We study the behavior of NQC generalized lc pairs with b-log abundant nef part. We show that this structure is preserved under the canonical bundle formula and sub-adjunction formulas, and is also compatible with the non-vanishing conjecture and the abundance conjecture in the classical minimal model program. As an application, we prove the non-vanishing theorem for rationally connected log canonical pairs, improving a result of Gongyo.

We prove a Miyaoka–Yau type inequality for threefolds such that −K_X is nef and of numerical dimension ≥ 2

where A is an ample line bundle and λ > 0 is a constant dependent on the characteristic p and the Cartier index n₀ of K_X. This is an analogue of the Miyaoka–Yau type inequality presented in [Algebra Number Theory, 2022, 16(10): 2339–2384], which treats minimal threefolds of general type. We attain this inequality by following the strategy of [Algebra Number Theory, 2022, 16(10): 2339–2384, Duke Math. J., 2019, 168(7): 1269–1301]. Applying a similar argument, we also show that in characteristic ≥ 5, if −K_X is nef and −K_X · c₂(X) < 0, then the nonvanishing theorem for anti-canonical divisor holds.

Let G be a graph. The Laplacian ratio of G is the permanent of the Laplacian matrix of G divided by the product of degrees of all vertices. The computational complexity of Laplacian ratio is #P-complete. Brualdi and Goldwasser studied systematically the properties of Laplacian ratios of graphs. And they proposed an open problem: what is the minimum value of the Laplacian ratios of trees with n vertices having diameter at least k? In this paper, we give a solution to the problem.

In this paper, we mainly prove some congruences conjectured by Z.H. Sun involving Almkvist-Zudilin numbers

Let p > 3 be a prime. Then

where B_n stands for the n-th Bernoulli number.

Let Δ₁ (x; φ) denote the Riesz mean error term in the classical Rankin-Selberg problem. In this paper, we study the higher power moments of Δ₁ (x; φ) and derive asymptotic formulas for 4-th and 5-th power moments.

The Galilean conformal algebra, a non-semisimple Lie algebra, is closely associated with the non-relativistic limit of the AdS/CFT correspondence. This paper investigates an infinite-dimensional Lie superalgebra called the 2D supersymmetric Galilean conformal algebra, which is obtained by the method of group contraction on 2D N = (2, 2) superconformal algebra. Physically, this superalgebra is relevant to superstring theory and the tricritical Ising model. We first construct a class of simple smooth modules induced from simple modules over finite-dimensional solvable Lie superalgebras. Furthermore, we provide several equivalent descriptions for simple smooth modules over the 2D supersymmetric Galilean conformal algebra. Additionally, examples of simple smooth modules such as the highest weight modules, Whittaker modules and high order Whittaker modules are presented.

In this paper, we first give a new interpretation of Jimbo’s boundary condition for the generic Painlevé VI transcendents, as the shrinking phenomenon in long time behavior of the Jimbo–Miwa–Mori–Sato equation with rank n = 3. We then interpret Jimbo’s asymptotic and monodromy formula from the viewpoint of the isomonodromy deformation with respect to irregular singularities.

In this paper, we study the mild ill-posedness problem of the two-dimensional MHD-Boussinesq system with the temperature-dependent thermal diffusivity and electrical conductivity near the background magnetic field in L^∞. Here we construct a sequence of initial data so that the L^∞-norm of vorticity of the corresponding solution is mildly ill-posed.

In this article, the authors introduce the Orlicz Calderón-Hardy spaces ${\cal{H}}_{q,2m}^{\Phi}(\mathbb{R}^{n})$ and investigate their properties. As an application, for m ∈ ℕ, the authors show that the iterated Laplacian Δ^m is bijective from Orlicz Calderón-Hardy spaces ${\cal{H}}_{q,2m}^{\Phi}(\mathbb{R}^{n})$ to corresponding Orlicz-Hardy spaces H^Φ(ℝⁿ).

We show that the spherically symmetric Einstein-scalar-field equations for small slowly particle-like decaying initial data at null infinity have unique global solutions.