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This paper classifies the global dynamics of a 4D competitive Lotka-Volterra system (1.3) with two positive parameters k_1,\ k_2 via carrying simplex. It is proved that the interior of the carrying simplex is filled with periodic orbits except equilibria and each interior trajectory is persistent and tends to either a periodic orbit or an equilibrium if k_1/k_2=2. Otherwise, the system admits two 2D carrying simplices \Delta _i on x_1=i for i=0,\ 1, which are filled with periodic orbits surrounding an equilibrium. All interior orbits go in the long run to \Delta _1 if k_1/k_2>2. If 0

In Theorem 0.1 of the paper "A Severi type theorem for surfaces in \mathbb {P}^6" [Proc. Amer. Math. Soc. 149 (2021), pp. 591–605], we claimed to have given a complete classification of smooth surfaces in \mathbb {P}^6 with one 4-secant plane through the general point of \mathbb {P}^6, but the classification is still incomplete. [ABSTRACT FROM AUTHOR]

In this paper, we prove that a compact quasi-Einstein manifold (Mn, g, u) of dimension n ≥ 4 with boundary ∂ M, nonnegative sectional curvature and zero radial Weyl tensor is either isometric, up to scaling, to the standard hemisphere Sn+, or g = dt2 + ψ2(t)gL and u = u(t), where gL is Einstein with nonnegative Ricci curvature. A similar classification result is obtained by assuming a fourth-order vanishing condition on the Weyl tensor. Moreover, a new example is presented in order to justify our assumptions. In addition, the case of dimension n = 3 is also discussed. [ABSTRACT FROM AUTHOR]

In this paper, we provide a unified framework to classify homogeneous stable solutions of arbitrary order polyharmonic Lane-Emden equations. The key idea is the spherical decomposition of polyharmonic operator into differential operators. [ABSTRACT FROM AUTHOR]

Let X ⊂ PN be a projective, non-degenerate, irreducible smooth variety of dimension n. After giving the definition of generalised OADP-variety (one apparent double point), i.e. varieties X such that: n(k + 1) − (N − r)(k − r) + r = N, there is one apparent (k + 1)-secant (r − 1)-space to a generic projection of X from a point, we concentrate in studying generalised OADP-surfaces in low dimensional projective spaces, and the main result of this paper is the classification of smooth surfaces in P6 with one 4-secant plane through the general point of P6. [ABSTRACT FROM AUTHOR]

Let X ⊂ PN be a nondegenerate irreducible closed subvariety of dimension n over the field of complex numbers and let SX ⊂ PN be its secant variety. X ⊂ PN is called 'secant defective' if dim(SX) is strictly less than the expected dimension 2n + 1. In a 1993 paper, F.L. Zak showed that for a secant defective manifold it is necessary that N ≤ (nn+2 - 1 and that the Veronese variety ν2(Pn) is the only boundary case. Recently R. Muñoz, J. C. Sierra, and L. E. Solá Conde classified secant defective varieties next to this extremal case. In this paper, we will consider secant defective manifolds X ⊂ PN of dimension n with N = (nn+2) - 1 - ∊ for ∊ ≥ 0. First, we will prove that X is an LQEL-manifold of type δ = 1 for ∊ ≤ n - 2 by showing that the tangential behavior of X is good enough to apply the Scorza lemma. Then we will completely describe the above manifolds by using the classification of conicconnected manifolds given by Ionescu and Russo. Our method generalizes previous results by Zak, and by Muñoz, Sierra, and Solá Conde. [ABSTRACT FROM AUTHOR]

We consider the Toda system Δui + Σj=1²aijeuj = 4πγiδ0 in R², ∫R² euidx < ∞, for i = 1,2, where γi > −1, δ0 is the Dirac measure at 0, and the coefficients aij are of the Cartan matrix of rank 2: A2,B2(= C2),G2. Previously, the authors have gotten the classification and non-degeneracy results of solutions for Cartan matrix A2 and B2. In this paper, we consider the G2 case, and we completely classify the solutions and obtain the quantization result as well as the nondegeneracy of solutions for the G2 Toda system. [ABSTRACT FROM AUTHOR]

This paper discusses the twisted conjugacy classes in loop groups. We restrict to classical groups and give some explicit classifications. [ABSTRACT FROM AUTHOR]

par This paper is a continuation and a complement of our previous work on isomorphic classification of some spaces of compact operators. We improve the main result concerning extensions of the classical isomorphic classification of the Banach spaces of continuous functions on ordinals. As an application, fixing an ordinal $alpha $ and denoting by $X^{xi }$, $omega _{alpha } leq xi < omega _{alpha +1}$, the Banach space of all $X$-valued continuous functions defined in the interval of ordinals $[0, xi ]$ and equipped with the supremum, we provide complete isomorphic classifications of some Banach spaces ${mathcal K}(X^{xi }, Y^eta )$ of compact operators from $X^xi $ to $Y^eta $, $eta geq omega $. par It is relatively consistent with ZFC (Zermelo-Fraenkel set theory with the axiom of choice) that these results include the following cases: par 1. $X^{*}$ contains no copy of $c_{0}$ and has the Mazur property, and $Y= c_{0}(J)$ for every set $J$. par 2. $X=c_{0}(I)$ and $Y=l_{q}(J)$ for any infinite sets $I$ and $J$ and $1 leq q < infty $. par 3. $X=l_{p}(I)$ and $Y=l_{q}(J)$ for any infinite sets $I$ and $J$ and $1 leq q