This article introduces an algebra $\mathscr{A}_{\mathsf{MTV}, c}$ of functions in one variable $c$ defined by iterated integrals of two specific differential forms depending on $c$, where the product is the shuffle product. The algebra $\mathscr{A}_{\mathsf{MTV}, c}$ can be seen as a common deformation of multiple zeta values and of Kaneko-Tsumura's recent multiple $T$-values and it satisfies the same duality relations. Its first graded dimensions, assuming that a grading by the weight does hold, are computed, and some relations not implied by duality are found. A generating function for specific elements in this algebra is described using Gamma and hypergeometric functions.

In this paper, we construct Frenet frame fields along a null curve when it is not a bi-null Cartan curve in a pseudo-hyperbolic space with index 2 and pseudo-sphere with index 2, respectively. Also, we find ruled surfaces with the above null curves which have non-diagonalizable shape operators, constant mean curvatures and constant scalar curvatures.

In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng [3]. Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on a Hom-Lie algebroid and the notion of Hom-Dirac structures on a Hom-Courant algebroid. We show that there exists a one-to-one correspondence between the pairs consisting of a Poisson structure on $M$ and a Poisson isomorphism for it, and Hom-Poisson structures on $M$ introduced in [3]. Moreover we show that these structures satisfy similar properties to structures non “Hom-”version. For example, there exists the hierarchy of a Hom-Poisson-Nijenhuis structure and we have a relation between Hom-Dirac structures and the Maurer-Cartan type equation.

In this paper, we study the complex vector space of holomorphic cross-sections of a homogeneous holomorphic vector bundle over an elliptic adjoint orbit, and give a sufficient condition for the vector space to be finite-dimensional.

For a regular uncountable cardinal $\kappa$, we discuss the order relationship between the unbounding and dominating numbers $\mathfrak{b}_{\kappa}$ and $\mathfrak{d}_{\kappa}$ on $\kappa$ and cardinal invariants of the higher meager ideal $\mathscr{M}_{\kappa}$. In particular, we obtain an almost complete characterization of $\mathsf{add}(\mathscr{M}_{\kappa})$ and $\mathsf{cof}(\mathscr{M}_{\kappa})$ in terms of $\mathsf{cov}(\mathscr{M}_{\kappa})$ and $\mathsf{non}(\mathscr{M}_{\kappa})$ and unbounding and dominating numbers, and we provide models showing that there are no restrictions on the value of $\mathsf{non}(\mathscr{M}_{\kappa})$ in the degenerate case $2^{\lt \kappa} > \kappa$ except $2^{\lt \kappa} \le \mathsf{non}(\mathscr{M}_{\kappa}) \le 2^{\kappa}$. The corresponding question for $\mathsf{cof}(\mathscr{M}_{\kappa})$ remains open. Our results answer questions of joint work of the author with Brooke-Taylor, Friedman, and Montoya [BBFM, Questions 29 and 32].