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Enrico Ducci
A Ducci sequence is a sequence of ''n''-tuples of integers, sometimes known as "the Diffy game", because it is based on sequences. Given an ''n''-tuple of integers (a_1,a_2,...,a_n), the next ''n''-tuple in the sequence is formed by taking the absolute differences of neighbouring integers: :(a_1,a_2,...,a_n) \rightarrow (, a_1-a_2, , , a_2-a_3, , ..., , a_n-a_1, )\, . Another way of describing this is as follows. Arrange ''n'' integers in a circle and make a new circle by taking the difference between neighbours, ignoring any minus signs; then repeat the operation. Ducci sequences are named after Enrico Ducci (1864 - 1940), the Italian mathematician who discovered in the 1930s that every such sequence eventually becomes periodic. Ducci sequences are also known as the Ducci map or the n-number game. Open problems in the study of these maps still remain. Properties From the second ''n''-tuple onwards, it is clear that every integer in each ''n''-tuple in a Ducci sequence is gre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
N-tuple
In mathematics, a tuple is a finite ordered list (sequence) of elements. An -tuple is a sequence (or ordered list) of elements, where is a non-negative integer. There is only one 0-tuple, referred to as ''the empty tuple''. An -tuple is defined inductively using the construction of an ordered pair. Mathematicians usually write tuples by listing the elements within parentheses "" and separated by a comma and a space; for example, denotes a 5-tuple. Sometimes other symbols are used to surround the elements, such as square brackets "nbsp; or angle brackets "⟨ ⟩". Braces "" are used to specify arrays in some programming languages but not in mathematical expressions, as they are the standard notation for sets. The term ''tuple'' can often occur when discussing other mathematical objects, such as vectors. In computer science, tuples come in many forms. Most typed functional programming languages implement tuples directly as product types, tightly associated with algebr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Difference Equation
In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter k that is independent of n; this number k is called the ''order'' of the relation. If the values of the first k numbers in the sequence have been given, the rest of the sequence can be calculated by repeatedly applying the equation. In ''linear recurrences'', the th term is equated to a linear function of the k previous terms. A famous example is the recurrence for the Fibonacci numbers, F_n=F_+F_ where the order k is two and the linear function merely adds the two previous terms. This example is a linear recurrence with constant coefficients, because the coefficients of the linear function (1 and 1) are constants that do not depend on n. For these recurrences, one can express the general term of the sequence as a closed-form expression of ... [...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] |
Finite Fields And Their Applications
Finite is the opposite of infinite. It may refer to: * Finite number (other) * Finite set, a set whose cardinality (number of elements) is some natural number * Finite verb, a verb form that has a subject, usually being inflected or marked for person and/or tense or aspect * "Finite", a song by Sara Groves from the album ''Invisible Empires'' See also * * Nonfinite (other) Nonfinite is the opposite of finite * a nonfinite verb A nonfinite verb is a derivative form of a verb unlike finite verbs. Accordingly, nonfinite verb forms are inflected for neither number nor person, and they cannot perform action as the root ... {{disambiguation fr:Fini it:Finito ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cyclotomic Polynomials
In mathematics, the ''n''th cyclotomic polynomial, for any positive integer ''n'', is the unique irreducible polynomial with integer coefficients that is a divisor of x^n-1 and is not a divisor of x^k-1 for any Its roots are all ''n''th primitive roots of unity e^ , where ''k'' runs over the positive integers not greater than ''n'' and coprime to ''n'' (and ''i'' is the imaginary unit). In other words, the ''n''th cyclotomic polynomial is equal to : \Phi_n(x) = \prod_\stackrel \left(x-e^\right). It may also be defined as the monic polynomial with integer coefficients that is the minimal polynomial over the field of the rational numbers of any primitive ''n''th-root of unity ( e^ is an example of such a root). An important relation linking cyclotomic polynomials and primitive roots of unity is :\prod_\Phi_d(x) = x^n - 1, showing that is a root of x^n - 1 if and only if it is a ''d''th primitive root of unity for some ''d'' that divides ''n''. Examples If ''n'' is a prime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods that attempt at finding approximate solutions of problems rather than the exact ones. Numerical analysis finds application in all fields of engineering and the physical sciences, and in the 21st century also the life and social sciences, medicine, business and even the arts. Current growth in computing power has enabled the use of more complex numerical analysis, providing detailed and realistic mathematical models in science and engineering. Examples of numerical analysis include: ordinary differential equations as found in celestial mechanics (predicting the motions of planets, stars and galaxies), numerical linear algebra in data analysis, and stochastic differential equations and Markov chains for simulating living ce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chaos Theory
Chaos theory is an interdisciplinary area of scientific study and branch of mathematics focused on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions, and were once thought to have completely random states of disorder and irregularities. Chaos theory states that within the apparent randomness of chaotic complex systems, there are underlying patterns, interconnection, constant feedback loops, repetition, self-similarity, fractals, and self-organization. The butterfly effect, an underlying principle of chaos, describes how a small change in one state of a deterministic nonlinear system can result in large differences in a later state (meaning that there is sensitive dependence on initial conditions). A metaphor for this behavior is that a butterfly flapping its wings in Brazil can cause a tornado in Texas. Small differences in initial conditions, such as those due to errors in measurements or due to rounding errors i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Non-linear Dynamics
In mathematics, a dynamical system is a system in which a function describes the time dependence of a point in an ambient space. Examples include the mathematical models that describe the swinging of a clock pendulum, the flow of water in a pipe, the random motion of particles in the air, and the number of fish each springtime in a lake. The most general definition unifies several concepts in mathematics such as ordinary differential equations and ergodic theory by allowing different choices of the space and how time is measured. Time can be measured by integers, by real or complex numbers or can be a more general algebraic object, losing the memory of its physical origin, and the space may be a manifold or simply a set, without the need of a smooth space-time structure defined on it. At any given time, a dynamical system has a state representing a point in an appropriate state space. This state is often given by a tuple of real numbers or by a vector in a geometrical manifo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language of mathematics, the set of integers is often denoted by the boldface or blackboard bold \mathbb. 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 natural numbers, \mathbb is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, , and are not. The integers form the smallest group and the smallest ring containing the natural numbers. In algebraic number theory, the integers are sometimes qualified as rational integers to distinguish them from the more general algebraic integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rule 90
In the mathematics, mathematical study of cellular automaton, cellular automata, Rule 90 is an elementary cellular automaton based on the exclusive or function. It consists of a one-dimensional array of cells, each of which can hold either a 0 or a 1 value. In each time step all values are simultaneously replaced by the exclusive or of their two neighboring values.. call it "the simplest non-trivial cellular automaton",. and it is described extensively in Stephen Wolfram's 2002 book ''A New Kind of Science''. When started from a single live cell, Rule 90 has a time-space diagram in the form of a Sierpiński triangle. The behavior of any other configuration can be explained as a superposition of copies of this pattern, combined using the exclusive or function. Any configuration with only finitely many nonzero cells becomes a replicator (cellular automaton), replicator that eventually fills the array with copies of itself. When Rule 90 is started from a random initial configuratio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wolfram Code
Wolfram code is a widely used numbering system for one-dimensional cellular automaton rules, introduced by Stephen Wolfram in a 1983 paper and popularized in his book ''A New Kind of Science''. The code is based on the observation that a table specifying the new state of each cell in the automaton, as a function of the states in its neighborhood, may be interpreted as a ''k''-digit number in the ''S''-ary positional number system, where ''S'' is the number of states that each cell in the automaton may have, ''k'' = S2''n'' + 1 is the number of neighborhood configurations, and ''n'' is the radius of the neighborhood. Thus, the Wolfram code for a particular rule is a number in the range from 0 to ''S''''S'' − 1, converted from ''S''-ary to decimal notation. It may be calculated as follows: # List all the ''S''2''n'' + 1 possible state configurations of the neighbourhood of a given cell. # Interpreting each configuration as a number as desc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
