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Constant-recursive Sequence
In mathematics and theoretical computer science, a constant-recursive sequence is an infinite sequence of numbers where each number in the sequence is equal to a fixed linear combination of one or more of its immediate predecessors. A constant-recursive sequence is also known as a linear recurrence sequence, linear-recursive sequence, linear-recurrent sequence, a C-finite sequence, or a solution to a linear recurrence with constant coefficients. The most famous example of a constant-recursive sequence is the Fibonacci sequence 0, 1, 1, 2, 3, 5, 8, 13, \ldots, in which each number is the sum of the previous two. The power of two sequence 1, 2, 4, 8, 16, \ldots is also constant-recursive because each number is the sum of twice the previous number. The square number sequence 0, 1, 4, 9, 16, 25, \ldots is also constant-recursive. However, not all sequences are constant-recursive; for example, the factorial number sequence 1, 1, 2, 6, 24, 120, \ldots is not constant-recursive. Al ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fibonacci Sequence
In mathematics, the Fibonacci numbers, commonly denoted , form a integer sequence, sequence, the Fibonacci sequence, in which each number is the sum of the two preceding ones. The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2. Starting from 0 and 1, the first few values in the sequence are: :0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144. The Fibonacci numbers were first described in Indian mathematics, as early as 200 BC in work by Pingala on enumerating possible patterns of Sanskrit poetry formed from syllables of two lengths. They are named after the Italian mathematician Leonardo of Pisa, later known as Fibonacci, who introduced the sequence to Western European mathematics in his 1202 book ''Liber Abaci''. Fibonacci numbers appear unexpectedly often in mathematics, so much so that there is an entire journal dedicated to their study, the ''Fibonacci Quarterly''. Applications of Fibonacci ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Difference
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for the numerical solution of differential equations, especially boundary value problems. The difference operator, commonly denoted \Delta is the operator that maps a function to the function \Delta /math> defined by :\Delta x)= f(x+1)-f(x). A difference equation is a functional equation that involves the finite difference operator in the same way as a differential equation involves derivatives. There are many similarities between difference equations and differential equations, specially in the solving methods. Certain recurrence relations can be written as difference equations by replacing iteration notation with finite differences. In numerical analysis, finite differences are widely used for approximating derivatives, and the term " ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skolem Problem
In mathematics, the Skolem problem is the problem of determining whether the values of a constant-recursive sequence include the number zero. The problem can be formulated for recurrences over different types of numbers, including integers, rational numbers, and algebraic numbers. It is not known whether there exists an algorithm that can solve this problem.. A linear recurrence relation expresses the values of a sequence of numbers as a linear combination of earlier values; for instance, the Fibonacci numbers may be defined from the recurrence relation : together with the initial values and . The Skolem problem is named after Thoralf Skolem, because of his 1933 paper proving the Skolem–Mahler–Lech theorem on the zeros of a sequence satisfying a linear recurrence with constant coefficients. This theorem states that, if such a sequence has zeros, then with finitely many exceptions the positions of the zeros repeat regularly. Skolem proved this for recurrences over the rational ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Zero Of A Function
In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or equivalently, x is the solution to the equation f(x) = 0. A "zero" of a function is thus an input value that produces an output of 0. A root of a polynomial is a zero of the corresponding polynomial function. The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities. For example, the polynomial f of degree two, defined by f(x)=x^2-5x+6 has the two roots (or zeros) that are 2 and 3. f(2)=2^2-5\times 2+6= 0\textf(3)=3^2-5\times 3+6=0. If the function maps real numbers to real numbers, then it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Skolem–Mahler–Lech Theorem
In additive and algebraic number theory, the Skolem–Mahler–Lech theorem states that if a sequence of numbers satisfies a linear difference equation, then with finitely many exceptions the positions at which the sequence is zero form a regularly repeating pattern. This result is named after Thoralf Skolem (who proved the theorem for sequences of rational numbers), Kurt Mahler (who proved it for sequences of algebraic numbers), and Christer Lech (who proved it for sequences whose elements belong to any field of characteristic 0). Its known proofs use p-adic analysis and are non-constructive. Theorem statement Let s(n)_ be a sequence of complex numbers satisfying s(n) = c_1 s(n-1) + c_2 s(n-2) + \cdots + c_d s(n-d) for all n \ge d, where c_i are complex number constants (i.e., a constant-recursive sequence of order d). Then the set of zeros \ is equal to the union of a finite set and finitely many arithmetic progressions. If we have c_d \ne 0 (excluding sequences such as 1, 0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Product
In mathematics, more specifically in mathematical analysis, the Cauchy product is the discrete convolution of two infinite series. It is named after the French mathematician Augustin-Louis Cauchy. Definitions The Cauchy product may apply to infinite series or power series. When people apply it to finite sequences or finite series, that can be seen merely as a particular case of a product of series with a finite number of non-zero coefficients (see discrete convolution). Convergence issues are discussed in the next section. Cauchy product of two infinite series Let \sum_^\infty a_i and \sum_^\infty b_j be two infinite series with complex terms. The Cauchy product of these two infinite series is defined by a discrete convolution as follows: :\left(\sum_^\infty a_i\right) \cdot \left(\sum_^\infty b_j\right) = \sum_^\infty c_k where c_k=\sum_^k a_l b_. Cauchy product of two power series Consider the following two power series :\sum_^\infty a_i x^i &n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction, and division. The result of a multiplication operation is called a ''product''. The multiplication of whole numbers may be thought of as repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the ''multiplicand'', as the quantity of the other one, the ''multiplier''. Both numbers can be referred to as ''factors''. :a\times b = \underbrace_ For example, 4 multiplied by 3, often written as 3 \times 4 and spoken as "3 times 4", can be calculated by adding 3 copies of 4 together: :3 \times 4 = 4 + 4 + 4 = 12 Here, 3 (the ''multiplier'') and 4 (the ''multiplicand'') are the ''factors'', and 12 is the ''product''. One of the main properties of multiplication is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. The addition of two Natural number, whole numbers results in the total amount or ''summation, sum'' of those values combined. The example in the adjacent image shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the Expression (mathematics), mathematical expression (that is, "3 ''plus'' 2 is Equality (mathematics), equal to 5"). Besides counting items, addition can also be defined and executed without referring to concrete objects, using abstractions called numbers instead, such as integers, real numbers and complex numbers. Addition belongs to arithmetic, a branch of mathematics. In algebra, another area of mathematics, addition can also be performed on abstract objects su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pointwise
In mathematics, the qualifier pointwise is used to indicate that a certain property is defined by considering each value f(x) of some function f. An important class of pointwise concepts are the ''pointwise operations'', that is, operations defined on functions by applying the operations to function values separately for each point in the domain of definition. Important relations can also be defined pointwise. Pointwise operations Formal definition A binary operation on a set can be lifted pointwise to an operation on the set of all functions from to as follows: Given two functions and , define the function by Commonly, ''o'' and ''O'' are denoted by the same symbol. A similar definition is used for unary operations ''o'', and for operations of other arity. Examples \begin (f+g)(x) & = f(x)+g(x) & \text \\ (f\cdot g)(x) & = f(x) \cdot g(x) & \text \\ (\lambda \cdot f)(x) & = \lambda \cdot f(x) & \text \end where f, g : X \to R. See also pointwise product, and scalar. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closure (mathematics)
In mathematics, a subset of a given set is closed under an operation of the larger set if performing that operation on members of the subset always produces a member of that subset. For example, the natural numbers are closed under addition, but not under subtraction: is not a natural number, although both 1 and 2 are. Similarly, a subset is said to be closed under a ''collection'' of operations if it is closed under each of the operations individually. The closure of a subset is the result of a closure operator applied to the subset. The ''closure'' of a subset under some operations is the smallest subset that is closed under these operations. It is often called the ''span'' (for example linear span) or the ''generated set''. Definitions Let be a set equipped with one or several methods for producing elements of from other elements of . Operations and (partial) multivariate function are examples of such methods. If is a topological space, the limit of a sequence of element ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Language
In theoretical computer science and formal language theory, a regular language (also called a rational language) is a formal language that can be defined by a regular expression, in the strict sense in theoretical computer science (as opposed to many modern regular expressions engines, which are augmented with features that allow recognition of non-regular languages). Alternatively, a regular language can be defined as a language recognized by a finite automaton. The equivalence of regular expressions and finite automata is known as Kleene's theorem (after American mathematician Stephen Cole Kleene). In the Chomsky hierarchy, regular languages are the languages generated by Type-3 grammars. Formal definition The collection of regular languages over an alphabet Σ is defined recursively as follows: * The empty language Ø is a regular language. * For each ''a'' ∈ Σ (''a'' belongs to Σ), the singleton language is a regular language. * If ''A'' is a regular language, ''A''* ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Language Theory
In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of symbols, letters, or tokens that concatenate into strings of the language. Each string concatenated from symbols of this alphabet is called a word, and the words that belong to a particular formal language are sometimes called ''well-formed words'' or ''well-formed formulas''. A formal language is often defined by means of a formal grammar such as a regular grammar or context-free grammar, which consists of its formation rules. In computer science, formal languages are used among others as the basis for defining the grammar of programming languages and formalized versions of subsets of natural languages in which the words of the language represent concepts that are associated with particular meanings or semantics. In computational complexity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
