Related papers: Commutator formulas for gradient Ricci shrinker an…

We discuss the generalization of the Dirac equations and spinors in momentum space to free unstable spin-$1/2$ fermions taking into account the fundamental requirement of Lorentz covariance. We derive the generalized adjoint Dirac equations…

Let $(M^n,g)$ be an n-dimensional complete Riemannian manifold. We consider gradient estimates and Liouville type theorems for positive solutions to the following nonlinear elliptic equation: $$\Delta u+au\log u=0,$$ where $a$ is a nonzero…

Recently, Sun-Zhang have developed an algebraic theory for K\"ahler-Ricci shrinkers showing that they admit the structure of a polarized Fano fibration $(\pi: X \to Y, \xi)$. In particular, they conjecture that existence of a K\"ahler-Ricci…

The radial Komatu-Loewner equation is a differential equation for certain normalized conformal mappings that can be used to describe the growth of slits within multiply connected domains. We show that it is possible to choose constant…

We prove global stability for a system of nonlinear wave equations satisfying a generalized null condition. The generalized null condition allows for null forms whose coefficients have bounded $C^k$ norms. We prove both pointwise decay and…

In this paper, we prove that any Ricci shrinker that is sufficiently close to $(\mathbb C P^N,g_{FS})$ in the Gromov-Hausdorff sense must itself be isometric to $(\mathbb C P^N,g_{FS})$.

Ostrogradsky instability generally appears in nondegenerate higher-order derivative theories and this issue can be resolved by removing any existing degeneracy present in such theories. We consider an action involving terms that are at most…

We provide explicit conditions for uniform stability, global asymptotic stability and uniform exponential stability for dynamic equations with a single delay and a nonnegative coefficient. Some examples on nonstandard time scales are also…

Let $(M,g)$ be a smooth Riemannian manifold and $\mathsf{G}$ a compact Lie group acting on $M$ effectively and by isometries. It is well known that a lower bound of the sectional curvature of $(M,g)$ is again a bound for the curvature of…

A new integrability condition of the Riccati equation $dy/dx=a(x)+b(x)y+c(x)y^{2}$ is presented. By introducing an auxiliary equation depending on a generating function $f(x)$, the general solution of the Riccati equation can be obtained if…

We show that Lorentzian manifolds whose isometry group is of dimension at least $\frac{1}{2}n(n-1)+1$ are expanding, steady and shrinking Ricci solitons and steady gradient Ricci solitons. This provides examples of complete locally…

We consider a family of conforming space-time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a CFL condition to guarantee stability. Recent works by O.…

We find new examples of steady gradient Ricci solitons with positive curvature operator in dimensions four and above. Utilising a procedure first introduced by Lai, we construct examples with $O(p) \times O(q)$ symmetry in dimension $p+q$…

In this paper we introduce the notion of rigidity for harmonic-Ricci solitons and we provide some characterizations of rigidity, generalizing some known results for Ricci solitons. In the compact case we are able to deal with not…

The Riccati equation method and an approach of the use of unknown factors is used to establish oscillation, suboscillation and nonoscillation criteria for linear systems of ordinary differential equations. A necessary condition for Lyapunov…

We prove local versions of the Ricci curvature and $\nu$-entropy gap theorems for Ricci shrinkers, which respectively generalize a previous result of Munteanu-Wang and a prior result of the authors with Ma. The key point is that these local…

We consider a problem of prescribing the partial Ricci curvature on a locally conformally flat manifold $(M^n, g)$ endowed with the complementary orthogonal distributions $D_1$ and $D_2$. We provide conditions for symmetric $(0,2)$-tensors…

We give degree formulas for Grothendieck polynomials indexed by vexillary permutations and $1432$-avoiding permutations via tableau combinatorics. These formulas generalize a formula for degrees of symmetric Grothendieck polynomials which…

We find estimates on the norms commutators of the form [f(x), y] in terms of the norm of [x, y] assuming that x and y are contractions in a C*-algebra A, with x normal and with spectrum within the domain of f. In particular we discuss [x^2,…

We study the stability of radial solutions of the semilinear elliptic equation $\Delta u +f(u)=0$ in ${\bf R^N}$, where $N \geq 3$ and $f$ is a general superciritical nonlinearity. We give a classification of the solution structures with…