(d) Investigate some other examples. Is there some fairly simple test to determine whether a given sequence A = (a,) is an alt-basis?

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Chapter2: Second-order Linear Odes
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Answer (d) only.

Problem 1
Suppose A = (an) = (a1, a2, az3, ...) is an increasing sequence of positive integers.
A number c is called A-erpressible if c is the alternating sum of a finite subsequence
of A. To form such a sum, choose a finite subset of the sequence A, list those numbers
in increasing order (no repetitions allowed), and combine them with alternating plus
and minus signs. We allow the trivial case of one-element subsequences, so that each
an is A-expressible.
Definition. Sequence A = (an) is an "alt-basis" if every positive integer is uniquely
A-expressible. That is, for every integer m > 0, there is exactly one way to express
m as an alternating sum of a finite subsequence of A.
Examples. Sequence B = (2"-1) = (1, 2, 4, 8, 16, ...) is not an alt-basis because
some numbers are B-expressible in more than one way. For instance 3 = -1+4 =
1- 2+4.
Sequence C = (3"-1) = (1, 3, 9, 27, 81, .) is not an alt-basis because some numbers
(like 4 and 5) are not C-expressible.
(a) Let D = (2" – 1) = (1, 3, 7, 15, 31,...). Note that:
1 = 1, 2 = -1+3, 3= 3, 4 = -3+7, 5 = 1 – 3+ 7,
6 = -1+7, 7 = 7, 8= -7+ 15, 9=1– 7+ 15,
Prove that D is an alt-basis.
(b) Can some E = (4,5, 7, . ) be an alt-basis? That is, does there exist an
alt-basis E = (en) with e1 = 4, e2 = 5, and e3 = 7? Justify your answer.
The first few values seem to work: 1 = -4+5, 2= -5+ 7, 3 = -4+7.
(c) Can F = (1, 4, . .. ) be an alt-basis? That is, does there exist an alt-basis F = (fn)
with fi = 1 and f2 = 4?
(d) Investigate some other examples. Is there some fairly simple test to determine
whether a given sequence A = (an) is an alt-basis?
Transcribed Image Text:Problem 1 Suppose A = (an) = (a1, a2, az3, ...) is an increasing sequence of positive integers. A number c is called A-erpressible if c is the alternating sum of a finite subsequence of A. To form such a sum, choose a finite subset of the sequence A, list those numbers in increasing order (no repetitions allowed), and combine them with alternating plus and minus signs. We allow the trivial case of one-element subsequences, so that each an is A-expressible. Definition. Sequence A = (an) is an "alt-basis" if every positive integer is uniquely A-expressible. That is, for every integer m > 0, there is exactly one way to express m as an alternating sum of a finite subsequence of A. Examples. Sequence B = (2"-1) = (1, 2, 4, 8, 16, ...) is not an alt-basis because some numbers are B-expressible in more than one way. For instance 3 = -1+4 = 1- 2+4. Sequence C = (3"-1) = (1, 3, 9, 27, 81, .) is not an alt-basis because some numbers (like 4 and 5) are not C-expressible. (a) Let D = (2" – 1) = (1, 3, 7, 15, 31,...). Note that: 1 = 1, 2 = -1+3, 3= 3, 4 = -3+7, 5 = 1 – 3+ 7, 6 = -1+7, 7 = 7, 8= -7+ 15, 9=1– 7+ 15, Prove that D is an alt-basis. (b) Can some E = (4,5, 7, . ) be an alt-basis? That is, does there exist an alt-basis E = (en) with e1 = 4, e2 = 5, and e3 = 7? Justify your answer. The first few values seem to work: 1 = -4+5, 2= -5+ 7, 3 = -4+7. (c) Can F = (1, 4, . .. ) be an alt-basis? That is, does there exist an alt-basis F = (fn) with fi = 1 and f2 = 4? (d) Investigate some other examples. Is there some fairly simple test to determine whether a given sequence A = (an) is an alt-basis?
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