We prove that quasiconvex functions always admit descent trajectories bypassing all nonminimizing critical points.

We study domination in zero-divisor graphs of matrix rings over a commutative ring with 1.

In this paper we define a space$VM{{O}_{P}}$ associated with a family $P$ of parabolic sections and show that the dual of $VM{{O}_{P}}$ is the Hardy space $H_{P}^{1}$. As an application, we prove that almost everywhere convergence of a bounded sequence in $H_{P}^{1}$ implies weak$^{\star }$ convergence

Let $p$ be a prime number and $F$ a field containing a root of unity of order $p$. We relate recent results on vanishing of triple Massey products in the $\bmod-p $ Galois cohomology of $F$, due to Hopkins, Wickelgren, Mináč, and Tân, to classical results in the theory of central simple algebras. We prove a stronger form of the vanishing property for global fields.

Let $\mathcal{E}$ be an injectively resolving subcategory of left $R$-modules. A left $R$-module $M$ (resp. right $R$-module $N$) is called $\mathcal{E}$-injective (resp. $\mathcal{E}$-flat) if $\text{Ext}_{R}^{1}\left( G,\,M \right)\,=\,0$ (resp. $\text{Tor}_{1}^{R}\left( N,\,G \right)\,=\,0$) for any $G\,\in \,\mathcal{E}$. Let $\mathcal{E}$ be a covering subcategory. We prove that a left $R$-module $M$ is $\mathcal{E}$-injective if and only if $M$ is a direct sum of an injective left $R$-module and a reduced $\mathcal{E}$-injective left $R$-module. Suppose $\mathcal{F}$ is a preenveloping subcategory of right $R$-modules such that ${{\mathcal{E}}^{+}}\,\subseteq \,\mathcal{F}$ and ${{\mathcal{F}}^{+}}\,\subseteq \,\mathcal{E}$. It is shown that a finitely presented right $R$-module $M$ is $\mathcal{E}$-flat if and only if $M$ is a cokernel of an $\mathcal{F}$-preenvelope of a right $R$-module. In addition, we introduce and investigate the $\mathcal{E}$-injective and $\mathcal{E}$-flat dimensions of modules and rings. We also introduce $\mathcal{E}$-(semi)hereditary rings and $\mathcal{E}$-von Neumann regular rings and characterize them in terms of $\mathcal{E}$-injective and $\mathcal{E}$-flat modules.

We define a refined motivic dimension for an algebraic variety by modifying the definition of motivic dimension by Arapura. We apply this to check and recheck the generalized Hodge conjecture for certain varieties, such as uniruled, rationally connected varieties and a rational surface fibration.

Given an essentially unitary contraction and an arbitrary unitary dilation of it, there is a naturally associated spectral flow that is shown to be equal to the index of the operator. This result is interpreted in terms of the K-theory of an associated mapping cone. It is then extended to connect Z2 indices of odd symmetric Fredholm operators to a Z2-valued spectral flow.

There exist injective Tauberian operators on ${{L}_{1}}\left( 0,\,1 \right)$ that have dense, nonclosed range. This gives injective nonsurjective operators on ${{\ell }_{\infty }}$ that have dense range. Consequently, there are two quasi-complementary noncomplementary subspaces of ${{\ell }_{\infty }}$ that are isometric to ${{\ell }_{\infty }}$.

There are several notions of Ricci curvature tensor in Finsler geometry and spray geometry. One of them is defined by the Hessian of the well-known Ricci curvature. In this paper we will introduce a new notion of Ricci curvature tensor and discuss its relationship with the Ricci curvature and some non-Riemannian quantities. Using this Ricci curvature tensor, we shall have a better understanding of these non-Riemannian quantities.

In this article we prove an embedding theorem for inhomogeneous Besov and Triebel–Lizorkin spaces on $\text{RD}$-spaces. The crucial idea is to use the geometric density condition on the measure.

A complex Lie supergroup can be described as a real Lie supergroup with integrable almost complex structure. The necessary and sufficient conditions on an almost complex structure on a real Lie supergroup for defining a complex Lie supergroup are deduced. The classification of real Lie supergroups with such almost complex structures yields a new approach to the known classification of complex Lie supergroups by complexHarish-Chandra superpairs. A universal complexification of a real Lie supergroup is constructed.

A group $G$ is self dual if every subgroup of $G$ is isomorphic to a quotient of $G$ and every quotient of $G$ is isomorphic to a subgroup of $G$. It is minimal non self dual if every proper subgroup of $G$ is self dual but $G$ is not self dual. In this paper, the structure of minimal non self dual groups is determined.

The normalized eigenvalues ${{\Lambda }_{i}}\left( M,\,g \right)$ of the Laplace–Beltrami operator can be considered as functionals on the space of all Riemannian metrics $g$ on a fixed surface $M$. In recent papers several explicit examples of extremal metrics were provided. These metrics are induced by minimal immersions of surfaces in ${{\mathbb{S}}^{3}}$ or ${{\mathbb{S}}^{4}}$. In this paper a family of extremal metrics induced by minimal immersions in ${{\mathbb{S}}^{5}}$ is investigated.

We prove character sum estimates for additive Bohr subsets modulo a prime. These estimates are analogous to the classical character sum bounds of Pólya–Vinogradov and Burgess. These estimates are applied to obtain results on recurrence $\bmod \,p$ by special elements.

Bibasic sequences are used to study relative weak compactness and relative norm compactness of limited sets.

In this paper, we investigate the common approximate fixed point sequences of nonexpansive semigroups of nonlinear mappings ${{\left\{ {{T}_{t}} \right\}}_{t\ge 0}}$, i.e., a family such that ${{T}_{0}}\left( x \right)\,=\,x,\,{{T}_{s+t}}\,=\,{{T}_{s}}\left( {{T}_{t}}\left( x \right) \right)$, where the domain is a metric space $\left( M,\,d \right)$. In particular, we prove that under suitable conditions the common approximate fixed point sequences set is the same as the common approximate fixed point sequences set of two mappings from the family. Then we use the Ishikawa iteration to construct a common approximate fixed point sequence of nonexpansive semigroups of nonlinear mappings.

Let $G$ be a complex semisimple linear algebraic group and let Pet be the associated Peterson variety in the flag variety $G/B$. The main theorem of this note gives an eõcient presentation of the equivariant cohomology ring $H_{S}^{*}$ (Pet) of the Peterson variety as a quotient of a polynomial ring by an ideal $J$ generated by quadratic polynomials, in the spirit of the Borel presentation of the cohomology of the flag variety. Here the group $S\,\cong \,{{\mathbb{C}}^{*}}$ is a certain subgroup of a maximal torus $T$ of $G$. Our description of the ideal $J$ uses the Cartan matrix and is uniform across Lie types. In our arguments we use the Monk formula and Giambelli formula for the equivariant cohomology rings of Peterson varieties for all Lie types, as obtained in the work of Drellich. Our result generalizes a previous theorem of Fukukawa–Harada–Masuda, which was only for Lie type $A$.

Let ${{S}_{k}}\left( \Gamma \right)$ be the space of holomorphic cusp forms of even integral weight $k$ for the full modular group $SL\left( 2,\mathbb{Z} \right)$. Let ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{g}}\left( n \right),{{\text{ }\!\!\lambda\!\!\text{ }}_{h}}\left( n \right)$ be the $n$-th normalized Fourier coefficients of three distinct holomorphic primitive cusp forms $f\left( z \right)\in {{S}_{{{k}_{1}}}}\left( \Gamma \right),g\left( z \right)\in {{S}_{{{k}_{2}}}}\left( \Gamma \right)$, and $h\left( z \right)\in {{S}_{{{k}_{3}}}}\left( \Gamma \right)$, respectively. In this paper we study the cancellations of sums related to arithmetic functions, such as $\text{ }{{\lambda }_{f}}{{\left( n \right)}^{4}}{{\lambda }_{g}}{{\left( n \right)}^{2}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{6}},\text{ }{{\lambda }_{g}}{{\left( n \right)}^{2}}{{\lambda }_{h}}{{\left( n \right)}^{4}}$, and ${{\text{ }\!\!\lambda\!\!\text{ }}_{g}}{{\left( {{n}^{3}} \right)}^{2}}$ twisted by the arithmetic function ${{\text{ }\!\!\lambda\!\!\text{ }}_{f}}\left( n \right)$.

In this paper, we generalize the finite generation result of Sormani to non-branching $RCD\left( 0,\,N \right)$ geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact $RCD\left( 0,\,N \right)$ spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.