Search results
Results from the WOW.Com Content Network
The word embedding approach is able to capture multiple different degrees of similarity between words. Mikolov et al. (2013) [26] found that semantic and syntactic patterns can be reproduced using vector arithmetic. Patterns such as "Man is to Woman as Brother is to Sister" can be generated through algebraic operations on the vector ...
A subset A of positive integers has natural density α if the proportion of elements of A among all natural numbers from 1 to n converges to α as n tends to infinity.. More explicitly, if one defines for any natural number n the counting function a(n) as the number of elements of A less than or equal to n, then the natural density of A being α exactly means that [1]
These sequences of natural numbers can again be represented by single natural numbers, facilitating their manipulation in formal theories of arithmetic. Since the publishing of Gödel's paper in 1931, the term "Gödel numbering" or "Gödel code" has been used to refer to more general assignments of natural numbers to mathematical objects.
Roth's theorem on arithmetic progressions (infinite version): A subset of the natural numbers with positive upper density contains a 3-term arithmetic progression. An alternate, more qualitative, formulation of the theorem is concerned with the maximum size of a Salem–Spencer set which is a subset of [ N ] = { 1 , … , N } {\displaystyle [N ...
In arithmetic combinatorics, Szemerédi's theorem is a result concerning arithmetic progressions in subsets of the integers. In 1936, Erdős and Turán conjectured [1] that every set of integers A with positive natural density contains a k-term arithmetic progression for every k. Endre Szemerédi proved the conjecture in 1975.
The older term for number theory is arithmetic. By the early twentieth century, it had been superseded by number theory. [note 1] (The word arithmetic is used by the general public to mean elementary calculations; it has also acquired other meanings in mathematical logic, as in Peano arithmetic, and computer science, as in floating-point ...
In 2022, Google Brain reported that chain-of-thought prompting improves reasoning ability by inducing the model to answer a multi-step problem with steps of reasoning that mimic a train of thought. [11] [16] Chain-of-thought techniques were developed to help LLMs handle multi-step reasoning tasks, such as arithmetic or commonsense reasoning ...
the arithmetic function counting a number's prime factors counted with multiplicity; the density parameter in cosmology [81] the first uncountable ordinal (also written as ω 1) [82] Chaitin's constant for a given computer program; the vacuum state in quantum field theory; represents: angular velocity / radian frequency (rad/sec) [83]