A. H. Copeland and P. Erdos. Note on normal numbers. Bull. Amer. Math. Soc., 52:857--860, 1946. Home/Search Document Not in Database Summary Related Articles Check |
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On the Random Character of Fundamental Constant Expansions - Bailey, Crandall (5 citations) (Correct)
....fractions, noting for the moment that the Champernowne constant has some gargantuan elements in its simple continued fraction, as can be seen by simple numerical experiments. Another example of a concoction known to be normal to base 10 is the Copeland Erdos number 0:23571113171923 : [Copeland and Erdos 1946], in which the primes are concatenated; this concatenation game can be generalized yet further to more general integer sequences for the digit construction. Theorem 2.8. If ff be normal to base b then ff is digitdense to base b. If ff be digit dense to some base b then ff is irrational . Proof. ....
A. H. Copeland and P. Erdos, "Note on normal numbers", Bull. Amer. Math. Soc. 52 (1946), 857--860.
Disjunctive Sequences: An Overview - Calude, Priese, Staiger (1997) (2 citations) (Correct)
....Champernowne proved that for b = 10 the above sequence is normal 3 in the scale of ten, so it is disjunctive. In fact, 2) is normal in every base b n , n # 2. It is not known whether this sequence is normal in any scale from b n , n # 2. A related example is due to Copeland and Erdos [13]: the sequence of primes 23571113171923 2 Words are arranged in increasing order of their length; words having the same length are arranged lexicographically. 3 Normality was introduced by Borel [1] see also Kuipers, Niederreiter [24] 3 is normal in the scale of ten, so it is ....
A. H. Copeland, P. Erdos, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857--860. 36
An Alternative Construction of Normal Numbers - Ugalde (1998) (Correct)
....numbers have been proposed following Champernowne s idea, i.e. a concatenation of blocks of digits of increasing length. Besicovitch [Be] proved that concatenating the sequence of squares of all the natural numbers produces the normal number 0:129142536496481100121 . Copeland and Erd os [CE] proved that the 1 concatenation of all prime numbers gives the normal 0:23571113171923293137 . More recently Schi er [Sc] and Nakai and Shiokawa [NS] proved that for a non constant, eventually increasing polynomial p, the number 0: p(1) p(2) p(3) is also a normal number. Here [x] ....
A. Copeland and P. Erdos, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857-860.
Normal Numbers and Sources for BPP - Strauss (1 citation) (Correct)
....noted that all reals except on a set of measure zero are normal numbers, i.e. have binary expansions which are normal in the scale of 2. In [4] it is shown that the decimal version of the sequence 1 10 11 100 101 110 111 1000 : formed by concatenating the binary numbers, is normal, and in [5] a criterion is given for a set fa i g of integers so that the concatenation of the a i s be normal. A language L will be identified with its characteristic sequence L : enumerate the strings, and set bit i of L to 1 iff the i th string is in L: Following [10] we can say that a sequence is in ....
A. Copeland and P. Erdos. Note on normal numbers. Bull. AMS, 52:857-- 860, 1946.
Borel Normality and Algorithmic Randomness - Calude (1994) (Correct)
....sequence x # A # and every string y # A # of length m, lim n## F (x, y, n) n = Q m . Remark. Of course, there exist many Borel normal sequences which are not random, e.g. Champernowne s sequence (see [15] 01234567891011121314151617181920212223242526 . or the sequence of primes (see [16]) 23571113171923 . over the alphabet A = 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 . The reason is simple: Both sequences are recursive, a property which excludes randomness in the sense used in this paper (see [6] On the other hand, it is still unknown whether the decimal representations of some ....
A. H. Copeland, P. Erdos. Note on normal numbers, Bull. Amer. Math. Soc. 52(1946), 857-860.
Dimensions of Copeland-Erdos Sequences - Xiaoyang Gu Jack Self-citation (Copeland Erdos) (Correct)
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A. H. Copeland and P. Erdos. Note on normal numbers. Bull. Amer. Math. Soc., 52:857--860, 1946.
Dimensions of Copeland-Erdos Sequences - Gu, Lutz, Moser (2005) Self-citation (Copeland Erdos) (Correct)
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A. H. Copeland and P. Erdos. Note on normal numbers. Bull. Amer. Math. Soc., 52:857--860, 1946.
Measuring Sets of Constructible Normal Numbers - Ugalde (Correct)
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A. H. Copeland and P. Erdos, Note on normal numbers, Bull. Amer. Math. Soc. 52 (1946), 857-860.
Bisection Hardly Ever Converges Linearly - Nievergelt (1995) (Correct)
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Copeland, A.H., Erdos, P. (1946): Note on Normal Numbers, Bull. Amer. Math. Soc. 52, 857--860
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