We consider Gabor systems generated by a Gaussian function and prove certain classical results of Paley and Wiener on nonharmonic Fourier series of complex exponentials for the Gabor expansion. In particular, we prove a version of Plancherel-Po ́ lya theorem for entire functions with finite order of growth and use the Hadamard factorization theorem to study regularity, exactness and deficiency of Gabor systems.

This paper studies impulsive dynamic Sturm-Liouville boundary value problems. The existence of a countably infinite set of eigenvalues and eigenfunctions is proved and a uniformly convergent expansion formula in the eigenfunctions is established.

In this paper, we introduce the notions of expansion of ideals in MV-algebras, ( ;  )􀀀 primary, ( ;  )-obstinate and ( ;  )-Boolean in MV 􀀀 algebras. We investigate the re-lations of them. For example, we show that every ( ;  )-obstinate ideal of an MV 􀀀 algebra is ( ;  )-primary and ( ;  )-Boolean. In particular, we de ne an expansion  y of ideals in an MV 􀀀 algebra. A characterization of expansion ideal with respect to  y is given. Finally, we show that the class C( y) of all constant ideals relative to  y is a Heyting algebra.

The vehicle routing problem (VRP) is one of the most important research issues in operations in industries and services, which is highly regarded today due to the high cost of transportation in the final price of goods. On the other hand, considering that the use of vehicles with different capacities will further reduce this cost, the issue of fleet size and mix vehicle routing (FSMVRP) was introduced and since then significant progress has been made on these issues and their types for use in Real tools were made. In this case, there are different types of vehicles with different capacities available in the unique depot to serve a group of customers with known geographical locations. In addition, in this case, each customer needs a certain amount of goods that must be delivered to them by a fixed fleet of vehicles. The goal is to determine the set of tours for the vehicles with the lowest cost, provided that: each vehicle starts from the depot and returns to it at the end, each customer is visited exactly once by one vehicle and the total customer demand of each tour exceeds the capacity do not exceed any type of vehicle, which is considered Qi. The purpose of this article is to categorize and review issues related to FSMVRP. This paper also provides a comparative analysis of meta-heuristic algorithms for these problems.