Results in Mathematics最新文献

A rational matrix is a matrix-valued function (R(lambda ): {mathbb {C}} rightarrow M_p) such that (R(lambda ) = begin{bmatrix} r_{ij}(lambda ) end{bmatrix} _{ptimes p}), where (r_{ij}(lambda )) are scalar complex rational functions in (lambda ) for (i,j=1,2,ldots ,p). The aim of this paper is to obtain bounds on the moduli of eigenvalues of rational matrices in terms of the moduli of their poles. To a given rational matrix (R(lambda )) we associate a block matrix ({mathcal {C}}_R) whose blocks consist of the coefficient matrices of (R(lambda )), as well as a scalar real rational function q(x) whose coefficients consist of the norm of the coefficient matrices of (R(lambda )). We prove that a zero of q(x) which is greater than the moduli of all the poles of (R(lambda )) will be an upper bound on the moduli of eigenvalues of (R(lambda )). Moreover, by using a block matrix associated with q(x), we establish bounds on the zeros of q(x), which in turn yields bounds on the moduli of eigenvalues of (R(lambda )).

In this paper, we establish a bialgebraic structure on a Coder pair of any weight (lambda ) (i.e., a coassociative coalgebra with a coderivation of weight (lambda )), which is consistent with the differential antisymmetric infinitesimal bialgebra in (SIGMA 19:018, 2023) when the weight (lambda =0).

The aim of this paper is to investigate the topological entropy for non-autonomous iterated function systems (NAIFSs) introduced by Ghane and Sarkooh. An inequality formula for two topological entropies with a factor map of NAIFSs is established. We extend the topological analogue of the Abramov–Rokhlin formula for the entropy of a skew product transformation. Finally, the partial variational principle is obtained about the measure-theoretic entropy and topological entropy for NAIFSs.

We prove that a normal subloop X of a Moufang loop Q induces an abelian congruence of Q if and only if (u(xy) = (uy)x) for all (x,yin X) and (uin Q). This characterization is then used to show that classically solvable finite 3-divisible Moufang loops are congruence solvable.

This paper aims to provide a thorough characterization of the family of all Cantor-Bendixson derivatives of the spectrum, Browder spectrum, and the Drazin spectrum of bounded linear operators using projections and invariant subspaces. Furthermore, our findings demonstrate that if two commuting operators, R and T, satisfy the conditions that R is Riesz and T is a direct sum of an invertible operator and an operator with an at most countable spectrum, then (T+R) can also be represented as a direct sum of an invertible operator and an operator with an at most countable spectrum.

In the present note we are interested in proving the counterpart of the (right-hand side of the) celebrated Hermite–Hadamard inequality for (varphi )-convex functions. In particular, we prove that the only (varphi )-convex function for which the Hermite–Hadamard inequality holds with the Lagrangian mean on the right-hand side is (up to an affine transformation) the (log )-convex function.

We study the behavior of a three-dimensional dynamical system with respect to some set (textbf{S}) given in 3-dimensional euclidean space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces that can be described by a real parameter (ain (0,1/2)), as for (textbf{S}) it represents the set of invariant Riemannian metrics of positive sectional curvature on the Wallach spaces. Establishing that (textbf{S}) is bounded by three conic surfaces and regarding the normalized Ricci flow as an abstract dynamical system we find out the character of interrelations between that system and (textbf{S}) for all (ain (0,1/2)). These results can cover some well-known results, in particular, they can imply that the normalized Ricci flow evolves all generic invariant Riemannian metrics with positive sectional curvature into metrics with mixed sectional curvature on the Wallach spaces corresponding to the cases (ain {1/9, 1/8, 1/6}) of generalized Wallach spaces.

Let ({mathbb {E}}={E_r(x):r>0,xin X}) be a family of open subsets of a topological space X equipped with a nonnegative Borel measure (mu ) satisfying some basic properties. We establish sharp quantitative weighted norm inequalities for the Hardy–Littlewood maximal operator (M_{{mathbb {E}}}) associated with ({mathbb {E}}) in terms of mixed (A_p)–(A_infty ) constants. The main ingredient to prove this result is a sharp form of a weak reverse Hölder inequality for the (A_{infty ,{mathbb {E}}}) weights. As an application of this inequality, we also provide a quantitative version of the open property for (A_{p,{mathbb {E}}}) weights. Next, we prove a covering lemma in this setting and using this lemma establish the endpoint Fefferman–Stein weighted inequality for the maximal operator (M_{{mathbb {E}}}). Moreover, vector-valued extensions for maximal inequalities are also obtained in this context.

Nonassociative differential extensions are generalizations of associative differential extensions, either of a purely inseparable field extension K of exponent one of a field F, F of characteristic p, or of a central division algebra over a purely inseparable field extension of F. Associative differential extensions are well known central simple algebras first defined by Amitsur and Jacobson. We explicitly compute the automorphisms of nonassociative differential extensions. These are canonically obtained by restricting automorphisms of the differential polynomial ring used in the construction of the algebra. In particular, we obtain descriptions for the automorphisms of associative differential extensions of D and K, which are known to be inner.