In a linear optimization problem, objective function, coefficients matrix, and the right-hand side might be perturbed with distinct parameters independently. For such a problem, we are interested in finding the region that contains the origin, and the optimal partition remains invariant. A computational methodology is presented here for detecting the boundary of this region. The cases where perturbation occurs only in the coefficients matrix and right-hand side vector or the objective function are specified as special cases. The findings are illustrated with some simple examples.

For a graph $G$ on $v(G)$ vertices let $m_k(G)$ denote the number of matchings of size $k$‎, ‎and consider the partition function $M_{G}(\lambda)=\sum_{k=0}^nm_k(G)\lambda^k$‎. ‎In this paper we show that if $G$ is a $d$--regular graph and $0<\lambda<(4d)^{-2}$‎, ‎then‎ ‎$$\frac{1}{v(G)}\ln M_G(\lambda)>\frac{1}{v(K_{d+1})}\ln M_{K_{d+1}}(\lambda).$$‎ ‎The same inequality holds true if $d=3$ and $\lambda<0.3575$‎. ‎More precise conjectures are also given‎.

This paper is about metric and partition dimension of a flower and a pencil graph. A metric dimension of G, denoted by dim(G), is the minimum cardinality of any resolving set of G. A partition dimension of G, denoted by pd(G), is the minimum number of sets in any resolving k-ordered partition for G. Here we give the exact value of the metric dimension of a flower graph fm×n for m ∈ {3, 4} and a pencil graph Pcm for any integer m ≥ 2. We also give the partition dimension of fm×n for m ∈ {3, 4, 5} and Pcm for any integer m ≥ 2.

Because the partition coefficient is one of the most important parameters affecting microsegregation, the aim of this research is to experimentally analyse the partition coefficient in Al-Mg alloys. In order to experimentally measure the partition coefficient, a series of quenching experiments during solidification were carried out. For this purpose binary Al-Mg alloys containing 6.7 and 10.2 wt-% Mg were melted and solidified in a DTA furnace capable of quenching samples during solidification. Cooling rates of 0.5 and 5 K/min were used and samples were quenched from predetermined temperatures during solidification. The fractions and compositions of the phases were measured by quantitative metallography and SEM/EDX analyses, respectively. These results were used to measure the experimental partition coefficients. The resultant partition coefficients were used to model the concentration profile in the primary phase and the results were compared with equilibrium calculations and experimental profiles. The results of calculations based on the experimental partition coefficients show better consistency with experimental concentration profiles than the equilibrium calculations.

There are many obstacles in quantum circuits implementation with large scales, so distributed quantum systems are appropriate solution for these quantum circuits. Therefore, reducing the number of quantum teleportation leads to improve the cost of implementing a quantum circuit. The minimum number of teleportations can be considered as a measure of the efficiency of distributed quantum systems. Due to the complexity of reducing teleportation costs in multi-partition quantum computing, a two-level optimization is proposed. In the first level, using graph partition methods, the quantum circuit is divided into k partitions. Secondly, using a series of proposed rules and functions, the number of teleportations for the partitioned quantum circuit is improved. In previous work, only optimization at the first level has been conisdered for the distributed quantum circuits. The implementation results show that the cost of teleportation in the proposed algorithm has been improved compared to previous work on benchmark circuits.