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Josephus Permutation
In computer science and mathematics, the Josephus problem (or Josephus permutation) is a theoretical problem related to a certain counting-out game. A number of people are standing in a circle waiting to be executed. Counting begins at a specified point in the circle and proceeds around the circle in a specified direction. After a specified number of people are skipped, the next person is executed. The procedure is repeated with the remaining people, starting with the next person, going in the same direction and skipping the same number of people, until only one person remains, and is freed. The problem—given the number of people, starting point, direction, and number to be skipped—is to choose the position in the initial circle to avoid execution. History The problem is named after Flavius Josephus, a Jewish historian living in the 1st century. According to Josephus' account of the siege of Yodfat, he and his 40 soldiers were trapped in a cave by Roman soldiers. They chose ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josephus Problem 41 3
Flavius Josephus (; grc-gre, Ἰώσηπος, ; 37 – 100) was a first-century Romano-Jewish historian and military leader, best known for '' The Jewish War'', who was born in Jerusalem—then part of Roman Judea—to a father of priestly descent and a mother who claimed royal ancestry. He initially fought against the Romans during the First Jewish–Roman War as head of Jewish forces in Galilee, until surrendering in 67 AD to Roman forces led by Vespasian after the six-week siege of Yodfat. Josephus claimed the Jewish Messianic prophecies that initiated the First Jewish–Roman War made reference to Vespasian becoming Emperor of Rome. In response, Vespasian decided to keep Josephus as a slave and presumably interpreter. After Vespasian became Emperor in 69 AD, he granted Josephus his freedom, at which time Josephus assumed the emperor's family name of Flavius.Simon Claude Mimouni, ''Le Judaïsme ancien du VIe siècle avant notre ère au IIIe siècle de notre ère : De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josephus Problem 30 9
Flavius Josephus (; grc-gre, Ἰώσηπος, ; 37 – 100) was a first-century Romano-Jewish historian and military leader, best known for '' The Jewish War'', who was born in Jerusalem—then part of Roman Judea—to a father of priestly descent and a mother who claimed royal ancestry. He initially fought against the Romans during the First Jewish–Roman War as head of Jewish forces in Galilee, until surrendering in 67 AD to Roman forces led by Vespasian after the six-week siege of Yodfat. Josephus claimed the Jewish Messianic prophecies that initiated the First Jewish–Roman War made reference to Vespasian becoming Emperor of Rome. In response, Vespasian decided to keep Josephus as a slave and presumably interpreter. After Vespasian became Emperor in 69 AD, he granted Josephus his freedom, at which time Josephus assumed the emperor's family name of Flavius.Simon Claude Mimouni, ''Le Judaïsme ancien du VIe siècle avant notre ère au IIIe siècle de notre ère : De ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Josephus
Flavius Josephus (; grc-gre, Ἰώσηπος, ; 37 – 100) was a first-century Romano-Jewish historian and military leader, best known for '' The Jewish War'', who was born in Jerusalem—then part of Roman Judea—to a father of priestly descent and a mother who claimed royal ancestry. He initially fought against the Romans during the First Jewish–Roman War as head of Jewish forces in Galilee, until surrendering in 67 AD to Roman forces led by Vespasian after the six-week siege of Yodfat. Josephus claimed the Jewish Messianic prophecies that initiated the First Jewish–Roman War made reference to Vespasian becoming Emperor of Rome. In response, Vespasian decided to keep Josephus as a slave and presumably interpreter. After Vespasian became Emperor in 69 AD, he granted Josephus his freedom, at which time Josephus assumed the emperor's family name of Flavius.Simon Claude Mimouni, ''Le Judaïsme ancien du VIe siècle avant notre ère au IIIe siècle de notre ère : ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Computational Problems
In theoretical computer science, a computational problem is a problem that may be solved by an algorithm. For example, the problem of factoring :"Given a positive integer ''n'', find a nontrivial prime factor of ''n''." is a computational problem. A computational problem can be viewed as a set of ''instances'' or ''cases'' together with a, possibly empty, set of ''solutions'' for every instance/case. For example, in the factoring problem, the instances are the integers ''n'', and solutions are prime numbers ''p'' that are the nontrivial prime factors of ''n''. Computational problems are one of the main objects of study in theoretical computer science. The field of computational complexity theory attempts to determine the amount of resources ( computational complexity) solving a given problem will require and explain why some problems are intractable or undecidable. Computational problems belong to complexity classes that define broadly the resources (e.g. time, space/memory, ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cut-the-knot
Alexander Bogomolny (January 4, 1948 July 7, 2018) was a Soviet-born Israeli-American mathematician. He was Professor Emeritus of Mathematics at the University of Iowa, and formerly research fellow at the Moscow Institute of Electronics and Mathematics, senior instructor at Hebrew University and software consultant at Ben Gurion University. He wrote extensively about arithmetic, probability, algebra, geometry, trigonometry and mathematical games. He was known for his contribution to heuristics and mathematics education, creating and maintaining the mathematically themed educational website ''Cut-the-Knot'' for the Mathematical Association of America (MAA) Online. He was a pioneer in mathematical education on the internet, having started ''Cut-the-Knot'' in October 1996.Interview with Alexander ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Introduction To Algorithms
''Introduction to Algorithms'' is a book on computer programming by Thomas H. Cormen, Charles E. Leiserson, Ronald L. Rivest, and Clifford Stein. The book has been widely used as the textbook for algorithms courses at many universities and is commonly cited as a reference for algorithms in published papers, with over 10,000 citations documented on CiteSeerX. The book sold half a million copies during its first 20 years. Its fame has led to the common use of the abbreviation "CLRS" (Cormen, Leiserson, Rivest, Stein), or, in the first edition, "CLR" (Cormen, Leiserson, Rivest). In the preface, the authors write about how the book was written to be comprehensive and useful in both teaching and professional environments. Each chapter focuses on an algorithm, and discusses its design techniques and areas of application. Instead of using a specific programming language, the algorithms are written in pseudocode. The descriptions focus on the aspects of the algorithm itself, its ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Big-O Notation
Big ''O'' notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for ''Ordnung'', meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dynamic Programming
Dynamic programming is both a mathematical optimization method and a computer programming method. The method was developed by Richard Bellman in the 1950s and has found applications in numerous fields, from aerospace engineering to economics. In both contexts it refers to simplifying a complicated problem by breaking it down into simpler sub-problems in a recursive manner. While some decision problems cannot be taken apart this way, decisions that span several points in time do often break apart recursively. Likewise, in computer science, if a problem can be solved optimally by breaking it into sub-problems and then recursively finding the optimal solutions to the sub-problems, then it is said to have '' optimal substructure''. If sub-problems can be nested recursively inside larger problems, so that dynamic programming methods are applicable, then there is a relation between the value of the larger problem and the values of the sub-problems.Cormen, T. H.; Leiserson, C. E.; R ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bitwise Operators
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 sign ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Strong Induction
Mathematical induction is a method for proving that a statement ''P''(''n'') is true for every natural number ''n'', that is, that the infinitely many cases ''P''(0), ''P''(1), ''P''(2), ''P''(3), ... all hold. Informal metaphors help to explain this technique, such as falling dominoes or climbing a ladder: A proof by induction consists of two cases. The first, the base case, proves the statement for ''n'' = 0 without assuming any knowledge of other cases. The second case, the induction step, proves that ''if'' the statement holds for any given case ''n'' = ''k'', ''then'' it must also hold for the next case ''n'' = ''k'' + 1. These two steps establish that the statement holds for every natural number ''n''. The base case does not necessarily begin with ''n'' = 0, but often with ''n'' = 1, and possibly with any fixed natural number ''n'' = ''N'', establishing the truth of the statement for all natu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counting
Counting is the process of determining the number of elements of a finite set of objects, i.e., determining the size of a set. The traditional way of counting consists of continually increasing a (mental or spoken) counter by a unit for every element of the set, in some order, while marking (or displacing) those elements to avoid visiting the same element more than once, until no unmarked elements are left; if the counter was set to one after the first object, the value after visiting the final object gives the desired number of elements. The related term '' enumeration'' refers to uniquely identifying the elements of a finite (combinatorial) set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one; for example, when counting money, counting out change, "counting by twos" (2, 4, 6, 8, 10, 12, ...), or "counting by fives" (5, 10, 15, 20, 25, ...). There is archaeological evidence suggesting that humans have been count ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
