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Reducing the range of any index to a single value effectively eliminates that index. This feature can be used, for example, to extract one-dimensional slices (vectors: in 3D, rows, columns, and tubes [1]) or two-dimensional slices (rectangular matrices) from a three-dimensional array. However, since the range can be specified at run-time, type ...
For example, a two-dimensional array A with three rows and four columns might provide access to the element at the 2nd row and 4th column by the expression A[1][3] in the case of a zero-based indexing system. Thus two indices are used for a two-dimensional array, three for a three-dimensional array, and n for an n-dimensional array. The number ...
The entry of a matrix A is written using two indices, say i and j, with or without commas to separate the indices: a ij or a i,j, where the first subscript is the row number and the second is the column number. Juxtaposition is also used as notation for multiplication; this may be a source of confusion. For example, if
It is common convention to use greek indices when writing expressions involving tensors in Minkowski space, while Latin indices are reserved for Euclidean space. Well-formulated expressions are constrained by the rules of Einstein summation : any index may appear at most twice and furthermore a raised index must contract with a lowered index.
That is, for any two symbols a and b in A that are not the same symbol, either a < b or b < a. The words of A are the finite sequences of symbols from A, including words of length 1 containing a single symbol, words of length 2 with 2 symbols, and so on, even including the empty sequence with no symbols at all. The lexicographical order on the ...
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. [a] Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time ...
It includes the F.F.1 list with 1,500 high-frequency words, completed by a later F.F.2 list with 1,700 mid-frequency words, and the most used syntax rules. [11] It is claimed that 70 grammatical words constitute 50% of the communicatives sentence, [12] [13] while 3,680 words make about 95~98% of coverage. [14] A list of 3,000 frequent words is ...
A defining vocabulary is a list of words used by lexicographers to write dictionary definitions. The underlying principle goes back to Samuel Johnson's notion that words should be defined using 'terms less abstruse than that which is to be explained', [1] and a defining vocabulary provides the lexicographer with a restricted list of high-frequency words which can be used for producing simple ...