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And the other three binary-expansion repeating sequences give the 3-cycle 110110110... → 101101101... → 011011011... → 110110110.... Either of these 3-cycles can be converted to fraction form: for example, the first-given 3-cycle can be written as 1 / 7 → 2 / 7 → 4 / 7 → 1 / 7 .
If X is a smooth complete curve (for example, P 1) and if f is a rational map from X to a projective space P m, then f is a regular map X → P m. [5] In particular, when X is a smooth complete curve, any rational function on X may be viewed as a morphism X → P 1 and, conversely, such a morphism as a rational function on X .
An example of a fraction that cannot be represented by a decimal expression (with a finite number of digits) is 1 / 3 , 3 not being a power of 10. More generally, a decimal with n digits after the separator (a point or comma) represents the fraction with denominator 10 n, whose numerator is the integer obtained by removing the separator.
The convergents of the continued fraction for φ are ratios of successive Fibonacci numbers: φ n = F n+1 / F n is the n-th convergent, and the (n + 1)-st convergent can be found from the recurrence relation φ n+1 = 1 + 1 / φ n. [31] The matrix formed from successive convergents of any continued fraction has a determinant of +1 or −1.
The base "Roman fraction" is S, indicating 1 ⁄ 2. The use of S (as in VIIS to indicate 7 1 ⁄ 2) is attested in some ancient inscriptions [45] and also in the now rare apothecaries' system (usually in the form SS): [44] but while Roman numerals for whole numbers are essentially decimal, S does not correspond to 5 ⁄ 10, as one might expect ...
For example, the monthly cost of a $250,000 home at 6% interest fixed over 30 years, with 1% property taxes, 0.75% maintenance costs, and a 30% federal income tax rate is approximately $1361 per month. The rental cost for an equivalent home may be less in many U.S. cities as of 2006.
Jean Swank joined the program in time for the beginning of our OSO-8 experiment (1975-1978), the first broadband (2-40 keV) orbiting observatory based on multi-anode multi-layer proportional chambers, one that showed the power of X-ray spectroscopy; for example, it established that iron K-line emission is a ubiquitous feature of clusters of ...
Claude is a family of large language models developed by Anthropic. [1] [2] The first model was released in March 2023.The Claude 3 family, released in March 2024, consists of three models: Haiku optimized for speed, Sonnet balancing capabilities and performance, and Opus designed for complex reasoning tasks.