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Topic Review
Biography
Topic Review
Hybrid Number
A hybrid number is a generalization of complex numbers [math]\displaystyle{ \left(a+\mathbf{i}b, \mathbf{i}^{2}=-1\right) }[/math], split-complex numbers (or "hyperbolic number") [math]\displaystyle{ \left(a+\mathbf{h}b, \mathbf{h}^2=1\right) }[/math] and dual numbers [math]\displaystyle{ \left(a+\mathbf{\varepsilon} b, \mathbf{\varepsilon}^2 = 0\right) }[/math]. Hybrid numbers form a noncommutative ring. Complex, hyperbolic and dual numbers are well known two-dimensional number systems. It is well known that, the set of complex numbers, hyperbolic numbers and dual numbers are respectively. The algebra of hybrid numbers is a noncommutative algebra which unifies all three number systems calls them hybrid numbers., , . A hybrid number is a number created with any combination of the complex, hyperbolic and dual numbers satisfying the relation Because these numbers are a composition of dual, complex and hyperbolic numbers, Ozdemir calls them hybrid numbers . A commutative two-dimensional unital algebra generated by a 2 by 2 matrix is isomorphic to either complex, dual or hyperbolic numbers . Due to the set of hybrid numbers is a two-dimensional commutative algebra spanned by 1 and [math]\displaystyle{ \mathbf{i}b+c\mathbf{\varepsilon }+d\mathbf{h} }[/math], it is isomorphic to one of the complex, dual or hyperbolic numbers. Especially in the last century, a lot of researchers deal with the geometric and physical applications of these numbers. Just as the geometry of the Euclidean plane can be described with complex numbers, the geometry of the Minkowski plane and Galilean plane can be described with hyperbolic numbers. The group of Euclidean rotations SO(2) is isomorphic to the group U(1) of unit complex numbers. The geometrical meaning of multiplying by [math]\displaystyle{ e^{\mathbf{i}\theta}=\cos \theta +\mathbf{i}\sin \theta }[/math] means a rotation of the plane. , . The group of Lorentzian rotations [math]\displaystyle{ SO(1,1) }[/math] is isomorphic to the group of unit spacelike hyperbolic numbers. This rotation can be viewed as hyperbolic rotation. Thus, multiplying by [math]\displaystyle{ e^{\mathbf{h}\theta }=\cosh \theta +\mathbf{h} \sinh \theta }[/math] means a map of hyperbolic numbers into itself which preserves the Lorentzian metric. , , , The Galilean rotations can be interpreted with dual numbers. The concept of a rotation in the dual number plane is equivalent to a vertical shear mapping since [math]\displaystyle{ \left( 1+x\mathbf{\varepsilon }\right) \left( 1+y\mathbf{\varepsilon }\right) =1+\left( x+y\right) \mathbf{\varepsilon } }[/math]. The Euler formula for dual numbers is [math]\displaystyle{ e^{\mathbf{\varepsilon }\theta }=1+\mathbf{\varepsilon }\theta }[/math]. Multiplying by [math]\displaystyle{ e^{\mathbf{\varepsilon \theta }} }[/math] is a map of dual numbers into itself which preserves the Galilean metric. This rotation can be named as parabolic rotation , , , , , . File:Planar rotations.tif In abstract algebra, the complex, the hyperbolic and the dual numbers can be described as the quotient of the polynomial ring [math]\displaystyle{ \mathbb{R}[x] }[/math] by the ideal generated by the polynomials [math]\displaystyle{ x^2+1, }[/math], [math]\displaystyle{ x^2-1 }[/math] and [math]\displaystyle{ x^{2} }[/math] respectively. That is, Matrix represantations of the units [math]\displaystyle{ \mathbf{i} }[/math], [math]\displaystyle{ \mathbf{\varepsilon } }[/math], [math]\displaystyle{ \mathbf{h} }[/math] are respectively.
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08 Nov 2022
Topic Review
Atari Microsoft BASIC
The Atari Microsoft BASIC and Atari Microsoft BASIC II variants of the 6502-version of Microsoft BASIC ported to the Atari 8-bit machines. The first version, released 1981, required 32 KB of RAM and was supplied on floppy disk. The second version, released the next year, had most of the code on a ROM cartridge with additional functions on an optional floppy. Atari Microsoft BASIC was highly compatible with contemporary BASICs like AppleSoft BASIC and Commodore BASIC, in contrast to their own Atari BASIC which was a somewhat different dialect and incompatible in many ways. Like Atari BASIC, Atari Microsoft BASIC included a number of new commands and features to take advantage of the Atari platform. These were only partially compatible with the similar features of Atari BASIC.
340
08 Nov 2022
Topic Review
Z4 (Computer)
The Z4 was arguably the world's first commercial digital computer. It was designed, and manufactured by early computer scientist Konrad Zuse's company Zuse Apparatebau, for an order placed by Henschel & Son, in 1942; though only partially assembled in Berlin, then completed in Göttingen, and not delivered by the defeat of Nazi Germany, in 1945. The Z4 was Zuse's final target for the Z3 design. Like the earlier Z2, it comprised a combination of mechanical memory and electromechanical logic, so was not a true electronic computer.
1.2K
08 Nov 2022
Topic Review
Karhunen–Loève Theorem
In the theory of stochastic processes, the Karhunen–Loève theorem (named after Kari Karhunen and Michel Loève), also known as the Kosambi–Karhunen–Loève theorem is a representation of a stochastic process as an infinite linear combination of orthogonal functions, analogous to a Fourier series representation of a function on a bounded interval. The transformation is also known as Hotelling transform and eigenvector transform, and is closely related to principal component analysis (PCA) technique widely used in image processing and in data analysis in many fields. Stochastic processes given by infinite series of this form were first considered by Damodar Dharmananda Kosambi. There exist many such expansions of a stochastic process: if the process is indexed over [a, b], any orthonormal basis of L2([a, b]) yields an expansion thereof in that form. The importance of the Karhunen–Loève theorem is that it yields the best such basis in the sense that it minimizes the total mean squared error.
1.7K
08 Nov 2022
Topic Review
SLinCA@Home
SLinCA@Home (Scaling Laws in Cluster Aggregation) was a research project that uses Internet-connected computers to do research in fields such as physics and materials science.
471
08 Nov 2022
Topic Review
Maze Solving Algorithm
There are a number of different maze solving algorithms, that is, automated methods for the solving of mazes. The random mouse, wall follower, Pledge, and Trémaux's algorithms are designed to be used inside the maze by a traveler with no prior knowledge of the maze, whereas the dead-end filling and shortest path algorithms are designed to be used by a person or computer program that can see the whole maze at once. Mazes containing no loops are known as "simply connected", or "perfect" mazes, and are equivalent to a tree in graph theory. Thus many maze solving algorithms are closely related to graph theory. Intuitively, if one pulled and stretched out the paths in the maze in the proper way, the result could be made to resemble a tree.
1.8K
08 Nov 2022
Topic Review
Go90
Go90 was an American Internet television service and mobile app owned and operated by Verizon Communications. The service was positioned as a mobile-oriented "social entertainment platform" targeted primarily towards millennials, featuring a mixture of new and acquired content from various providers. The service was available exclusively within the United States, although as of March 2018 some of its content became available internationally via Tumblr. Following an unsuccessful launch (which content partners credited to a lack of firm content strategies), resulting managerial turnover (including a relaunch of the service by the staff of Verizon acquisition and former competitor Vessel), Including acquisitions that became the basis of the service (such as Intel's OnCue project, and later Vessel), it was estimated that Verizon had spent $1.2 billion on the service.
1.9K
08 Nov 2022
Topic Review
The Incoherence of the Philosophers
The Incoherence of the Philosophers (تهافت الفلاسفة Tahāfut al-Falāsifaʰ in Arabic) is the title of a landmark 11th-century work by the Persian theologian Abū Ḥāmid Muḥammad ibn Muḥammad al-Ghazali and a student of the Asharite school of Islamic theology criticizing the Avicennian school of early Islamic philosophy. Muslim philosophers such as Ibn Sina (Avicenna) and Al-Farabi (Alpharabius) are denounced in this book, as they follow Greek philosophy even when it contradicts Islam. The text was dramatically successful, and marked a milestone in the ascendance of the Asharite school within Islamic philosophy and theological discourse. The book favors faith over philosophy in matters specifically concerning metaphysics or knowledge of the divine.
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08 Nov 2022
Topic Review
Content-Control Software
Content-control software, commonly referred to as an Internet filter, is software that restricts or controls the content an Internet user is capable to access, especially when utilised to restrict material delivered over the Internet via the Web, Email, or other means. Content-control software determines what content will be available or be blocked. Such restrictions can be applied at various levels: a government can attempt to apply them nationwide (see Internet censorship), or they can, for example, be applied by an ISP to its clients, by an employer to its personnel, by a school to its students, by a library to its visitors, by a parent to a child's computer, or by an individual user to their own computer. The motive is often to prevent access to content which the computer's owner(s) or other authorities may consider objectionable. When imposed without the consent of the user, content control can be characterised as a form of internet censorship. Some content-control software includes time control functions that empowers parents to set the amount of time that child may spend accessing the Internet or playing games or other computer activities. In some countries, such software is ubiquitous. In Cuba, if a computer user at a government-controlled Internet cafe types certain words, the word processor or web browser is automatically closed, and a "state security" warning is given.
2.3K
08 Nov 2022
Topic Review
Country Code Top-Level Domain
A country code top-level domain (ccTLD) is an Internet top-level domain generally used or reserved for a country, sovereign state, or dependent territory identified with a country code. All ASCII ccTLD identifiers are two letters long, and all two-letter top-level domains are ccTLDs. In 2018, the Internet Assigned Numbers Authority (IANA) began implementing internationalized country code top-level domains, consisting of language-native characters when displayed in an end-user application. Creation and delegation of ccTLDs is described in RFC 1591, corresponding to ISO 3166-1 alpha-2 country codes. While gTLD have to obey international regulations, ccTLD are subjected to requirements that are determined by each country’s domain name regulation corporation. With over 150 million domain name registrations today, ccTLDs make up 40% of the total domain name industry. Country code extension applications began in 1985. The registered first extensions that year were .us (United States), .uk (United Kingdom), and .il (Israel). There are 312 ccTLDs in active use totally. .cn, .tk, .de and .uk contain the highest number of domains.
4.6K
08 Nov 2022
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