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This definition covers several different uses of the word "sequence", including one-sided infinite sequences, bi-infinite sequences, and finite sequences (see below for definitions of these kinds of sequences). However, many authors use a narrower definition by requiring the domain of a sequence to be the set of natural numbers. This narrower ...
An example is the sequence of theorems: Fermat's little theorem, Euler's theorem, Lagrange's theorem, each of which is stronger than the last; another is that a sharp upper bound (see sharp above) is a stronger result than a non-sharp one.
For example, the sequence ,, is a subsequence of ,,,,, obtained after removal of elements ,, and . The relation of one sequence being the subsequence of another is a partial order . Subsequences can contain consecutive elements which were not consecutive in the original sequence.
Also in 2016, Quizlet launched "Quizlet Live", a real-time online matching game where teams compete to answer all 12 questions correctly without an incorrect answer along the way. [15] In 2017, Quizlet created a premium offering called "Quizlet Go" (later renamed "Quizlet Plus"), with additional features available for paid subscribers.
A training data set is a data set of examples used during the learning process and is used to fit the parameters (e.g., weights) of, for example, a classifier. [9] [10]For classification tasks, a supervised learning algorithm looks at the training data set to determine, or learn, the optimal combinations of variables that will generate a good predictive model. [11]
Conversely, any net whose domain is the natural numbers is a sequence because by definition, a sequence in is just a function from = {,, …} into . It is in this way that nets are generalizations of sequences: rather than being defined on a countable linearly ordered set ( N {\displaystyle \mathbb {N} } ), a net is defined on an arbitrary ...
For example, the sequence 2, 6, 18, 54, ... is a geometric progression with a common ratio of 3. Similarly 10, 5, 2.5, 1.25, ... is a geometric sequence with a common ratio of 1/2. Examples of a geometric sequence are powers r k of a fixed non-zero number r, such as 2 k and 3 k. The general form of a geometric sequence is
A similar definition can be made for other algebraic structures. For example, one could have an exact sequence of vector spaces and linear maps, or of modules and module homomorphisms. More generally, the notion of an exact sequence makes sense in any category with kernels and cokernels, and more specially in abelian categories, where it is ...