In the mathematical theory of random processes, the Markov chain central limit theorem has a conclusion somewhat similar in form to that of the classic central limit theorem (CLT) of probability theory, but the quantity in the role taken by the variance in the classic CLT has a more complicated definition. See also the general form of Bienaymé's identity. == Statement == Suppose that: the sequence X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } of random elements of some set is a Markov chain that has a stationary probability distribution; and the initial distribution of the process, i.e. the distribution of X 1 {\textstyle X_{1}} , is the stationary distribution, so that X 1 , X 2 , X 3 , … {\textstyle X_{1},X_{2},X_{3},\ldots } are identically distributed. In the classic central limit theorem these random variables would be assumed to be independent, but here we have only the weaker assumption that the process has the Markov property; and g {\textstyle g} is some (measurable) real-valued function for which var ( g ( X 1 ) ) < + ∞ . {\textstyle \operatorname {var} (g(X_{1}))<+\infty .} Now let μ = E ( g ( X 1 ) ) , μ ^ n = 1 n ∑ k = 1 n g ( X k ) σ 2 := lim n → ∞ var ( n μ ^ n ) = lim n → ∞ n var ( μ ^ n ) = var ( g ( X 1 ) ) + 2 ∑ k = 1 ∞ cov ( g ( X 1 ) , g ( X 1 + k ) ) . {\displaystyle {\begin{aligned}\mu &=\operatorname {E} (g(X_{1})),\\{\widehat {\mu }}_{n}&={\frac {1}{n}}\sum _{k=1}^{n}g(X_{k})\\\sigma ^{2}&:=\lim _{n\to \infty }\operatorname {var} ({\sqrt {n}}{\widehat {\mu }}_{n})=\lim _{n\to \infty }n\operatorname {var} ({\widehat {\mu }}_{n})=\operatorname {var} (g(X_{1}))+2\sum _{k=1}^{\infty }\operatorname {cov} (g(X_{1}),g(X_{1+k})).\end{aligned}}} Then as n → ∞ , {\textstyle n\to \infty ,} we have n ( μ ^ n − μ ) → D Normal ( 0 , σ 2 ) , {\displaystyle {\sqrt {n}}({\hat {\mu }}_{n}-\mu )\ {\xrightarrow {\mathcal {D}}}\ {\text{Normal}}(0,\sigma ^{2}),} where the decorated arrow indicates convergence in distribution. == Monte Carlo Setting == The Markov chain central limit theorem can be guaranteed for functionals of general state space Markov chains under certain conditions. In particular, this can be done with a focus on Monte Carlo settings. An example of the application in a MCMC (Markov Chain Monte Carlo) setting is the following: Consider a simple hard spheres model on a grid. Suppose X = { 1 , … , n 1 } × { 1 , … , n 2 } ⊆ Z 2 {\displaystyle X=\{1,\ldots ,n_{1}\}\times \{1,\ldots ,n_{2}\}\subseteq Z^{2}} . A proper configuration on X {\displaystyle X} consists of coloring each point either black or white in such a way that no two adjacent points are white. Let χ {\displaystyle \chi } denote the set of all proper configurations on X {\displaystyle X} , N χ ( n 1 , n 2 ) {\displaystyle N_{\chi }(n_{1},n_{2})} be the total number of proper configurations and π be the uniform distribution on χ {\displaystyle \chi } so that each proper configuration is equally likely. Suppose our goal is to calculate the typical number of white points in a proper configuration; that is, if W ( x ) {\displaystyle W(x)} is the number of white points in x ∈ χ {\displaystyle x\in \chi } then we want the value of E π W = ∑ x ∈ χ W ( x ) N χ ( n 1 , n 2 ) {\displaystyle E_{\pi }W=\sum _{x\in \chi }{\frac {W(x)}{N_{\chi }{\bigl (}n_{1},n_{2}{\bigr )}}}} If n 1 {\displaystyle n_{1}} and n 2 {\displaystyle n_{2}} are even moderately large then we will have to resort to an approximation to E π W {\displaystyle E_{\pi }W} . Consider the following Markov chain on χ {\displaystyle \chi } . Fix p ∈ ( 0 , 1 ) {\displaystyle p\in (0,1)} and set X 1 = x 1 {\displaystyle X_{1}=x_{1}} where x 1 ∈ χ {\displaystyle x_{1}\in \chi } is an arbitrary proper configuration. Randomly choose a point ( x , y ) ∈ X {\displaystyle (x,y)\in X} and independently draw U ∼ U n i f o r m ( 0 , 1 ) {\displaystyle U\sim \mathrm {Uniform} (0,1)} . If u ≤ p {\displaystyle u\leq p} and all of the adjacent points are black then color ( x , y ) {\displaystyle (x,y)} white leaving all other points alone. Otherwise, color ( x , y ) {\displaystyle (x,y)} black and leave all other points alone. Call the resulting configuration X 1 {\displaystyle X_{1}} . Continuing in this fashion yields a Harris ergodic Markov chain { X 1 , X 2 , X 3 , … } {\displaystyle \{X_{1},X_{2},X_{3},\ldots \}} having π {\displaystyle \pi } as its invariant distribution. It is now a simple matter to estimate E π W {\displaystyle E_{\pi }W} with w n ¯ = ∑ i = 1 n W ( X i ) / n {\displaystyle {\overline {w_{n}}}=\sum _{i=1}^{n}W(X_{i})/n} . Also, since χ {\displaystyle \chi } is finite (albeit potentially large) it is well known that X {\displaystyle X} will converge exponentially fast to π {\displaystyle \pi } which implies that a CLT holds for w n ¯ {\displaystyle {\overline {w_{n}}}} . == Implications == Not taking into account the additional terms in the variance which stem from correlations (e.g. serial correlations in markov chain monte carlo simulations) can result in the problem of pseudoreplication when computing e.g. the confidence intervals for the sample mean.
Rapid PHP Editor
rapid PHP Editor is a PHP Editor that incorporates many functions such as AutoComplete, Syntax checker, debugger and many other tools for fast PHP development. Rapid PHP Editor also contain other development tools for helping on HTML, CSS, JavaScript and many other languages. Is part of a family of products covering most aspects of modern web development integrating as well many other capabilities used by developers. Some features: (X)HTML to HTML5 CSS to CSS3 Code intelligence Powerful search and replace Support for several frameworks Code beautifier FTP Explorer (FTP/SFTP/FTPS) File explorer Database explorer Code snippets Validators and Debuggers FAST, real fast Many other tools available (many more to describe all here) == History == Rapid PHP Editor was built using the Delphi programming language.
Strategic Air Command Digital Information Network
The Strategic Air Command DIgital Network (SACDIN) was a United States military computer network that provided computerized record communications, replacing the Data Transmission Subsystem and part of the Data Display Subsystem of the SAC Automated Command and Control System. SACDIN enabled a rapid flow of communications from headquarters SAC to its fielded forces, such as B-52 bases and ICBM Launch Control Centers. == Logistics == Major portions of SACDIN were developed, engineered and installed by the International Telephone and Telegraph (ITT) company, under contract to the Electronic Systems Center. == Chronology == 1969 - Headquarters SAC submits a request to the Joint Chiefs of Staff to study an expanded communications system, known as the SAC Total Information Network (SATIN). It would interconnect Air Force Satellite Communications (AFSATCOM), Advanced Airborne Command Post (AABNCP), Airborne Command Post (ABNCP), high frequency/single sideband radio HF/SSB radio, SAC Automated Command and Control System (SACCS), Automatic Digital Information Network (AUTODIN), Survivable Low Frequency Communications System (SLFCS) and Command Data Buffer (CDB) 1977 1 November - SATIN IV was effectively terminated by Congress. The restructured program was renamed SAC Digital Network (SACDIN), and was formulated to meet SAC's minimum essential data communications requirements, but also had the capability to grow in a modular fashion. 1986 ?? ??? - SACDIN replaces much of the SAC Automated Command and Control System (SACCS) and the SAC Automated Total Information Network (SATIN)
Conjugate coding
Conjugate coding is a cryptographic tool, introduced by Stephen Wiesner in the late 1960s. It is part of the two applications Wiesner described for quantum coding, along with a method for creating fraud-proof banking notes. The application that the concept was based on was a method of transmitting multiple messages in such a way that reading one destroys the others. This is called quantum multiplexing and it uses photons polarized in conjugate bases as "qubits" to pass information. Conjugate coding also is a simple extension of a random number generator. At the behest of Charles Bennett, Wiesner published the manuscript explaining the basic idea of conjugate coding with a number of examples but it was not embraced because it was significantly ahead of its time. Because its publication has been rejected, it was developed to the world of public-key cryptography in the 1980s as oblivious transfer, first by Michael Rabin and then by Shimon Even. It is used in the field of quantum computing. The initial concept of quantum cryptography developed by Bennett and Gilles Brassard was also based on this concept.
Server-Gated Cryptography
Server-Gated Cryptography (SGC), also known as International Step-Up by Netscape, is a defunct mechanism that was used to step up from 40-bit or 56-bit to 128-bit cipher suites with SSL. It was created in response to United States federal legislation on the export of strong cryptography in the 1990s. The legislation had limited encryption to weak algorithms and shorter key lengths in software exported outside of the United States of America. When the legislation added an exception for financial transactions, SGC was created as an extension to SSL with the certificates being restricted to financial organisations. In 1999, this list was expanded to include online merchants, healthcare organizations, and insurance companies. This legislation changed in January 2000, resulting in vendors no longer shipping export-grade browsers and SGC certificates becoming available without restriction. Internet Explorer supported SGC starting with patched versions of Internet Explorer 3. SGC became obsolete when Internet Explorer 5.01 SP1 and Internet Explorer 5.5 started supporting strong encryption without the need for a separate high encryption pack (except on Windows 2000, which needs its own high encryption pack that was included in Service Pack 2 and later). "Export-grade" browsers are unusable on the modern Web due to many servers disabling export cipher suites. Additionally, these browsers are incapable of using SHA-2 family signature hash algorithms like SHA-256. Certification authorities are trying to phase out the new issuance of certificates with the older SHA-1 signature hash algorithm. The continuing use of SGC facilitates the use of obsolete, insecure Web browsers with HTTPS. However, while certificates that use the SHA-1 signature hash algorithm remain available, some certificate authorities continue to issue SGC certificates (often charging a premium for them) although they are obsolete. The reason certificate authorities can charge a premium for SGC certificates is that browsers only allowed a limited number of roots to support SGC. When an SSL handshake takes place, the software (e.g. a web browser) would list the ciphers that it supports. Although the weaker exported browsers would only include weaker ciphers in its initial SSL handshake, the browser also contained stronger cryptography algorithms. There are two protocols involved to activate them. Netscape Communicator 4 used International Step-Up, which used the now obsolete insecure renegotiation to change to a stronger cipher suite. Microsoft used SGC, which sends a new Client Hello message listing the stronger cipher suites on the same connection after the certificate is determined to be SGC capable, and also supported Netscape Step-Up for compatibility (though this support in the NT 4.0 SP6 and IE 5.01 version had a bug where changing MAC algorithms during Step-Up did not work properly).
Carrier cloud
In cloud computing, a carrier cloud is a class of cloud that integrates wide area networks (WAN) and other attributes of communications service providers’ carrier-grade networks to enable the deployment of highly-complex applications in the cloud. In contrast, classic cloud computing focuses on the data center and does not address the network connecting data centers and cloud users. This may result in unpredictable response times and security issues when business-critical data are transferred over the Internet. == History == The advent of virtualization technology, cost-effective computing hardware, and ubiquitous Internet connectivity have enabled the first wave of cloud services starting in the early years of the 21st century. But many businesses and other organizations hesitated to move to more demanding applications, from on-premises dedicated hardware to private or public clouds. As a response, communications service providers started in the 2010/2011 time frame to develop carrier clouds that address perceived weaknesses in existing cloud services. Cited weaknesses vary but often include possible downtime, security issues, high cost of custom software and data transfer, inflexibility of some cloud apps, poor customer and nonfulfillment of service level agreements (SLAs). == Characteristics == To enable the deployment of time-sensitive and business critical applications in the cloud, the carrier cloud is designed to match or even exceed the characteristics of on-premises deployments. Therefore, the carrier cloud is characterized by some or all of the following items: Configurable, elastic network performance: Typical cloud computing solutions use the best effort of the public Internet to connect cloud users and data centers. This approach provides instant connectivity but does not offer control over network capacities, latencies, and jitter. Carrier clouds address these gaps with content delivery networks and/or dedicated virtual private networks (VPN) at OSI layers 1 (optical wavelengths), 2 (data link layer), and 3 (network layer). These VPNs can be configured to offer the desired performance parameters and exhibit the same type of elasticity for the network that regular clouds provide for servers and storage. To achieve the requested performance parameters, such as low latency, cloud applications can be (automatically) allocated to distributed data centers that are close enough to the cloud users. Automatic resource placement: For a cloud with multiple data centers, information about both the data center and the connecting network is relevant for a decision of where to place cloud images and storage volumes. For this decision, carrier clouds can obtain relevant information about the network, e.g., using the Application-Layer Traffic Optimization (ALTO) protocol. High level of security and governance: Cloud application providers are subject to general and domain specific security, privacy, and governance requirements and regulations, such as the European Data Protection Directive and the U.S. Health Insurance Portability and Accountability Act. For added security, the wide area network of the carrier cloud can provide segregated encrypted or unencrypted network links that are not accessible from the general Internet. At the data center, the carrier cloud provides e.g. virtual private servers, management processes, logs, and documentation to fulfill security and governance rules. Location control: Fundamentally, cloud users should not be concerned with the geographic location of their cloud resources. However, privacy and other regulations may mandate that certain types of data must not be sent outside a national jurisdiction or other geographical region. Open APIs: Carrier clouds provide graphical user interfaces and Web application programming interfaces that allow cloud application providers to set up, manage, and monitor both, the data center and the WAN, of their cloud services. == Architecture == Carrier clouds encompass data centers at different network tiers and wide area networks that connect multiple data centers to each other as well as to the cloud users. Links between data centers are used for failover, overflow, backup, and geographic diversity. Carrier clouds can be set up as public, private, or hybrid clouds. The carrier cloud federates these cloud entities by using a single management system to orchestrate, manage, and monitor data center and network resources as a single system.
Backup
In information technology, a backup, or data backup is a copy of computer data taken and stored elsewhere so that it may be used to restore the original after a data loss event. The verb form, referring to the process of doing so, is "back up", whereas the noun and adjective form is "backup". Backups can be used to recover data after its loss from data deletion or corruption, or to recover data from an earlier time. Backups provide a simple form of IT disaster recovery; however not all backup systems are able to reconstitute a computer system or other complex configuration such as a computer cluster, active directory server, or database server. A backup system contains at least one copy of all data considered worth saving. The data storage requirements can be large. An information repository model may be used to provide structure to this storage. There are different types of data storage devices used for copying backups of data that is already in secondary storage onto archive files. There are also different ways these devices can be arranged to provide geographic dispersion, data security, and portability. Data is selected, extracted, and manipulated for storage. The process can include methods for dealing with live data, including open files, as well as compression, encryption, and de-duplication. Additional techniques apply to enterprise client-server backup. Backup schemes may include dry runs that validate the reliability of the data being backed up. There are limitations and human factors involved in any backup scheme. == Storage == A backup strategy requires an information repository, "a secondary storage space for data" that aggregates backups of data "sources". The repository could be as simple as a list of all backup media (DVDs, etc.) and the dates produced, or could include a computerized index, catalog, or relational database. === 3-2-1 Backup Rule === The backup data needs to be stored, requiring a backup rotation scheme, which is a system of backing up data to computer media that limits the number of backups of different dates retained separately, by appropriate re-use of the data storage media by overwriting of backups no longer needed. The scheme determines how and when each piece of removable storage is used for a backup operation and how long it is retained once it has backup data stored on it. The 3-2-1 rule can aid in the backup process. It states that there should be at least 3 copies of the data, stored on 2 different types of storage media, and one copy should be kept offsite, in a remote location (this can include cloud storage). 2 or more different media should be used to eliminate data loss due to similar reasons (for example, optical discs may tolerate being underwater while LTO tapes may not, and SSDs cannot fail due to head crashes or damaged spindle motors since they do not have any moving parts, unlike hard drives). An offsite copy protects against fire, theft of physical media (such as tapes or discs) and natural disasters like floods and earthquakes. Physically protected hard drives are an alternative to an offsite copy, but they have limitations like only being able to resist fire for a limited period of time, so an offsite copy still remains as the ideal choice. Because there is no perfect storage, many backup experts recommend maintaining a second copy on a local physical device, even if the data is also backed up offsite. === Backup methods === ==== Unstructured ==== An unstructured repository may simply be a stack of tapes, DVD-Rs or external HDDs with minimal information about what was backed up and when. This method is the easiest to implement, but unlikely to achieve a high level of recoverability as it lacks automation. ==== Full only/System imaging ==== A repository using this backup method contains complete source data copies taken at one or more specific points in time. Copying system images, this method is frequently used by computer technicians to record known good configurations. However, imaging is generally more useful as a way of deploying a standard configuration to many systems rather than as a tool for making ongoing backups of diverse systems. ==== Incremental ==== An incremental backup stores data changed since a reference point in time. Duplicate copies of unchanged data are not copied. Typically a full backup of all files is made once or at infrequent intervals, serving as the reference point for an incremental repository. Subsequently, a number of incremental backups are made after successive time periods. Restores begin with the last full backup and then apply the incrementals. Some backup systems can create a synthetic full backup from a series of incrementals, thus providing the equivalent of frequently doing a full backup. When done to modify a single archive file, this speeds restores of recent versions of files. ==== Near-CDP ==== Continuous Data Protection (CDP) refers to a backup that instantly saves a copy of every change made to the data. This allows restoration of data to any point in time and is the most comprehensive and advanced data protection. Near-CDP backup applications—often marketed as "CDP"—automatically take incremental backups at a specific interval, for example every 15 minutes, one hour, or 24 hours. They can therefore only allow restores to an interval boundary. Near-CDP backup applications use journaling and are typically based on periodic "snapshots", read-only copies of the data frozen at a particular point in time. Near-CDP (except for Apple Time Machine) intent-logs every change on the host system, often by saving byte or block-level differences rather than file-level differences. This backup method differs from simple disk mirroring in that it enables a roll-back of the log and thus a restoration of old images of data. Intent-logging allows precautions for the consistency of live data, protecting self-consistent files but requiring applications "be quiesced and made ready for backup." Near-CDP is more practicable for ordinary personal backup applications, as opposed to true CDP, which must be run in conjunction with a virtual machine or equivalent and is therefore generally used in enterprise client-server backups. Software may create copies of individual files such as written documents, multimedia projects, or user preferences, to prevent failed write events caused by power outages, operating system crashes, or exhausted disk space, from causing data loss. A common implementation is an appended ".bak" extension to the file name. ==== Reverse incremental ==== A Reverse incremental backup method stores a recent archive file "mirror" of the source data and a series of differences between the "mirror" in its current state and its previous states. A reverse incremental backup method starts with a non-image full backup. After the full backup is performed, the system periodically synchronizes the full backup with the live copy, while storing the data necessary to reconstruct older versions. This can either be done using hard links—as Apple Time Machine does, or using binary diffs. ==== Differential ==== A differential backup saves only the data that has changed since the last full backup. This means a maximum of two backups from the repository are used to restore the data. However, as time from the last full backup (and thus the accumulated changes in data) increases, so does the time to perform the differential backup. Restoring an entire system requires starting from the most recent full backup and then applying just the last differential backup. A differential backup copies files that have been created or changed since the last full backup, regardless of whether any other differential backups have been made since, whereas an incremental backup copies files that have been created or changed since the most recent backup of any type (full or incremental). Changes in files may be detected through a more recent date/time of last modification file attribute, and/or changes in file size. Other variations of incremental backup include multi-level incrementals and block-level incrementals that compare parts of files instead of just entire files. === Storage media === Regardless of the repository model that is used, the data has to be copied onto an archive file data storage medium. The medium used is also referred to as the type of backup destination. ==== Magnetic tape ==== Magnetic tape was for a long time the most commonly used medium for bulk data storage, backup, archiving, and interchange. It was previously a less expensive option, but this is no longer the case for smaller amounts of data. Tape is a sequential access medium, so the rate of continuously writing or reading data can be very fast. While tape media itself has a low cost per space, tape drives are typically dozens of times as expensive as hard disk drives and optical drives. Tape media are generally rotated on a schedule so at least one set is off-site in case something should happe