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Komornik–Loreti Constant
In the mathematical theory of non-standard positional numeral systems, the Komornik–Loreti constant is a mathematical constant that represents the smallest base ''q'' for which the number 1 has a unique representation, called its ''q''-development. The constant is named after Vilmos Komornik and Paola Loreti, who defined it in 1998. Definition Given a real number ''q'' > 1, the series : x = \sum_^\infty a_n q^ is called the ''q''-expansion, or \beta-expansion, of the positive real number ''x'' if, for all n \ge 0, 0 \le a_n \le \lfloor q \rfloor, where \lfloor q \rfloor is the floor function and a_n need not be an integer. Any real number x such that 0 \le x \le q \lfloor q \rfloor /(q-1) has such an expansion, as can be found using the greedy algorithm. The special case of x = 1, a_0 = 0, and a_n = 0 or 1 is sometimes called a q-development. a_n = 1 gives the only 2-development. However, for almost all 1 < q < 2, there are an infinite number of different [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-integer Representation
A non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix ''β'' > 1, the value of :x = d_n \dots d_2d_1d_0.d_d_\dots d_ is :\begin x &= \beta^nd_n + \cdots + \beta^2d_2 + \beta d_1 + d_0 \\ &\qquad + \beta^d_ + \beta^d_ + \cdots + \beta^d_. \end The numbers ''d''''i'' are non-negative integers less than ''β''. This is also known as a ''β''-expansion, a notion introduced by and first studied in detail by . Every real number has at least one (possibly infinite) ''β''-expansion. The set of all ''β''-expansions that have a finite representation is a subset of the ring Z 'β'', ''β''−1 There are applications of ''β''-expansions in coding theory and models of quasicrystals. Construction ''β''-expansions are a generalization of decimal expansions. While infinite decimal expansions are not unique (for example, 1.000... = 0.999...), all finite decimal expansions are unique. However, even fin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Constant
A mathematical constant is a number whose value is fixed by an unambiguous definition, often referred to by a special symbol (e.g., an Letter (alphabet), alphabet letter), or by mathematicians' names to facilitate using it across multiple mathematical problems. Constants arise in many areas of mathematics, with constants such as and pi, occurring in such diverse contexts as geometry, number theory, statistics, and calculus. Some constants arise naturally by a fundamental principle or intrinsic property, such as the ratio between the circumference and diameter of a circle (). Other constants are notable more for historical reasons than for their mathematical properties. The more popular constants have been studied throughout the ages and computed to many decimal places. All named mathematical constants are Definable real number, definable numbers, and usually are also computable numbers (Chaitin's constant being a significant exception). Basic mathematical constants These a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vilmos Komornik
Vilmos () is a masculine given name, the Hungarian form of the Germanic Wilhelm gained through the Latin Vilhelmus. People named Vilmos In sport * Vilmos Szabó (1964–), Hungarian fencer * Vilmos Orbán (1992–), Hungarian footballer * Vilmos Vanczák (1983–), Hungarian footballer * Vilmos Göttler (1951–), Hungarian equestrian * Vilmos Sebők (1973–), Hungarian footballer * Vilmos Tölgyesi (1931–1970), Hungarian runner * Vilmos Galló (1996–), Hungarian ice hockey player * Vilmos Földes (1984–), Hungarian pool player * Vilmos Énekes (1915–1990), Hungarian boxer * Vilmos Kohut (1906–1986), Hungarian footballer * Vilmos Radasics (1983–), Hungarian BMX racer * Vilmos Iváncsó (1939–1997), Hungarian volleyball player * Vilmos Jakab (1952–2024), Hungarian boxer * Vilmos Telinger (1950–2013), Hungarian footballer * Vilmos Zombori (1906–1993), Hungarian footballer * Vilmos Lóczi (1925–1991), Hungarian basketball player * Vilmos Var ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paola Loreti
Paola Loreti is an Italian mathematician, and a professor of mathematical analysis at Sapienza University of Rome. She is known for her research on Fourier analysis, control theory, and non-integer representations. The Komornik–Loreti constant, the smallest non-integer base for which the representation of 1 is unique, is named after her and Vilmos Komornik. Loreti earned a laurea In Italy, the ''laurea'' is the main post-secondary academic degree. The name originally referred literally to the laurel wreath, since ancient times a sign of honor and now worn by Italian students right after their official graduation ceremo ... from Sapienza University in 1984. Her dissertation, ''Programmazione dinamica ed equazione di Bellman'' dynamic_programming.html" ;"title="'dynamic programming">'dynamic programming and the Bellman equation''] was supervised by Italo Capuzzo-Dolcetta. With Vilmos Komornik, Loreti is the author of the book ''Fourier Series in Control Theory'' (Springer, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Floor And Ceiling Functions
In mathematics, the floor function is the function that takes as input a real number , and gives as output the greatest integer less than or equal to , denoted or . Similarly, the ceiling function maps to the least integer greater than or equal to , denoted or . For example, for floor: , , and for ceiling: , and . The floor of is also called the integral part, integer part, greatest integer, or entier of , and was historically denoted (among other notations). However, the same term, ''integer part'', is also used for truncation towards zero, which differs from the floor function for negative numbers. For an integer , . Although and produce graphs that appear exactly alike, they are not the same when the value of is an exact integer. For example, when , . However, if , then , while . Notation The ''integral part'' or ''integer part'' of a number ( in the original) was first defined in 1798 by Adrien-Marie Legendre in his proof of the Legendre's formula. Carl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Greedy Algorithm
A greedy algorithm is any algorithm that follows the problem-solving heuristic of making the locally optimal choice at each stage. In many problems, a greedy strategy does not produce an optimal solution, but a greedy heuristic can yield locally optimal solutions that approximate a globally optimal solution in a reasonable amount of time. For example, a greedy strategy for the travelling salesman problem (which is of high computational complexity) is the following heuristic: "At each step of the journey, visit the nearest unvisited city." This heuristic does not intend to find the best solution, but it terminates in a reasonable number of steps; finding an optimal solution to such a complex problem typically requires unreasonably many steps. In mathematical optimization, greedy algorithms optimally solve combinatorial problems having the properties of matroids and give constant-factor approximations to optimization problems with the submodular structure. Specifics Greedy algori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Thue–Morse Sequence
In mathematics, the Thue–Morse or Prouhet–Thue–Morse sequence is the binary sequence (an infinite sequence of 0s and 1s) that can be obtained by starting with 0 and successively appending the Boolean complement of the sequence obtained thus far. It is sometimes called the fair share sequence because of its applications to fair division or parity sequence. The first few steps of this procedure yield the strings 0, 01, 0110, 01101001, 0110100110010110, and so on, which are the prefixes of the Thue–Morse sequence. The full sequence begins: :01101001100101101001011001101001.... The sequence is named after Axel Thue, Marston Morse and (in its extended form) Eugène Prouhet. Definition There are several equivalent ways of defining the Thue–Morse sequence. Direct definition To compute the ''n''th element ''tn'', write the number ''n'' in binary. If the number of ones in this binary expansion is odd then ''tn'' = 1, if even then ''tn'' = 0. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Number
A binary number is a number expressed in the Radix, base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computer, computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thoma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transcendental Number
In mathematics, a transcendental number is a real or complex number that is not algebraic: that is, not the root of a non-zero polynomial with integer (or, equivalently, rational) coefficients. The best-known transcendental numbers are and . The quality of a number being transcendental is called transcendence. Though only a few classes of transcendental numbers are known, partly because it can be extremely difficult to show that a given number is transcendental. Transcendental numbers are not rare: indeed, almost all real and complex numbers are transcendental, since the algebraic numbers form a countable set, while the set of real numbers and the set of complex numbers are both uncountable sets, and therefore larger than any countable set. All transcendental real numbers (also known as real transcendental numbers or transcendental irrational numbers) are irrational numbers, since all rational numbers are algebraic. The converse is not true: Not all irrational numbers are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler-Mascheroni Constant
Euler's constant (sometimes called the Euler–Mascheroni constant) is a mathematical constant, usually denoted by the lowercase Greek letter gamma (), defined as the limiting difference between the harmonic series and the natural logarithm, denoted here by : \begin \gamma &= \lim_\left(-\log n + \sum_^n \frac1\right)\\ px&=\int_1^\infty\left(-\frac1x+\frac1\right)\,\mathrm dx. \end Here, represents the floor function. The numerical value of Euler's constant, to 50 decimal places, is: History The constant first appeared in a 1734 paper by the Swiss mathematician Leonhard Euler, titled ''De Progressionibus harmonicis observationes'' (Eneström Index 43), where he described it as "worthy of serious consideration". Euler initially calculated the constant's value to 6 decimal places. In 1781, he calculated it to 16 decimal places. Euler used the notations and for the constant. The Italian mathematician Lorenzo Mascheroni attempted to calculate the constant to 32 dec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Word
A Fibonacci word is a specific sequence of Binary numeral system, binary digits (or symbols from any two-letter Alphabet (formal languages), alphabet). The Fibonacci word is formed by repeated concatenation in the same way that the Fibonacci numbers are formed by repeated addition. It is a paradigmatic example of a Sturmian word and specifically, a morphic word. The name "Fibonacci word" has also been used to refer to the members of a formal language ''L'' consisting of strings of zeros and ones with no two repeated ones. Any prefix of the specific Fibonacci word belongs to ''L'', but so do many other strings. ''L'' has a Fibonacci number of members of each possible length. Definition Let S_0 be "0" and S_1 be "01". Now S_n = S_S_ (the concatenation of the previous sequence and the one before that). The infinite Fibonacci word is the limit S_, that is, the (unique) infinite sequence that contains each S_n, for finite n, as a prefix. Enumerating items from the above definit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
