منابع مشابه
Combinatorial sums and finite differences
We present a new approach to evaluating combinatorial sums by using finite differences. Let {ak}k=0 and {bk}k=0 be sequences with the property that ∆bk = ak for k ≥ 0. Let gn = ∑n k=0 ( n k ) ak, and let hn = ∑n k=0 ( n k ) bk. We derive expressions for gn in terms of hn and for hn in terms of gn. We then extend our approach to handle binomial sums of the form ∑n k=0 ( n k ) (−1)ak, ∑ k ( n 2k ...
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Many fundamental questions in additive number theory (such as Goldbach’s conjecture, Fermat’s last theorem, and the Twin Primes conjecture) can be expressed in the language of sum and difference sets. As a typical pair of elements contributes one sum and two differences, we expect that |A−A| > |A+A| for a finite set A. However, in 2006 Martin and O’Bryant showed that a positive proportion of su...
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A more sums than differences (MSTD) set is a finite subset S of the integers such |S + S| > |S− S|. We show that the probability that a uniform random subset of {0, 1, . . . , n} is an MSTD set approaches some limit ρ > 4.28× 10−4. This improves the previous result of Martin and O’Bryant that there is a lower limit of at least 2 × 10−7. Monte Carlo experiments suggest that ρ ≈ 4.5× 10−4. We pre...
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Let $(X, N)$ be a fuzzy normed space and $A$ be a fuzzy boundedsubset of $X$. We define fuzzy $ell^infty$-sums and fuzzy $c_0$-sums offuzzy normed spaces. Then we will show that in these spaces, all fuzzyuniquely remotal sets are singletons.
متن کاملSets with More Sums than Differences
Let A be a finite subset of the integers or, more generally, of any abelian group, written additively. The set A has more sums than differences if |A + A| > |A − A|. A set with this property is called an MSTD set. This paper gives explicit constructions of families of MSTD sets of integers.
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ژورنال
عنوان ژورنال: Functiones et Approximatio Commentarii Mathematici
سال: 2007
ISSN: 0208-6573
DOI: 10.7169/facm/1229618749