Romeo Meštrović

Mathematical Models of Multiserver Queuing System for Dynamic Performance Evaluation in Port

We discuss dynamic system performance evaluation in the river port utilizing queuing models with batch arrivals. The general models of the system are developed. This system is modelled by queue with finite waiting areas and identical and independent cargo-handling capacities. The models are considered with whole and part batch acceptance (or whole and part batch rejections) and the interarrival and service times are exponentially distributed. Results related to the batch blocking probability and the blocking probability of an arbitrary vessel in nonstationary and stationary states have been obtained. Numerical results and computational experiments are reported to evaluate the efficiency of the models for the real system.

PROOF OF A CONGRUENCE FOR HARMONIC NUMBERS CONJECTURED BY Z.-W. SUN

For a positive integer n let be the nth harmonic number. In this paper, we prove that for any prime p ≥ 7, which confirms the conjecture recently proposed by Z.-W. Sun. Furthermore, we also prove two similar congruences modulo p2.

WEAKLY DENSE IDEALS IN PRIVALOV SPACES OF HOLOMORPHIC FUNCTIONS

In this paper we study the structure of closed weakly dense ideals in Privalov spaces (1 ) of holomorphic functions on the disk : |z| with the topology given by Stolls metric [21] becomes an F-algebra. N. Mochizuki [16] proved that a closed ideal in is a principal ideal generated by an inner function. Consequently, a closed subspace E of is invariant under multiplication by z if and only if it has the form for some inner function I. We prove that if is a closed ideal in that is dense in the weak topology of , then is generated by a singular inner function. On the other hand, if is a singular inner function whose associated singular measure has the modulus of continuity , then we prove that the ideal is weakly dense in . Consequently, for such singular inner function , the quotient space is an F-space with trivial dual, and hence does not have the separation property.

An extension of Babbage’s criterion for primality

AbstractLet n > 1 and k > 1 be positive integers. We show that if

A search for primes such that the Euler number _{-3} is divisible by

\left( {\begin{array}{*{20}c} {n + m} \\ n \\ \end{array} } \right) \equiv 1 (\bmod k)

An approach to traffic modelling at seaport automobile terminals: the case of the Port of Bar

for each integer m with 0 ≤ m ≤ n − 1, then k is a prime and n is a power of this prime. In particular, this assertion under the hypothesis that n = k implies that n is a prime. This was proved by Babbage, and thus our result may be considered as a generalization of this criterion for primality.

An Identity in Commutative Rings with Unity with Applications to Various Sums of Powers

We consider the distribution of the number of customers that arrive in an arbitrary bulk arrival queue system. Under certain conditions on the distributions of the time of arrival of an arriving group ( ) and its size ( ) with respect to the considered bulk queue, we derive a general expression for the probability mass function of the random variable which expresses the number of customers that arrive in this bulk queue during any considered period . Notice that can be considered as a well-known compound random variable. Using this expression, without the use of generating function, we establish the expressions for probability mass function for some compound distributions concerning certain pairs of discrete random variables which play an important role in application of batch arrival queues which have a wide range of applications in different forms of transportation. In particular, we consider the cases when and/or are some of the following distributions: Poisson, shifted-Poisson, geometrical, or uniform random variable.

A Very Short Proof of the Infinitude of Primes

Let p > 3 be a prime. Euler numbers Ep−3 first appeared in H. S. Vandiver’s work (1940) in connection with the first case of Fermat’s Last Theorem. Vandiver proved that if xp + yp = zp has a solution for integers x, y, z with gcd(xyz, p) = 1, then it must be that Ep−3 ≡ 0 (mod p). Numerous combinatorial congruences recently obtained by Z.-W. Sun and Z.-H. Sun involve the Euler numbers Ep−3. This gives a new significance to the primes p for which Ep−3 ≡ 0 (mod p). For the computation of residues of Euler numbers Ep−3 modulo a prime p, we use a congruence which runs significantly faster than other known congruences involving Ep−3. Applying this, congruence, via a computation in Mathematica 8, shows that there are only three primes less than 107 that satisfy the condition Ep−3 ≡ 0 (mod p) (these primes are 149, 241 and 2946901). By using related computational results and statistical considerations similar to those used for Wieferich, Fibonacci-Wieferich and Wolstenholme primes, we conjecture that there are infinitely many primes p such that Ep−3 ≡ 0 (mod p).