Aequationes Mathematicae最新文献

The note is concerned with the functional equation

which is a generalised form of the so-called polynomial-like iterative equation. We investigate the existence of nondecreasing convex (both usual and higher order) solutions to this equation on open intervals using the Schauder fixed point theorem. The results supplement those proved by Trif (Aquat Math, 79:315–327, 2010) for the polynomial-like iterative equation by generalising them to a greater extent. This assertion is supported by some examples illustrating their applicability.

Let (G=(V(G), E(G), F(G))) be a plane graph with vertex, edge, and region sets V(G), E(G),  and F(G) respectively. A zonal labeling of a plane graph G is a labeling (ell : V(G)rightarrow {1,2}subset mathbb {Z}_3) such that for every region (Rin F(G)) with boundary (B_R), (sum _{vin V(B_R)}ell (v)=0) in (mathbb {Z}_3). It has been proven by Chartrand, Egan, and Zhang that a cubic map has a zonal labeling if and only if it has a 3-edge coloring, also known as a Tait coloring. A dual notion of cozonal labelings is defined, and an alternate proof of this theorem is given. New features of cozonal labelings and their utility are highlighted along the way. Potential extensions of results to related problems are presented.

It was found that a function with exactly one discontinuity may have a continuous iterate of second order, indicating that a discontinuity may be repaired to be a continuous one by its adjacent pair of functions of second order, called second order sui-repair. If a function has more than one discontinuities, examples show that some discontinuities may be repaired to be continuous ones by the other’s adjacent pair of functions of second order, called second order (C^{0}) homi-repair. In this paper we investigate second order (C^{0}) homi-repairs of removable and jumping discontinuities for functions having more than one but finitely many discontinuities. We give necessary and sufficient conditions for removable and jumping discontinuities to be (C^0) repaired by the second order iteration.

A double Roman dominating function (DRDF) on a graph (G=(V,E)) is a function (f:Vrightarrow {0,1,2,3}) having the property that if (f(v)=0), then vertex v must have at least two neighbors assigned 2 under f or one neighbor w with (f(w)=3), and if (f(v)=1), then vertex v must have at least one neighbor w with (f(w)ge 2). The weight of a DRDF is the sum of its function values over all vertices, and the double Roman domination number (gamma _{dR}(G)) is the minimum weight of a DRDF on G. Khoeilar et al. (Discrete Appl. Math. 270:159–167, 2019) proved that if G is a connected graph of order n with minimum degree two different from (C_{5}) and (C_{7}), then (gamma _{dR}(G)le frac{11}{10}n.) Moreover, they presented an infinite family of graphs ({mathcal {G}}) attaining the upper bound, and conjectured that ({mathcal {G}}) is the only family of extremal graphs reaching the bound. In this paper, we disprove this conjecture by characterizing all extremal graphs for this bound.

In this research paper, we investigate a generalization of Vincze’s type functional equations involving several (up to four) unknown functions in connection with the maximum functional equation as

where G is an arbitrary group, (x, y in G), and (psi , eta , chi , phi :G rightarrow mathbb {R}) are unknown functions.

In this work, we focus on the evolution of the vortex filament flow (frac{partial gamma }{partial t} = frac{partial gamma }{partial s} wedge frac{D}{ds}frac{partial gamma }{partial s}) for spacelike and timelike curves in a 3-dimensional pseudo-Riemannian manifold. We study the relations between a partial differential equation and the vortex filament flow for spacelike and timelike curves. As a result, we prove that the vortex filament flow of the spacelike curve in a 3-dimensional pseudo-Riemannian manifold with constant sectional curvature is equivalent to the heat equation, and the flow of the timelike curve is equivalent to the non-linear Schrödinger equation. Also, we give some examples to illustrate the vortex filament flow.

We show that every (varepsilon )-isometry of the unit ball in (l^n_1) can be uniformly approximated by an affine surjective isometry to within (Cnvarepsilon ) for some absolute constant C.

In this paper, we give a matrix version of an equivalent form of the classical arithmetic–geometric mean inequality for two positive scalars. Applications and generalizations of our results are also given.