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Monotone Function, Monotonic
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See pure tone and monotonic scale. Monotone or monotonicity may also refer to: In economics *Monotone preferences, a property of a consumer's preference ordering. *Monotonicity (mechanism design), a property of a social choice function. *Monotonicity criterion, a property of a voting system. *Resource monotonicity, a property of resource allocation rules and bargaining systems. In mathematics *Monotone class theorem, in measure theory *Monotone convergence theorem, in mathematics *Monotone polygon, a property of a geometric object *Monotonic function, a property of a mathematical function *Monotonicity of entailment, a property of some logical systems *Monotonically increasing, a property of number sequence Other uses *Monotone (software), an open source revision control system *Monotonic orthography, simplified spelling of modern Greek *The Monotones The Monotones were a six-member Am ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pure Tone
In psychoacoustics, a pure tone is a sound with a sinusoidal waveform; that is, a sine wave of constant frequency, phase-shift, and amplitude. By extension, in signal processing a single-frequency tone or pure tone is a purely sinusoidal signal (e.g., a voltage). A pure tone has the property – unique among real-valued wave shapes – that its wave shape is unchanged by linear time-invariant systems; that is, only the phase and amplitude change between such a system's pure-tone input and its output. Sine and cosine waves can be used as basic building blocks of more complex waves. As additional sine waves having different frequencies are combined, the waveform transforms from a sinusoidal shape into a more complex shape. When considered as part of a whole spectrum, a pure tone may also be called a ''spectral component''. In clinical audiology, pure tones are used for pure-tone audiometry to characterize hearing thresholds at different frequencies. Sound localization is often ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonicity Of Entailment
Monotonicity of entailment is a property of many logical systems such that if a sentence follows deductively from a given set of sentences then it also follows deductively from any superset of those sentences. A corollary is that if a given argument is deductively valid, it cannot become invalid by the addition of extra premises. Logical systems with this property are called monotonic logics in order to differentiate them from non-monotonic logics. Classical logic and intuitionistic logic are examples of monotonic logics. Weakening rule Monotonicity may be stated formally as a rule called weakening, or sometimes thinning. A system is monotonic if and only if the rule is admissible. The weakening rule may be expressed as a natural deduction sequent: :\frac This can be read as saying that if, on the basis of a set of assumptions \Gamma, one can prove C, then by adding an assumption A, one can still prove C. Example The following argument is valid: "All men are mortal. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generalized Quantifier
In formal semantics, a generalized quantifier (GQ) is an expression that denotes a set of sets. This is the standard semantics assigned to quantified noun phrases. For example, the generalized quantifier ''every boy'' denotes the set of sets of which every boy is a member: \ This treatment of quantifiers has been essential in achieving a compositional semantics for sentences containing quantifiers. Type theory A version of type theory is often used to make the semantics of different kinds of expressions explicit. The standard construction defines the set of types recursively as follows: #''e'' and ''t'' are types. #If ''a'' and ''b'' are both types, then so is \langle a,b\rangle #Nothing is a type, except what can be constructed on the basis of lines 1 and 2 above. Given this definition, we have the simple types ''e'' and ''t'', but also a countable infinity of complex types, some of which include: \langle e,t\rangle;\qquad \langle t,t\rangle;\qquad \langle\langle e,t\rangl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Monotones
The Monotones were a six-member American doo-wop vocal group in the 1950s. They are considered a one-hit wonder, as their only hit single was " The Book of Love", which peaked at No. 5 on the Billboard Top 100 in 1958. Biography The Monotones formed in 1955 when the seven original singers, all residents of the Baxter Terrace housing project in Newark, New Jersey, began performing covers of popular songs. They were: * Lead singer Charles Howard Patrick (September 11, 1938 - September 11, 2020) * First tenor Warren Davis (born March 1, 1939 - April 17, 2016) * Second tenor George Malone (January 5, 1940 – October 5, 2007) * Bass singer Frankie Smith (May 13, 1938 – November 26, 2000) * Second bass singer John Ryanes (November 16, 1940 – May 30, 1972) * Baritone Warren Ryanes (December 14, 1937 – June 16, 1982) Charles Patrick's brother James was originally a member, but he left soon after the group's formation. John Ryanes and Warren Ryanes were also brothers. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Orthography
Greek orthography has used a variety of diacritics starting in the Hellenistic period. The more complex polytonic orthography (), which includes five diacritics, notates Ancient Greek phonology. The simpler monotonic orthography (), introduced in 1982, corresponds to Modern Greek phonology, and requires only two diacritics. Polytonic orthography () is the standard system for Ancient Greek and Medieval Greek and includes: * acute accent () * circumflex accent () * grave accent (); these 3 accents indicate different kinds of pitch accent * rough breathing () indicates the presence of the sound before a letter * smooth breathing () indicates the absence of . Since in Modern Greek the pitch accent has been replaced by a dynamic accent (stress), and was lost, most polytonic diacritics have no phonetic significance, and merely reveal the underlying Ancient Greek etymology. Monotonic orthography () is the standard system for Modern Greek. It retains two diacritics: * single ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone (software)
Monotone is an open source software tool for distributed revision control. It tracks revisions to files, groups sets of revisions into changesets, and tracks history across renames. The focus of the project is on integrity over performance. Monotone is designed for distributed operation, and makes heavy use of cryptographic primitives to track file revisions (via the SHA-1 secure hash) and to authenticate user actions (via RSA cryptographic signatures). History Milestones Monotone version 0.26 introduced major changes to the internal database structures, including a new structure known by Monotone developers as a ''roster''. Monotone databases created with version 0.26 can not exchange revisions with older Monotone databases. Older databases must first be upgraded to the new format. The new netsync protocol is incompatible with earlier versions of Monotone. As Git inspiration In April 2005, Monotone became the subject of increased interest in the FOSS community a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sequences
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be '' finite'', as in these examples, or '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonically Increasing Sequence
In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is called the ''length'' of the sequence. Unlike a set, the same elements can appear multiple times at different positions in a sequence, and unlike a set, the order does matter. Formally, a sequence can be defined as a function from natural numbers (the positions of elements in the sequence) to the elements at each position. The notion of a sequence can be generalized to an indexed family, defined as a function from an ''arbitrary'' index set. For example, (M, A, R, Y) is a sequence of letters with the letter "M" first and "Y" last. This sequence differs from (A, R, M, Y). Also, the sequence (1, 1, 2, 3, 5, 8), which contains the number 1 at two different positions, is a valid sequence. Sequences can be ''finite'', as in these examples, or '' inf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if it is either entirely non-decreasing, or entirely non-increasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is termed ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\right), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonic Scale
A monotonic scale is a scale (music), musical scale consisting of only one Musical note, note in the octave. Having a deliberate fixed note, the monotonic is still a musical form rather than a total absence of melody. The monotonic stands in contrast to more common musical scales, such as the penatonic scale, pentatonic (five notes) and modern, common Western heptatonic scale, heptatonic and chromatic scales. Liturgical usage Early Christian liturgical recitation may have been monotonic. Charles William Pearce speculated that the monotonic Reciting tone, psalm tone might have been an intermediary step between spoken recitation of the Psalter and melodic singing: The ''Annotated Book of Common Prayer'' similarly notes that (according to Saint Augustine) Athanasius of Alexandria, Saint Athanasius discouraged variance in note in liturgical recitation, but that eventual modulation of the note led to the development of plainsong.John Henry Blunt (ed.), Digitized reprint by Forgotten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Polygon
In geometry, a polygon in the plane is called monotone with respect to a straight line , if every line orthogonal to intersects the boundary of at most twice. Similarly, a polygonal chain is called monotone with respect to a straight line , if every line orthogonal to intersects at most once. For many practical purposes this definition may be extended to allow cases when some edges of are orthogonal to , and a simple polygon may be called monotone if a line segment that connects two points in and is orthogonal to lies completely in . Following the terminology for monotone functions, the former definition describes polygons strictly monotone with respect to . Properties Assume that ''L'' coincides with the ''x''-axis. Then the leftmost and rightmost vertices of a monotone polygon decompose its boundary into two monotone polygonal chains such that when the vertices of any chain are being traversed in their natural order, their X-coordinates are monotonically increa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Convergence Theorem
In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its simplest form, it says that a non-decreasing bounded-above sequence of real numbers a_1 \le a_2 \le a_3 \le ...\le K converges to its smallest upper bound, its supremum. Likewise, a non-increasing bounded-below sequence converges to its largest lower bound, its infimum. In particular, infinite sums of non-negative numbers converge to the supremum of the partial sums if and only if the partial sums are bounded. For sums of non-negative increasing sequences 0 \le a_ \le a_ \le \cdots , it says that taking the sum and the supremum can be interchanged. In more advanced mathematics the monotone convergence theorem usually refers to a fundamental result in measure theory due to Lebesgue and Beppo Levi that says that for sequences of non ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
