AlphabetAn alphabet is a standardized set of basic written graphemes (called letters) representing phonemes, units of sounds that distinguish words, of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a syllable, and logographic systems use characters to represent words, morphemes, or other semantic units. The Egyptians have created the first alphabet in a technical sense.
Mathematical proofA mathematical proof is a deductive argument for a mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning which establish logical certainty, to be distinguished from empirical arguments or non-exhaustive inductive reasoning which establish "reasonable expectation".
Latin alphabetThe Latin alphabet, also known as the Roman alphabet, is the collection of letters originally used by the ancient Romans to write the Latin language. Largely unaltered with the exception of extensions (such as diacritics), it forms the Latin script that is used to write many modern European languages, including English. With modifications, it is also used for other alphabets, such as the Vietnamese alphabet. Its modern repertoire is standardised as the ISO basic Latin alphabet.
Mathematical inductionMathematical induction is a method for proving that a statement is true for every natural number , that is, that the infinitely many cases all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: Mathematical induction proves that we can climb as high as we like on a ladder, by proving that we can climb onto the bottom rung (the basis) and that from each rung we can climb up to the next one (the step). A proof by induction consists of two cases.
Phoenician alphabetThe Phoenician alphabet is an alphabet (more specifically, an abjad) known in modern times from the Canaanite and Aramaic inscriptions found across the Mediterranean region. The name comes from the Phoenician civilization. The Phoenician alphabet is also called the Early Linear script (in a Semitic context, not connected to Minoan writing systems), because it is an early development of the Proto- or Old Canaanite or Proto-Sinaitic script, into a linear, purely alphabetic script, also marking the transfer from a multi-directional writing system, where a variety of writing directions occurred, to a regulated horizontal, right-to-left script.
Proof theoryProof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.
F-divergenceIn probability theory, an -divergence is a function that measures the difference between two probability distributions and . Many common divergences, such as KL-divergence, Hellinger distance, and total variation distance, are special cases of -divergence. These divergences were introduced by Alfréd Rényi in the same paper where he introduced the well-known Rényi entropy. He proved that these divergences decrease in Markov processes.
Kullback–Leibler divergenceIn mathematical statistics, the Kullback–Leibler divergence (also called relative entropy and I-divergence), denoted , is a type of statistical distance: a measure of how one probability distribution P is different from a second, reference probability distribution Q. A simple interpretation of the KL divergence of P from Q is the expected excess surprise from using Q as a model when the actual distribution is P.
Montenegrin alphabetThe Montenegrin alphabet is the collective name given to "Abeceda" (Montenegrin Latin alphabet) and "Азбука" (Montenegrin Cyrillic alphabet), the writing systems used to write the Montenegrin language. It was adopted on 9 June 2009 by the Montenegrin Minister of Education, Sreten Škuletić and replaced the Serbian Cyrillic and Gaj's Latin alphabets in use at the time.
Divergence (statistics)In information geometry, a divergence is a kind of statistical distance: a binary function which establishes the separation from one probability distribution to another on a statistical manifold. The simplest divergence is squared Euclidean distance (SED), and divergences can be viewed as generalizations of SED. The other most important divergence is relative entropy (also called Kullback–Leibler divergence), which is central to information theory.
History of the Greek alphabetThe history of the Greek alphabet starts with the adoption of Phoenician letter forms in the 9th–8th centuries BC during early Archaic Greece and continues to the present day. The Greek alphabet was developed during the Iron Age, centuries after the loss of Linear B, the syllabic script that was used for writing Mycenaean Greek until the Late Bronze Age collapse and Greek Dark Age. This article concentrates on the development of the alphabet before the modern codification of the standard Greek alphabet.
Serbian Cyrillic alphabetThe Serbian Cyrillic alphabet (Српска ћирилица / Srpska ćirilica, sr̩̂pskaː tɕirǐlitsa) is a variation of the Cyrillic script used to write the Serbian language, adapted in 1818 by the Serbian philologist and linguist Vuk Karadžić. It is one of the two alphabets used to write modern standard Serbian, the other being Gaj's Latin alphabet. Karadžić based his alphabet on the previous Slavonic-Serbian script, following the principle of "write as you speak and read as it is written", removing obsolete letters and letters representing iotated vowels, introducing from the Latin alphabet instead, and adding several consonant letters for sounds specific to Serbian phonology.
Computer-assisted proofA computer-assisted proof is a mathematical proof that has been at least partially generated by computer. Most computer-aided proofs to date have been implementations of large proofs-by-exhaustion of a mathematical theorem. The idea is to use a computer program to perform lengthy computations, and to provide a proof that the result of these computations implies the given theorem. In 1976, the four color theorem was the first major theorem to be verified using a computer program.
IOSiOS (formerly iPhone OS) is a mobile operating system developed by Apple Inc. exclusively for its hardware. It is the operating system that powers many of the company's mobile devices, including the iPhone; the term also includes the system software for iPads (predating iPadOS, which was introduced in 2019) as well as on the iPod Touch devices (which were discontinued in mid-2022). It is the world's second-most widely installed mobile operating system, after Android.
Proofs and RefutationsProofs and Refutations: The Logic of Mathematical Discovery is a 1976 book by philosopher Imre Lakatos expounding his view of the progress of mathematics. The book is written as a series of Socratic dialogues involving a group of students who debate the proof of the Euler characteristic defined for the polyhedron. A central theme is that definitions are not carved in stone, but often have to be patched up in the light of later insights, in particular failed proofs. This gives mathematics a somewhat experimental flavour.
MacOSmacOS (ˌmækoʊˈɛs; previously OS X and originally Mac OS X) is an operating system developed and marketed by Apple Inc. since 2001. It is the primary operating system for Apple's Mac computers. Within the market of desktop and laptop computers, it is the second most widely used desktop OS, after Microsoft Windows and ahead of Linux (including ChromeOS). macOS succeeded the classic Mac OS, a Mac operating system with nine releases from 1984 to 1999.
Mac OS 8Mac OS 8 is an operating system that was released by Apple Computer on July 26, 1997. It includes the largest overhaul of the classic Mac OS experience since the release of System 7, approximately six years before. It places a greater emphasis on color than prior versions. Released over a series of updates, Mac OS 8 represents an incremental integration of many of the technologies which had been developed from 1988 to 1996 for Apple's overly ambitious OS named Copland.
Electronic band structureIn solid-state physics, the electronic band structure (or simply band structure) of a solid describes the range of energy levels that electrons may have within it, as well as the ranges of energy that they may not have (called band gaps or forbidden bands). Band theory derives these bands and band gaps by examining the allowed quantum mechanical wave functions for an electron in a large, periodic lattice of atoms or molecules.
Finite setIn mathematics, particularly set theory, a finite set is a set that has a finite number of elements. Informally, a finite set is a set which one could in principle count and finish counting. For example, is a finite set with five elements. The number of elements of a finite set is a natural number (possibly zero) and is called the cardinality (or the cardinal number) of the set. A set that is not a finite set is called an infinite set.
Band gapIn solid-state physics and solid-state chemistry, a band gap, also called a bandgap or energy gap, is an energy range in a solid where no electronic states exist. In graphs of the electronic band structure of solids, the band gap refers to the energy difference (often expressed in electronvolts) between the top of the valence band and the bottom of the conduction band in insulators and semiconductors. It is the energy required to promote an electron from the valence band to the conduction band.
