The purpose of this paper is to give an explicit formula of the Łojasiewicz exponent of an isolated weighted homogeneous singularity in terms of its weights.

Let f,g:(ℝn, 0) → (ℝ, 0) be Cr+1 functions, r ∈ ℕ. We will show that if ∇f(0)=0 and there exist a neighbourhood U of 0 ∈ ℝn and a constant C > 0 such that

$$\begin{equation*} \left|\partial^m(g-f)(x)\right| ≤ C \left|\nabla f(x)\right|^{r+2-|m|} \quad \textrm{ for } x\in U, \end{equation*} $$

In this paper, we study the poset of basic tilting kQ-modules when Q is a Dynkin quiver, and the poset of basic support τ-tilting kQ-modules when Q is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if Q is of types $\mathbb{A}_{1}$, $\mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a non-linear orientation and the second poset is a distributive lattice if and only if Q is of type $\mathbb{A}_{1}$.

Let (A, ${\mathfrak{m}$) be a Cohen–Macaulay local ring of dimension d and let I ⊆ J be two ${\mathfrak{m}$-primary ideals with I a reduction of J. For i = 0,. . .,d, let eiJ(A) (eiI(A)) be the ith Hilbert coefficient of J (I), respectively. We call the number ci(I, J) = eiJ(A) − eiI(A) the ith relative Hilbert coefficient of J with respect to I. If GI(A) is Cohen–Macaulay, then ci(I, J) satisfy various constraints. We also show that vanishing of some ci(I, J) has strong implications on depth GJn(A) for n ≫ 0.

Let (M, g(t)) be a compact Riemannian manifold and the metric g(t) evolve by the Ricci flow. In the paper, we prove that the eigenvalues of geometric operator −Δφ + $\frac{R}{2}$ are non-decreasing under the Ricci flow for manifold M with some curvature conditions, where Δφ is the Witten Laplacian operator, φ ∈ C2(M), and R is the scalar curvature with respect to the metric g(t). We also derive the evolution of eigenvalues under the normalized Ricci flow. As a consequence, we show that compact steady Ricci breather with these curvature conditions must be trivial.