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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 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]

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 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]

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]

We give some conditions on positive braids with at least two full twists that ensure their closure is a hyperbolic knot, with applications to the geometric classification of T-links, arising from dynamics, and twisted torus knots. [ABSTRACT FROM AUTHOR]

In 2021, Dzhunusov and Zaitseva classified two-dimensional normal affine commutative algebraic monoids. In this work, we extend this classification to noncommutative monoid structures on normal affine surfaces. We prove that two-dimensional algebraic monoids are toric. We also show how to find all monoid structures on a normal toric surface. Every such structure is induced by a comultiplication formula involving Demazure roots. We also give descriptions of opposite monoids, quotient monoids, and boundary divisors. [ABSTRACT FROM AUTHOR]

We refer to the discussion on different characterizations of the A_\infty class of weights, initiated by Duoandikoetxea, Martín-Reyes, and Ombrosi [Math. Z. 282 (2016), pp. 955–972]. Twelve definitions of the A_\infty condition are considered. For cubes in \mathbb {R}^d every two conditions are known to be equivalent, while for general bases we have a trichotomy: equivalence, one-way implication, or no dependency may occur. In most cases the relations between different conditions have already been established. Here all the unsolved cases are treated and, as a result, a full diagram of the said relations is presented. [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]

Given a collection of boundary divisors in the moduli space \overline {\mathcal {M}}_{0,n} of stable genus-zero n-pointed curves, Giansiracusa proved that their intersection is nonempty if and only if all pairwise intersections are nonempty. We give a complete classification of the pairs (g,n) for which the analogous statement holds in \overline {\mathcal {M}}_{g,n}. [ABSTRACT FROM AUTHOR]

A group \Gamma is said to be uniformly HS-stable if any map \varphi : \Gamma \to U(n) that is almost a unitary representation (w.r.t. the Hilbert Schmidt norm) is close to a genuine unitary representation of the same dimension. We present a complete classification of uniformly HS-stable groups among finitely generated residually finite ones. Necessity of the residual finiteness assumption is discussed. A similar result is shown to hold assuming only amenability. [ABSTRACT FROM AUTHOR]

By the algebraization of affine Nash groups, a connected affine Nash group is an abelian Nash manifold if and only if its algebraization is a real abelian variety. We first classify real abelian varieties up to isomorphisms. Then with a bit more efforts, we classify abelian Nash manifolds up to Nash equivalences. [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

We prove a functoriality result for the full C*-algebras of right-LCM monoids with respect to monoid inclusions that are closed under factorization and preserve orthogonality, and use this to show that if a right-LCM monoid is amenable in the sense of Nica, then so are its submonoids. As applications, we complete the classification of Artin monoids with respect to Nica amenability by showing that only the right-angled ones are amenable in the sense of Nica and we show that the Nica amenability of a graph product of right-LCM semigroups having no nontrivial units is inherited by the factors. [ABSTRACT FROM AUTHOR]

Noncommutative Berezin kernels are used to obtain Wold decompositions for *-representations of C*-algebras associated with noncommutative polydomains. We study the structure of these representations and C*-algebras and obtain some classification results. [ABSTRACT FROM AUTHOR]

As in the classical setting, we classify a large class of exterior fibrations in the based exterior homotopy category. More precisely, the fibration sequences F → E → X of path-connected based exterior CW-complexes are in bijective correspondence with the based exterior homotopy classes of maps from X to a universal exterior CW-complex YF. As a result we classify proper fibrations between CW-complexes. [ABSTRACT FROM AUTHOR]

A linear group G ≤ GL(V ), where V is a finite vector space, is called 1/2 -transitive if all the G-orbits on the set of nonzero vectors have the same size. We complete the classification of all the 1/2 -transitive linear groups. As a consequence we complete the determination of the finite 3/2 -transitive permutation groups - the transitive groups for which a point-stabilizer has all its nontrivial orbits of the same size. We also determine the (k + 1/2 )-transitive groups for integers k ≥ 2. [ABSTRACT FROM AUTHOR]

We give a complete algebraic characterization of the non-degenerated centers for planar trigonometric Cherkas systems. The main tools used in our proof are the classical results of Cherkas on planar analytic Liénard systems, the characterization of some subfields of the quotient field of the ring of trigonometric polynomials, and results due to Rosenlicht concerning algebraic solutions to transcendental equations. The results obtained are reminiscent of the ones for planar polynomial Cherkas systems, but the proofs are different. [ABSTRACT FROM AUTHOR]

We generalize a classification result for self-shrinkers of the mean curvature flow with nonnegative mean curvature, which was obtained by Colding and Minicozzi, by replacing the assumption on polynomial volume growth with a weighted L2 condition on the norm of the second fundamental form. Our approach adopts the viewpoint of weighted manifolds and also permits us to recover and to extend some other recent classification and gap results for self-shrinkers. [ABSTRACT FROM AUTHOR]

Let H3F denote the three dimensional hyperbolic space over F, where F denotes either the complex numbers C or the quaternions H. We offer an algebraic characterization of isometries of H3F. [ABSTRACT FROM AUTHOR]

It is a well-known result of Etingof, Nikshych and Ostrik that there are finitely many inequivalent integral modular categories of any fixed rank n. This follows from a double-exponential bound on the maximal denominator in an Egyptian fraction representation of 1. A naïve computer search approach to the classification of rank n integral modular categories using this bound quickly overwhelms the computer's memory (for n ≥ 7). We use a modified strategy: find general conditions on modular categories that imply integrality and study the classification problem in these limited settings. The first such condition is that the order of the twist matrix is 2, 3, 4 or 6, and we obtain a fairly complete description of these classes of modular categories. The second condition is that the unit object is the only simple non-self-dual object, which is equivalent to odd-dimensionality. In this case we obtain a (linear) improvement on the bounds and employ number-theoretic techniques to obtain a classification for rank at most 11 for odd-dimensional modular categories. [ABSTRACT FROM AUTHOR]