Double factorialIn mathematics, the double factorial of a number n, denoted by n!!, is the product of all the positive integers up to n that have the same parity (odd or even) as n. That is, Restated, this says that for even n, the double factorial is while for odd n it is For example, 9!! = 9 × 7 × 5 × 3 × 1 = 945. The zero double factorial 0!! = 1 as an empty product. The sequence of double factorials for even n = 0, 2, 4, 6, 8,... starts as The sequence of double factorials for odd n = 1, 3, 5, 7, 9,...
Common logarithmIn mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered its use, as well as standard logarithm. Historically, it was known as logarithmus decimalis or logarithmus decadis. It is indicated by log(x), log10 (x), or sometimes Log(x) with a capital L (however, this notation is ambiguous, since it can also mean the complex natural logarithmic multi-valued function).
TreapIn computer science, the treap and the randomized binary search tree are two closely related forms of binary search tree data structures that maintain a dynamic set of ordered keys and allow binary searches among the keys. After any sequence of insertions and deletions of keys, the shape of the tree is a random variable with the same probability distribution as a random binary tree; in particular, with high probability its height is proportional to the logarithm of the number of keys, so that each search, insertion, or deletion operation takes logarithmic time to perform.
Exponential functionThe exponential function is a mathematical function denoted by or (where the argument x is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, although it can be extended to the complex numbers or generalized to other mathematical objects like matrices or Lie algebras. The exponential function originated from the notion of exponentiation (repeated multiplication), but modern definitions (there are several equivalent characterizations) allow it to be rigorously extended to all real arguments, including irrational numbers.
Fibonacci wordA Fibonacci word is a specific sequence of binary digits (or symbols from any two-letter 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.
Falling and rising factorialsIn mathematics, the falling factorial (sometimes called the descending factorial, falling sequential product, or lower factorial) is defined as the polynomial The rising factorial (sometimes called the Pochhammer function, Pochhammer polynomial, ascending factorial, rising sequential product, or upper factorial) is defined as The value of each is taken to be 1 (an empty product) when These symbols are collectively called factorial powers. The Pochhammer symbol, introduced by Leo August Pochhammer, is the notation (x)_n , where n is a non-negative integer.
Binary logarithmIn mathematics, the binary logarithm (log2n) is the power to which the number 2 must be raised to obtain the value n. That is, for any real number x, For example, the binary logarithm of 1 is 0, the binary logarithm of 2 is 1, the binary logarithm of 4 is 2, and the binary logarithm of 32 is 5. The binary logarithm is the logarithm to the base 2 and is the inverse function of the power of two function. As well as log2, an alternative notation for the binary logarithm is lb (the notation preferred by ISO 31-11 and ISO 80000-2).
Fibonacci codingIn mathematics and computing, Fibonacci coding is a universal code which encodes positive integers into binary code words. It is one example of representations of integers based on Fibonacci numbers. Each code word ends with "11" and contains no other instances of "11" before the end. The Fibonacci code is closely related to the Zeckendorf representation, a positional numeral system that uses Zeckendorf's theorem and has the property that no number has a representation with consecutive 1s.
Generalizations of Fibonacci numbersIn mathematics, the Fibonacci numbers form a sequence defined recursively by: That is, after two starting values, each number is the sum of the two preceding numbers. The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects other than numbers. Using , one can extend the Fibonacci numbers to negative integers. So we get: −8, 5, −3, 2, −1, 1, 0, 1, 1, 2, 3, 5, 8, .
Fibonacci polynomialsIn mathematics, the Fibonacci polynomials are a polynomial sequence which can be considered as a generalization of the Fibonacci numbers. The polynomials generated in a similar way from the Lucas numbers are called Lucas polynomials. These Fibonacci polynomials are defined by a recurrence relation: The Lucas polynomials use the same recurrence with different starting values: They can be defined for negative indices by The Fibonacci polynomials form a sequence of orthogonal polynomials with and .
General recursive functionIn mathematical logic and computer science, a general recursive function, partial recursive function, or μ-recursive function is a partial function from natural numbers to natural numbers that is "computable" in an intuitive sense – as well as in a formal one. If the function is total, it is also called a total recursive function (sometimes shortened to recursive function). In computability theory, it is shown that the μ-recursive functions are precisely the functions that can be computed by Turing machines (this is one of the theorems that supports the Church–Turing thesis).
Search algorithmIn computer science, a search algorithm is an algorithm designed to solve a search problem. Search algorithms work to retrieve information stored within particular data structure, or calculated in the search space of a problem domain, with either discrete or continuous values. Although search engines use search algorithms, they belong to the study of information retrieval, not algorithmics. The appropriate search algorithm to use often depends on the data structure being searched, and may also include prior knowledge about the data.
