Consider partial maps Σ∗ −→ R with a rational domain. We show that two families of such series are actually the same: the unambiguous rational series on the one hand, and the max-plus and min-plus rational series on the other hand. The decidability of equality was known to hold in both families with different proofs, so the above unifies the picture. We give an effective procedure to build an u...

A function is said to be bi-univalent on the open unit disk D if both the function and its inverse are univalent in D. Not much is known about the behavior of the classes of bi-univalent functions let alone about their coefficients. In this paper we use the Faber polynomial expansions to find coefficient estimates for four well-known classes of bi-univalent functions which are defined by subord...

Abstract When classes of structures are not first-order definable, we might still try to find a nice description. There two common ways for doing this. One is expand the language, leading notions pseudo-elementary classes, and other allow infinite conjuncts disjuncts. In this paper examine intersection. Namely, address question: Which both ${\mathcal {L}}_{\omega _1, \omega }$ -elementary? We t...

In this paper, we propose a numerical method based on the generalized hat functions (GHFs) and improved hat functions (IHFs) to find numerical solutions for stochastic Volterra-Fredholm integral equation. To do so, all known and unknown functions are expanded in terms of basic functions and replaced in the original equation. The operational matrices of both basic functions are calculated and em...