We develop closed form expressions for various finite binomial Fibonacci and Lucas sums depending on the modulo 5 nature of the upper summation limit. Our expressions are inferred from some trigonometric identities.

In this study, the resolvent of the impulsive singular Hahn-Sturm-Liouville operator is considered. An integral representation for the resolvent of this operator is obtained.

We give a proof of the uniform boundedness principle for linear continuous maps from F-spaces into topological vector spaces which is elementary and also quite simple.

This paper presents the concept of a partial idempotent valued S-metric space, abbreviated as PIV-S-metric space, as a generalization of both the PIV-metric space and S-metric space. The study utilizes this new framework to establish a fixed point theorem and a best proximity point theorem. Additionally, the paper proves the existence and uniqueness of the best proximity point within this context. Several illustrative examples are provided to demonstrate the practical applications of the main findings.

In this paper, we prove Talenti’s comparison theorem for mixed local/nonlocal elliptic operators and derive the Faber–Krahn inequality for the first eigenvalue of the Dirichlet mixed local/nonlocal problem. Our findings are relevant to the fractional p&q−Laplacian operator.

A transfunction is a function which maps between sets of finite measures on measurable spaces. In this paper we characterize transfunctions that correspond to Markov operators and to plans; such a transfunction will contain the “instructions” common to several Markov operators and plans. We also define the adjoint of transfunctions in two settings and provide conditions for existence of adjoints. Finally, we develop approximations of identity in each setting and use them to approximate weakly-continuous transfunctions with simple transfunctions; one of these results can be applied to some optimal transport problems to approximate the optimal cost with simple Markov transfunctions.

The independent domination number i(G) of a graph G is the minimum cardinality of a maximal independent set of G, also called an i(G)-set. The i-graph of G, denoted ℐ (G), is the graph whose vertices correspond to the i(G)-sets, and where two i(G)-sets are adjacent if and only if they differ by two adjacent vertices. Not all graphs are i-graph realizable, that is, given a target graph H, there does not necessarily exist a source graph G such that H = ℐ (G). We consider a class of graphs called “theta graphs”: a theta graph is the union of three internally disjoint nontrivial paths with the same two distinct end vertices. We characterize theta graphs that are i-graph realizable, showing that there are only finitely many that are not. We also characterize those line graphs and claw-free graphs that are i-graphs, and show that all 3-connected cubic bipartite planar graphs are i-graphs.

In this paper, we introduce Mersenne and Mersenne–Lucas bihyperbolic numbers, i.e. bihyperbolic numbers whose coefficients are consecutive Mersenne and Mersenne–Lucas numbers. Moreover, we study one parameter generalizations of Mersenne and Mersenne–Lucas bihyperbolic numbers. We present some properties of these numbers and relations between them.

In this paper we determine the complex-valued solutions of the Kannappan-cosine functional equation g(xyz₀) = g(x)g(y) − f(x)f(y), x,y∈S, where S is a semigroup and z₀ is a fixed element in S.

We prove a new irreducibility criterion for certain septinomials in ℤ[x], and we use this result to construct infinite families of reciprocal septinomials of degree 2ⁿ3 that are monogenic for all n ≥ 1.

Non-nil abelian groups are classified on which every ring, different from the zero-ring, is unital. It is shown that the assumption on the associativity of the considered rings does not influence the obtained classification. A significant mistake made by other authors studied this topic is corrected.