Aequationes Mathematicae最新文献

A set of vertices (Ssubseteq V) in a graph (G=(V,E)) is called an internal minority set if for every vertex (vin S), a minority of the neighbors of v are in S, or equivalently, every vertex (vin S) has strictly more neighbors in (V-S) than it has in S. As we will show, minority sets in graphs are closely related to, but different than, a variety of sets that have been studied, such as defensive and offensive alliances, cost effective and very cost effective sets, unfriendly partitions in graphs, and independent and dominating sets in graphs. Sets similar to minority sets can also be defined by specifying that similar conditions apply to every vertex (win V-S), giving rise to external minority sets, and to all vertices (uin V), giving rise to total minority sets in graphs. In this paper we introduce the study of these types of sets. Various properties and results are obtained, a corollary of which is a new lower bound for the chromatic number of a graph. Moreover, the complexity issues of two minority related problems are addressed.

Assume ( (Omega , {mathcal {A}}, {mathbb {P}}) ) is a probability space, ((X,rho )) is a complete and separable metric space with the ( sigma )–algebra ( {mathcal {B}} ) of all its Borel subsets and ( f: X times Omega rightarrow X ) is measurable for ( {mathcal {B}} otimes {mathcal {A}}) and such that

with a concave (beta : [0,infty ) rightarrow [0,infty )) satisfying (beta (t)<t) for (t in (0,infty )), and (int _{Omega } rho big (f(x_0, omega ), x_0big ) {mathbb {P}}(d omega ) <infty ) for an (x_0 in X.) We consider the weak limit of the sequence of iterates of f and problems of the existence and uniqueness of solutions (varphi ) of the equations

in some classes of continuous functions mapping X into a separable Banach space.

Letting (L_{n}(N, u)) denote a polylogarithm ladder of weight n and index N with u as an algebraic number, there is a rich history surrounding how mathematical objects of this form can be constructed for a given weight or index. This raises questions as to what minimal polynomials for u are permissible in such constructions. Classical relations for the dilogarithm (text {Li}_{2}) provide families of weight-2 ladders in such a way so that the base equations for u consist of a fixed number of terms, and subsequent constructions for dilogarithm ladders rely on sporadic cases whereby u is defined via a cyclotomic equation, as in the supernumary ladders due to Abouzahra and Lewin. This motivates our construction of an infinite family of dilogarithm ladders so as to obtain arbitrarily many terms with nonzero coefficients for the minimal polynomials for u. Our construction relies on a derivation of a dilogarithm identity introduced by Khoi in 2014 via the Seifert volumes of manifolds obtained from the use of Dehn surgery.

Let (T_n(mathbb {K})) be the ring of all (ntimes n) upper triangular matrices over a field (mathbb {K}). For fixed positive integers n, s satisfying (frac{n}{2}le s<n), it is proved that (f: T_n(mathbb {K})rightarrow T_n(mathbb {K})) is additive if and only if (f(A+B)=f(A)+f(B)) for all rank-s matrices (A,Bin T_n(mathbb {K})), which has been proved to be true for (M_n(mathbb {K})) the ring of all (ntimes n) full matrices over (mathbb {K}) [Xu X., Liu H., Additive maps on rank-s matrices, Linear Multilinear Algebra 2017; 65: 806-812].

Let S be a semigroup, Z(S) be the center of S and (sigma :Srightarrow S) is an involutive automorphism. In this paper, we describe the complex-valued solutions of one of d’Alembert’s functional equations

where (tau :Srightarrow {mathbb {C}}) is a multiplicative function such that (tau (xsigma (x))=1) for all (xin S). This allows us to solve Van Vleck’s functional equation

where (mu ) is a measure that is a linear combination of Dirac measures ((delta _{z_i})_{iin I}), such that (z_iin Z(S)) for all (iin I), and I is a finite set. Many consequences of these results are presented.

We study Wilson’s functional equation with an anti-endomorphism on semigroups and derive its optimal links to d’Alembert’s functional equation. As an application, we solve d’Alembert’s functional equation on semigroups with both an involutive endomorphism and an anti-endomorphism.

We construct a measure which determines a two variable mean in a very natural way. Using that measure we can extend the mean to infinite sets as well. E.g. we can calculate the geometric mean of any set with positive Lebesgue measure. We also study the properties and behavior of such generalized means that are obtained by a measure, and we provide some applications as well.

In this paper we consider a semiclassical version of the fractional Klein-Gordon equation on the lattice (hbar mathbb {Z}^{n}.) Contrary to the Euclidean case that was considered in [2], the discrete fractional Klein-Gordon equation is well-posed in (ell ^{2}left( hbar mathbb {Z}^{n}right) .) However, we also recover the well-posedness results in the certain Sobolev spaces in the limit of the semiclassical parameter (hbar rightarrow 0).

In this study, the Ulam stability of quantum equations on time scales that alternate between two quanta is considered. We show that linear equations of first order with constant coefficient or of Euler type are Ulam stable across large regions of the complex plane, and give the best Ulam constants for those regions. We also show, however, that linear equations of first order of period-1 type are not Ulam stable for any parameter value in the complex plane. This is due to the importance of pre-positioning the non-autonomous term for Ulam stability.