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Topic Review
Biography
Topic Review
Gauss's Lemma (Polynomial)
In algebra, Gauss's lemma, named after Carl Friedrich Gauss, is a statement about polynomials over the integers, or, more generally, over a unique factorization domain (that is, a ring that has a unique factorization property similar to the fundamental theorem of arithmetic). Gauss's lemma underlies all the theory of factorization and greatest common divisors of such polynomials. Gauss's lemma asserts that the product of two primitive polynomials is primitive (a polynomial with integer coefficients is primitive if it has 1 as a greatest common divisor of its coefficients). A corollary of Gauss's lemma, sometimes also called Gauss's lemma, is that a primitive polynomial is irreducible over the integers if and only if it is irreducible over the rational numbers. More generally, a primitive polynomial has the same complete factorization over the integers and over the rational numbers. In the case of coefficients in a unique factorization domain R, "rational numbers" must be replaced by "field of fractions of R". This implies that, if R is either a field, the ring of integers, or a unique factorization domain, then every polynomial ring (in one or several indeterminates) over R is a unique factorization domain. Another consequence is that factorization and greatest common divisor computation of polynomials with integers or rational coefficients may be reduced to similar computations on integers and primitive polynomials. This is systematically used (explicitly or implicitly) in all implemented algorithms (see Polynomial greatest common divisor and Factorization of polynomials). Gauss's lemma, and all its consequences that do not involve the existence of a complete factorization remain true over any GCD domain (an integral domain over which greatest common divisors exist). In particular, a polynomial ring over a GCD domain is also a GCD domain. If one calls primitive a polynomial such that the coefficients generate the unit ideal, Gauss's lemma is true over every commutative ring. However, some care must be taken, when using this definition of primitive, as, over a unique factorization domain that is not a principal ideal domain, there are polynomials that are primitive in the above sense and not primitive in this new sense.
3.2K
28 Nov 2022
Topic Review
Qt
Qt (pronounced "cute") is cross-platform software for creating graphical user interfaces as well as cross-platform applications that run on various software and hardware platforms such as Linux, Windows, macOS, Android or embedded systems with little or no change in the underlying codebase while still being a native application with native capabilities and speed. Qt is currently being developed by The Qt Company, a publicly listed company, and the Qt Project under open-source governance, involving individual developers and organizations working to advance Qt. Qt is available under both commercial licenses and open-source GPL 2.0, GPL 3.0, and LGPL 3.0 licenses.
2.0K
28 Nov 2022
Topic Review
Ko Fight
A ko (Japanese: コウ, 劫, kō, from the translation of the Sanskrit term kalpa) fight is a tactical and strategic phase that can arise in the game of go.
483
28 Nov 2022
Topic Review
Bidirectional Text
Bidirectional text is the text containing both text directionalities, both right-to-left (RTL or dextrosinistral) and left-to-right (LTR or sinistrodextral). It generally involves text containing different types of alphabets, but may also refer to boustrophedon, which is changing text directionality in each row. Some writing systems including the Arabic and Hebrew scripts or derived systems such as the Persian, Urdu, and Yiddish scripts, are written in a form known as right-to-left (RTL), in which writing begins at the right-hand side of a page and concludes at the left-hand side. This is different from the left-to-right (LTR) direction used by the dominant Latin script. When LTR text is mixed with RTL in the same paragraph, each type of text is written in its own direction, which is known as bidirectional text. This can get rather complex when multiple levels of quotation are used. Many computer programs fail to display bidirectional text correctly. For example, the Hebrew name Sarah (שרה) is spelled: sin (ש) (which appears rightmost), then resh (ר), and finally heh (ה) (which should appear leftmost). Note: Some web browsers may display the Hebrew text in this article in the opposite direction.
618
28 Nov 2022
Topic Review
Property
In logic and philosophy (especially metaphysics), a property is a characteristic of an object; a red object is said to have the property of redness. The property may be considered a form of object in its own right, able to possess other properties. A property, however, differs from individual objects in that it may be instantiated, and often in more than one object. It differs from the logical/mathematical concept of class by not having any concept of extensionality, and from the philosophical concept of class in that a property is considered to be distinct from the objects which possess it. Understanding how different individual entities (or particulars) can in some sense have some of the same properties is the basis of the problem of universals.
687
28 Nov 2022
Topic Review
Windows 10 Version History (Version 1703)
Windows 10 Creators Update (also known as version 1703 and codenamed "Redstone 2") is the third major update to Windows 10 and the second in a series of updates under the Redstone codenames. It carries the build number 10.0.15063.
1.4K
28 Nov 2022
Topic Review
Cryptography
Cryptography, or cryptology (from Ancient Greek: "hidden, secret"; and γράφειν graphein, "to write", or -λογία -logia, "study", respectively), is the practice and study of techniques for secure communication in the presence of adversarial behavior. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, information security, electrical engineering, digital signal processing, physics, and others. Core concepts related to information security (data confidentiality, data integrity, authentication, and non-repudiation) are also central to cryptography. Practical applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications. Cryptography prior to the modern age was effectively synonymous with encryption, converting readable information (plaintext) to unintelligible nonsense text (ciphertext), which can only be read by reversing the process (decryption). The sender of an encrypted (coded) message shares the decryption (decoding) technique only with intended recipients to preclude access from adversaries. The cryptography literature often uses the names "Alice" (or "A") for the sender, "Bob" (or "B") for the intended recipient, and "Eve" (or "E") for the eavesdropping adversary. Since the development of rotor cipher machines in World War I and the advent of computers in World War II, cryptography methods have become increasingly complex and their applications more varied. Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions, making such algorithms hard to break in actual practice by any adversary. While it is theoretically possible to break into a well-designed system, it is infeasible in actual practice to do so. Such schemes, if well designed, are therefore termed "computationally secure"; theoretical advances (e.g., improvements in integer factorization algorithms) and faster computing technology require these designs to be continually reevaluated, and if necessary, adapted. Information-theoretically secure schemes that cannot be broken even with unlimited computing power, such as the one-time pad, are much more difficult to use in practice than the best theoretically breakable, but computationally secure, schemes. The growth of cryptographic technology has raised a number of legal issues in the Information Age. Cryptography's potential for use as a tool for espionage and sedition has led many governments to classify it as a weapon and to limit or even prohibit its use and export. In some jurisdictions where the use of cryptography is legal, laws permit investigators to compel the disclosure of encryption keys for documents relevant to an investigation. Cryptography also plays a major role in digital rights management and copyright infringement disputes in regard to digital media.
3.1K
28 Nov 2022
Topic Review
Topology of Uniform Convergence
In mathematics, a linear map is a mapping V → W between two modules (including vector spaces) that preserves the operations of addition and scalar multiplication. By studying the linear maps between two modules one can gain insight into their structures. If the modules have additional structure, like topologies or bornologies, then one can study the subspace of linear maps that preserve this structure.
1.6K
28 Nov 2022
Topic Review
Children's Way
Children's Way is a non-profit organization that teaches children through an online program called Woogi World. Scott Dow is the founder and president of Children's Way. Some notable members of the advisory board for Children's Way include 2000 Olympic Gold Medalist Wrestler Rulon Gardner, entertainer and philanthropist Alan Osmond, and former U.S. Senator Jake Garn.
403
28 Nov 2022
Topic Review
Latin-1 Supplement (Unicode Block)
The Latin-1 Supplement (also called C1 Controls and Latin-1 Supplement) is the second Unicode block in the Unicode standard. It encodes the upper range of ISO 8859-1: 80 (U+0080) - FF (U+00FF). Controls C1 (0080–009F) are not graphic. This block ranges from U+0080 to U+00FF, contains 128 characters and includes the C1 controls, Latin-1 punctuation and symbols, 30 pairs of majuscule and minuscule accented Latin characters and 2 mathematical operators. The C1 controls and Latin-1 Supplement block has been included in its present form, with the same character repertoire since version 1.0 of the Unicode Standard. Its block name in Unicode 1.0 was simply Latin1.
646
28 Nov 2022
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