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The word problem for an algebra is then to determine, given two expressions (words) involving the generators and operations, whether they represent the same element of the algebra modulo the identities. The word problems for groups and semigroups can be phrased as word problems for algebras. [1]
Computing the position of a particular unique tuple or matrix in a de Bruijn sequence or torus is known as the de Bruijn decoding problem. Efficient O ( n log n ) {\displaystyle \color {Blue}O(n\log n)} decoding algorithms exist for special, recursively constructed sequences [ 17 ] and extend to the two-dimensional case.
Other problems for which the greedy algorithm gives a strong guarantee, but not an optimal solution, include Set cover; The Steiner tree problem; Load balancing [11] Independent set; Many of these problems have matching lower bounds; i.e., the greedy algorithm does not perform better than the guarantee in the worst case.
Shannon's diagram of a general communications system, showing the process by which a message sent becomes the message received (possibly corrupted by noise). seq2seq is an approach to machine translation (or more generally, sequence transduction) with roots in information theory, where communication is understood as an encode-transmit-decode process, and machine translation can be studied as a ...
Formally, a parity check matrix H of a linear code C is a generator matrix of the dual code, C ⊥.This means that a codeword c is in C if and only if the matrix-vector product Hc ⊤ = 0 (some authors [1] would write this in an equivalent form, cH ⊤ = 0.)
The simplification in a word problem induced by subgroup distortion suffices to construct a cryptosystem, algorithms for encoding and decoding secret messages. [4] Formally, the plaintext message is any object (such as text, images, or numbers) that can be encoded as a number n. The transmitter then encodes n as an element g ∈ H with word ...
As with ideal observer decoding, a convention must be agreed to for non-unique decoding. The maximum likelihood decoding problem can also be modeled as an integer programming problem. [1] The maximum likelihood decoding algorithm is an instance of the "marginalize a product function" problem which is solved by applying the generalized ...
Word problem from the Līlāvatī (12th century), with its English translation and solution. In science education, a word problem is a mathematical exercise (such as in a textbook, worksheet, or exam) where significant background information on the problem is presented in ordinary language rather than in mathematical notation.