A partial flag variety is a smooth projective homogeneous variety admitting an action of a maximal torus T. Schubert varieties form an interesting family of T-invariant subvarieties of the partial flag varieties. We study toric Schubert varieties in Grassmannians with respect to the action of a quotient of the torus T. Indeed, we present an explicit description of the fan of a toric Schubert variety in a Grassmannian, and as a corollary, we show that any Gorenstein toric Schubert variety in a Grassmannian is Fano.

In this paper, we assume that q > 0, p > 1 and s ∈ (0, 1), and consider the following nonlinear fractional p-Laplacian equations on finite graphs: $$\{{\partial}_tu^q(x,t)+(-{\Delta})^s_pu(x,t)=0,\\u(x,t){\mid}_{t=0}=u_0>0,$$ where (-∆)^s_p is fractional Laplace operator on finite graphs. We establish the existence of solutions to the above parabolic equation using an iterative approach, which is different from previous works on graphs. Furthermore, we also derive some energy estimates of the solution. The difficulties lie in the estimates and the convergence of the nonlinear terms ∂_tu^q(x, t) and (-∆)^s_pu(x, t).

If ξ is a Killing vector field of the hyperbolic space ℍ³ whose flow are parabolic isometries, a surface Σ ⊂ ℍ³ is a ξ-translator if its mean curvature H satisfies H = ⟨N, ξ⟩, where N is the unit normal of Σ. We classify all ξ-translators invariant by a one-parameter group of rotations of ℍ³, exhibiting the existence of a new family of grim reapers. We use these grim reapers as barriers to prove a half-space theorem of the non-existence of ξ-translators included in vertical half-spaces.

In this paper, we prove that every C¹ generic diffeomorphism f on a compact smooth manifold M is partially hyperbolic if it has the pseudo-orbit tracing property. Moreover we show that every C¹ generic vector filed 𝜑 on M with dimM = 3 is singular hyperbolic if it has the pseudo-orbit tracing property. This is a partial answer of the conjecture proposed by Abdenur and Díaz in [1].

In this paper, we give a proof that it is undecidable whether a set of five polyominoes can tile the plane by translation. The proof involves a new method of labeling the edges of polyominoes, making it possible to assign whether two edges can match for any set of two edges chosen. This is achieved by dedicating 1 polyomino to the labeling process.

The aim of this paper is to establish the boundedness of a bilinear singular integral operator $\tilde{T}$ associated with generalized kernels and its corresponding commutator $\tilde{T}_{b_1,b_2}$ formed by b₁, b₂ ∈ BMO(ℝⁿ) and $\tilde{T}$ on product mixed Lebesgue spaces $L^u_{\vec{p}}({\mathbb{R}}^n)$ and product mixed generalized Morrey spaces ${\mathcal{L}}^u_{\vec{p}}({\mathbb{R}}^n)$. Via some known results for the operators $\tilde{T}$ and $\tilde{T}_{b_1,b_2}$, the authors prove that $\tilde{T}$ and $\tilde{T}_{b_1,b_2}$ are bounded from product spaces $L^{\vec{p}_1}({\mathbb{R}}^n){\times}L^{{\vec{p}}_2}({\mathbb{R}}^n)$ into spaces $L^{\vec{p}}({\mathbb{R}}^n)$. Furthermore, under assumption that Lebesgue measurable functions u₁, u₂ and u belong to the class ${\mathbb{W}}_{\vec{p}}$ and meet u₁u₂ = u, the authors show that $\tilde{T}$ and $\tilde{T}_{b_1,b_2}$ are bounded from product spaces ${\mathcal{L}}^{u_1}_{{\vec{p}}_1}({\mathbb{R}}^n){\times}{\mathcal{L}}^{u_2}_{{\vec{p}}_2}({\mathbb{R}}^n)$ into spaces ${\mathcal{L}}^u_{\vec{p}}({\mathbb{R}}^n)$, respectively.

We study the mean-field limit of the relativistic Cucker-Smale (RCS) model on complete smooth Riemannian manifolds. The RCS model describes the collective dynamics of the relativistic Cucker-Smale particles on abstract manifolds, and it was first introduced as the generalization of the Cucker-Smale model in special relativity framework via a suitable ansatz for entropy, and using Boillat and Ruggeri's principle of subsystem [9] from the Euler equations for a gas mixture on Riemannian manifolds. In this paper, we derive a Vlasov-type kinetic RCS model on Riemannian manifolds using manifold counterpart of particlein-cell method and study its emergent dynamics. For the proposed kinetic model, we provide a priori velocity alignment estimate via the dissipation of total energy. We also adopt the concept of measure-valued solution to the kinetic RCS model in [4,32] and show the global existence of a unique measure-valued solution.

In this paper, we will investigate the boundedness of the commutators of the fractional Hardy operator 𝓗_Ω,β(·) of variable order β(·) with some rough kernel Ω ∈ L^s (𝕊 ^n-1 )(s > 1) and its dual operator 𝓗^*_Ω,β(·) generated with the function b ∈ CMO^q(·) (ℝⁿ) on the grand variable Herz-Morrey spaces $M\dot{K}^{{\alpha}({\cdot}),u),{\theta}}_{{\lambda},p({\cdot})}(\mathbb{R}^n)$, respectively. It's worth noting that, compared with spaces BMO(ℝⁿ), the spaces CMO^q(·) (ℝⁿ) are not equipped with the property BMOq(·) (ℝⁿ) ≈ BMO(ℝⁿ), which brings some difficulties to establish main results. Moreover, the corresponding results are also new even on the grand variable Herz spaces $M\dot{K}^{{\alpha}({\cdot}),u),{\theta}}_{p({\cdot})}(\mathbb{R}^n)$.

Let T₁M² be the unit tangent sphere bundle of an oriented Riemannian two-manifold M². In this paper we show that T₁M² admits a one parameter family of bi-contact metric structures (η_1c, η_2c, ${\tilde{g}}_c$), c > -1, where ${\tilde{g}}_c$ is a Kaluza-Klein metric and (η_1c, η_2c) is a taut contact circle. If M² has Gaussian curvature K = constant < 0, we get that T₁M² admits a one parameter family of bi-contact metric structures (${\tilde{\eta}}_{1_a}$, ${\tilde{\eta}}_{2_a}$, ${\tilde{g}}_a$), a > 0, ${\tilde{g}}_a$ is a metric of Kaluza-Klein type and (${\tilde{\eta}}_{1_a}$, ${\tilde{\eta}}_{2_a}$) is a taut contact hyperbola. Furthermore, we find that the Kaluza-Klein metric ${\tilde{g}}_c$ is a critical bi-contact metric, with respect to the Chern-Hamilton functional, if and only if c + 1 = |K| > 0. Finally, when the Gaussian curvature K = constant < 0 and M² is compact, we show that T₁M² admits two ℍ₁¹-families of volume preserving contact Anosov vector fields, where ℍ₁¹ denotes an equilateral hyperbola.

Let 𝔽^d_q be a d-dimensional vector space over a finite field 𝔽_q with q elements. For x ∈ 𝔽^d_q, let ‖x‖ = x²₁ + · · · + x²_d. By abuse of terminology, we shall call ‖·‖ a norm on 𝔽^d_q. For a subset E ⊂ 𝔽^d_q, let Δ(E) be the distance set on E defined as Δ(E) := {‖x - y‖ : x, y ∈ E}. The Mattila-Sjölin problem seeks the smallest exponent α > 0 such that Δ(E) = 𝔽_q for all subsets E ⊂ 𝔽^d_q with |E| ≥ Cq^α. In this article, we consider this problem for a variant of this norm, which generates a smaller distance set than the norm ‖·‖. Namely, we replace the norm ‖·‖ by the so-called k-norm (1 ≤ k ≤ d), which can be viewed as a kind of deformation of ‖·‖. To derive our result on the Mattila-Sjölin problem for the k-norm, we use a combinatorial method to analyze various summations arising from the discrete Fourier machinery. Even though our distance set is smaller than the one in the Mattila-Sjölin problem, for some k we still obtain the same result as that of Iosevich and Rudnev [15], which deals with the Mattila-Sjölin problem. Furthermore, our result is sharp in all odd dimensions.