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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

In this paper, we study the local existence of weak solutions for parabolic problem involving the fractional p-Laplacian. Our technique is based on the Galerkin method combined with the theory of Young measures. In addition, an example is given to illustrate the main results.

Given S a semigroup. We study two Pexider-type functional equations
f(xy) + g(xy) = f(x) + f(y) + g(x)g(y),   x,y ∈ S,
and
∫_Sf(xyt)dμ(t) + ∫_Sg(xyt)dμ(t) = f(x) + f(y) + g(x)g(y),   x,y ∈ S,
for unknown functions f and g mapping S into ℂ, where μ is a linear combination of Dirac measures (δ_{z_i}})_i∈I for some fixed elements (z_i)_i∈I contained in S such that ∫_Sdμ(t) = 1.
The main goal of this paper is to solve the above two functional equations and examine whether or not they are equivalent to the systems of equations
f(xy) = f(x) + f(y),  g(xy) = g(x)g(y),   x,y ∈ S,
and
∫_Sf(xyt)dμ(t) = f(x) + f(y),  ∫_Sg(xyt)dμ(t) = g(x)g(y),   x,y ∈ S,
respectively.

Let S be a semigroup, let (H, +) be a uniquely 2-divisible, abelian group and let ϕ, ψ be two endomorphisms of S that need not be involutive. In this paper, we express the solutions f : S→H of the following quadratic functional equation
f(xϕ(y)) + f(ψ(y)x) = 2f(x) + 2f(y),   x,y ∈ S,
in terms of bi-additive maps and solutions of the symmetrized additive Cauchy equation. Some applications of this result are presented.

In this paper we are interested in the existence of solutions for the Dirichlet problem associated with the degenerate nonlinear elliptic equations
-div[A(x,u,∇u)ω₁ + B(x,u,∇u)ν₁] + ????(x,u,∇u)ν₂ + |u|^p-2uω₂ - Σ_i,j=1ⁿD_j(a_ij(x)D_iu(x)) = f₀(x) - Σ_j=1ⁿD_jf_j(x)   in Ω,
u(x) = 0   on ∂Ω,
in the setting of the weighted Sobolev spaces.

Let S be a semigroup, ℍ be the skew field of quaternions, and ψ: S→S be an anti-endomorphism. We determine the general solution of the functional equation
g(xy) - g(xψ(y)) = 2g(x)g(y),    x,y ∈ S,
where g: S→ℂ is the unknown function. And when S = M is a monoid, we solve the functional equation
g(xy) + g(xψ(y)) = 2g(x)g(y),   x,y ∈ M,
where g: M→ℍ is the unknown function.

The sequence spaces ru`∞(O, ∇q), ru`p(O, ∇q), ruc(O, ∇q), ruc0(O, ∇q), rumφ(O, ∇q, p), runφ(O, ∇q, p), rumφ(O, ∇q), runφ(O, ∇q) are defined by the Orlicz function in this article. We examine all of its characteristics, including symmetry, solidity, and completeness. A few geometric properties on convexity on the space rumφ(O, ∇q, p) are also examined in this article.

In this paper, the almost everywhere convergence of Cesàro means of Walsh–Kaczmarz–Fourier series in a varying parameter setting is investigated. In particular, we define subsequence ℕ_{α_n,q} of natural numbers and prove that the maximal operator
sup_{n∈ℕ_{α_n,q}}|σ_n^α_nf|
is of strong type (H¹,L¹), where H¹ is a Hardy space.

In this study, we conduct a literature review on normed linear spaces whose strengths are between rotundity and uniform rotundity. In this discourse, we also explore inter-relationships and juxtapositions between the subjects under consideration. There has been some discussion on the extent to which the geometry of the factor spaces has an impact on the geometry of the product spaces, as well as the degree to which the quotient spaces and subspaces inherit the geometry of the space itself. A comprehensive review has been conducted on the applications of most of these rotundities to some fields within the realm of approximation theory. In addition, some open problems are enumerated in the paper.

In this paper, by Schauder's fixed point theorem and the Banach contraction principle, we consider the existence, uniqueness, and stability of convex solutions of a nonhomogeneous iterative functional differential equation. Finally, some examples were considered by our results.

In this work we focus on a dynamical system with jumps, where the intensity of the jumps depends on the system’s state. By verifying the assumptions of the theorem from [4], we show that our model satisfies the central limit theorem.