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Monotone Dualization
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See: monophony. 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 American doo-wop voca ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monophony
In music, monophony is the simplest of musical textures, consisting of a melody (or "tune"), typically sung by a single singer or played by a single instrument player (e.g., a flute player) without accompanying harmony or chords. Many folk songs and traditional songs are monophonic. A melody is also considered to be monophonic if a group of singers (e.g., a choir) sings the same melody together at the unison (exactly the same pitch) or with the same melody notes duplicated at the octave (such as when men and women sing together). If an entire melody is played by two or more instruments or sung by a choir with a fixed interval, such as a perfect fifth, it is also said to be monophony (or "monophonic"). The musical texture of a song or musical piece is determined by assessing whether varying components are used, such as an accompaniment part or polyphonic melody lines (two or more independent lines). In the Early Middle Ages, the earliest Christian songs, called plainchant (a we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Preferences
In economics, an agent's preferences are said to be weakly monotonic if, given a consumption bundle x, the agent prefers all consumption bundles y that have more of all goods. That is, y \gg x implies y\succ x. An agent's preferences are said to be strongly monotonic if, given a consumption bundle x, the agent prefers all consumption bundles y that have more of at least one good, and not less in any other good. That is, y\geq x and y\neq x imply y\succ x. This definition defines monotonic increasing preferences. Monotonic decreasing preferences can often be defined to be compatible with this definition. For instance, an agent's preferences for pollution may be monotonic decreasing (less pollution is better). In this case, the agent's preferences for lack of pollution are monotonic increasing. Much of consumer theory relies on a weaker assumption, local nonsatiation. An example of preferences which are weakly monotonic but not strongly monotonic are those represented by a Leontief ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonicity (mechanism Design)
In mechanism design, monotonicity is a property of a social choice function. It is a necessary condition for being able to implement the function using a strategyproof mechanism. Its verbal description is: In other words: Notation There is a set X of possible outcomes. There are n agents which have different valuations for each outcome. The valuation of agent i is represented as a function: v_i : X \longrightarrow R_+ which expresses the value it assigns to each alternative. The vector of all value-functions is denoted by v. For every agent i, the vector of all value-functions of the ''other'' agents is denoted by v_. So v \equiv (v_i,v_). A social choice function is a function that takes as input the value-vector v and returns an outcome x\in X. It is denoted by \text(v) or \text(v_i,v_). In mechanisms without money A social choice function satisfies the strong monotonicity property (SMON) if for every agent i and every v_i,v_i',v_, if: x = \text(v_i, v_) x' = \text ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonicity Criterion
The monotonicity criterion is a voting system criterion used to evaluate both single and multiple winner ranked voting systems. A ranked voting system is monotonic if it is neither possible to prevent the election of a candidate by ranking them higher on some of the ballots, nor possible to elect an otherwise unelected candidate by ranking them lower on some of the ballots (while nothing else is altered on any ballot).D R Woodall"Monotonicity and Single-Seat Election Rules" ''Voting matters'', Issue 6, 1996 That is to say, in single winner elections no winner is harmed by up-ranking and no loser is helped by down-ranking. Douglas Woodall called the criterion mono-raise. Raising a candidate on some ballots ''while changing'' the orders of other candidates does ''not'' constitute a failure of monotonicity. E.g., harming candidate by changing some ballots from to would violate the monotonicity criterion, while harming candidate by changing some ballots from to would not. The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Resource Monotonicity
Resource monotonicity (RM; aka aggregate monotonicity) is a principle of fair division. It says that, if there are more resources to share, then all agents should be weakly better off; no agent should lose from the increase in resources. The RM principle has been studied in various division problems. Allocating divisible resources Single homogeneous resource, general utilities Suppose society has m units of some homogeneous divisible resource, such as water or flour. The resource should be divided among n agents with different utilities. The utility of agent i is represented by a function u_i; when agent i receives y_i units of resource, he derives from it a utility of u_i(y_i). Society has to decide how to divide the resource among the agents, i.e, to find a vector y_1,\dots,y_n such that: y_1+\cdots+y_n = m. Two classic allocation rules are the egalitarian rule - aiming to equalize the utilities of all agents (equivalently: maximize the minimum utility), and the utilitari ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Class Theorem
In measure theory and probability, the monotone class theorem connects monotone classes and sigma-algebras. The theorem says that the smallest monotone class containing an algebra of sets G is precisely the smallest -algebra containing G. It is used as a type of transfinite induction to prove many other theorems, such as Fubini's theorem. Definition of a monotone class A ' is a family (i.e. class) M of sets that is closed under countable monotone unions and also under countable monotone intersections. Explicitly, this means M has the following properties: # if A_1, A_2, \ldots \in M and A_1 \subseteq A_2 \subseteq \cdots then \bigcup_^ A_i \in M, and # if B_1, B_2, \ldots \in M and B_1 \supseteq B_2 \supseteq \cdots then \bigcap_^ B_i \in M. Monotone class theorem for sets Monotone class theorem for functions Proof The following argument originates in Rick Durrett's Probability: Theory and Examples. Results and applications As a corollary, if G is a r ... [...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 convergence of monotonic sequences (sequences that are non-decreasing or non-increasing) that are also bounded. Informally, the theorems state that if a sequence is increasing and bounded above by a supremum, then the sequence will converge to the supremum; in the same way, if a sequence is decreasing and is bounded below by an infimum, it will converge to the infimum. Convergence of a monotone sequence of real numbers Lemma 1 If a sequence of real numbers is increasing and bounded above, then its supremum is the limit. Proof Let (a_n)_ be such a sequence, and let \ be the set of terms of (a_n)_ . By assumption, \ is non-empty and bounded above. By the least-upper-bound property of real numbers, c = \sup_n \ exists and is finite. Now, for every \varepsilon > 0, there exists N such that a_N > c - \varepsilon , since otherwise c - \varepsilon is an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotone Polygon
In geometry, a polygon ''P'' in the plane is called monotone with respect to a straight line ''L'', if every line orthogonal to ''L'' intersects the boundary of ''P'' at most twice. Similarly, a polygonal chain ''C'' is called monotone with respect to a straight line ''L'', if every line orthogonal to ''L'' intersects ''C'' at most once. For many practical purposes this definition may be extended to allow cases when some edges of ''P'' are orthogonal to ''L'', and a simple polygon may be called monotone if a line segment that connects two points in ''P'' and is orthogonal to ''L'' lies completely in ''P''. Following the terminology for monotone functions, the former definition describes polygons strictly monotone with respect to ''L''. 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 th ... [...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 and only if it is either entirely non-increasing, or entirely non-decreasing. 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 called ''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\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monotonicity Of Entailment
Monotonicity of entailment is a property of many logical systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. In sequent calculi this property can be captured by an inference rule called weakening, or sometimes thinning, and in such systems one may say that entailment is monotone if and only if the rule is admissible. Logical systems with this property are occasionally called ''monotonic logics'' in order to differentiate them from non-monotonic logics. Weakening rule To illustrate, consider the natural deduction sequent: Γ \vdash C That is, on the basis of a list of assumptions Γ, one can prove C. Weakening, by adding an assumption A, allows one to conclude: Γ, A \vdash C For example, the syllogism "All men are mortal. Socrates is a man. Therefore Socrates is mortal." can be weakened by adding a premise: "All men are mortal. Socrates is a man. Cows produce milk. Therefore Socrates is mortal." The validity of th ... [...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 ''infinit ... [...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 ''infinit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
