On Chebyshev-Type Inequalities for Primes
TL;DR: In this paper, Chebyshev-type inequalities for prime numbers were studied in the context of the American Mathematical Monthly: Vol. 89, No. 2, pp. 126-129.
Abstract: (1982). On Chebyshev-Type Inequalities for Primes. The American Mathematical Monthly: Vol. 89, No. 2, pp. 126-129.
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TL;DR: In this paper, an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite is presented. But the algorithm is not deterministic in the sense that
Abstract: We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite.
1,022 citations
TL;DR: This work proposes a theoretical framework that also covers updates as a repair primitive, and shows the construct of nucleus: a single database that yields consistent answers to a class of queries, without the need for query rewriting.
Abstract: Repairing a database means bringing the database in accordance with a given set of integrity constraints by applying some minimal change. If a database can be repaired in more than one way, then the consistent answer to a query is defined as the intersection of the query answers on all repaired versions of the database.Earlier approaches have confined the repair work to deletions and insertions of entire tuples. We propose a theoretical framework that also covers updates as a repair primitive. Update-based repairing is interesting in that it allows rectifying an error within a tuple without deleting the tuple, thereby preserving consistent values in the tuple. Another novel idea is the construct of nucleus: a single database that yields consistent answers to a class of queries, without the need for query rewriting. We show the construction of nuclei for full dependencies and conjunctive queries. Consistent query answering and constructing nuclei is generally intractable under update-based repairing. Nevertheless, we also show some tractable cases of practical interest.
212 citations
Cites background from "On Chebyshev-Type Inequalities for ..."
...Since it is known [Nair 1982a; Nair 1982b] that lcm(1, 2, 3, ..., n) =2n if n =7, |G|=2n....
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...Since it is known [Nair 1982a; Nair 1982b] that lcm(1, 2, 3, ....
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23 Jan 2013
TL;DR: B bisimulation up to congruence is introduced as a technique for proving language equivalence of non-deterministic finite automata and an optimisation of the classical algorithm by Hopcroft and Karp is devised.
Abstract: We introduce bisimulation up to congruence as a technique for proving language equivalence of non-deterministic finite automata. Exploiting this technique, we devise an optimisation of the classical algorithm by Hopcroft and Karp. We compare our approach to the recently introduced antichain algorithms, by analysing and relating the two underlying coinductive proof methods. We give concrete examples where we exponentially improve over antichains; experimental results moreover show non negligible improvements.
166 citations
Cites background from "On Chebyshev-Type Inequalities for ..."
...n], the least common multiplier of the first n natural numbers (that is greater than 2 for n > 7 [20])....
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TL;DR: In this paper, the authors present a table of table of tables of this paper : Table of Table 1.3.1.1-3.2.0.1]
Abstract: Table of
102 citations
TL;DR: In this article, the least common multiple of some finite sequences of integers was studied and lower bounds for the arithmetic progressions were derived for quadratic sequences whose general term has the form u n = a n (n + t ) + b with (a, t, b ) ∈ Z 3, a ⩾ 5, t ⩽ 0, gcd (a, b ) = 1.
69 citations
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