A construction sequence for a graph is a listing of the elements of the graph (the set of vertices and edges) such that each edge follows both its endpoints. The construction number of the graph is the number of such sequences. We determine this number for various graph families.

In this paper, the concept of interval-valued intuitionistic fuzzy deductive systems (IVIF deductive systems) of Hilbert algebras is introduced. The relationship between deductive systems and IVIF deductive systems is studied in terms of upper and lower-level subsets. We also find a relationship between an IVIF deductive system and its fuzzy deductive system. The homomorphic inverse image of IVIF deductive systems in Hilbert algebras is studied, and some related properties are investigated. Equivalence relations on IVIF deductive systems are discussed.

This article continues in a vein explored by the first author, E.Rarity, and J.Z. Schroeder, in which the smallest self-dual embeddable graphs in a pseudosurface were shown to have 7 vertices and 13 edges. We show here that the next-smallest self-dual embeddable graphs in a pseudosurface have 7 vertices and 14 edges, and we also establish a set of 11 candidate graphs for analysis; we establish the possible pseudosurfaces in which they can be embedded, which must have zero Euler characteristic. We conduct a computer search, which implicitly uses homology theory, to find that 7 of the 11 candidate graphs have self-dual embeddings in pseudosurfaces with pinchpoints, and we include embeddings in the pinched projective plane and in the twice pinched sphere with 2 distinct pinchpoints. We explore some properties of these graph embeddings, including examples of self-dual embeddings of the same graphs in the same pseudosurfaces that are not equivalent. We close with a brief discussion of ideas for further investigation.

The aim of this paper is to study some nonlinear elliptic problems with data in L1(Ω) in variable Lebesgue spaces. The existence of entropy solutions is established and an improved regularity result in Musielak-Orlicz spaces is given.

In 2012, the author discovered that the base of the natural logarithm, e, can be derived by operating on the products of the rows of Pascal’s triangle. The unexpected result raised the question as to what other sequences through Pascal’s triangle might be connected to e. Employing a property of Sidi polynomials related to the factorial function, we prove that there are an infinite number of such sequences whose growth rates are asymptotic to powers of e.

We introduce the notion of an unfair sequential game of perfect information as a general form of the EN-gammon, which is a concrete model of our discussion. We will discuss the unfairness of the EN-gammon when it is played for an odd number of times. We recall the notion of the EN-gammon, which is a backgammon variant and it follows exactly all the rules of the standard backgammon with the exception that both players using the entangled numbers of the dice for their turns, respectively. That is, the player who rolls the six-sided dice, plays with the face-up numbers showing on the dice and the opponent plays with the face-down numbers of the same rolling of those dice at his/her turn, respectively. In this game, only one of the players (by convention) rolls the dice till end of the game.