platform-specific models are indispensable for the actual implementation of a system.
For example, a need to implement an online shop. The system will need to store information regarding users, goods, credit cards, etc. The designer might decide to use for this purpose an Oracle database. For this to work, the designer will need to express concepts (e.g. the concept of a user) in a relational model using the Oracle's SQL dialect. This Oracle's specific relational model is an example of a Platform-specific model.
The term platform-specific model is most frequently used in the context of the MDA approach. This MDA approach corresponds the OMG vision of Model Driven Engineering. The main idea is that it should be possible to use a MTL to transform a Platform-independent model into a Platform-specific model. In order to achieve this transformation, one can use a language compliant to the newly defined QVT standard. Examples of such languages are AndroMDA, VIATRA or ATL.
Related Concepts
ATLAS Transformation Language (ATL)
Domain Specific Language (DSL)
Domain-specific modelling (DSM)
Eclipse Modeling Framework (EMF)
Generic Modeling Environment (GME)
Graphical Modeling Framework (GMF)
Meta-Object Facility (MOF)
Meta-modeling
Model-based testing (MBT)
Model-driven architecture (MDA)
Model Transformation Language (MTL)
Object Constraint Language (OCL)
Object-oriented analysis and design (OOAD)
Visual Automated model Transformations VIATRA
XML Metadata Interchange (XMI)
See also
Platform-independent model
References
Software architecture
Systems engineering |
https://en.wikipedia.org/wiki/Model-driven%20architecture | Model Driven Architecture (MDA) is a software design approach for the development of software systems. It provides a set of guidelines for the structuring of specifications, which are expressed as models. Model Driven Architecture is a kind of domain engineering, and supports model-driven engineering of software systems. It was launched by the Object Management Group (OMG) in 2001.
Overview
Model Driven Architecture® (MDA®) "provides an approach for deriving value from models and architecture in support of the full life cycle of physical, organizational and I.T. systems". A model is a (representation of) an abstraction of a system. MDA® provides value by producing models at varying levels of abstraction, from a conceptual view down to the smallest implementation detail. OMG literature speaks of three such levels of abstraction, or architectural viewpoints: the Computation-independent Model (CIM), the Platform-independent model (PIM), and the Platform-specific model (PSM). The CIM describes a system conceptually, the PIM describes the computational aspects of a system without reference to the technologies that may be used to implement it, and the PSM provides the technical details necessary to implement the system. The OMG Guide notes, though, that these three architectural viewpoints are useful, but are just three of many possible viewpoints.
The OMG organization provides specifications rather than implementations, often as answers to Requests for Proposals (RFPs). Implementations come from private companies or open source groups.
Related standards
The MDA model is related to multiple standards, including the Unified Modeling Language (UML), the Meta-Object Facility (MOF), XML Metadata Interchange (XMI), Enterprise Distributed Object Computing (EDOC), the Software Process Engineering Metamodel (SPEM), and the Common Warehouse Metamodel (CWM). Note that the term “architecture” in Model Driven Architecture does not refer to the architecture of the system being mo |
https://en.wikipedia.org/wiki/Plessey%20System%20250 | Plessey System 250, also known as PP250, was the first operational computer to implement capability-based addressing, to check and balance the computation as a pure Church–Turing machine. Plessey built the systems for a British Army message routing project.
Description
A Church–Turing machine is a digital computer that encapsulates the symbols in a thread of computation as a chain of protected abstractions by enforcing the dynamic binding laws of Alonzo Church's lambda calculus Other capability based computers, which include CHERI and CAP computers, are hybrids. They retain default instructions that can access every word of accessible physical or logical (paged) memory.
It is an unavoidable characteristic of the von Neumann architecture that is founded on shared random access memory and trust in the sharing default access rights. For example, every word in every page managed by the virtual memory manager in an operating system using a memory management unit (MMU) must be trusted. Using a default privilege among many compiled programs allows corruption to grow without any method of error detection. However, the range of virtual addresses given to the MMU or the range of physical addresses produced by the MMU is shared undetected corruption flows across the shared memory space from one software function to another. PP250 removed not only virtual memory or any centralized, precompiled operating system, but also the superuser, removing all default machine privileges.
It is default privileges that empower undetected malware and hacking in a computer. Instead, the pure object capability model of PP250 always requires a limited capability key to define the authority to operate. PP250 separated binary data from capability data to protect access rights, simplify the computer and speed garbage collection. The Church machine encapsulates and context limits the Turing machine by enforcing the laws of the lambda calculus. The typed digital media is program controlled by di |
https://en.wikipedia.org/wiki/Cross-ratio | In geometry, the cross-ratio, also called the double ratio and anharmonic ratio, is a number associated with a list of four collinear points, particularly points on a projective line. Given four points , , , on a line, their cross ratio is defined as
where an orientation of the line determines the sign of each distance and the distance is measured as projected into Euclidean space. (If one of the four points is the line's point at infinity, then the two distances involving that point are dropped from the formula.)
The point is the harmonic conjugate of with respect to and precisely if the cross-ratio of the quadruple is , called the harmonic ratio. The cross-ratio can therefore be regarded as measuring the quadruple's deviation from this ratio; hence the name anharmonic ratio.
The cross-ratio is preserved by linear fractional transformations. It is essentially the only projective invariant of a quadruple of collinear points; this underlies its importance for projective geometry.
The cross-ratio had been defined in deep antiquity, possibly already by Euclid, and was considered by Pappus, who noted its key invariance property. It was extensively studied in the 19th century.
Variants of this concept exist for a quadruple of concurrent lines on the projective plane and a quadruple of points on the Riemann sphere.
In the Cayley–Klein model of hyperbolic geometry, the distance between points is expressed in terms of a certain cross-ratio.
Terminology and history
Pappus of Alexandria made implicit use of concepts equivalent to the cross-ratio in his Collection: Book VII. Early users of Pappus included Isaac Newton, Michel Chasles, and Robert Simson. In 1986 Alexander Jones made a translation of the original by Pappus, then wrote a commentary on how the lemmas of Pappus relate to modern terminology.
Modern use of the cross ratio in projective geometry began with Lazare Carnot in 1803 with his book Géométrie de Position. Chasles coined the French term [a |
https://en.wikipedia.org/wiki/Prime%20Time%20Entertainment%20Network | The Prime Time Entertainment Network (PTEN) was an American television network that was operated by the Prime Time Consortium, a joint venture between the Warner Bros. Domestic Television subsidiary of Time Warner and Chris-Craft Industries. First launched on January 20, 1993, and operating until 1997, the network mainly aired drama programs aimed at adults between the ages of 18 and 54. At its peak, PTEN's programming was carried on 177 television stations, covering 93% of the country.
History
Origins
At the time of PTEN's founding, co-owner Chris-Craft Industries owned independent television stations in several large and mid-sized U.S. cities (among them its two largest stations, WWOR-TV in New York City and KCOP-TV in Los Angeles) through its BHC Communications and United Television divisions, which formed the nuclei of the network.
PTEN was launched as a potential fifth television network, and was created in reaction to the success of the Fox network (which debuted in October 1986, seven years before PTEN launched) as well as the successes of first-run syndicated programming during the late 1980s and early 1990s. It offered packaged nights of programming to participating television stations, beginning with a two-hour block on Wednesday evenings, with a second block (originally airing on Saturday, before moving to Monday for the 1994-95 season) being added in September 1993. Originally, the station groups involved in the Prime Time Consortium helped finance PTEN's programs; however, that deal was restructured at the beginning of the network's second year.
The service sought affiliations with various television stations not affiliated with the Big Three television networks. However, close to half of PTEN's initial affiliates were stations that were already affiliated with Fox; as a result, these stations usually scheduled PTEN programming around Fox's then five-night prime time schedule (although Fox would expand its schedule to seven nights with the addition |
https://en.wikipedia.org/wiki/Plessey | The Plessey Company plc was a British electronics, defence and telecommunications company. It originated in 1917, growing and diversifying into electronics. It expanded after World War II by acquisition of companies and formed overseas companies.
It was listed on the London Stock Exchange and was a constituent of the FTSE 100 Index. In 1989, it was taken over by a consortium formed by GEC and Siemens which split the assets of the Plessey group.
The majority of Plessey's defence assets were amalgamated into BAE Systems in 1999 when British Aerospace merged with the defence arm of GEC, Marconi Electronic Systems (MES). The Plessey Microsystems division was the subject of a management buyout in 1988 becoming Radstone Technology, which survives today as part of Abaco Systems based in Towcester, Northamptonshire. The bulk of Plessey's telecommunications assets were acquired by Ericsson through its 2005 acquisition of Marconi Communications, a successor company of GEC.
History
Early history
The Plessey company was founded in 1917 in Marylebone, central London. The original shareholders were Thomas Hurst Hodgson, C. H. Whitaker, Raymond Parker and his brother Plessey Parker. A talented German engineer, William Oscar Heyne, was employed by the company. Heyne later became the managing director and chairman of Plessey and was one of the key figures in the development of Plessey during the 1920s and 1930s. The company moved to Cottenham Road in Ilford early in 1919. In 1925, the original company was wound up and a new one was formed with a greater share capital. Most of the early work carried out by the company was in mechanical engineering.
The Clark connection
An early customer of Plessey was a galvanising company called British Electro Chemists. One of that company's shareholders was Byron G. Clark, an American, who was also a business associate of T. H. Hodgson, one of the founders of Plessey. The Clark family would eventually dominate the management of Plessey for mo |
https://en.wikipedia.org/wiki/GeForce%204%20series | The GeForce 4 series (codenames below) refers to the fourth generation of GeForce-branded graphics processing units (GPUs) manufactured by Nvidia. There are two different GeForce4 families, the high-performance Ti family, and the budget MX family. The MX family spawned a mostly identical GeForce4 Go (NV17M) family for the laptop market. All three families were announced in early 2002; members within each family were differentiated by core and memory clock speeds. In late 2002, there was an attempt to form a fourth family, also for the laptop market, the only member of it being the GeForce4 4200 Go (NV28M) which was derived from the Ti line.
GeForce4 Ti
Architecture
The GeForce4 Ti (NV25) was launched in February 2002 and was a revision of the GeForce 3 (NV20). It was very similar to its predecessor; the main differences were higher core and memory clock rates, a revised memory controller (known as Lightspeed Memory Architecture II), updated pixel shaders with new instructions for Direct3D 8.0a support, an additional vertex shader (the vertex and pixel shaders were now known as nFinite FX Engine II), hardware anti-aliasing (Accuview AA), and DVD playback. Legacy Direct3D 7-class fixed-function T&L was now implemented as vertex shaders. Proper dual-monitor support (TwinView) was also brought over from the GeForce 2 MX. The GeForce 4 Ti was superior to the GeForce 4 MX in virtually every aspect save for production cost, although the MX had the Nvidia VPE (video processing engine) which the Ti lacked.
Lineup
The initial two models were the Ti4400 and the top-of-the-range Ti4600. At the time of their introduction, Nvidia's main products were the entry-level GeForce 2 MX, the midrange GeForce4 MX models (released the same time as the Ti4400 and Ti4600), and the older but still high-performance GeForce 3 (demoted to the upper mid-range or performance niche). However, ATI's Radeon 8500LE was somewhat cheaper than the Ti4400, and outperformed its price competitors, the Ge |
https://en.wikipedia.org/wiki/Multistage%20rocket | A multistage rocket or step rocket is a launch vehicle that uses two or more rocket stages, each of which contains its own engines and propellant. A tandem or serial stage is mounted on top of another stage; a parallel stage is attached alongside another stage. The result is effectively two or more rockets stacked on top of or attached next to each other. Two-stage rockets are quite common, but rockets with as many as five separate stages have been successfully launched.
By jettisoning stages when they run out of propellant, the mass of the remaining rocket is decreased. Each successive stage can also be optimized for its specific operating conditions, such as decreased atmospheric pressure at higher altitudes. This staging allows the thrust of the remaining stages to more easily accelerate the rocket to its final speed and height.
In serial or tandem staging schemes, the first stage is at the bottom and is usually the largest, the second stage and subsequent upper stages are above it, usually decreasing in size. In parallel staging schemes solid or liquid rocket boosters are used to assist with launch. These are sometimes referred to as "stage 0". In the typical case, the first-stage and booster engines fire to propel the entire rocket upwards. When the boosters run out of fuel, they are detached from the rest of the rocket (usually with some kind of small explosive charge or explosive bolts) and fall away. The first stage then burns to completion and falls off. This leaves a smaller rocket, with the second stage on the bottom, which then fires. Known in rocketry circles as staging, this process is repeated until the desired final velocity is achieved. In some cases with serial staging, the upper stage ignites before the separation—the interstage ring is designed with this in mind, and the thrust is used to help positively separate the two vehicles.
A multistage rocket is required to reach orbital speed. Single-stage-to-orbit designs are sought, but have not yet |
https://en.wikipedia.org/wiki/Closed%20system | A closed system is a natural physical system that does not allow transfer of matter in or out of the system, althoughin the contexts of physics, chemistry, engineering, etc.the transfer of energy (e.g. as work or heat) is allowed.
Physics
In classical mechanics
In nonrelativistic classical mechanics, a closed system is a physical system that doesn't exchange any matter with its surroundings, and isn't subject to any net force whose source is external to the system. A closed system in classical mechanics would be equivalent to an isolated system in thermodynamics. Closed systems are often used to limit the factors that can affect the results of a specific problem or experiment.
In thermodynamics
In thermodynamics, a closed system can exchange energy (as heat or work) but not matter, with its surroundings.
An isolated system cannot exchange any heat, work, or matter with the surroundings, while an open system can exchange energy and matter. (This scheme of definition of terms is not uniformly used, though it is convenient for some purposes. In particular, some writers use 'closed system' where 'isolated system' is used here.)
For a simple system, with only one type of particle (atom or molecule), a closed system amounts to a constant number of particles. However, for systems which are undergoing a chemical reaction, there may be all sorts of molecules being generated and destroyed by the reaction process. In this case, the fact that the system is closed is expressed by stating that the total number of each elemental atom is conserved, no matter what kind of molecule it may be a part of. Mathematically:
where is the number of j-type molecules, is the number of atoms of element in molecule and is the total number of atoms of element in the system, which remains constant, since the system is closed. There will be one such equation for each different element in the system.
In thermodynamics, a closed system is important for solving complicated thermodynamic p |
https://en.wikipedia.org/wiki/Thyratron | A thyratron is a type of gas-filled tube used as a high-power electrical switch and controlled rectifier. Thyratrons can handle much greater currents than similar hard-vacuum tubes. Electron multiplication occurs when the gas becomes ionized, producing a phenomenon known as Townsend discharge. Gases used include mercury vapor, xenon, neon, and (in special high-voltage applications or applications requiring very short switching times) hydrogen. Unlike a vacuum tube (valve), a thyratron cannot be used to amplify signals linearly.
In the 1920s, thyratrons were derived from early vacuum tubes such as the UV-200, which contained a small amount of argon gas to increase its sensitivity as a radio signal detector, and the German LRS relay tube, which also contained argon gas. Gas rectifiers, which predated vacuum tubes, such as the argon-filled General Electric "Tungar bulb" and the Cooper-Hewitt mercury-pool rectifier, also provided an influence. Irving Langmuir and G. S. Meikle of GE are usually cited as the first investigators to study controlled rectification in gas tubes, about 1914. The first commercial thyratrons appeared circa 1928.
The term "thyratron" is derived from Ancient Greek "θύρα" ("thyra"), meaning "door" or "valve". The term "thyristor" was further derived from a combination of "thyratron" and "transistor". Since the 1960s thyristors have replaced thyratrons in most low- and medium-power applications.
Description
Thyratrons resemble vacuum tubes both in appearance and construction but differ in behavior and operating principle. In a vacuum tube, conduction is dominated by free electrons because the distance between anode and cathode is small compared to the mean free path of electrons. A thyratron, on the other hand, is intentionally filled with gas so that the distance between anode and cathode is comparable with the mean free path of electrons. This causes conduction in a thyratron to be dominated by plasma conductivity. Due to the high conducti |
https://en.wikipedia.org/wiki/Omega%20%28video%20game%29 | Omega is a video game developed and published by Origin Systems in 1989. It was directed by Stuart B. Marks.
The player assumes the role of a cyber-tank designer and programmer, with the objective of creating tanks to defeat increasingly difficult opponents. The game emphasizes programming the tank, using a built-in text editor with artificial intelligence script commands similar to BASIC. Tanks can communicate and coordinate actions, and successful designs tend to be automated. Code is cross-platform, allowing Apple, Commodore, and IBM users to compete against each other.
The game received positive reviews, with Compute! praising its ease of use for newcomers to programming. Computer Gaming World acknowledged its similarities to RobotWar, while noting its improvements. Games International magazine awarded it 4 stars out of 5, highlighting its unique gameplay and requirement for strategic thinking.
Gameplay
The game puts the player in the role of a cyber-tank designer and programmer. Given a limited budget, the player must design a tank that can defeat a series of ever more challenging opponent tanks. Each successful design yields a higher security clearance and a larger budget, ultimately resulting in an OMEGA clearance and an unlimited budget. The focus of the game is not on the combat but on game programming the tank itself.
Tanks are programmed using a built-in text editor that allows the player to use various artificial intelligence script commands, similar in structure to BASIC. These commands permit control of various aspects of the tank, and also allows teams of tanks to communicate and coordinate actions. While commands exist that enable a range of control over the tank, successful designs tend to be automated. Decision making is an important part of the design process, as the programming must reflect the equipment placed on the tank.
Code was cross-platform, so Apple, Commodore, and IBM users could compete against each other. Origin operated a bulleti |
https://en.wikipedia.org/wiki/ISO/IEC%205218 | ISO/IEC 5218 Information technology — Codes for the representation of human sexes is an international standard that defines a representation of human sexes through a language-neutral single-digit code. It can be used in information systems such as database applications.
The four codes specified in ISO/IEC 5218 are:
0 = Not known;
1 = Male;
2 = Female;
9 = Not applicable.
The standard specifies that its use may be referred to by the designator "SEX".
The standard explicitly states that no significance is to be placed on the encoding of male as 1 and female as 2; the encoding merely reflects existing practice in the countries that initiated this standard. The standard also explains that it "meets the requirements of most applications that need to code human sexes. It does not provide codes for sexes that may be required in specific medical and scientific applications or in applications that need to code sex information other than for human beings." Since its 2022 revision, the standard also states that its scope does not cover human gender identities and therefore does not provide codes for those.
ISO/IEC 5218 was created by ISO's Data Management and Interchange Technical Committee, proposed in November 1976, and updated in June 2022. The standard is currently maintained by
the ISO/IEC Joint Technical Committee (ISO/IEC JTC 1) subcommittee on Data management and interchange (ISO/IEC JTC 1/SC 32).
This standard is used in several national identification numbers. For example, the first digit of the French INSEE number and the first digit of the Republic of China National Identification Card (Chinese: 中華民國國民身分證) are based on ISO/IEC 5218 values.
References
2004 introductions
05218
Gender |
https://en.wikipedia.org/wiki/List%20of%20equations | This is a list of equations, by Wikipedia page under appropriate bands of their field.
Eponymous equations
The following equations are named after researchers who discovered them.
Mathematics
Cauchy–Riemann equations
Chapman–Kolmogorov equation
Maurer–Cartan equation
Pell's equation
Poisson's equation
Riccati equation
sine-Gordon equation
Verhulst equation
Physics
Ampère's circuital law
Bernoulli's equation
Bogoliubov–Born–Green–Kirkwood–Yvon hierarchy of equations
Bessel's differential equation
Boltzmann equation
Borda–Carnot equation
Burgers' equation
Darcy–Weisbach equation
Dirac equation
Dirac equation in the algebra of physical space
Dirac–Kähler equation
Doppler equations
Drake equation (aka Green Bank equation)
Einstein's field equations
Euler equations (fluid dynamics)
Euler's equations (rigid body dynamics)
Relativistic Euler equations
Euler–Lagrange equation
Faraday's law of induction
Fokker–Planck equation
Fresnel equations
Friedmann equations
Gauss's law for electricity
Gauss's law for gravity
Gauss's law for magnetism
Gibbs–Helmholtz equation
Gross–Pitaevskii equation
Hamilton–Jacobi–Bellman equation
Helmholtz equation
Karplus equation
Kepler's equation
Kepler's laws of planetary motion
Kirchhoff's diffraction formula
Klein–Gordon equation
Korteweg–de Vries equation
Landau–Lifshitz–Gilbert equation
Lane–Emden equation
Langevin equation
Levy–Mises equations
Lindblad equation
Lorentz equation
Maxwell's equations
Maxwell's relations
Newton's laws of motion
Navier–Stokes equations
Reynolds-averaged Navier–Stokes equations
Prandtl–Reuss equations
Prony equation
Rankine–Hugoniot equation
Roothaan equations
Saha ionization equation
Sackur–Tetrode equation
Samik Hazra equation
Schrödinger equation
screened Poisson equation
Schwinger–Dyson equation
Sellmeier equation
Stokes–Einstein relation
Tsiolkovsky rocket equation
Van der Waals equation
Vlasov equation
Wiener equation
Chemistry
A |
https://en.wikipedia.org/wiki/Interface%20bloat | In software design, interface bloat (also called fat interfaces by Bjarne Stroustrup and Refused Bequests by Martin Fowler) is when an interface incorporates too many operations on some data into an interface, only to find that most of the objects cannot perform the given operations.
Interface bloat is an example of an anti-pattern. One might consider using visitor pattern, Adapter Pattern, or interface segregation instead.
Anti-patterns
Computer programming folklore
Software engineering folklore |
https://en.wikipedia.org/wiki/Metalanguage | In logic and linguistics, a metalanguage is a language used to describe another language, often called the object language. Expressions in a metalanguage are often distinguished from those in the object language by the use of italics, quotation marks, or writing on a separate line. The structure of sentences and phrases in a metalanguage can be described by a metasyntax. For example, to say that the word "noun" can be used as a noun in a sentence, one could write "noun" is a <noun>.
Types of metalanguage
There are a variety of recognized types of metalanguage, including embedded, ordered, and nested (or hierarchical) metalanguages.
Embedded
An embedded metalanguage is a language formally, naturally and firmly fixed in an object language. This idea is found in Douglas Hofstadter's book, Gödel, Escher, Bach, in a discussion of the relationship between formal languages and number theory: "... it is in the nature of any formalization of number theory that its metalanguage is embedded within it."
It occurs in natural, or informal, languages, as well—such as in English, where words such as noun, verb, or even word describe features and concepts pertaining to the English language itself.
Ordered
An ordered metalanguage is analogous to an ordered logic. An example of an ordered metalanguage is the construction of one metalanguage to discuss an object language, followed by the creation of another metalanguage to discuss the first, etc.
Nested
A nested (or hierarchical) metalanguage is similar to an ordered metalanguage in that each level represents a greater degree of abstraction. However, a nested metalanguage differs from an ordered one in that each level includes the one below.
The paradigmatic example of a nested metalanguage comes from the Linnean taxonomic system in biology. Each level in the system incorporates the one below it. The language used to discuss genus is also used to discuss species; the one used to discuss orders is also used to discuss gener |
https://en.wikipedia.org/wiki/Petkau%20effect | The Petkau effect is an early counterexample to linear-effect assumptions usually made about radiation exposure. It was found by Dr. Abram Petkau at the Atomic Energy of Canada Whiteshell Nuclear Research Establishment, Manitoba and published in Health Physics March 1972. The Petkau effect was coined by Swiss nuclear hazards commentator Ralph Graeub in 1985 in this book Der Petkau-Effekt und unsere strahlende Zukunft (The Petkau effect and our Radiating Future).
Petkau had been measuring, in the usual way, the radiation dose that would rupture a simulated artificial cell membrane. He found that 3500 rads delivered in hours (26 rad/min = 15.5 Sv/h) would do it. Then, almost by chance, Petkau repeated the experiment with much weaker radiation and found that 0.7 rad delivered in hours (1 millirad/min = 0.61 mSv/h) also ruptured the membrane. This was counter to the prevailing assumption of a linear relationship between total dose or dose rate and the consequences.
The radiation was of ionizing nature, and produced negative oxygen ions (free radicals). Those ions were more damaging to the simulated membrane in lower concentrations than higher (a somewhat counter-intuitive result in itself) because in the latter, they more readily recombine with each other instead of interfering with the membrane. The ion concentration directly correlated with the radiation dose rate and the composition had non-monotonic consequences.
Radio-protective effects of superoxide dismutase
Petkau conducted further experiments with simulated cells in 1976 and found that the enzyme superoxide dismutase protected the cells from free radicals generated by ionizing radiation, obviating the effects seen in his earlier experiment. Petkau also discovered that superoxide dismutase was elevated in the leukocytes (white blood cells) in a sub-population of nuclear workers occupationally exposed to elevated radiation (ca. 10 mSv in 6 months), further supporting the hypothesis that superoxide dismuta |
https://en.wikipedia.org/wiki/Point%20at%20infinity | In geometry, a point at infinity or ideal point is an idealized limiting point at the "end" of each line.
In the case of an affine plane (including the Euclidean plane), there is one ideal point for each pencil of parallel lines of the plane. Adjoining these points produces a projective plane, in which no point can be distinguished, if we "forget" which points were added. This holds for a geometry over any field, and more generally over any division ring.
In the real case, a point at infinity completes a line into a topologically closed curve. In higher dimensions, all the points at infinity form a projective subspace of one dimension less than that of the whole projective space to which they belong. A point at infinity can also be added to the complex line (which may be thought of as the complex plane), thereby turning it into a closed surface known as the complex projective line, CP1, also called the Riemann sphere (when complex numbers are mapped to each point).
In the case of a hyperbolic space, each line has two distinct ideal points. Here, the set of ideal points takes the form of a quadric.
Affine geometry
In an affine or Euclidean space of higher dimension, the points at infinity are the points which are added to the space to get the projective completion. The set of the points at infinity is called, depending on the dimension of the space, the line at infinity, the plane at infinity or the hyperplane at infinity, in all cases a projective space of one less dimension.
As a projective space over a field is a smooth algebraic variety, the same is true for the set of points at infinity. Similarly, if the ground field is the real or the complex field, the set of points at infinity is a manifold.
Perspective
In artistic drawing and technical perspective, the projection on the picture plane of the point at infinity of a class of parallel lines is called their vanishing point.
Hyperbolic geometry
In hyperbolic geometry, points at infinity are typical |
https://en.wikipedia.org/wiki/Line%20at%20infinity | In geometry and topology, the line at infinity is a projective line that is added to the real (affine) plane in order to give closure to, and remove the exceptional cases from, the incidence properties of the resulting projective plane. The line at infinity is also called the ideal line.
Geometric formulation
In projective geometry, any pair of lines always intersects at some point, but parallel lines do not intersect in the real plane. The line at infinity is added to the real plane. This completes the plane, because now parallel lines intersect at a point which lies on the line at infinity. Also, if any pair of lines do not intersect at a point on the line, then the pair of lines are parallel.
Every line intersects the line at infinity at some point. The point at which the parallel lines intersect depends only on the slope of the lines, not at all on their y-intercept.
In the affine plane, a line extends in two opposite directions. In the projective plane, the two opposite directions of a line meet each other at a point on the line at infinity. Therefore, lines in the projective plane are closed curves, i.e., they are cyclical rather than linear. This is true of the line at infinity itself; it meets itself at its two endpoints (which are therefore not actually endpoints at all) and so it is actually cyclical.
Topological perspective
The line at infinity can be visualized as a circle which surrounds the affine plane. However, diametrically opposite points of the circle are equivalent—they are the same point. The combination of the affine plane and the line at infinity makes the real projective plane, .
A hyperbola can be seen as a closed curve which intersects the line at infinity in two different points. These two points are specified by the slopes of the two asymptotes of the hyperbola. Likewise, a parabola can be seen as a closed curve which intersects the line at infinity in a single point. This point is specified by the slope of the axis of the par |
https://en.wikipedia.org/wiki/Plane%20at%20infinity | In projective geometry, a plane at infinity is the hyperplane at infinity of a three dimensional projective space or to any plane contained in the hyperplane at infinity of any projective space of higher dimension. This article will be concerned solely with the three-dimensional case.
Definition
There are two approaches to defining the plane at infinity which depend on whether one starts with a projective 3-space or an affine 3-space.
If a projective 3-space is given, the plane at infinity is any distinguished projective plane of the space. This point of view emphasizes the fact that this plane is not geometrically different than any other plane. On the other hand, given an affine 3-space, the plane at infinity is a projective plane which is added to the affine 3-space in order to give it closure of incidence properties. Meaning that the points of the plane at infinity are the points where parallel lines of the affine 3-space will meet, and the lines are the lines where parallel planes of the affine 3-space will meet. The result of the addition is the projective 3-space, . This point of view emphasizes the internal structure of the plane at infinity, but does make it look "special" in comparison to the other planes of the space.
If the affine 3-space is real, , then the addition of a real projective plane at infinity produces the real projective 3-space .
Analytic representation
Since any two projective planes in a projective 3-space are equivalent, we can choose a homogeneous coordinate system so that any point on the plane at infinity is represented as (X:Y:Z:0).
Any point in the affine 3-space will then be represented as (X:Y:Z:1). The points on the plane at infinity seem to have three degrees of freedom, but homogeneous coordinates are equivalent up to any rescaling:
,
so that the coordinates (X:Y:Z:0) can be normalized, thus reducing the degrees of freedom to two (thus, a surface, namely a projective plane).
Proposition: Any line which passes through t |
https://en.wikipedia.org/wiki/Eve%20Online | Eve Online (stylised EVE Online) is a space-based, persistent world massively multiplayer online role-playing game (MMORPG) developed and published by CCP Games. Players of Eve Online can participate in a number of in-game professions and activities, including mining, piracy, manufacturing, trading, exploration, and combat (both player versus environment and player versus player). The game contains a total of 7,800 star systems that can be visited by players.
The game is renowned for its scale and complexity with regards to player interactions. In its single, shared game world, players engage in unscripted economic competition, warfare, and political schemes with other players. The Bloodbath of B-R5RB, a battle involving thousands of players in a single star system, took 21 hours and was recognized as one of the largest and most expensive battles in gaming history. Eve Online was exhibited at the Museum of Modern Art with a video including the historical events and accomplishments of the playerbase.
Eve Online was released in North America and Europe in May 2003. It was published from May to December 2003 by Simon & Schuster Interactive, after which CCP purchased the rights and began to self-publish via a digital distribution scheme. On January 22, 2008, it was announced that Eve Online would be distributed via Steam. On March 10, 2009, the game was again made available in boxed form in stores, released by Atari. In February 2013, Eve Online reached over 500,000 subscribers. On November 11, 2016, Eve Online added a limited free-to-play version.
Background
Set more than 21,000 years in the future, the background story of Eve Online explains that humanity, having used up most of Earth's resources through centuries of explosive population growth, began colonizing the rest of the Milky Way. As on Earth, this expansion also led to competition and fighting over available resources, but everything changed with the discovery of a natural wormhole leading to an unexplored |
https://en.wikipedia.org/wiki/61%20%28number%29 | 61 (sixty-one) is the natural number following 60 and preceding 62.
In mathematics
61 is the 18th prime number, and a twin prime with 59. It is the sum of two consecutive squares, It is also a centered decagonal number, a centered hexagonal number, and a centered square number.
61 is the fourth cuban prime of the form where , and the forth Pillai prime since is divisible by 61, but 61 is not one more than a multiple of 8. It is also a Keith number, as it recurs in a Fibonacci-like sequence started from its base 10 digits: 6, 1, 7, 8, 15, 23, 38, 61, ...
61 is a unique prime in base 14, since no other prime has a 6-digit period in base 14, and palindromic in bases 6 (1416) and 60 (1160). It is the sixth up/down or Euler zigzag number.
61 is the smallest proper prime, a prime which ends in the digit 1 in decimal and whose reciprocal in base-10 has a repeating sequence of length where each digit (0, 1, ..., 9) appears in the repeating sequence the same number of times as does each other digit (namely, times).
In the list of Fortunate numbers, 61 occurs thrice, since adding 61 to either the tenth, twelfth or seventeenth primorial gives a prime number (namely 6,469,693,291; 7,420,738,134,871; and 1,922,760,350,154,212,639,131).
61 is the exponent of the ninth Mersenne prime, and the next candidate exponent for a potential fifth double Mersenne prime:
The exotic sphere is the last odd-dimensional sphere to contain a unique smooth structure; , and are the only other such spheres.
In science
The chemical element with the atomic number 61 is promethium.
Astronomy
Messier object M61, a magnitude 10.5 galaxy in the constellation Virgo
The New General Catalogue object NGC 61, a double spiral galaxy in the constellation Cetus
61 Ursae Majoris is located about 31.1 light-years from the Sun.
61 Cygni was christened the "Flying Star" in 1792 by Giuseppe Piazzi (1746–1826) for its unusually large proper motion.
In other fields
Sixty-one is:
The number of the |
https://en.wikipedia.org/wiki/62%20%28number%29 | 62 (sixty-two) is the natural number following 61 and preceding 63.
In mathematics
62 is:
the eighteenth discrete semiprime () and tenth of the form (2.q), where q is a higher prime.
with an aliquot sum of 34; itself a semiprime, within an aliquot sequence of seven composite numbers (62,34,20,22,14,10,8,7,1,0) to the Prime in the 7-aliquot tree. This is the longest aliquot sequence for a semiprime up to 118 which has one more sequence member. 62 is the tenth member of the 7-aliquot tree (7, 8, 10, 14, 20, 22, 34, 38, 49, 62, 75, 118, 148, etc).
a nontotient.
palindromic and a repdigit in bases 5 (2225) and 30 (2230)
the sum of the number of faces, edges and vertices of icosahedron or dodecahedron.
the number of faces of two of the Archimedean solids, the rhombicosidodecahedron and truncated icosidodecahedron.
the smallest number that is the sum of three distinct positive squares in two (or more) ways,
the only number whose cube in base 10 (238328) consists of 3 digits each occurring 2 times.
The 20th & 21st, 72nd & 73rd, 75th & 76th digits of pi.
In science
Sixty-two is the atomic number of samarium, a lanthanide.
In other fields
62 is the code for international direct dial calls to Indonesia.
In the 1998 Home Run Race, Mark McGwire hit his 62nd home run on September 8, breaking the single-season record. Sammy Sosa hit his 62nd home run just days later on September 13.
Under Social Security (United States), the earliest age at which a person may begin receiving retirement benefits (other than disability).
References
Integers |
https://en.wikipedia.org/wiki/63%20%28number%29 | 63 (sixty-three) is the natural number following 62 and preceding 64.
Mathematics
63 is the sum of the first six powers of 2 (20 + 21 + ... 25). It is the eighth highly cototient number, and the fourth centered octahedral number; after 7 and 25. For five unlabeled elements, there are 63 posets.
Sixty-three is the seventh square-prime of the form and the second of the form . It contains a prime aliquot sum of 41, the thirteenth indexed prime; and part of the aliquot sequence (63, 41, 1, 0) within the 41-aliquot tree.
Zsigmondy's theorem states that where are coprime integers for any integer , there exists a primitive prime divisor that divides and does not divide for any positive integer , except for when
, with having no prime divisors,
, a power of two, where any odd prime factors of are contained in , which is even;
and for a special case where with and , which yields .
63 is a Mersenne number of the form with an of , however this does not yield a Mersenne prime, as 63 is the forty-fourth composite number. It is the only number in the Mersenne sequence whose prime factors are each factors of at least one previous element of the sequence (3 and 7, respectively the first and second Mersenne primes). In the list of Mersenne numbers, 63 lies between Mersenne primes 31 and 127, with 127 the thirty-first prime number. The thirty-first odd number, of the simplest form , is 63. It is also the fourth Woodall number of the form with , with the previous members being 1, 7 and 23 (they add to 31, the third Mersenne prime).
In the integer positive definite quadratic matrix representative of all (even and odd) integers, the sum of all nine terms is equal to 63.
63 is the third Delannoy number, which represents the number of pathways in a grid from a southwest corner to a northeast corner, using only single steps northward, eastward, or northeasterly.
Finite simple groups
63 holds thirty-six integers that are relatively prime with itself (and up to |
https://en.wikipedia.org/wiki/64%20%28number%29 | 64 (sixty-four) is the natural number following 63 and preceding 65.
In mathematics
Sixty-four is the square of 8, the cube of 4, and the sixth-power of 2. It is the smallest number with exactly seven divisors. 64 is the first non-unitary sixth-power prime of the form p6 where p is a prime number.
The aliquot sum of a 2-power (2n) is always one less than the 2-power itself therefore the aliquot sum of 64 is 63, within an aliquot sequence of two composite members ( 64,63,41,1,0) to the prime 41 in the 41-aliquot tree.
It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers. It is also a dodecagonal number and a centered triangular number. 64 is also the first whole number (greater than 1) that is both a perfect square and a perfect cube.
Since it is possible to find sequences of 65 consecutive integers (intervals of length 64) such that each inner member shares a factor with either the first or the last member, 64 is an Erdős–Woods number.
In base 10, no integer added to the sum of its own digits yields 64; hence 64 is a self number.
64 is a superperfect number—a number such that σ(σ(n)) = 2n.
64 is the index of Graham's number in the rapidly growing sequence 3↑↑↑↑3, 3 ↑ 3,…
In the fourth dimension, there are 64 uniform polychora aside from two infinite families of duoprisms and antiprismatic prisms, and 64 Bravais lattices.
In science
The atomic number of gadolinium, a lanthanide
In astronomy
Messier object M64, a magnitude 9.0 galaxy in the constellation Coma Berenices, also known as the Black Eye Galaxy.
The New General Catalogue object NGC 64, a barred spiral galaxy in the constellation Cetus.
In technology
In some computer programming languages, the size in bits of certain data types
64-bit computing
A 64-bit integer can represent up to 18,446,744,073,709,551,616 values.
Base 64 is used in with Base64 encoding and other dat |
https://en.wikipedia.org/wiki/65%20%28number%29 | 65 (sixty-five) is the natural number following 64 and preceding 66.
In mathematics
65 is the nineteenth distinct semiprime, (5.13); and the third of the form (5.q), where q is a higher prime.
65 has a prime aliquot sum of 19 within an aliquot sequence of one composite numbers (65,19,1,0) to the prime; as the first member' of the 19-aliquot tree.
It is an octagonal number. It is also a Cullen number. Given 65, the Mertens function returns 0.
This number is the magic constant of a 5x5 normal magic square:
This number is also the magic constant of n-Queens Problem for n = 5.
65 is the smallest integer that can be expressed as a sum of two distinct positive squares in two (or more) ways, 65 = 82 + 12 = 72 + 42.
It appears in the Padovan sequence, preceded by the terms 28, 37, 49 (it is the sum of the first two of these).
65 is a Stirling number of the second kind, the number of ways of dividing a set of six objects into four non-empty subsets.
65 = 15 + 24 + 33 + 42 + 51.
65 is the length of the hypotenuse of 4 different Pythagorean triangles, the lowest number to have more than 2: 652 = 162 + 632 = 332 + 562 = 392 + 522 = 252 + 602. The first two are "primitive", and 65 is the lowest number to be the largest side of more than one such triple.
65 is the number of compositions of 11 into distinct parts.
In science
The atomic number of terbium, a lanthanide
Astronomy
Messier object M65, a galaxy of magnitude 10.5 in the constellation Leo
The New General Catalogue object NGC 65, a spiral galaxy in the constellation Cetus
In music
"65 Love Affair", singer Paul Davis' hit song in 1982
Sammy Hagar re-recorded his hit "I Can't Drive 55", with the 55 changed to 65, in 2001 for NBC's NASCAR broadcasts to reflect higher speed limits; the song was used from 2001 until 2004 to introduce Budweiser Pole Award winners on NBC and TNT broadcasts
65 is a commonly used abbreviation for the Sheffield, UK, post-rock band 65daysofstatic
Referenced in "Heroes and Villains" |
https://en.wikipedia.org/wiki/66%20%28number%29 | 66 (sixty-six) is the natural number following 65 and preceding 67.
Usages of this number include:
In mathematics
66 is:
a sphenic number.
a triangular number.
a hexagonal number.
a semi-meandric number.
a semiperfect number, being a multiple of a perfect number.
an Erdős–Woods number, since it is possible to find sequences of 66 consecutive integers such that each inner member shares a factor with either the first or the last member.
palindromic and a repdigit in bases 10 (6610), 21 (3321) and 32 (2232)
In science
Astronomy
Messier object Spiral Galaxy M66, a magnitude 10.0 galaxy in the constellation Leo.
The New General Catalogue object NGC 66, a peculiar barred spiral galaxy in the constellation Cetus.
66 Maja, a carbonaceous background asteroid from the central regions of the asteroid belt.
Physics
The atomic number of dysprosium, a lanthanide.
In computing
66 (more specifically 66.667) megahertz (MHz) is a common divisor for the front side bus (FSB) speed, overall central processing unit (CPU) speed, and base bus speed. On a Core 2 CPU, and a Core 2 motherboard, the FSB is 1066 MHz (~16 × 66 MHz), the memory speed is usually 666.67 MHz (~10 × 66 MHz), and the processor speed ranges from 1.86 gigahertz (GHz) (~66 MHz × 28) to 2.93 GHz (~66 MHz × 44), in 266 MHz (~66 MHz × 4) increments.
In motor vehicle transportation
The designation of the historic U.S. Route 66, dubbed the "Mother Road" by novelist John Steinbeck, and other roads.
Phillips 66, a brand of gasoline and service station in the United States.
In religion
The total number of chapters in the Bible Book of Isaiah.
The number of verses in Chapter 3 of the Book of Lamentations in the Old Testament.
The total number of books in the Protestant edition of the Bible (Old Testament and New Testament) combined.
In Abjad numerals, The Name Of Allah (الله) numeric value is 66.
In sports
The number of the laps of the Spanish Grand Prix.
The longest field goal made in NFL history was 66 yards and kic |
https://en.wikipedia.org/wiki/67%20%28number%29 | 67 (sixty-seven) is the natural number following 66 and preceding 68. It is an odd number.
In mathematics
67 is:
the 19th prime number (the next is 71).
a Chen prime.
an irregular prime.
a lucky prime.
the sum of five consecutive primes (7 + 11 + 13 + 17 + 19).
a Heegner number.
a Pillai prime since 18! + 1 is divisible by 67, but 67 is not one more than a multiple of 18.
palindromic in quinary (2325) and senary (1516).
a super-prime. (19 is prime)
an isolated prime. (65 and 69 are not prime)
In science
The atomic number of holmium, a lanthanide.
Astronomy
Messier object M67, a magnitude 7.5 open cluster in the constellation Cancer.
The New General Catalogue object NGC 67, an elliptical galaxy in the constellation Andromeda.
In music
"Car 67", a song by the band Driver 67
Chicago's song "Questions 67 and 68"
Elton John's song "Old '67" on The Captain & The Kid CD, (2006)
British rap group called 67
Rapper Drake released the song named "Star67" off his album If You're Reading This It's Too Late
In other fields
Sixty-seven is:
The registry of the U.S. Navy's aircraft carrier , named after U.S. President John F. Kennedy.
The number of the French department Bas-Rhin.
The number of counties in Alabama, Florida, and Pennsylvania.
The province/traffic code of Zonguldak Province in Turkey.
In the U.S., *67 is a common prefix-code for blocking caller ID info on the subsequent call.
In sports
Buddy Arrington's best-known NASCAR car number.
The Ottawa 67's, founded in 1967.
Pekka Koskela skated the 1000 metres in 1:07:00 (67 seconds) on 10 November 2007, a world record at the time.
The number of the laps of the German Grand Prix since 2002 if the race was held at Hockenheimring.
External links
References
Integers |
https://en.wikipedia.org/wiki/68%20%28number%29 | 68 (sixty-eight) is the natural number following 67 and preceding 69. It is an even number.
In mathematics
68 is a composite number; a square-prime, of the form (p2, q) where q is a higher prime. It is the eighth of this form and the sixth of the form (22.q).
68 is a Perrin number.
It has an aliquot sum of 58 within an aliquot sequence of two composite numbers (68, 58,32,31,1,0) to the Prime in the 31-aliquot tree.
It is the largest known number to be the sum of two primes in exactly two different ways: 68 = 7 + 61 = 31 + 37. All higher even numbers that have been checked are the sum of three or more pairs of primes; the conjecture that 68 is the largest number with this property is closely related to the Goldbach conjecture and, like it, remains unproven.
Because of the factorization of 68 as , a 68-sided regular polygon may be constructed with compass and straightedge.
There are exactly 68 10-bit binary numbers in which each bit has an adjacent bit with the same value, exactly 68 combinatorially distinct triangulations of a given triangle with four points interior to it, and exactly 68 intervals in the Tamari lattice describing the ways of parenthesizing five items. The largest graceful graph on 14 nodes has exactly 68 edges. There are 68 different undirected graphs with six edges and no isolated nodes, 68 different minimally 2-connected graphs on seven unlabeled nodes, 68 different degree sequences of four-node connected graphs, and 68 matroids on four labeled elements.
Størmer's theorem proves that, for every number p, there are a finite number of pairs of consecutive numbers that are both p-smooth (having no prime factor larger than p). For p = 13 this finite number is exactly 68. On an infinite chessboard, there are 68 squares three knight's moves away from any cell.
As a decimal number, 68 is the last two-digit number to appear for the first time in the digits of pi. It is a happy number, meaning that repeatedly summing the squares of its digits even |
https://en.wikipedia.org/wiki/Radeon | Radeon () is a brand of computer products, including graphics processing units, random-access memory, RAM disk software, and solid-state drives, produced by Radeon Technologies Group, a division of AMD. The brand was launched in 2000 by ATI Technologies, which was acquired by AMD in 2006 for US$5.4 billion.
Radeon Graphics
Radeon Graphics is the successor to the Rage line. Three different families of microarchitectures can be roughly distinguished, the fixed-pipeline family, the unified shader model-families of TeraScale and Graphics Core Next. ATI/AMD have developed different technologies, such as TruForm, HyperMemory, HyperZ, XGP, Eyefinity for multi-monitor setups, PowerPlay for power-saving, CrossFire (for multi-GPU) or Hybrid Graphics. A range of SIP blocks is also to be found on certain models in the Radeon products line: Unified Video Decoder, Video Coding Engine and TrueAudio.
The brand was previously only known as "ATI Radeon" until August 2010, when it was renamed to increase AMD's brand awareness on a global scale. Products up to and including the HD 5000 series are branded as ATI Radeon, while the HD 6000 series and beyond use the new AMD Radeon branding.
On 11 September 2015, AMD's GPU business was split into a separate unit known as Radeon Technologies Group, with Raja Koduri as Senior Vice President and chief architect.
Radeon Graphics card brands
AMD does not distribute Radeon cards directly to consumers (though some exceptions can be found). Instead, it sells Radeon GPUs to third-party manufacturers, who build and sell the Radeon-based video cards to the OEM and retail channels. Manufacturers of the Radeon cards—some of whom also make motherboards—include ASRock, Asus, Biostar, Club 3D, Diamond, Force3D, Gainward, Gigabyte, HIS, MSI, PowerColor, Sapphire, VisionTek, and XFX.
Graphics processor generations
Early generations were identified with a number and major/minor alphabetic prefix. Later generations were assigned code names. New or hea |
https://en.wikipedia.org/wiki/Genetic%20diversity | Genetic diversity is the total number of genetic characteristics in the genetic makeup of a species, it ranges widely from the number of species to differences within species and can be attributed to the span of survival for a species. It is distinguished from genetic variability, which describes the tendency of genetic characteristics to vary.
Genetic diversity serves as a way for populations to adapt to changing environments. With more variation, it is more likely that some individuals in a population will possess variations of alleles that are suited for the environment. Those individuals are more likely to survive to produce offspring bearing that allele. The population will continue for more generations because of the success of these individuals.
The academic field of population genetics includes several hypotheses and theories regarding genetic diversity. The neutral theory of evolution proposes that diversity is the result of the accumulation of neutral substitutions. Diversifying selection is the hypothesis that two subpopulations of a species live in different environments that select for different alleles at a particular locus. This may occur, for instance, if a species has a large range relative to the mobility of individuals within it. Frequency-dependent selection is the hypothesis that as alleles become more common, they become more vulnerable. This occurs in host–pathogen interactions, where a high frequency of a defensive allele among the host means that it is more likely that a pathogen will spread if it is able to overcome that allele.
Within-species diversity
A study conducted by the National Science Foundation in 2007 found that genetic diversity (within-species diversity) and biodiversity are dependent upon each other — i.e. that diversity within a species is necessary to maintain diversity among species, and vice versa. According to the lead researcher in the study, Dr. Richard Lankau, "If any one type is removed from the system, the cycl |
https://en.wikipedia.org/wiki/Algebraic%20equation | In mathematics, an algebraic equation or polynomial equation is an equation of the form , where P is a polynomial with coefficients in some field, often the field of the rational numbers.
For example, is an algebraic equation with integer coefficients and
is a multivariate polynomial equation over the rationals.
For many authors, the term algebraic equation refers only to the univariate case, that is polynomial equations that involve only one variable. On the other hand, a polynomial equation may involve several variables (the multivariate case), in which case the term polynomial equation is usually preferred.
Some but not all polynomial equations with rational coefficients have a solution that is an algebraic expression that can be found using a finite number of operations that involve only those same types of coefficients (that is, can be solved algebraically). This can be done for all such equations of degree one, two, three, or four; but for degree five or more it can only be done for some equations, not all. A large amount of research has been devoted to compute efficiently accurate approximations of the real or complex solutions of a univariate algebraic equation (see Root-finding algorithm) and of the common solutions of several multivariate polynomial equations (see System of polynomial equations).
Terminology
The term "algebraic equation" dates from the time when the main problem of algebra was to solve univariate polynomial equations. This problem was completely solved during the 19th century; see Fundamental theorem of algebra, Abel–Ruffini theorem and Galois theory.
Since then, the scope of algebra has been dramatically enlarged. In particular, it includes the study of equations that involve th roots and, more generally, algebraic expressions. This makes the term algebraic equation ambiguous outside the context of the old problem. So the term polynomial equation is generally preferred when this ambiguity may occur, specially when considering multiv |
https://en.wikipedia.org/wiki/List%20of%20mathematics-based%20methods | This is a list of mathematics-based methods.
Adams' method (differential equations)
Akra–Bazzi method (asymptotic analysis)
Bisection method (root finding)
Brent's method (root finding)
Condorcet method (voting systems)
Coombs' method (voting systems)
Copeland's method (voting systems)
Crank–Nicolson method (numerical analysis)
D'Hondt method (voting systems)
D21 – Janeček method (voting system)
Discrete element method (numerical analysis)
Domain decomposition method (numerical analysis)
Epidemiological methods
Euler's forward method
Explicit and implicit methods (numerical analysis)
Finite difference method (numerical analysis)
Finite element method (numerical analysis)
Finite volume method (numerical analysis)
Highest averages method (voting systems)
Method of exhaustion
Method of infinite descent (number theory)
Information bottleneck method
Inverse chain rule method (calculus)
Inverse transform sampling method (probability)
Iterative method (numerical analysis)
Jacobi method (linear algebra)
Largest remainder method (voting systems)
Level-set method
Linear combination of atomic orbitals molecular orbital method (molecular orbitals)
Method of characteristics
Least squares method (optimization, statistics)
Maximum likelihood method (statistics)
Method of complements (arithmetic)
Method of moving frames (differential geometry)
Method of successive substitution (number theory)
Monte Carlo method (computational physics, simulation)
Newton's method (numerical analysis)
Pemdas method (order of operation)
Perturbation methods (functional analysis, quantum theory)
Probabilistic method (combinatorics)
Romberg's method (numerical analysis)
Runge–Kutta method (numerical analysis)
Sainte-Laguë method (voting systems)
Schulze method (voting systems)
Sequential Monte Carlo method
Simplex method
Spectral method (numerical analysis)
Variational methods (mathematical analysis, differential equations)
Welch's method
See also
Automatic basis function construction
List of graphi |
https://en.wikipedia.org/wiki/Circular%20segment | In geometry, a circular segment (symbol: ⌓), also known as a disk segment, is a region of a disk which is "cut off" from the rest of the disk by a secant or a chord. More formally, a circular segment is a region of two-dimensional space that is bounded by a circular arc (of less than π radians by convention) and by the circular chord connecting the endpoints of the arc.
Formulae
Let R be the radius of the arc which forms part of the perimeter of the segment, θ the central angle subtending the arc in radians, c the chord length, s the arc length, h the sagitta (height) of the segment, d the apothem of the segment, and a the area of the segment.
Usually, chord length and height are given or measured, and sometimes the arc length as part of the perimeter, and the unknowns are area and sometimes arc length. These can't be calculated simply from chord length and height, so two intermediate quantities, the radius and central angle are usually calculated first.
Radius and central angle
The radius is:
The central angle is
Chord length and height
The chord length and height can be back-computed from radius and central angle by:
The chord length is
The sagitta is
The apothem is
Arc length and area
The arc length, from the familiar geometry of a circle, is
The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of ):
In terms of and ,
In terms of and ,
What can be stated is that as the central angle gets smaller (or alternately the radius gets larger), the area a rapidly and asymptotically approaches . If , is a substantially good approximation.
If is held constant, and the radius is allowed to vary, then we have
As the central angle approaches π, the area of the segment is converging to the area of a semicircle, , so a good approximation is a delta offset from the latter area:
for h>.75R
As an example, the area is one quart |
https://en.wikipedia.org/wiki/Brauer%20group | In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras over K, with addition given by the tensor product of algebras. It was defined by the algebraist Richard Brauer.
The Brauer group arose out of attempts to classify division algebras over a field. It can also be defined in terms of Galois cohomology. More generally, the Brauer group of a scheme is defined in terms of Azumaya algebras, or equivalently using projective bundles.
Construction
A central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A such that A is a simple ring and the center of A is equal to K. Note that CSAs are in general not division algebras, though CSAs can be used to classify division algebras.
For example, the complex numbers C form a CSA over themselves, but not over R (the center is C itself, hence too large to be CSA over R). The finite-dimensional division algebras with center R (that means the dimension over R is finite) are the real numbers and the quaternions by a theorem of Frobenius, while any matrix ring over the reals or quaternions – or – is a CSA over the reals, but not a division algebra (if n > 1).
We obtain an equivalence relation on CSAs over K by the Artin–Wedderburn theorem (Wedderburn's part, in fact), to express any CSA as a M(n, D) for some division algebra D. If we look just at D, that is, if we impose an equivalence relation identifying with for all positive integers m and n, we get the Brauer equivalence relation on CSAs over K. The elements of the Brauer group are the Brauer equivalence classes of CSAs over K.
Given central simple algebras A and B, one can look at their tensor product A ⊗ B as a K-algebra (see tensor product of R-algebras). It turns out that this is always central simple. A slick way to see this is to use a characterization: a central simple algebra A over K is a K-algebra that becomes a matrix ring when we extend the fie |
https://en.wikipedia.org/wiki/Central%20simple%20algebra | In ring theory and related areas of mathematics a central simple algebra (CSA) over a field K is a finite-dimensional associative K-algebra A which is simple, and for which the center is exactly K. (Note that not every simple algebra is a central simple algebra over its center: for instance, if K is a field of characteristic 0, then the Weyl algebra is a simple algebra with center K, but is not a central simple algebra over K as it has infinite dimension as a K-module.)
For example, the complex numbers C form a CSA over themselves, but not over the real numbers R (the center of C is all of C, not just R). The quaternions H form a 4-dimensional CSA over R, and in fact represent the only non-trivial element of the Brauer group of the reals (see below).
Given two central simple algebras A ~ M(n,S) and B ~ M(m,T) over the same field F, A and B are called similar (or Brauer equivalent) if their division rings S and T are isomorphic. The set of all equivalence classes of central simple algebras over a given field F, under this equivalence relation, can be equipped with a group operation given by the tensor product of algebras. The resulting group is called the Brauer group Br(F) of the field F. It is always a torsion group.
Properties
According to the Artin–Wedderburn theorem a finite-dimensional simple algebra A is isomorphic to the matrix algebra M(n,S) for some division ring S. Hence, there is a unique division algebra in each Brauer equivalence class.
Every automorphism of a central simple algebra is an inner automorphism (this follows from the Skolem–Noether theorem).
The dimension of a central simple algebra as a vector space over its centre is always a square: the degree is the square root of this dimension. The Schur index of a central simple algebra is the degree of the equivalent division algebra: it depends only on the Brauer class of the algebra.
The period or exponent of a central simple algebra is the order of its Brauer class as an element of the |
https://en.wikipedia.org/wiki/Normal%20function | In axiomatic set theory, a function f : Ord → Ord is called normal (or a normal function) if and only if it is continuous (with respect to the order topology) and strictly monotonically increasing. This is equivalent to the following two conditions:
For every limit ordinal γ (i.e. γ is neither zero nor a successor), it is the case that f(γ) = sup {f(ν) : ν < γ}.
For all ordinals α < β, it is the case that f(α) < f(β).
Examples
A simple normal function is given by (see ordinal arithmetic). But is not normal because it is not continuous at any limit ordinal; that is, the inverse image of the one-point open set is the set , which is not open when λ is a limit ordinal. If β is a fixed ordinal, then the functions , (for ), and (for ) are all normal.
More important examples of normal functions are given by the aleph numbers , which connect ordinal and cardinal numbers, and by the beth numbers .
Properties
If f is normal, then for any ordinal α,
f(α) ≥ α.
Proof: If not, choose γ minimal such that f(γ) < γ. Since f is strictly monotonically increasing, f(f(γ)) < f(γ), contradicting minimality of γ.
Furthermore, for any non-empty set S of ordinals, we have
f(sup S) = sup f(S).
Proof: "≥" follows from the monotonicity of f and the definition of the supremum. For "≤", set δ = sup S and consider three cases:
if δ = 0, then S = {0} and sup f(S) = f(0);
if δ = ν + 1 is a successor, then there exists s in S with ν < s, so that δ ≤ s. Therefore, f(δ) ≤ f(s), which implies f(δ) ≤ sup f(S);
if δ is a nonzero limit, pick any ν < δ, and an s in S such that ν < s (possible since δ = sup S). Therefore, f(ν) < f(s) so that f(ν) < sup f(S), yielding f(δ) = sup {f(ν) : ν < δ} ≤ sup f(S), as desired.
Every normal function f has arbitrarily large fixed points; see the fixed-point lemma for normal functions for a proof. One can create a normal function f' : Ord → Ord, called the derivative of f, such that f' (α) is the α-th fixed point of f. For a hierarchy of normal |
https://en.wikipedia.org/wiki/Fixed-point%20lemma%20for%20normal%20functions | The fixed-point lemma for normal functions is a basic result in axiomatic set theory stating that any normal function has arbitrarily large fixed points (Levy 1979: p. 117). It was first proved by Oswald Veblen in 1908.
Background and formal statement
A normal function is a class function from the class Ord of ordinal numbers to itself such that:
is strictly increasing: whenever .
is continuous: for every limit ordinal (i.e. is neither zero nor a successor), .
It can be shown that if is normal then commutes with suprema; for any nonempty set of ordinals,
.
Indeed, if is a successor ordinal then is an element of and the equality follows from the increasing property of . If is a limit ordinal then the equality follows from the continuous property of .
A fixed point of a normal function is an ordinal such that .
The fixed point lemma states that the class of fixed points of any normal function is nonempty and in fact is unbounded: given any ordinal , there exists an ordinal such that and .
The continuity of the normal function implies the class of fixed points is closed (the supremum of any subset of the class of fixed points is again a fixed point). Thus the fixed point lemma is equivalent to the statement that the fixed points of a normal function form a closed and unbounded class.
Proof
The first step of the proof is to verify that for all ordinals and that commutes with suprema. Given these results, inductively define an increasing sequence by setting , and for . Let , so . Moreover, because commutes with suprema,
The last equality follows from the fact that the sequence increases.
As an aside, it can be demonstrated that the found in this way is the smallest fixed point greater than or equal to .
Example application
The function f : Ord → Ord, f(α) = ωα is normal (see initial ordinal). Thus, there exists an ordinal θ such that θ = ωθ. In fact, the lemma shows that there is a closed, unbounded class of such θ.
References
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https://en.wikipedia.org/wiki/Finitary | In mathematics and logic, an operation is finitary if it has finite arity, i.e. if it has a finite number of input values. Similarly, an infinitary operation is one with an infinite number of input values.
In standard mathematics, an operation is finitary by definition. Therefore these terms are usually only used in the context of infinitary logic.
Finitary argument
A finitary argument is one which can be translated into a finite set of symbolic propositions starting from a finite set of axioms. In other words, it is a proof (including all assumptions) that can be written on a large enough sheet of paper.
By contrast, infinitary logic studies logics that allow infinitely long statements and proofs. In such a logic, one can regard the existential quantifier, for instance, as derived from an infinitary disjunction.
History
Logicians in the early 20th century aimed to solve the problem of foundations, such as, "What is the true base of mathematics?" The program was to be able to rewrite all mathematics using an entirely syntactical language without semantics. In the words of David Hilbert (referring to geometry), "it does not matter if we call the things chairs, tables and beer mugs or points, lines and planes."
The stress on finiteness came from the idea that human mathematical thought is based on a finite number of principles and all the reasonings follow essentially one rule: the modus ponens. The project was to fix a finite number of symbols (essentially the numerals 1, 2, 3, ... the letters of alphabet and some special symbols like "+", "⇒", "(", ")", etc.), give a finite number of propositions expressed in those symbols, which were to be taken as "foundations" (the axioms), and some rules of inference which would model the way humans make conclusions. From these, regardless of the semantic interpretation of the symbols the remaining theorems should follow formally using only the stated rules (which make mathematics look like a game with symbols more than |
https://en.wikipedia.org/wiki/Well-formed%20formula | In mathematical logic, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language.
A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic.
Introduction
A key use of formulas is in propositional logic and predicate logic such as first-order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven.
Although the term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence of symbols being expressed, with the marks being a token instance of formula. This distinction between the vague notion of "property" and the inductively-defined notion of well-formed formula has roots in Weyl's 1910 paper "Uber die Definitionen der mathematischen Grundbegriffe". Thus the same formula may be written more than once, and a formula might in principle be so long that it cannot be written at all within the physical universe.
Formulas themselves are syntactic objects. They are given meanings by interpretations. For example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not be interpreted, however, to be considered solely as a formula.
Propositional calculus
The formulas of propositional calculus, also called propositional formulas, are expressions such as . Their definition beg |
https://en.wikipedia.org/wiki/Logical%20block%20addressing | Logical block addressing (LBA) is a common scheme used for specifying the location of blocks of data stored on computer storage devices, generally secondary storage systems such as hard disk drives. LBA is a particularly simple linear addressing scheme; blocks are located by an integer index, with the first block being LBA 0, the second LBA 1, and so on.
The IDE standard included 22-bit LBA as an option, which was further extended to 28-bit with the release of ATA-1 (1994) and to 48-bit with the release of ATA-6 (2003), whereas the size of entries in on-disk and in-memory data structures holding the address is typically 32 or 64 bits. Most hard disk drives released after 1996 implement logical block addressing.
Overview
In logical block addressing, only one number is used to address data, and each linear base address describes a single block.
The LBA scheme replaces earlier schemes which exposed the physical details of the storage device to the software of the operating system. Chief among these was the cylinder-head-sector (CHS) scheme, where blocks were addressed by means of a tuple which defined the cylinder, head, and sector at which they appeared on the hard disk. CHS did not map well to devices other than hard disks (such as tapes and networked storage), and was generally not used for them. CHS was used in early MFM and RLL drives, and both it and its successor, extended cylinder-head-sector (ECHS), were used in the first ATA drives. However, current disk drives use zone bit recording, where the number of sectors per track depends on the track number. Even though the disk drive will report some CHS values as sectors per track (SPT) and heads per cylinder (HPC), they have little to do with the disk drive's true geometry.
LBA was first introduced in SCSI as an abstraction. While the drive controller still addresses data blocks by their CHS address, this information is generally not used by the SCSI device driver, the OS, filesystem code, or any applic |
https://en.wikipedia.org/wiki/New%20Math | New Mathematics or New Math was a dramatic but temporary change in the way mathematics was taught in American grade schools, and to a lesser extent in European countries and elsewhere, during the 1950s1970s.
Overview
In 1957, the U.S. National Science Foundation funded the development of several new curricula in the sciences, such as the Physical Science Study Committee high school physics curriculum, Biological Sciences Curriculum Study in biology, and CHEM Study in chemistry. Several mathematics curriculum development efforts were also funded as part of the same initiative, such as the Madison Project, School Mathematics Study Group, and University of Illinois Committee on School Mathematics.
These curricula were quite diverse, yet shared the idea that children's learning of arithmetic algorithms would last past the exam only if memorization and practice were paired with teaching for comprehension. More specifically, elementary school arithmetic beyond single digits makes sense only on the basis of understanding place value. This goal was the reason for teaching arithmetic in bases other than ten in the New Math, despite critics' derision: In that unfamiliar context, students couldn't just mindlessly follow an algorithm, but had to think why the place value of the "hundreds" digit in base seven is 49. Keeping track of non-decimal notation also explains the need to distinguish numbers (values) from the numerals that represent them.
Topics introduced in the New Math include set theory, modular arithmetic, algebraic inequalities, bases other than 10, matrices, symbolic logic, Boolean algebra, and abstract algebra.
All of the New Math projects emphasized some form of discovery learning. Students worked in groups to invent theories about problems posed in the textbooks. Materials for teachers described the classroom as "noisy." Part of the job of the teacher was to move from table to table assessing the theory that each group of students had developed and " |
https://en.wikipedia.org/wiki/Multiply%E2%80%93accumulate%20operation | In computing, especially digital signal processing, the multiply–accumulate (MAC) or multiply-add (MAD) operation is a common step that computes the product of two numbers and adds that product to an accumulator. The hardware unit that performs the operation is known as a multiplier–accumulator (MAC unit); the operation itself is also often called a MAC or a MAD operation. The MAC operation modifies an accumulator a:
When done with floating point numbers, it might be performed with two roundings (typical in many DSPs), or with a single rounding. When performed with a single rounding, it is called a fused multiply–add (FMA) or fused multiply–accumulate (FMAC).
Modern computers may contain a dedicated MAC, consisting of a multiplier implemented in combinational logic followed by an adder and an accumulator register that stores the result. The output of the register is fed back to one input of the adder, so that on each clock cycle, the output of the multiplier is added to the register. Combinational multipliers require a large amount of logic, but can compute a product much more quickly than the method of shifting and adding typical of earlier computers. Percy Ludgate was the first to conceive a MAC in his Analytical Machine of 1909, and the first to exploit a MAC for division (using multiplication seeded by reciprocal, via the convergent series ). The first modern processors to be equipped with MAC units were digital signal processors, but the technique is now also common in general-purpose processors.
In floating-point arithmetic
When done with integers, the operation is typically exact (computed modulo some power of two). However, floating-point numbers have only a certain amount of mathematical precision. That is, digital floating-point arithmetic is generally not associative or distributive. (See .)
Therefore, it makes a difference to the result whether the multiply–add is performed with two roundings, or in one operation with a single rounding (a fused mul |
https://en.wikipedia.org/wiki/IEEE%20754 | The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably. Many hardware floating-point units use the IEEE 754 standard.
The standard defines:
arithmetic formats: sets of binary and decimal floating-point data, which consist of finite numbers (including signed zeros and subnormal numbers), infinities, and special "not a number" values (NaNs)
interchange formats: encodings (bit strings) that may be used to exchange floating-point data in an efficient and compact form
rounding rules: properties to be satisfied when rounding numbers during arithmetic and conversions
operations: arithmetic and other operations (such as trigonometric functions) on arithmetic formats
exception handling: indications of exceptional conditions (such as division by zero, overflow, etc.)
IEEE 754-2008, published in August 2008, includes nearly all of the original IEEE 754-1985 standard, plus the IEEE 854-1987 Standard for Radix-Independent Floating-Point Arithmetic. The current version, IEEE 754-2019, was published in July 2019. It is a minor revision of the previous version, incorporating mainly clarifications, defect fixes and new recommended operations.
History
The first standard for floating-point arithmetic, IEEE 754-1985, was published in 1985. It covered only binary floating-point arithmetic.
A new version, IEEE 754-2008, was published in August 2008, following a seven-year revision process, chaired by Dan Zuras and edited by Mike Cowlishaw. It replaced both IEEE 754-1985 (binary floating-point arithmetic) and IEEE 854-1987 Standard for Radix-Independent Floating-Point Arithmetic. The binary formats in the original standard are included in this new standard along with three new basic formats, one b |
https://en.wikipedia.org/wiki/Simutrans | Simutrans is a cross-platform simulation game in which the player strives to run a successful transport system by constructing and managing transportation systems for passengers, mail and goods by land (rail, road, tram, monorail, maglev), air (airplanes) and water (ship) between places. Like OpenTTD, Simutrans is an open-source transportation game based on the Transport Tycoon idea.
Development history
Simutrans was originally written by Hansjörg Malthaner in 1997. Around 2004 he retired from development, and an international community of volunteers took over the development. Simutrans was developed internally as a closed source game until 2007, when the software was relicensed under the Artistic License.
Simutrans has been ported to Microsoft Windows, Linux, BeOS/Haiku, Mac OS X, and AmigaOS 4.x, which make use of several graphics libraries such as GDI (Windows only), SDL (all versions) or Allegro (BeOS only). It is portable to any architecture using GCC and one of the aforementioned libraries. Simutrans has also multilingual support.
The current stable release of Simutrans is version 123.0.1 as of January 30, 2022. There is a popular branch of the code called Simutrans-Extended, which aims to extend the basic game. Simutrans-Extended was formerly called Simutrans-Experimental, but changed its name to Simutrans-Extended on February 13, 2017 to make clear that it is a distinct fork of Simutrans and not a testing branch. Nightly builds for Simutrans and the main PakSets are also released for both standard and Extended versions.
Overview and features
The main goal in Simutrans is to provide an efficient transport system for passengers, mail and goods to be transported to their desired destinations fast and with minimal transfers and at the same time making the company grow avoiding bankruptcy or excess administration.
Simutrans has a number of factory chains that are interconnected with other chains, for example, a coal mine produces coal for a coal power plan |
https://en.wikipedia.org/wiki/Almost%20periodic%20function | In mathematics, an almost periodic function is, loosely speaking, a function of a real number that is periodic to within any desired level of accuracy, given suitably long, well-distributed "almost-periods". The concept was first studied by Harald Bohr and later generalized by Vyacheslav Stepanov, Hermann Weyl and Abram Samoilovitch Besicovitch, amongst others. There is also a notion of almost periodic functions on locally compact abelian groups, first studied by John von Neumann.
Almost periodicity is a property of dynamical systems that appear to retrace their paths through phase space, but not exactly. An example would be a planetary system, with planets in orbits moving with periods that are not commensurable (i.e., with a period vector that is not proportional to a vector of integers). A theorem of Kronecker from diophantine approximation can be used to show that any particular configuration that occurs once, will recur to within any specified accuracy: if we wait long enough we can observe the planets all return to within a second of arc to the positions they once were in.
Motivation
There are several inequivalent definitions of almost periodic functions. The first was given by Harald Bohr. His interest was initially in finite Dirichlet series. In fact by truncating the series for the Riemann zeta function ζ(s) to make it finite, one gets finite sums of terms of the type
with s written as (σ + it) – the sum of its real part σ and imaginary part it. Fixing σ, so restricting attention to a single vertical line in the complex plane, we can see this also as
Taking a finite sum of such terms avoids difficulties of analytic continuation to the region σ < 1. Here the 'frequencies' log n will not all be commensurable (they are as linearly independent over the rational numbers as the integers n are multiplicatively independent – which comes down to their prime factorizations).
With this initial motivation to consider types of trigonometric polynomial with indepe |
https://en.wikipedia.org/wiki/Solomonoff%27s%20theory%20of%20inductive%20inference | Solomonoff's theory of inductive inference is a mathematical theory of induction introduced by Ray Solomonoff, based on probability theory and theoretical computer science. In essence, Solomonoff's induction derives the posterior probability of any computable theory, given a sequence of observed data. This posterior probability is derived from Bayes' rule and some universal prior, that is, a prior that assigns a positive probability to any computable theory.
Solomonoff's induction naturally formalizes Occam's razor by assigning larger prior credences to theories that require a shorter algorithmic description.
Origin
Philosophical
The theory is based in philosophical foundations, and was founded by Ray Solomonoff around 1960. It is a mathematically formalized combination of Occam's razor and the Principle of Multiple Explanations. All computable theories which perfectly describe previous observations are used to calculate the probability of the next observation, with more weight put on the shorter computable theories. Marcus Hutter's universal artificial intelligence builds upon this to calculate the expected value of an action.
Principle
Solomonoff's induction has been argued to be the computational formalization of pure Bayesianism. To understand, recall that Bayesianism derives the posterior probability of a theory given data by applying Bayes rule, which yields , where theories are alternatives to theory . For this equation to make sense, the quantities and must be well-defined for all theories and . In other words, any theory must define a probability distribution over observable data . Solomonoff's induction essentially boils down to demanding that all such probability distributions be computable.
Interestingly, the set of computable probability distributions is a subset of the set of all programs, which is countable. Similarly, the sets of observable data considered by Solomonoff were finite. Without loss of generality, we can thus consider that an |
https://en.wikipedia.org/wiki/EarthStation%205 | Earth Station 5 (ES5) was a peer-to-peer network active between 2003 and 2005, operated by a company of the same name. The user client application also shared this name. Earth Station 5 was notable for its strong, if overstated, emphasis on user anonymity, and for its bold advocacy of piracy and copyright infringement. ES5's highly antagonistic position toward copyright advocacy and enforcement organizations garnered the group significant attention and peaked with an ES5 press release announcing a "declaration of war" against the Motion Picture Association of America. ES5 claimed to operate out of the Jenin in the Palestinian Authority-controlled West Bank, a region where they argued that copyright laws were unenforceable. Investigative journalism cast serious doubts on the company's Palestinian origin as well as many of its other claims. To this day, much about the company and its leadership remains uncertain or unknown.
Peer-to-peer services
Earth Station 5 was based around a peer-to-peer (P2P) file sharing service and a standalone Earth Station 5 file-sharing client. Initial versions of the software could only share or download files by using the ES5 network.
ES5's P2P network and client were announced on June 9, 2003. People associated with ES5 claimed in media reports that the network had more than 16,000,000 participants at its peak, but these numbers were unsupported and viewed very skeptically. The actual number of participants was probably several orders of magnitude smaller.
Largely due to the low availability of files on the small ES5 network, later versions of the ES5 client included the free software/open source giFT daemon which provided ES5 users access to the larger Gnutella and FastTrack networks. While Gnutella and FastTrack offered access to many more files, the functionality that let users access these networks did not take advantage of any of ES5's anonymity features, which decreased the advantages of ES5 over other P2P clients — in particu |
https://en.wikipedia.org/wiki/Exact%20differential | In multivariate calculus, a differential or differential form is said to be exact or perfect (exact differential), as contrasted with an inexact differential, if it is equal to the general differential for some differentiable function in an orthogonal coordinate system (hence is a multivariable function whose variables are independent, as they are always expected to be when treated in multivariable calculus).
An exact differential is sometimes also called a total differential, or a full differential, or, in the study of differential geometry, it is termed an exact form.
The integral of an exact differential over any integral path is path-independent, and this fact is used to identify state functions in thermodynamics.
Overview
Definition
Even if we work in three dimensions here, the definitions of exact differentials for other dimensions are structurally similar to the three dimensional definition. In three dimensions, a form of the type
is called a differential form. This form is called exact on an open domain in space if there exists some differentiable scalar function defined on such that
throughout , where are orthogonal coordinates (e.g., Cartesian, cylindrical, or spherical coordinates). In other words, in some open domain of a space, a differential form is an exact differential if it is equal to the general differential of a differentiable function in an orthogonal coordinate system.
Note: In this mathematical expression, the subscripts outside the parenthesis indicate which variables are being held constant during differentiation. Due to the definition of the partial derivative, these subscripts are not required, but they are explicitly shown here as reminders.
Integral path independence
The exact differential for a differentiable scalar function defined in an open domain is equal to , where is the gradient of , represents the scalar product, and is the general differential displacement vector, if an orthogonal coordinate system is u |
https://en.wikipedia.org/wiki/Analog%20signal%20processing | Analog signal processing is a type of signal processing conducted on continuous analog signals by some analog means (as opposed to the discrete digital signal processing where the signal processing is carried out by a digital process). "Analog" indicates something that is mathematically represented as a set of continuous values. This differs from "digital" which uses a series of discrete quantities to represent signal. Analog values are typically represented as a voltage, electric current, or electric charge around components in the electronic devices. An error or noise affecting such physical quantities will result in a corresponding error in the signals represented by such physical quantities.
Examples of analog signal processing include crossover filters in loudspeakers, "bass", "treble" and "volume" controls on stereos, and "tint" controls on TVs. Common analog processing elements include capacitors, resistors and inductors (as the passive elements) and transistors or opamps (as the active elements).
Tools used in analog signal processing
A system's behavior can be mathematically modeled and is represented in the time domain as h(t) and in the frequency domain as H(s), where s is a complex number in the form of s=a+ib, or s=a+jb in electrical engineering terms (electrical engineers use "j" instead of "i" because current is represented by the variable i). Input signals are usually called x(t) or X(s) and output signals are usually called y(t) or Y(s).
Convolution
Convolution is the basic concept in signal processing that states an input signal can be combined with the system's function to find the output signal. It is the integral of the product of two waveforms after one has reversed and shifted; the symbol for convolution is *.
That is the convolution integral and is used to find the convolution of a signal and a system; typically a = -∞ and b = +∞.
Consider two waveforms f and g. By calculating the convolution, we determine how much a reversed functio |
https://en.wikipedia.org/wiki/Mobile%20data%20terminal | A mobile data terminal (MDT) or mobile digital computer (MDC) is a computerized device used in emergency services, public transport, taxicabs, package delivery, roadside assistance, and logistics, among other fields, to communicate with a central dispatcher. They are also used to display mapping and information relevant to the tasks and actions performed by the vehicle such as CAD drawings, diagrams and safety information.
Mobile data terminals feature a screen on which to view information and a keyboard or keypad for entering information, and may be connected to various peripheral devices. Standard peripherals include two-way radios and taximeters, both of which predate computer-aided dispatching. MDTs may be simple display and keypad units, intended to be connected to a separate black-box or AVL (see below) computer. While MDTs were originally thin clients, most have been replaced with fully functional PC hardware, known as MDCs (Mobile Digital Computers). While the MDC term is more correct, MDT is still widely used. Other common terms include MVC (Motor Vehicle Computer) and names of manufacturers such as iMobile or KDT.
Technology
In the earlier days of computer-aided dispatching (CAD), many MDT's were custom devices, used with specialized point to point radios, particularly in applications such as police dispatching. While applications like taxi and package delivery often still use custom designed terminals, the majority of CAD systems have switched to ruggedized laptops and Wide-Area Wireless IP communications, utilizing the Internet or private IP networks connected to and over it.
For industrial applications such as commercial trucking, GIS, agriculture, mobile asset management, and other industries, custom electronic hardware is still preferred. Custom terminals use I/O interfaces that connect directly to industry-specific equipment. They are usually environmentally hardened packages with power supply protection and robust memory file systems that greatly |
https://en.wikipedia.org/wiki/Cockade | A cockade is a knot of ribbons, or other circular- or oval-shaped symbol of distinctive colours which is usually worn on a hat or cap.
The word cockade derives from the French cocarde, from Old French coquarde, feminine of coquard (vain, arrogant), from coc (cock), of imitative origin. The earliest documented use was in 1709.
Eighteenth century
In the 18th and 19th centuries, coloured cockades were used in Europe to show the allegiance of their wearers to some political faction, or to show their rank or to indicate a servant's livery. Because individual armies might wear a variety of differing regimental uniforms, cockades were used as an effective and economical means of national identification.
A cockade was pinned on the side of a man's tricorne or cocked hat, or on his lapel. Women could also wear it on their hat or in their hair.
In pre-revolutionary France, the cockade of the Bourbon dynasty was all white. In the Kingdom of Great Britain supporters of a Jacobite restoration wore white cockades, while the recently established Hanoverian monarchy used a black cockade. The Hanoverians also accorded the right to all German nobility to wear the black cockade in the United Kingdom.
During the 1780 Gordon Riots in London, the blue cockade became a symbol of anti-government feelings and was worn by most of the rioters.
During the American Revolution, the Continental Army initially wore cockades of various colors as an ad hoc form of rank insignia, as General George Washington wrote:
Before long however, the Continental Army reverted to wearing the black cockade they inherited from the British. Later, when France became an ally of the United States, the Continental Army pinned the white cockade of the French Ancien Régime onto their old black cockade; the French reciprocally pinned the black cockade onto their white cockade, as a mark of the French-American alliance. The black-and-white cockade thus became known as the "Union Cockade".
In the Storming of the B |
https://en.wikipedia.org/wiki/GameLine | GameLine was a dialup game distribution service for the Atari 2600, developed and operated by Control Video Corporation (CVC, now AOL). Subscribers could install the proprietary modem and storage cartridge in their home game console, accessing the GameLine service to download games over a telephone line. GameLine had an exclusive selection of games, and its pioneering business model eventually gave rise to America Online. Despite being ahead of its time, it wasn't very popular, possibly due to its price of $60 for the hardware, $15 for the membership fee, and $1 per game, which you could only keep for a week.
In 1983, cable pioneer William von Meister was looking for a way to use his modem transmission technology, which was previously acquired in ill-fated attempts of sending music to cable companies. Legal issues caused cable providers to step away from the service, leaving Von Meister with a delivery tool and no content.
He then converted his variable speed adaptive modem technology to download games from central servers to individual households. This allowed users to dial a system and, for a fee, download games to their GameLine modules. The game would typically work for 5-10 plays, after which the user would have to connect to GameLine again and pay for another download.
The "Master Module" has 8 KB of RAM and a 1,200 Bps modem, and physically resembles an oversized silver Atari cartridge. It has a phone jack on the side that was used to link the Master Module with the CVC computers. The module is able to transmit with pulse or tone dialing, allowing the unit to be versatile in the field. The games available on the GameLine service were all from third-party gamemakers, the largest of which was Imagic. CVC tried, but failed to obtain licensing agreements from the largest game makers, such as Atari, Activision, Coleco, Mattel, and Parker Brothers.
Membership benefits
When a user registered with the service, they were given a PIN. This PIN was used to log into |
https://en.wikipedia.org/wiki/Reye%20syndrome | Reye syndrome is a rapidly worsening brain disease. Symptoms of Reye syndrome may include vomiting, personality changes, confusion, seizures, and loss of consciousness. While liver toxicity typically occurs in the syndrome, jaundice usually does not. Death occurs in 20–40% of those affected with Reye syndrome, and about a third of those who survive are left with a significant degree of brain damage.
The cause of Reye syndrome is unknown. It usually begins shortly after recovery from a viral infection, such as influenza or chickenpox. About 90% of cases in children are associated with aspirin (salicylate) use. Inborn errors of metabolism are also a risk factor. The syndrome is associated with changes on blood tests such as a high blood ammonia level, low blood sugar level, and prolonged prothrombin time. Often, the liver is enlarged in the syndrome.
Prevention is typically by avoiding the use of aspirin in children. When aspirin was withdrawn for use in children in the US and UK in the 1980s, a decrease of more than 90% in rates of Reye syndrome was seen. Early diagnosis of the syndrome improves outcomes. Treatment is supportive; mannitol may be used to help with the brain swelling.
The first detailed description of Reye syndrome was in 1963 by Australian pathologist Douglas Reye. The syndrome most commonly affects children. It affects fewer than one in a million children a year. The general recommendation to use aspirin in children was withdrawn because of Reye syndrome, with use only recommended in Kawasaki disease.
Signs and symptoms
Reye syndrome progresses through five stages:
Stage I
Vasoconstrictive rash on palms of hands and feet
Persistent, heavy vomiting that is not relieved by not eating
Generalized lethargy
Confusion
Nightmares (possible symptom)
No fever usually present
Headaches
Stage II
Deep lethargy
Delirium
Confusion
Combative behavior
Stupor
Hyperventilation
Fatty liver (found on biopsy)
Hyperactive reflexes
Stage III
Continua |
https://en.wikipedia.org/wiki/Time%20series | In mathematics, a time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average.
A time series is very frequently plotted via a run chart (which is a temporal line chart). Time series are used in statistics, signal processing, pattern recognition, econometrics, mathematical finance, weather forecasting, earthquake prediction, electroencephalography, control engineering, astronomy, communications engineering, and largely in any domain of applied science and engineering which involves temporal measurements.
Time series analysis comprises methods for analyzing time series data in order to extract meaningful statistics and other characteristics of the data. Time series forecasting is the use of a model to predict future values based on previously observed values. While regression analysis is often employed in such a way as to test relationships between one or more different time series, this type of analysis is not usually called "time series analysis", which refers in particular to relationships between different points in time within a single series.
Time series data have a natural temporal ordering. This makes time series analysis distinct from cross-sectional studies, in which there is no natural ordering of the observations (e.g. explaining people's wages by reference to their respective education levels, where the individuals' data could be entered in any order). Time series analysis is also distinct from spatial data analysis where the observations typically relate to geographical locations (e.g. accounting for house prices by the location as well as the intrinsic characteristics of the houses). A stochastic model for a time series will generally reflect the fact t |
https://en.wikipedia.org/wiki/Airy%20function | In the physical sciences, the Airy function (or Airy function of the first kind) is a special function named after the British astronomer George Biddell Airy (1801–1892). The function and the related function , are linearly independent solutions to the differential equation
known as the Airy equation or the Stokes equation.
Because the solution of the linear differential equation
is oscillatory for and exponential for , the Airy functions are oscillatory for and exponential for . In fact, the Airy equation is the simplest second-order linear differential equation with a turning point (a point where the character of the solutions changes from oscillatory to exponential).
Definitions
For real values of , the Airy function of the first kind can be defined by the improper Riemann integral:
which converges by Dirichlet's test. For any real number there is a positive real number such that function is increasing, unbounded and convex with continuous and unbounded derivative on interval The convergence of the integral on this interval can be proven by Dirichlet's test after substitution
satisfies the Airy equation
This equation has two linearly independent solutions.
Up to scalar multiplication, is the solution subject to the condition as .
The standard choice for the other solution is the Airy function of the second kind, denoted Bi(x). It is defined as the solution with the same amplitude of oscillation as as which differs in phase by :
Properties
The values of and and their derivatives at are given by
Here, denotes the Gamma function. It follows that the Wronskian of and is .
When is positive, is positive, convex, and decreasing exponentially to zero, while is positive, convex, and increasing exponentially. When is negative, and oscillate around zero with ever-increasing frequency and ever-decreasing amplitude. This is supported by the asymptotic formulae below for the Airy functions.
The Airy functions are orthogonal in the sense t |
https://en.wikipedia.org/wiki/Dawson%20function | In mathematics, the Dawson function or Dawson integral
(named after H. G. Dawson)
is the one-sided Fourier–Laplace sine transform of the Gaussian function.
Definition
The Dawson function is defined as either:
also denoted as or or alternatively
The Dawson function is the one-sided Fourier–Laplace sine transform of the Gaussian function,
It is closely related to the error function erf, as
where erfi is the imaginary error function, Similarly,
in terms of the real error function, erf.
In terms of either erfi or the Faddeeva function the Dawson function can be extended to the entire complex plane:
which simplifies to
for real
For near zero, For large, More specifically, near the origin it has the series expansion
while for large it has the asymptotic expansion
More precisely
where is the double factorial.
satisfies the differential equation
with the initial condition Consequently, it has extrema for
resulting in x = ±0.92413887... (), F(x) = ±0.54104422... ().
Inflection points follow for
resulting in x = ±1.50197526... (), F(x) = ±0.42768661... (). (Apart from the trivial inflection point at )
Relation to Hilbert transform of Gaussian
The Hilbert transform of the Gaussian is defined as
P.V. denotes the Cauchy principal value, and we restrict ourselves to real can be related to the Dawson function as follows. Inside a principal value integral, we can treat as a generalized function or distribution, and use the Fourier representation
With we use the exponential representation of and complete the square with respect to to find
We can shift the integral over to the real axis, and it gives Thus
We complete the square with respect to and obtain
We change variables to
The integral can be performed as a contour integral around a rectangle in the complex plane. Taking the imaginary part of the result gives
where is the Dawson function as defined above.
The Hilbert transform of is also related to the Dawson function. We |
https://en.wikipedia.org/wiki/Debye%20function | In mathematics, the family of Debye functions is defined by
The functions are named in honor of Peter Debye, who came across this function (with n = 3) in 1912 when he analytically computed the heat capacity of what is now called the Debye model.
Mathematical properties
Relation to other functions
The Debye functions are closely related to the polylogarithm.
Series expansion
They have the series expansion
where is the n-th Bernoulli number.
Limiting values
If is the gamma function and is the Riemann zeta function, then, for ,
Derivative
The derivative obeys the relation
where is the Bernoulli function.
Applications in solid-state physics
The Debye model
The Debye model has a density of vibrational states
for
with the Debye frequency ωD.
Internal energy and heat capacity
Inserting g into the internal energy
with the Bose–Einstein distribution
.
one obtains
.
The heat capacity is the derivative thereof.
Mean squared displacement
The intensity of X-ray diffraction or neutron diffraction at wavenumber q is given by
the Debye-Waller factor or the Lamb-Mössbauer factor.
For isotropic systems it takes the form
).
In this expression, the mean squared displacement refers to just once Cartesian component
ux of the vector u that describes the displacement of atoms from their equilibrium positions.
Assuming harmonicity and developing into normal modes,
one obtains
Inserting the density of states from the Debye model, one obtains
.
From the above power series expansion of follows that the mean square displacement at high temperatures is linear in temperature
.
The absence of indicates that this is a classical result. Because goes to zero for it follows that for
(zero-point motion).
References
Further reading
"Debye function" entry in MathWorld, defines the Debye functions without prefactor n/xn
Implementations
Fortran 77 code
Fortran 90 version
C version of the GNU Scientific Library
Special functions
Peter Debye |
https://en.wikipedia.org/wiki/Legendre%20form | In mathematics, the Legendre forms of elliptic integrals are a canonical set of three elliptic integrals to which all others may be reduced. Legendre chose the name elliptic integrals because the second kind gives the arc length of an ellipse of unit semi-major axis and eccentricity (the ellipse being defined parametrically by , ).
In modern times the Legendre forms have largely been supplanted by an alternative canonical set, the Carlson symmetric forms. A more detailed treatment of the Legendre forms is given in the main article on elliptic integrals.
Definition
The incomplete elliptic integral of the first kind is defined as,
the second kind as
and the third kind as
The argument n of the third kind of integral is known as the characteristic, which in different notational conventions can appear as either the first, second or third argument of Π and furthermore is sometimes defined with the opposite sign. The argument order shown above is that of Gradshteyn and Ryzhik as well as Numerical Recipes. The choice of sign is that of Abramowitz and Stegun as well as Gradshteyn and Ryzhik, but corresponds to the of Numerical Recipes.
The respective complete elliptic integrals are obtained by setting the amplitude, , the upper limit of the integrals, to .
The Legendre form of an elliptic curve is given by
Numerical evaluation
The classic method of evaluation is by means of Landen's transformations. Descending Landen transformation decreases the modulus towards zero, while increasing the amplitude . Conversely, ascending transformation increases the modulus towards unity, while decreasing the amplitude. In either limit of approaching zero or one, the integral is readily evaluated.
Most modern authors recommend evaluation in terms of the Carlson symmetric forms, for which there exist efficient, robust and relatively simple algorithms. This approach has been adopted by Boost C++ Libraries, GNU Scientific Library and Numerical Recipes.
References
See also
|
https://en.wikipedia.org/wiki/Complete%20Fermi%E2%80%93Dirac%20integral | In mathematics, the complete Fermi–Dirac integral, named after Enrico Fermi and Paul Dirac, for an index j is defined by
This equals
where is the polylogarithm.
Its derivative is
and this derivative relationship is used to define the Fermi-Dirac integral for nonpositive indices j. Differing notation for appears in the literature, for instance some authors omit the factor . The definition used here matches that in the NIST DLMF.
Special values
The closed form of the function exists for j = 0:
For x = 0, the result reduces to
where is the Dirichlet eta function.
See also
Incomplete Fermi–Dirac integral
Gamma function
Polylogarithm
References
External links
GNU Scientific Library - Reference Manual
Fermi-Dirac integral calculator for iPhone/iPad
Notes on Fermi-Dirac Integrals
Section in NIST Digital Library of Mathematical Functions
npplus: Python package that provides (among others) Fermi-Dirac integrals and inverses for several common orders.
Wolfram's MathWorld: Definition given by Wolfram's MathWorld.
Special functions |
https://en.wikipedia.org/wiki/Incomplete%20Fermi%E2%80%93Dirac%20integral | In mathematics, the incomplete Fermi–Dirac integral for an index j is given by
This is an alternate definition of the incomplete polylogarithm.
See also
Complete Fermi–Dirac integral
External links
GNU Scientific Library - Reference Manual
Special functions |
https://en.wikipedia.org/wiki/Virulence | Virulence is a pathogen's or microorganism's ability to cause damage to a host.
In most, especially in animal systems, virulence refers to the degree of damage caused by a microbe to its host. The pathogenicity of an organism—its ability to cause disease—is determined by its virulence factors. In the specific context of gene for gene systems, often in plants, virulence refers to a pathogen's ability to infect a resistant host.
The noun virulence derives from the adjective virulent, meaning disease severity. The word virulent derives from the Latin word virulentus, meaning "a poisoned wound" or "full of poison."
From an ecological standpoint, virulence is the loss of fitness induced by a parasite upon its host. Virulence can be understood in terms of proximate causes—those specific traits of the pathogen that help make the host ill—and ultimate causes—the evolutionary pressures that lead to virulent traits occurring in a pathogen strain.
Virulent bacteria
The ability of bacteria to cause disease is described in terms of the number of infecting bacteria, the route of entry into the body, the effects of host defense mechanisms, and intrinsic characteristics of the bacteria called virulence factors. Many virulence factors are so-called effector proteins that are injected into the host cells by specialized secretion apparati, such as the type three secretion system. Host-mediated pathogenesis is often important because the host can respond aggressively to infection with the result that host defense mechanisms do damage to host tissues while the infection is being countered (e.g., cytokine storm).
The virulence factors of bacteria are typically proteins or other molecules that are synthesized by enzymes. These proteins are coded for by genes in chromosomal DNA, bacteriophage DNA or plasmids. Certain bacteria employ mobile genetic elements and horizontal gene transfer. Therefore, strategies to combat certain bacterial infections by targeting these specific virulence f |
https://en.wikipedia.org/wiki/Transport%20function | In mathematics and the field of transportation theory, the transport functions J(n,x) are defined by
Note that
See also
Incomplete gamma function
Special functions
Transportation theory |
https://en.wikipedia.org/wiki/Synchrotron%20function | In mathematics the synchrotron functions are defined as follows (for x ≥ 0):
First synchrotron function
Second synchrotron function
where Kj is the modified Bessel function of the second kind.
Use in astrophysics
In astrophysics, x is usually a ratio of frequencies, that is, the frequency over a critical frequency (critical frequency is the frequency at which most synchrotron radiation is radiated). This is needed when calculating the spectra for different types of synchrotron emission. It takes a spectrum of electrons (or any charged particle) generated by a separate process (such as a power law distribution of electrons and positrons from a constant injection spectrum) and converts this to the spectrum of photons generated by the input electrons/positrons.
References
Further reading
Special functions |
https://en.wikipedia.org/wiki/Divisor%20function | In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of divisors of an integer (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important congruences and identities; these are treated separately in the article Ramanujan's sum.
A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function.
Definition
The sum of positive divisors function σz(n), for a real or complex number z, is defined as the sum of the zth powers of the positive divisors of n. It can be expressed in sigma notation as
where is shorthand for "d divides n".
The notations d(n), ν(n) and τ(n) (for the German Teiler = divisors) are also used to denote σ0(n), or the number-of-divisors function (). When z is 1, the function is called the sigma function or sum-of-divisors function, and the subscript is often omitted, so σ(n) is the same as σ1(n) ().
The aliquot sum s(n) of n is the sum of the proper divisors (that is, the divisors excluding n itself, ), and equals σ1(n) − n; the aliquot sequence of n is formed by repeatedly applying the aliquot sum function.
Example
For example, σ0(12) is the number of the divisors of 12:
while σ1(12) is the sum of all the divisors:
and the aliquot sum s(12) of proper divisors is:
σ-1(n) is sometimes called the abundancy index of n, and we have:
Table of values
The cases x = 2 to 5 are listed in through , x = 6 to 24 are listed in through .
Properties
Formulas at prime powers
For a prime number p,
because by definition, the factors of a prime number are 1 and itself. Also, where pn# denotes the primorial,
since n prime factors allow a sequence of binary s |
https://en.wikipedia.org/wiki/Video%20game%20modding | Video game modding (short for "modification") is the process of alteration by players or fans of one or more aspects of a video game, such as how it looks or behaves, and is a sub-discipline of general modding. Mods may range from small changes and tweaks to complete overhauls, and can extend the replay value and interest of the game.
Modding a game can also be understood as the act of seeking and installing mods to the player's game, but the act of tweaking pre-existing settings and preferences is not truly modding.
Mods have arguably become an increasingly important factor in the commercial success of some games, as they add depth to the original work, and can be both fun for players playing the mods and as a means of self-expression for mod developers.
People can become fans of specific mods, in addition to fans of the game they are for, such as requesting features and alterations for these mods. In cases where mods are very popular, players might have to clarify that they are referring to the unmodified game when talking about playing a game. The term vanilla is often used to make this distinction. "Vanilla Minecraft", for example, refers to the original, unmodified game.
As early as the 1980s, video game mods have also been used for the sole purpose of creating art, as opposed to an actual game. This can include recording in-game actions as a film, as well as attempting to reproduce real-life areas inside a game with no regard for game play value. This has led to the rise of artistic video game modification, as well as machinima and the demoscene.
Popular games can have tens of thousands of mods created for them. Popular websites dedicated to modding include Nexus Mods, Mod DB, and Steam Workshop.
Development
Many mods are not publicly released to the gaming community by their creators. Some are very limited and just include some gameplay changes or even a different loading screen, while others are total conversions and can modify content and gameplay e |
https://en.wikipedia.org/wiki/91%20%28number%29 | 91 (ninety-one) is the natural number following 90 and preceding 92.
In mathematics
91 is:
the twenty-seventh distinct semiprime and the second of the form (7.q), where q is a higher prime.
the aliquot sum of 91 is 21 33; itself a semiprime, within an aliquot sequence of two composite numbers (91,21,11, 1,0) to the prime in the 11-aliquot tree. 91 is the fourth composite number in the 11-aliquot tree.(91,51,21,18).
a triangular number.
a hexagonal number, one of the few such numbers to also be a centered hexagonal number.
a centered nonagonal number.
a centered cube number.
a square pyramidal number, being the sum of the squares of the first six integers.
the smallest positive integer expressible as a sum of two cubes in two different ways if negative roots are allowed (alternatively the sum of two cubes and the difference of two cubes): . (See 1729 for more details). This implies that 91 is the second cabtaxi number.
the smallest positive integer expressible as a sum of six distinct squares: .
The only other ways to write 91 as a sum of distinct squares are: and
.
the smallest pseudoprime satisfying the congruence .
a repdigit in base 9 (1119).
palindromic in bases 3 (101013), 9 (1119), and 12 (7712).
a Riordan number.
The decimal equivalent of the fraction can be obtained by using powers of 9.
In science
91 is the atomic number of protactinium, an actinide.
McCarthy 91 function, a recursive function in discrete mathematics
Messier object M91, a magnitude 11.5 spiral galaxy in the constellation Coma Berenices
The New General Catalogue object NGC 91, a single star in the constellation Andromeda
In other fields
Ninety-one is also:
The code for international direct dial phone calls to India
In cents of a U.S. dollar, the amount of money one has if one has one each of the coins of denominations less than a dollar (penny, nickel, dime, quarter and half dollar)
The ISBN Group Identifier for books published in Sweden.
Psalm 91 is known as |
https://en.wikipedia.org/wiki/92%20%28number%29 | 92 (ninety-two) is the natural number following 91 and preceding 93.
In mathematics
92 is a composite number; a square-prime, of the general form (p2, q) where q is a higher prime. It is the tenth of this form and the eighth of the form (22.q).
92 is the eighth pentagonal number, and an Erdős–Woods number, since it is possible to find sequences of 92 consecutive integers such that each inner member shares a factor with either the first or the last member.
With an aliquot sum of 76; itself a square-prime, within an aliquot sequence of five composite numbers (92,76,64,63,1,0) to the prime in the 63-aliquot tree.
For , there are 92 solutions in the n-Queens Problem.
There are 92 "atomic elements" in John Conway's look-and-say sequence, corresponding to the 92 non-transuranic elements in the chemist's periodic table.
92 is palindromic in bases 6 (2326), 7 (1617), 22 (4422), and 45 (2245).
The most faces or vertices an Archimedean or Catalan solid can have is 92: the snub dodecahedron has 92 faces while its dual polyhedron, the pentagonal hexecontahedron, has 92 vertices.
As a simple polyhedron, the final stellation of the icosahedron has 92 vertices.
There are 92 Johnson solids.
In science
The atomic number of uranium, an actinide.
Messier object M92, a magnitude 7.5 globular cluster in the constellation Hercules
The New General Catalogue object NGC 92, a magnitude 13.1 peculiar spiral galaxy in the constellation Phoenix, and a member of Robert's Quartet
In other fields
Ninety-two is also:
The code for international direct dial phone calls to Pakistan.
The numeric code for the Hauts-de-Seine department of France. The number is reflected in the department's postal code, plus the names of at least three local sports clubs, specifically Racing 92 in rugby union and Metropolitans 92 and Nanterre 92 in basketball.
In the title of the book Ninety-two in the Shade, by Thomas McGuane.
The 92nd Tiger book by Michael Gilbert.
The House on 92nd Street, a 19 |
https://en.wikipedia.org/wiki/93%20%28number%29 | 93 (ninety-three) is the natural number following 92 and preceding 94.
In mathematics
93 is:
the 28th distinct semiprime and the 9th of the form (3.q) where q is a higher prime.
the first number in the 3rd triplet of consecutive semiprimes, 93, 94, 95.
with an aliquot sum of 35; itself a semiprime, within an aliquot sequence (93,35,13,1,0) of three numbers to the Prime 13 in the 13-Aliquot tree.
a Blum integer, since its two prime factors, 3 and 31 are both Gaussian primes.
a repdigit in base 5 (3335), and 30 (3330).
palindromic in bases 2, 5, and 30.
a lucky number.
a cake number.
an idoneal number.
There are 93 different cyclic Gilbreath permutations on 11 elements, and therefore there are 93 different real periodic points of order 11 on the Mandelbrot set.
In other fields
Ninety-three is:
The atomic number of neptunium, an actinide.
The code for international direct dial phone calls to Afghanistan.
One of two ISBN Group Identifiers for books published in India.
The number of the French department Seine-Saint-Denis, a Paris suburb with high proportions of immigrants and low-income people, and as such used by many French rappers and those emulating their speech.
In classical Persian finger counting, the number 93 is represented by a closed fist. Because of this, classical Arab and Persian poets around 1 CE referred to someone's lack of generosity by saying that the person's hand made "ninety-three".
See also
AD 93, a year in the Julian calendar
List of highways numbered 93
Ninety-Three (Quatrevingt-treize), a novel concerning the French Revolution by Victor Hugo
93 (Thelema), a greeting among Thelemites based on the numerological (gematric) value of Thelema (Will) and Agape (Love) in Greek letters.
Babia 93, an album from a Pakistani pop singer Sajjad Ali
London's 93 Feet East music venue
Current 93, a musical project of David Tibet
Los Angeles 93 KHJ radio
United Airlines Flight 93, one of the airplanes hijacked on September 11, 2001.
93 |
https://en.wikipedia.org/wiki/94%20%28number%29 | 94 (ninety-four) is the natural number following 93 and preceding 95.
In mathematics
94 is:
the twenty-ninth distinct semiprime and the fourteenth of the form (2.q).
the ninth composite number in the 43-aliquot tree. The aliquot sum of 94 is 50 within the aliquot sequence; (94,50,43,1,0).
the second number in the third triplet of three consecutive distinct semiprimes, 93, 94 and 95
a 17-gonal number and a nontotient.
an Erdős–Woods number, since it is possible to find sequences of 94 consecutive integers such that each inner member shares a factor with either the first or the last member.
a Smith number in decimal.
In computing
The ASCII character set (and, more generally, ISO 646) contains exactly 94 graphic non-whitespace characters, which form a contiguous range of code points. These codes (0x21–0x7E, as corresponding high bit set bytes 0xA1–0xFE) also used in various multi-byte encoding schemes for languages of East Asia, such as ISO 2022, EUC and GB 2312. For this reason, code pages of 942 and even 943 code points were common in East Asia in 1980s–1990s.
In astronomy
Messier 94, a spiral galaxy in the constellation Canes Venatici
The New General Catalogue object NGC 94, a galaxy in the constellation Andromeda
In other fields
Ninety-four is:
The atomic number of plutonium, an actinide.
The designation of STS-94 Space Shuttle Columbia launched July 1, 1997
The code for international direct dial phone calls to Sri Lanka.
Part of the model number of
AN-94, a Russian assault rifle.
M-94, a piece of cryptographic equipment used by the United States army in use from 1922 to 1943.
The number of Haydn's Surprise Symphony (Symphony No. 94).
Used as a nonsense number by the British satire magazine Private Eye. Most commonly used in spoof articles end halfway through a sentence with "(continued p. 94)". The magazine never extends to 94 pages: this was originally a reference to the enormous size of some Sunday newspapers.
Each February, Respiratory Health Associatio |
https://en.wikipedia.org/wiki/95%20%28number%29 | 95 (ninety-five) is the natural number following 94 and preceding 96.
In mathematics
95 is:
the 30th distinct semiprime and the fifth of the form (5.q).
the third composite number in the 6-aliquot tree. The aliquot sum of 95 is 25, within the aliquot sequence (95,25,6).
the last member in the third triplet of distinct semiprimes 93, 94, and 95.
an 11-gonal number.
a Thabit number.
the lowest integer for which the Mertens function is greater than 1. (The lowest integer producing a Merten's value greater than that of 95 is 218).
In astronomy
The Messier object M95, a magnitude 11.0 spiral galaxy in the constellation Leo
The New General Catalogue object NGC 95, a magnitude 12.6 peculiar spiral galaxy in the constellation Pisces
In sports
NBA record for Most Assists in a 7-game playoff series, 95, Los Angeles Lakers Magic Johnson, 1984
NBA record for Most Free Throw Attempts in a 6-game playoff series, 95, Los Angeles Lakers Jerry West, 1965
In other uses
Ninety-five is also:
The atomic number of americium, an actinide.
The number of theses in Martin Luther's 95 Theses.
"95 Poems" by E.E. Cummings (1958)
The book The Prince, Utopia, Ninety-Five Thesis by Sir Thomas More
The designation of American
Interstate 95, a freeway that runs from Florida to Maine.
U.S. Highway 95, a freeway that runs through the western part of the United States.
Bay Ridge–95th Street subway station, Brooklyn, on the R Train
York Mills 95E, a bus route number from the Toronto Transit Commission
95th Street is a major east–west thoroughfare on Chicago's South Side, designated as 9500 South in the address system
Part of the name of:
Windows 95, a version of the Microsoft Windows graphical interface.
CommSuite 95 was a communications software suite of products launched by Delrina in 1995, created for use with Windows 95
Dogme 95, a movement in filmmaking developed in 1995
The model number of the automobile Saab 95 introduced in 1959, and Saab 9-5 introduced 1997
The racing number |
https://en.wikipedia.org/wiki/96%20%28number%29 | 96 (ninety-six) is the natural number following 95 and preceding 97. It is a number that appears the same when turned upside down.
In mathematics
96 is:
an octagonal number.
a refactorable number.
an untouchable number.
a semiperfect number since it is a multiple of 6.
an abundant number since the sum of its proper divisors is greater than 96.
the fourth Granville number and the second non-perfect Granville number. The next Granville number is 126, the previous being 24.
the sum of Euler's totient function φ(x) over the first seventeen integers.
strobogrammatic in bases 10 (9610), 11 (8811) and 95 (1195).
palindromic in bases 11 (8811), 15 (6615), 23 (4423), 31 (3331), 47 (2247) and 95 (1195).
an Erdős–Woods number, since it is possible to find sequences of 96 consecutive integers such that each inner member shares a factor with either the first or the last member.
divisible by the number of prime numbers (24) below 96.
the smallest natural number that can be expressed as the difference of two nonzero squares in more than three ways: , , or .
Skilling's figure, a degenerate uniform polyhedron, has a Euler characteristic
Every integer greater than 96 may be represented as a sum of distinct super-prime numbers.
In geography
Ninety Six, South Carolina
Ninety Six District, a historical judicial and military district of colonial America which extended through North and South Carolina
Ninety Six National Historic Site, in Ninety Six, South Carolina, derives its name from the original settlement's distance in miles from a Cherokee village in the Blue Ridge Mountains
In music
96Neko is a female Japanese singer
The song "96 Tears" by garage rock band Question Mark and the Mysterians
"96", a song by Uverworld, a Japanese band.
"96 Quite Bitter Beings", a song recorded by rock band CKY
"96 Degrees In The Shade", a song on an album with the same title (official song title "1865", recorded by Jamaican reggae band Third World.
In science
The atomic n |
https://en.wikipedia.org/wiki/97%20%28number%29 | 97 (ninety-seven) is the natural number following 96 and preceding 98. It is a prime number and the only prime in the nineties.
In mathematics
97 is:
the 25th prime number (the largest two-digit prime number in base 10), following 89 and preceding 101.
a Proth prime and a Pierpont prime as it is 3 × 25 + 1.
the eleventh member of the Mian–Chowla sequence.
a self number in base 10, since there is no integer that added to its own digits, adds up to 97.
the smallest odd prime that is not a cluster prime.
the highest two-digit number where the sum of its digits is a square.
the number of primes <= 29.
The numbers 97, 907, 9007, 90007 and 900007 are all primes, and they are all happy primes. However, 9000007 (read as nine million seven) is composite and has the factorisation 277 × 32491.
an emirp
an isolated prime, since 95 and 99 aren't prime.
In science
Ninety-seven is:
The atomic number of berkelium, an actinide.
In astronomy
Messier object M97, a magnitude 12.0 planetary nebula in the constellation Ursa Major, also known as the Owl Nebula
The New General Catalogue object NGC 97, an elliptical galaxy in the constellation Andromeda
In other fields
Ninety-seven is:
The 97th United States Congress met during the Ronald Reagan administration, from January 1981 to January 1983
The 10-97 police code means "arrived on the scene"
STS-97 Space Shuttle Endeavour mission launched November 30, 2000
The 97th Infantry Division was a unit of the United States Army in World War I and World War II
Madden NFL 97 was the first John Madden NFL American football game to be created in the 32-bit gaming era
Radio stations broadcasting on frequencies near 97, such as Hot 97, New York City and 97X, Tampa, Florida
The decimal unicode number representing the Latin lowercase "a"
In music
A song "Baby Boy / Saturday Night '97" by Whigfield
The number of the Southern Railway train in the Wreck of the Old 97, a ballad recorded by numerous artists, including Flatt and Scru |
https://en.wikipedia.org/wiki/98%20%28number%29 | 98 (ninety-eight) is the natural number following 97 and preceding 99.
In mathematics
98 is:
Wedderburn–Etherington number
nontotient
number of non-isomorphic set-systems of weight 7
In astronomy
98 Ianthe, a main-belt asteroid
Messier 98, a magnitude 11.0 spiral galaxy in the constellation Coma Berenices.
The New General Catalogue object NGC 98, a magnitude 12.7 spiral galaxy in the constellation Phoenix.
In computing
Windows 98, a Microsoft operating system for personal computers
Microsoft Flight Simulator 98, a flight simulator program
In space travel
MPTA-098, the Main Propulsion Test Article built as a systems testbed for the Space Shuttle program
Pathfinder (OV-098), a Space Shuttle simulator built by NASA in 1977
STS-98, Space Shuttle Atlantis mission launched February 7, 2001
In other fields
Ninety-eight is:
The atomic number of californium, an actinide
+98, the code for international direct dial phone calls to Iran
98 Degrees, an American adult contemporary boy band
98.6 degrees Fahrenheit is normal body temperature
"98.6", a 1967 hit song by Keith
10-98 code in police code means "Assignment Completed"
The number of sons of Ater in the census of men of Israel upon return from exile (Bible, Ezra 2:16)
Beach 98th Street, often referred as Beach 98th Street–Playland, a station on the New York City Subway's IND Rockaway Line
Expo '98, a World's Fair held in Lisbon, Portugal, from May to September 1998
Oldsmobile 98, a full-size automobile and the highest-end of the Oldsmobile division of General Motors
Power 98 (radio station), an English radio station of So Drama! Entertainment in Singapore
Power 98 (film), a 1996 film starring Eric Roberts about a Los Angeles talk radio station.
"Power 98", official nickname of radio station WPEG, in Charlotte, North Carolina
Saab 98, a project by automaker Saab for a combi coupé based on the Saab 95
Spirit of '98, a ship owned by Cruise West built to accommodate 98 passengers
The Trail of '98, a 1928 western |
https://en.wikipedia.org/wiki/101%20%28number%29 | 101 (one hundred [and] one) is the natural number following 100 and preceding 102.
It is variously pronounced "one hundred and one" / "a hundred and one", "one hundred one" / "a hundred one", and "one oh one". As an ordinal number, 101st (one hundred [and] first), rather than 101th, is the correct form.
In mathematics
101 is:
the 26th prime number and the smallest above 100.
a palindromic number in decimal, and so a palindromic prime.
a Chen prime since 103 is also prime, with which it makes a twin prime pair.
a sexy prime since 107 and 113 are also prime, with which it makes a sexy prime triplet.
a unique prime because the period length of its reciprocal is unique among primes.
an Eisenstein prime with no imaginary part and real part of the form .
the fifth alternating factorial.
a centered decagonal number.
the only existing prime with alternating 1s and 0s in decimal and the largest known prime of the form .
the number of compositions of 12 into distinct parts.
the smallest number that can be expressed as the sum of three distinct nonzero squares in more than two ways: , or (see image).
Given 101, the Mertens function returns 0. It is the second prime to have this property after 2.
For a 3-digit number in decimal, this number has a relatively simple divisibility test. The candidate number is split into groups of four, starting with the rightmost four, and added up to produce a 4-digit number. If this 4-digit number is of the form (where and are integers from 0 to 9), such as 3232 or 9797, or of the form , such as 707 and 808, then the number is divisible by 101.
On the seven-segment display of a calculator, 101 is both a strobogrammatic prime and a dihedral prime.
In science
In mineralogy, a Miller index of 101 is a crystal face that crosses the horizontal axis (a) and 3D vertical axis (c) but does not cross the 2D vertical axis (b).
In physics and chemistry, it is the atomic number of mendelevium, an actinide.
In astronomy it is the Messier des |
https://en.wikipedia.org/wiki/102%20%28number%29 | 102 (one hundred [and] two) is the natural number following 101 and preceding 103.
In mathematics
102 is an abundant number and a semiperfect number. It is a sphenic number.
The sum of Euler's totient function φ(x) over the first eighteen integers is 102.
102 is the first three-digit base 10 polydivisible number, since 1 is divisible by 1, 10 is divisible by 2 and 102 is divisible by 3. This also shows that 102 is a Harshad number. 102 is the first 3-digit number divisible by the numbers 3, 6, 17, 34 and 51.
10264 + 1 is a prime number
There are 102 vertices in the Biggs-Smith graph.
In science
The atomic number of nobelium, an actinide.
In other fields
102 is also:
The emergency telephone number for police in Azerbaijan, Ukraine and Belarus
The emergency telephone number for fire in Israel
The emergency telephone number for ambulance in parts of India
The emergency telephone number for ambulance in Maldives
See also
List of highways numbered 102
One Hundred and Two, a song by The Judds
References
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 133
Integers |
https://en.wikipedia.org/wiki/103%20%28number%29 | 103 (one hundred [and] three) is the natural number following 102 and preceding 104.
In mathematics
103 is a prime number, the largest prime factor of . The previous prime is 101, making them both twin primes. It is the fifth irregular prime, because it divides the numerator of the Bernoulli number
The equation makes 103 part of a "Fermat near miss".
There are 103 different connected series-parallel partial orders on exactly six unlabeled elements.
103 is conjectured to be the smallest number for which repeatedly reversing the digits of its ternary representation, and adding the number to its reversal, does not eventually reach a ternary palindrome.
See also
103 (disambiguation)
References
Integers |
https://en.wikipedia.org/wiki/104%20%28number%29 | 104 (one hundred [and] four) is the natural number following 103 and preceding 105.
In mathematics
104 forms the fifth Ruth-Aaron pair with 105, since the distinct prime factors of 104 (2 and 13) and 105 (3, 5, and 7) both add up to 15. Also, the sum of the divisors of 104 aside from unitary divisors, is 105. With eight total divisors where 8 is the fourth largest, 104 is the seventeenth refactorable number. 104 is also the twenty-fifth primitive semiperfect number.
The sum of all its divisors is σ(104) = 210, which is the sum of the first twenty nonzero integers, as well as the product of the first four prime numbers (2 × 3 × 5 × 7).
Its Euler totient, or the number of integers relatively prime with 104, is 48. This value is also equal to the totient of its sum of divisors, φ(104) = φ(σ(104)).
The smallest known 4-regular matchstick graph has 104 edges and 52 vertices, where four unit line segments intersect at every vertex.
A row of four adjacent congruent rectangles can be divided into a maximum of 104 regions, when extending diagonals of all possible rectangles.
Regarding the second largest sporadic group , its McKay–Thompson series representative of a principal modular function is , with constant term :
The Tits group , which is the only finite simple group to classify as either a non-strict group of Lie type or sporadic group, holds a minimal faithful complex representation in 104 dimensions. This is twice the dimensional representation of exceptional Lie algebra in 52 dimensions, whose associated lattice structure forms the ring of Hurwitz quaternions that is represented by the vertices of the 24-cell — with this regular 4-polytope one of 104 total four-dimensional uniform polychora, without taking into account the infinite families of uniform antiprismatic prisms and duoprisms.
In other fields
104 is also:
The atomic number of rutherfordium.
The number of Corinthian columns in the Temple of Olympian Zeus, the largest temple ever built in Greec |
https://en.wikipedia.org/wiki/105%20%28number%29 | 105 (one hundred [and] five) is the natural number following 104 and preceding 106.
In mathematics
105 is a triangular number, a dodecagonal number, and the first Zeisel number. It is the first odd sphenic number and is the product of three consecutive prime numbers. 105 is the double factorial of 7. It is also the sum of the first five square pyramidal numbers.
105 comes in the middle of the prime quadruplet (101, 103, 107, 109). The only other such numbers less than a thousand are 9, 15, 195, and 825.
105 is also the middle of the only prime sextuplet (97, 101, 103, 107, 109, 113) between the ones occurring at 7-23 and at 16057–16073. As the product of the first three odd primes () and less than the square of the next prime (11) by > 8, for , n ± 2, ± 4, and ± 8 must be prime, and n ± 6, ± 10, ± 12, and ± 14 must be composite (prime gap).
105 is also a pseudoprime to the prime bases 13, 29, 41, 43, 71, 83, and 97. The distinct prime factors of 105 add up to 15, and so do those of 104; hence, the two numbers form a Ruth-Aaron pair under the first definition.
105 is also a number n for which is prime, for . (This even works up to , ignoring the negative sign.)
105 is the smallest integer such that the factorization of over Q includes non-zero coefficients other than . In other words, the 105th cyclotomic polynomial, Φ105, is the first with coefficients other than .
105 is the number of parallelogram polyominoes with 7 cells.
In science
The atomic number of dubnium.
In other fields
105 is also:
A Shimano Road groupset since 1984
See also
List of highways numbered 105
References
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers London: Penguin Group. (1987): 134
Integers |
https://en.wikipedia.org/wiki/106%20%28number%29 | 106 (one hundred [and] six) is the natural number following 105 and preceding 107.
In mathematics
106 is a centered pentagonal number, a centered heptagonal number, and a regular 19-gonal number.
There are 106 mathematical trees with ten vertices.
See also
106 (disambiguation)
References
Integers |
https://en.wikipedia.org/wiki/107%20%28number%29 | 107 (one hundred [and] seven) is the natural number following 106 and preceding 108.
In mathematics
107 is the 28th prime number. The next prime is 109, with which it comprises a twin prime, making 107 a Chen prime.
Plugged into the expression , 107 yields 162259276829213363391578010288127, a Mersenne prime. 107 is itself a safe prime.
It is the fourth Busy beaver number, the maximum number of steps that any Turing machine with 2 symbols and 4 states can make before eventually halting.
It is the number of triangle-free graphs on 7 vertices.
It is the ninth emirp, because reversing its digits gives another prime number (701)
In other fields
As "one hundred and seven", it is the smallest positive integer requiring six syllables in English (without the "and" it only has five syllables and seventy-seven is a smaller 5-syllable number).
107 is also:
The atomic number of bohrium.
The emergency telephone number in Argentina and Cape Town.
The telephone of the police in Hungary.
A common designation for the fair use exception in copyright law (from 17 U.S.C. 107)
Peugeot 107 model of car
In sports
The 107% rule, a Formula One Sporting Regulation in operation from 1996 to 2002 and 2011 onward.
The number 107 is also associated with the Timbers Army supporters group of the Portland Timbers soccer team, in reference to the stadium seating section where the group originally congregated.
See also
List of highways numbered 107
References
Integers |
https://en.wikipedia.org/wiki/108%20%28number%29 | 108 (one hundred [and] eight) is the natural number following 107 and preceding 109.
In mathematics
108 is:
an abundant number.
a semiperfect number.
a tetranacci number.
the hyperfactorial of 3 since it is of the form .
divisible by the value of its φ function, which is 36.
divisible by the total number of its divisors (12), hence it is a refactorable number.
the angle in degrees of the interior angles of a regular pentagon in Euclidean space.
palindromic in bases 11 (9911), 17 (6617), 26 (4426), 35 (3335) and 53 (2253)
a Harshad number in bases 2, 3, 4, 6, 7, 9, 10, 11, 12, 13 and 16
a self number.
an Achilles number because it is a powerful number but not a perfect power.
nine dozen
There are 108 free polyominoes of order 7.
The equation results in the golden ratio.
This could be restated as saying that the "chord" of 108 degrees is , the golden ratio.
Religion and the arts
The number 108 is considered sacred by the Dharmic Religions, such as Hinduism, Buddhism, and Jainism.
Hinduism
In Hindu tradition, the Mukhya Shivaganas (attendants of Shiva) are 108 in number and hence Shaiva religions, particularly Lingayats, use malas of 108 beads for prayer and meditation.
Similarly, in Gaudiya Vaishnavism, Lord Krishna in Brindavan had 108 followers known as gopis. Recital of their names, often accompanied by the counting of a 108-beaded mala, is often done during religious ceremonies.
The Sri Vaishnavite Tradition has 108 Divya Desams (temples of Vishnu) that are revered by the 12 Alvars in the Divya Prabandha, a collection of 4,000 Tamil verses. There are also 18 pithas (sacred places).
The Sudarshana Chakra is a spinning, discus weapon with 108 serrated edges, generally portrayed on the right rear hand of the four hands of Vishnu.
The total number of Upanishads is 108 as per Muktikā canon.
Jainism
In Jainism, the total number of ways of Karma influx (Aasrav). 4 Kashays (anger, pride, conceit, greed) x 3 karanas (mind, speech, bodily action) x 3 sta |
https://en.wikipedia.org/wiki/109%20%28number%29 | 109 (one hundred [and] nine) is the natural number following 108 and preceding 110.
In mathematics
109 is the 29th prime number. As 29 is itself prime, 109 is the tenth super-prime. The previous prime is 107, making them both twin primes.
109 is a centered triangular number.
There are exactly:
109 different families of subsets of a three-element set whose union includes all three elements.
109 different loops (invertible but not necessarily associative binary operations with an identity) on six elements.
109 squares on an infinite chessboard that can be reached by a knight within three moves.
There are 109 uniform edge-colorings to the 11 regular and semiregular (or Archimedean) tilings.
The decimal expansion of 1/109 can be computed using the alternating series, with the Fibonacci number:
The decimal expansion of 1/109 has 108 digits, making 109 a full reptend prime in decimal. The last six digits of the 108-digit cycle are 853211, the first six Fibonacci numbers in descending order.
See also
109 (disambiguation)
References
Integers |
https://en.wikipedia.org/wiki/110%20%28number%29 | 110 (one hundred [and] ten) is the natural number following 109 and preceding 111.
In mathematics
110 is a sphenic number and a pronic number. Following the prime quadruplet (101, 103, 107, 109), at 110, the Mertens function reaches a low of −5.
110 is the sum of three consecutive squares, .
RSA-110 is one of the RSA numbers, large semiprimes that are part of the RSA Factoring Challenge.
In base 10, the number 110 is a Harshad number and a self number.
In science
The atomic number of darmstadtium.
In religion
According to the Bible, the figures Joseph and Joshua both died aged 110.
In sports
Olympic male track and field athletics run 110 metre hurdles. (Female athletes run the 100 metre hurdles instead.)
The International 110, or the 110, is a one-design racing sailboat designed in 1939 by C. Raymond Hunt.
In other fields
110 is also:
The year AD 110 or 110 BC
A common name for mains electricity in North America, despite the nominal voltage actually being 120 V (range 110–120 V). Normally spoken as "one-ten".
1-1-0, the emergency telephone number used to reach police services in Iran, Germany, Estonia, China, Indonesia, and Japan. Also used to reach the fire and rescue services in Norway and Turkey.
The age a person must attain in order to be considered a supercentenarian.
A card game related to Forty-five (card game).
A percentage in the expression "To give 110%", meaning to give a little more effort than one's maximum effort
The number of stories of each of the towers of the former World Trade Center in New York.
The number of stories (by common reckoning) of the Sears Tower in Chicago.
The TCP port used for POP3 email protocol
A 110 block is a type of punch block used to connect sets of wires in a structured cabling system.
The abjad (ابجد) translation of word "علی" (Ali) in Arabic and Persian.
It is also known as "eleventy", a term made famous by linguist and author J. R. R. Tolkien (Bilbo Baggins celebrates his eleventy-first birthday at th |
https://en.wikipedia.org/wiki/112%20%28number%29 | 112 (one hundred [and] twelve) is the natural number following 111 and preceding 113.
Mathematics
112 is an abundant number, a heptagonal number, and a Harshad number.
112 is the number of connected graphs on 6 unlabeled nodes.
If an equilateral triangle has sides of length 112, then it contains an interior point at integer distances 57, 65, and 73 from its vertices. This is the smallest possible side length of an equilateral triangle that contains a point at integer distances from the vertices.
See also
112 (disambiguation)
References
Integers |
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