Perfect Ruler

	Perfect Ruler A perfect ruler of length \ell is a ruler with integer markings a_1=0 < a_2 < \dots < a_n=\ell, for which there exists an integer $m$ such that any positive integer $k\leq m$ is uniquely expressed as the difference $k=a_i-a_j$ for some $i,j$ . This is referred to as an $m$ -perfect ruler. An optimal perfect ruler is one of the smallest length for fixed values of $m$ and $n$ . Example A 4-perfect ruler of length $7$ is given by $(a_1,a_2,a_3,a_4 ... [...More Info...] [...Related Items...]$ OR: [Wikipedia] [Google] [Baidu]
picture info	Ruler A ruler, sometimes called a rule, scale, line gauge, or metre/meter stick, is an instrument used to make length measurements, whereby a length is read from a series of markings called "rules" along an edge of the device. Usually, the instrument is rigid and the edge itself is a straightedge ("ruled straightedge"), which additionally allows one to draw straighter lines. Rulers are an important tool in geometry, geography and mathematics. They have been used since at least 2650 BC. Variants Rulers have long been made from different materials and in multiple sizes. Historically, they were mainly wood but plastics have also been used. They can be created with length markings instead of being wikt:scribe, scribed. Metal is also used for more durable rulers for use in the workshop; sometimes a metal edge is embedded into a wooden desk ruler to preserve the edge when used for straight-line cutting. Typically in length, though some can go up to 100 cm, it is useful for a ruler to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set (mathematics), set of all integers is often denoted by the boldface or blackboard bold The set of natural numbers \mathbb is a subset of \mathbb, which in turn is a subset of the set of all rational numbers \mathbb, itself a subset of the real numbers \mathbb. Like the set of natural numbers, the set of integers \mathbb is Countable set, countably infinite. An integer may be regarded as a real number that can be written without a fraction, fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , 5/4, and Square root of 2, are not. The integers form the smallest Group (mathematics), group and the smallest ring (mathematics), ring containing the natural numbers. In algebraic number theory, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Positive Integer In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like jersey numbers on a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Difference (mathematics) Subtraction (which is signified by the minus sign, –) is one of the four Arithmetic#Arithmetic operations, arithmetic operations along with addition, multiplication and Division (mathematics), division. Subtraction is an operation that represents removal of objects from a collection. For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken away, resulting in a total of 3 peaches. Therefore, the ''difference'' of 5 and 2 is 3; that is, . While primarily associated with natural numbers in arithmetic, subtraction can also represent removing or decreasing physical and abstract quantities using different kinds of objects including negative numbers, Fraction (mathematics), fractions, irrational numbers, Euclidean vector, vectors, decimals, functions, and matrices. In a sense, subtraction is the inverse of addition. That is, if and only if . In words: the difference of two numbers is the number that gives the first one when added to the second one. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Optimization (mathematics) Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries. In the more general approach, an optimization problem consists of maxima and minima, maximizing or minimizing a Function of a real variable, real function by systematically choosing Argument of a function, input values from within an allowed set and computing the Value (mathematics), value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics. Optimization problems Opti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Golomb Ruler In mathematics, a Golomb ruler is a set (mathematics), set of marks at integer positions along a ruler such that no two pairs of marks are the same distance apart. The number of marks on the ruler is its ''order'', and the largest distance between two of its marks is its ''length''. Translation and reflection of a Golomb ruler are considered trivial, so the smallest mark is customarily put at 0 and the next mark at the smaller of its two possible values. Golomb rulers can be viewed as a one-dimensional special case of Costas arrays. The Golomb ruler was named for Solomon W. Golomb and discovered independently by and . Sophie Piccard also published early research on these sets, in 1939, stating as a theorem the claim that two Golomb rulers with the same distance set must be congruence (geometry), congruent. This turned out to be false for six-point rulers, but true otherwise. There is no requirement that a Golomb ruler be able to measure ''all'' distances up to its length, but ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Sparse Ruler A sparse ruler is a ruler in which some of the distance marks may be missing. More abstractly, a sparse ruler of length L with m marks is a sequence of integers a_1, a_2, ..., a_m where 0 = a_1 < a_2 < ... < a_m = L. The marks $a_1$ and $a_m$ correspond to the ends of the ruler. In order to measure the distance $K$ , with $0\le K\le L$ there must be marks $a_i$ and $a_j$ such that $a_j-a_i=K$ . A ''complete'' sparse ruler allows one to measure any integer distance up to its full length. A complete sparse ruler is called ''minimal'' if there is no complete sparse ruler of length $L$ with $m-1$ marks. In other words, if any of the marks is removed one can no longer measure all of the distances, even if the marks could be rearranged. A complete sparse ruler is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	All-interval Tetrachord An all-interval tetrachord is a tetrachord, a collection of four pitch classes, containing all six interval classes. There are only two possible all-interval tetrachords (to within inversion), when expressed in prime form. In set theory notation, these are ,1,4,6(4-Z15) and ,1,3,7(4-Z29). Their inversions are ,2,5,6(4-Z15b) and ,4,6,7(4-Z29b). The interval vector for all all-interval tetrachords is ,1,1,1,1,1 Table of interval classes as relating to all-interval tetrachords In the examples below, the tetrachords ,1,4,6and ,1,3,7are built on E. Use in modern music The unique qualities of the all-interval tetrachord have made it very popular in 20th-century music. Composers including Frank Bridge, Elliott Carter ( First String Quartet) and George Perle used it extensively. See also * All-interval twelve-tone row * All-trichord hexachord * Perfect ruler Serialism Trichord References External links The All-Interval Tetrachord, A Musical Application of Almost D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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