You are looking at all articles with the topic "Mathematics". We found 245 matches.

Hint: To view all topics, click here. Too see the most popular topics, click here instead.

🔗 Mathematics

The medieval Cistercian numerals, or "ciphers" in nineteenth-century parlance, were developed by the Cistercian monastic order in the early thirteenth century at about the time that Arabic numerals were introduced to northwestern Europe. They are more compact than Arabic or Roman numerals, with a single character able to indicate any integer from 1 to 9,999.

Digits are based on a horizontal or vertical stave, with the position of the digit on the stave indicating its place value (units, tens, hundreds or thousands). These digits are compounded on a single stave to indicate more complex numbers. The Cistercians eventually abandoned the system in favor of the Arabic numerals, but marginal use outside the order continued until the early twentieth century.

🔗 Computing 🔗 Mathematics

The Bibi-binary system for numeric notation (in French système Bibi-binaire, or abbreviated "système Bibi") is a hexadecimal numeral system first described in 1968 by singer/mathematician Robert "Boby" Lapointe (1922–1972). At the time, it attracted the attention of André Lichnerowicz, then engaged in studies at the University of Lyon. It found some use in a variety of unforeseen applications: stochastic poetry, stochastic art, colour classification, aleatory music, architectural symbolism, etc.

The notational system directly and logically encodes the binary representations of the digits in a hexadecimal (base sixteen) numeral. In place of the Arabic numerals 0–9 and letters A–F currently used in writing hexadecimal numerals, it presents sixteen newly devised symbols (thus evading any risk of confusion with the decimal system). The graphical and phonetic conception of these symbols is meant to render the use of the Bibi-binary "language" simple and fast.

The description of the language first appeared in Les Cerveaux non-humains ("Non-human brains"), and the system can also be found in Boby Lapointe by Huguette Long Lapointe.

🔗 Mathematics 🔗 Lists

The mathematical statements discussed below are provably independent of ZFC (the canonical axiomatic set theory of contemporary mathematics, consisting of the Zermelo–Fraenkel axioms plus the axiom of choice), assuming that ZFC is consistent. A statement is independent of ZFC (sometimes phrased "undecidable in ZFC") if it can neither be proven nor disproven from the axioms of ZFC.

🔗 Mathematics 🔗 Toys

In geometry, flexagons are flat models, usually constructed by folding strips of paper, that can be flexed or folded in certain ways to reveal faces besides the two that were originally on the back and front.

Flexagons are usually square or rectangular (tetraflexagons) or hexagonal (hexaflexagons). A prefix can be added to the name to indicate the number of faces that the model can display, including the two faces (back and front) that are visible before flexing. For example, a hexaflexagon with a total of six faces is called a hexahexaflexagon.

In hexaflexagon theory (that is, concerning flexagons with six sides), flexagons are usually defined in terms of pats.

Two flexagons are equivalent if one can be transformed to the other by a series of pinches and rotations. Flexagon equivalence is an equivalence relation.

🔗 Mathematics 🔗 Economics 🔗 Politics 🔗 Elections and Referendums

In social choice theory, Arrow's impossibility theorem, the general possibility theorem or Arrow's paradox is an impossibility theorem stating that when voters have three or more distinct alternatives (options), no ranked voting electoral system can convert the ranked preferences of individuals into a community-wide (complete and transitive) ranking while also meeting a specified set of criteria: unrestricted domain, non-dictatorship, Pareto efficiency, and independence of irrelevant alternatives. The theorem is often cited in discussions of voting theory as it is further interpreted by the Gibbard–Satterthwaite theorem. The theorem is named after economist and Nobel laureate Kenneth Arrow, who demonstrated the theorem in his doctoral thesis and popularized it in his 1951 book Social Choice and Individual Values. The original paper was titled "A Difficulty in the Concept of Social Welfare".

In short, the theorem states that no rank-order electoral system can be designed that always satisfies these three "fairness" criteria:

• If every voter prefers alternative X over alternative Y, then the group prefers X over Y.
• If every voter's preference between X and Y remains unchanged, then the group's preference between X and Y will also remain unchanged (even if voters' preferences between other pairs like X and Z, Y and Z, or Z and W change).
• There is no "dictator": no single voter possesses the power to always determine the group's preference.

Cardinal voting electoral systems are not covered by the theorem, as they convey more information than rank orders. However, Gibbard's theorem extends Arrow's theorem for that case. The theorem can also be sidestepped by weakening the notion of independence.

The axiomatic approach Arrow adopted can treat all conceivable rules (that are based on preferences) within one unified framework. In that sense, the approach is qualitatively different from the earlier one in voting theory, in which rules were investigated one by one. One can therefore say that the contemporary paradigm of social choice theory started from this theorem.

The practical consequences of the theorem are debatable: Arrow has said "Most systems are not going to work badly all of the time. All I proved is that all can work badly at times."

🔗 Mathematics

Sexy primes are prime numbers that differ from each other by 6. For example, the numbers 5 and 11 are both sexy primes, because 11 - 5 = 6.

The term "sexy prime" is a pun stemming from the Latin word for six: sex.

If p + 2 or p + 4 (where p is the lower prime) is also prime, then the sexy prime is part of a prime triplet.

🔗 Computing 🔗 Mathematics 🔗 Computing/Software 🔗 Computing/Computer science

The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. It reduces the multiplication of two n-digit numbers to at most ${\displaystyle n^{\log _{2}3}\approx n^{1.58}}$ single-digit multiplications in general (and exactly ${\displaystyle n^{\log _{2}3}}$ when n is a power of 2). It is therefore faster than the classical algorithm, which requires ${\displaystyle n^{2}}$ single-digit products. For example, the Karatsuba algorithm requires 3¹⁰ = 59,049 single-digit multiplications to multiply two 1024-digit numbers (n = 1024 = 2¹⁰), whereas the classical algorithm requires (2¹⁰)² = 1,048,576 (a speedup of 17.75 times).

The Karatsuba algorithm was the first multiplication algorithm asymptotically faster than the quadratic "grade school" algorithm. The Toom–Cook algorithm (1963) is a faster generalization of Karatsuba's method, and the Schönhage–Strassen algorithm (1971) is even faster, for sufficiently large n.

🔗 Computing 🔗 Mathematics

In mathematical logic and type theory, the λ-cube is a framework introduced by Henk Barendregt to investigate the different dimensions in which the calculus of constructions is a generalization of the simply typed λ-calculus. Each dimension of the cube corresponds to a new kind of dependency between terms and types. Here, "dependency" refers to the capacity of a term or type to bind a term or type. The respective dimensions of the λ-cube correspond to:

• y-axis (${\displaystyle \uparrow }$): terms that can bind types, corresponding to polymorphism.
• x-axis (${\displaystyle \rightarrow }$): types that can bind terms, corresponding to dependent types.
• z-axis (${\displaystyle \nearrow }$): types that can bind types, corresponding to (binding) type operators.

The different ways to combine these three dimension yield the 8 vertices of the cube, each corresponding to a different kind of typed system. The λ-cube can be generalized into the concept of a pure type system.

🔗 Computing 🔗 Mathematics 🔗 Computing/Software 🔗 Cryptography 🔗 Cryptography/Computer science 🔗 Computing/Computer Security

Homomorphic encryption is a form of encryption that allows computation on ciphertexts, generating an encrypted result which, when decrypted, matches the result of the operations as if they had been performed on the plaintext.

Homomorphic encryption can be used for privacy-preserving outsourced storage and computation. This allows data to be encrypted and out-sourced to commercial cloud environments for processing, all while encrypted. In highly regulated industries, such as health care, homomorphic encryption can be used to enable new services by removing privacy barriers inhibiting data sharing. For example, predictive analytics in health care can be hard to apply due to medical data privacy concerns, but if the predictive analytics service provider can operate on encrypted data instead, these privacy concerns are diminished.

🔗 Mathematics

The dangerous bend or caution symbol ☡ (U+2621 ☡ CAUTION SIGN) was created by the Nicolas Bourbaki group of mathematicians and appears in the margins of mathematics books written by the group. It resembles a road sign that indicates a "dangerous bend" in the road ahead, and is used to mark passages tricky on a first reading or with an especially difficult argument.