Sparse Ruler

	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 ''maximal'' if there is no complete sparse ruler of length $L+1$ with $m$ marks. A sparse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Golomb Ruler In mathematics, a Golomb ruler is a 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 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 if it does, it is called a '' perfect'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Gauge Block Gauge blocks (also known as gage blocks, Johansson gauges, slip gauges, or Jo blocks) are a system for producing precision lengths. The individual gauge block is a metal or ceramic block that has been precision ground and lapped to a specific thickness. Gauge blocks come in sets of blocks with a range of standard lengths. In use, the blocks are stacked to make up a desired length (or height). An important feature of gauge blocks is that they can be joined together with very little dimensional uncertainty. The blocks are joined by a sliding process called ''wringing'', which causes their ultra-flat surfaces to cling together. A small number of gauge blocks can be used to create accurate lengths within a wide range. By using three blocks at a time taken from a set of 30 blocks, one may create any of the 1000 lengths from 3.000 to 3.999 mm in 0.001 mm steps (or .3000 to .3999 inches in 0.0001 inch steps). Gauge blocks were invented in 1896 by Swedish machinist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	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 $(a_1,a_2,a_3,a_4)=(0,1,3,7)$ . To verify this, we ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Number Theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Combinatorics Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many applications ranging from logic to statistical physics and from evolutionary biology to computer science. Combinatorics is well known for the breadth of the problems it tackles. Combinatorial problems arise in many areas of pure mathematics, notably in algebra, probability theory, topology, and geometry, as well as in its many application areas. Many combinatorial questions have historically been considered in isolation, giving an ''ad hoc'' solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general theoretical methods were developed, making combinatorics into an independent branch of mathematics in its own right. One of the oldest and most accessible parts of combinatorics is gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]

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