Abstract Let 1 < k 14 / 5 1\lt k\lt 14\hspace{-0.08em}\text{/}\hspace{-0.08em}5 , λ , 2 3 {\lambda }_{1},{\lambda }_{2},{\lambd...

T integrated production and transportation scheduling problem (PTSP) with capacity constraints is common in many industries. An optimal solution to PTSP requires one to simultaneously solve the production scheduling and the transportation routing problems, which requires excessive computational time, even for relatively small problems. In this study, we consider a variation of PTSP that involve...

Solution: The ring F [x] of polynomials with coefficients in a field F is a P.I.D. Each prime ideal is generated by a monic, irreducible polynomial. Assume there are only a finite number of prime ideals generated by the polynomials f1, . . . , fn and let f(x) = 1+f1(x) · · · fn(x). No fi divides f , hence f is also irreducible. This contradicts the assumption that all the prime ideals were gene...

At the 1912 Fifth International Congress of Mathematicians (Cambridge), Landau mentioned four unattackable problems: Goldbach Conjecture; Twin Prime Conjecture; for each n, there is a prime p such that n2 < p < (n+1)2; there are infinitely many primes p of the form p = n2+1. As Erdős [6, preface] maintained five years ago, we still do not have a satisfactory solution for each of them. For large...

Abstract. A precise estimation of the computational complexity in Shor’s factoring algorithm under the condition that the large integer we want to factorize is composed by the product of two prime numbers, is derived by the results related to number theory. Compared with Shor’s original estimation, our estimation shows that one can obtain the solution under such a condition, by less computation...

Let $G$ be a finite group. We construct the prime graph of $ G $,which is denoted by $ Gamma(G) $ as follows: the vertex set of thisgraph is the prime divisors of $ |G| $ and two distinct vertices $ p$ and $ q $ are joined by an edge if and only if $ G $ contains anelement of order $ pq $.In this paper, we determine finite groups $ G $ with $ Gamma(G) =Gamma(L_3(q)) $, $2 leq q < 100 $ and prov...

Prime factorization is useful and crucial for public-key cryptography, and its application in public-key cryptography is possible only because prime factorization has been presumed to be difficult. A polynomial-time algorithm for prime factorization on a quantum computer was given by P. W. Shor in 1997. In this work, it is considered as a function problem, and in the framework of tissue P syste...

PURPOSE To evaluate the effect of an acid containing 2% chlorhexidine (Ac/CHX) or a 2% CHX aqueous solution (Aq/CHX) on the immediate and 5-year bonding properties of resin/dentin interfaces produced by two adhesives. The presence of CHX in these interfaces was also evaluated under micro-Raman spectroscopy. MATERIALS AND METHODS Forty-two molars were ground to expose a flat dentin surface. In...