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Moser–de Bruijn Sequence
In number theory, the Moser–de Bruijn sequence is an integer sequence named after Leo Moser and Nicolaas Govert de Bruijn, consisting of the sums of distinct powers of 4, or equivalently the numbers whose binary representations are nonzero only in even positions. These numbers grow in proportion to the square numbers, and are the squares for a modified form of arithmetic without Carry (arithmetic), carrying. When the values in the sequence are doubled, their differences are all non-square. Every non-negative integer has a unique representation as the sum of a sequence member and a doubled sequence member. This decomposition into sums can be used to define a bijection between the integers and pairs of integers, to define coordinates for the Z-order curve, and to construct inverse pairs of transcendental numbers with simple decimal representations. A simple recurrence relation allows values of the Moser–de Bruijn sequence to be calculated from earlier values, and can be used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Bruijn Sequence
In combinatorial mathematics, a de Bruijn sequence of order ''n'' on a size-''k'' alphabet ''A'' is a cyclic sequence in which every possible length-''n'' string on ''A'' occurs exactly once as a substring (i.e., as a ''contiguous'' subsequence). Such a sequence is denoted by and has length , which is also the number of distinct strings of length ''n'' on ''A''. Each of these distinct strings, when taken as a substring of , must start at a different position, because substrings starting at the same position are not distinct. Therefore, must have ''at least'' symbols. And since has ''exactly'' symbols, De Bruijn sequences are optimally short with respect to the property of containing every string of length ''n'' at least once. The number of distinct de Bruijn sequences is :\dfrac. The sequences are named after the Dutch mathematician Nicolaas Govert de Bruijn, who wrote about them in 1946. As he later wrote, the existence of de Bruijn sequences for each order together ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Negabinary
A negative base (or negative radix) may be used to construct a non-standard positional numeral system. Like other place-value systems, each position holds multiples of the appropriate power of the system's base; but that base is negative—that is to say, the base is equal to for some natural number (). Negative-base systems can accommodate all the same numbers as standard place-value systems, but both positive and negative numbers are represented without the use of a minus sign (or, in computer representation, a sign bit); this advantage is countered by an increased complexity of arithmetic operations. The need to store the information normally contained by a negative sign often results in a negative-base number being one digit longer than its positive-base equivalent. The common names for negative-base positional numeral systems are formed by prefixing ''nega-'' to the name of the corresponding positive-base system; for example, negadecimal (base −10) corresponds to decima ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Irrational Number
In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integers. When the ratio of lengths of two line segments is an irrational number, the line segments are also described as being '' incommensurable'', meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself. Among irrational numbers are the ratio of a circle's circumference to its diameter, Euler's number ''e'', the golden ratio ''φ'', and the square root of two. In fact, all square roots of natural numbers, other than of perfect squares, are irrational. Like all real numbers, irrational numbers can be expressed in positional notation, notably as a decimal number. In the cas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subtract A Square
Subtract-a-square (also referred to as take-a-square) is a two-player mathematical subtraction game. It is played by two people with a pile of coins (or other tokens) between them. The players take turns removing coins from the pile, always removing a non-zero square number of coins. The game is usually played as a '' normal play'' game, which means that the player who removes the last coin wins. It is an impartial game, meaning that the set of moves available from any position does not depend on whose turn it is. Solomon W. Golomb credits the invention of this game to Richard A. Epstein.. Example A normal play game starting with 13 coins is a win for the first player provided they start with a subtraction of 1: player 1: 13 - 1*1 = 12 Player 2 now has three choices: subtract 1, 4 or 9. In each of these cases, player 1 can ensure that within a few moves the number 2 gets passed on to player 2: player 2: 12 - 1*1 = 11 player 2: 12 - 2*2 = 8 player 2: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subtraction Game
In combinatorial game theory, a subtraction game is an abstract strategy game whose state can be represented by a natural number or vector of numbers (for instance, the numbers of game tokens in piles of tokens, or the positions of pieces on board) and in which the allowed moves reduce these numbers., "Subtraction games", pp. 83–86. Often, the moves of the game allow any number to be reduced by subtracting a value from a specified ''subtraction set'', and different subtraction games vary in their subtraction sets. These games also vary in whether the last player to move wins (the normal play convention) or loses (misère play convention). Another winning convention that has also been used is that a player who moves to a position with all numbers zero wins, but that any other position with no moves possible is a draw. Examples Examples of notable subtraction games include the following: * Nim is a game whose state consists of multiple piles of tokens, such as coins or matchsticks, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
P-adic Number
In mathematics, the -adic number system for any prime number extends the ordinary arithmetic of the rational numbers in a different way from the extension of the rational number system to the real and complex number systems. The extension is achieved by an alternative interpretation of the concept of "closeness" or absolute value. In particular, two -adic numbers are considered to be close when their difference is divisible by a high power of : the higher the power, the closer they are. This property enables -adic numbers to encode congruence information in a way that turns out to have powerful applications in number theory – including, for example, in the famous proof of Fermat's Last Theorem by Andrew Wiles. These numbers were first described by Kurt Hensel in 1897, though, with hindsight, some of Ernst Kummer's earlier work can be interpreted as implicitly using -adic numbers.Translator's introductionpage 35 "Indeed, with hindsight it becomes apparent that a d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ordered Pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In contrast, the unordered pair equals the unordered pair .) Ordered pairs are also called 2-tuples, or sequences (sometimes, lists in a computer science context) of length 2. Ordered pairs of scalars are sometimes called 2-dimensional vectors. (Technically, this is an abuse of terminology since an ordered pair need not be an element of a vector space.) The entries of an ordered pair can be other ordered pairs, enabling the recursive definition of ordered ''n''-tuples (ordered lists of ''n'' objects). For example, the ordered triple (''a'',''b'',''c'') can be defined as (''a'', (''b'',''c'')), i.e., as one pair nested in another. In the ordered pair (''a'', ''b''), the object ''a'' is called the ''first entry'', and the object ''b'' the '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitwise Operation
In computer programming, a bitwise operation operates on a bit string, a bit array or a binary numeral (considered as a bit string) at the level of its individual bits. It is a fast and simple action, basic to the higher-level arithmetic operations and directly supported by the processor. Most bitwise operations are presented as two-operand instructions where the result replaces one of the input operands. On simple low-cost processors, typically, bitwise operations are substantially faster than division, several times faster than multiplication, and sometimes significantly faster than addition. While modern processors usually perform addition and multiplication just as fast as bitwise operations due to their longer instruction pipelines and other architectural design choices, bitwise operations do commonly use less power because of the reduced use of resources. Bitwise operators In the explanations below, any indication of a bit's position is counted from the right (least signi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sidon Sequence
In number theory, a Sidon sequence is a sequence A=\ of natural numbers in which all pairwise sums a_i+a_j (for i\le j) are different. Sidon sequences are also called Sidon sets; they are named after the Hungarian mathematician Simon Sidon, who introduced the concept in his investigations of Fourier series. The main problem in the study of Sidon sequences, posed by Sidon, is to find the maximum number of elements that a Sidon sequence can contain, up to some bound x. Despite a large body of research, the question remained unsolved. Early results Paul Erdős and Pál Turán proved that, for every x>0, the number of elements smaller than x in a Sidon sequence is at most \sqrt+O(\sqrt . Several years earlier, James Singer had constructed Sidon sequences with \sqrt(1-o(1)) terms less than ''x''. Infinite Sidon sequences Erdős also showed that, for any particular infinite Sidon sequence A with A(x) denoting the number of its elements up to x, \liminf_ \frac\leq 1. That is, infinit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imre Z
Imre is a Hungarian masculine first name, which is also in Estonian use, where the corresponding name day is 10 April. It has been suggested that it relates to the name Emeric, Emmerich or Heinrich. Its English equivalents are Emery and Henry. Bearers of the name include the following (who generally held Hungarian nationality, unless otherwise noted): *Imre Antal (1935–2008), pianist *Imre Bajor (1957–2014), actor * Imre Bebek (d. 1395), baron *Imre Bródy (1891–1944), physicist * Imre Bujdosó (b. 1959), Olympic fencer *Imre Csáky (cardinal) (1672–1732), Roman Catholic cardinal * Imre Csermelyi (b. 1988), football player *Imre Cseszneky (1804–1874), agriculturist and patriot *Imre Csiszár (b. 1938), mathematician * Imre Csösz (b. 1969), Olympic judoka *Imre Czobor (1520–1581), Noble and statesman *Imre Czomba (b. 1972), Composer and musician *Imre Deme (b. 1983), football player *Imre Erdődy (1889–1973), Olympic gymnast * Imre Farkas (1879–1976), musician ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Furstenberg–Sárközy Theorem
In mathematics, a square-difference-free set is a set of natural numbers, no two of which differ by a square number. Hillel Furstenberg and András Sárközy proved in the late 1970s the Furstenberg–Sárközy theorem of additive number theory showing that, in a certain sense, these sets cannot be very large. In the game of subtract a square, the positions where the next player loses form a square-difference-free set. Another square-difference-free set is obtained by doubling the Moser–de Bruijn sequence. The best known upper bound on the size of a square-difference-free set of numbers up to n is only slightly sublinear, but the largest known sets of this form are significantly smaller, of size \approx n^. Closing the gap between these upper and lower bounds remains an open problem. The sublinear size bounds on square-difference-free sets can be generalized to sets where certain other polynomials are forbidden as differences between pairs of elements. Example An example of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logical Conjunction
In logic, mathematics and linguistics, And (\wedge) is the truth-functional operator of logical conjunction; the ''and'' of a set of operands is true if and only if ''all'' of its operands are true. The logical connective that represents this operator is typically written as \wedge or . A \land B is true if and only if A is true and B is true, otherwise it is false. An operand of a conjunction is a conjunct. Beyond logic, the term "conjunction" also refers to similar concepts in other fields: * In natural language, the denotation of expressions such as English "and". * In programming languages, the short-circuit and control structure. * In set theory, intersection. * In lattice theory, logical conjunction ( greatest lower bound). * In predicate logic, universal quantification. Notation And is usually denoted by an infix operator: in mathematics and logic, it is denoted by \wedge, or ; in electronics, ; and in programming languages, &, &&, or and. In Jan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
