We first prove that if a is both left (b; c)-invertible and left (c; b)- invertible, then a is both (b; c)-invertible and (c; b)-invertible in a *-monoid, which generalizes the recent result about the inverse along an element by L. Wang and D. Mosić [Linear Multilinear Algebra, Doi.org/10.1080/03081087. 2019.1679073], under the conditions (ab)* = ab and (ac)* = ac: In addition, we consider that ba is (c; b)-invertible, and at the same time ca is (b; c)-invertible under the same conditions, which extend the related results about Moore- Penrose inverses studied by J. Chen, H. Zou, H. Zhu, and P. Patrício [Mediterr J. Math., 2017, 14: 208] to (b; c)-inverses. As applications, we obtain that under condition (a²)* = a²; a is an EP element if and only if a is one-sided core invertible, if and only if a is group invertible.

Let \mathbb{S}(m; d; k) be the set of k-uniform supertrees with m edges and diameter d; and S₁(m; d; k) be the k-uniform supertree obtained from a loose path u₁; e₁; u₂; e₂,..., u_d; e_d; u_d+1 with length d by attaching m — d edges at vertex u_{\lfloor d/2\rfloor +1}: In this paper, we mainly determine S₁(m; d; k) with the largest signless Laplacian spectral radius in \mathbb{S}(m; d; k) for 3≤d≤m –1: We also determine the supertree with the second largest signless Laplacian spectral radius in \mathbb{S}(m; 3; k): Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices).

The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrödinger equation u_t={\mathrm{i}\alpha u}_{xx}+\beta u^{2-}{u}_x+\gamma {\left|u\right|}^2{u}_x+\mathrm{i}{\left|u\right|}^{2}u on the half-line with inhomogeneous boundary condition. We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces. Moreover, we show that the nonlinear part of the solution on the half-line is smoother than the initial data.

We prove that a Finsler manifold with vanishing Berwald scalar curvature has zero E-curvature. As a consequence, Landsberg manifolds with vanishing Berwald scalar curvature are Berwald manifolds. For (α,β)-metrics on manifold of dimension greater than 2, if the mean Landsberg curvature and the Berwald scalar curvature both vanish, then the Berwald curvature also vanishes.

Let k>2 be an integer and P be a 2n×2n symplectic orthogonal matrix satisfying P^k = I_2n and ker(P^j - I_2n) = 0; 1≤j <k: For any compact convex hypersurface \sum \subset {ℝ}^{2n} with n≥2 which is P-cyclic symmetric, i.e., x\mathrm{\in }\sum implies P_x\mathrm{\in }\sum ; we prove that if \sumis (r;R)-pinched with ,then there exist at least n geometrically distinct P-cyclic symmetric closed characteristics on \sum for a broad class of matrices P:

Interval hypermatrices (tensors) are introduced and interval \alpha-hypermatrices are uniformly characterized using a finite set of 'extreme' hypermatrices, where \alpha can be strong P, semi-positive, or positive definite, among many others. It is shown that a symmetric interval is an interval (strictly) copositive-hypermatrix if and only if it is an interval (E) E₀-hypermatrix. It is also shown that an even-order, symmetric interval is an interval positive (semi-) definite-hypermatrix if and only if it is an interval P (P₀)-hypermatrix. Interval hypermatrices are generalized to sets of hyper-matrices, several slice-properties of a set of hypermatrices are introduced and sets of hypermatrices with various slice-properties are uniformly characterized. As a consequence, several slice-properties of a compact, convex set of hyper-matrices are characterized by its extreme points.

This paper is concerned with the Schrödinger-Poisson equation-\mathrm{\Delta }u+V(x)u+\phi (x)u=f(x,u),x\in {ℝ}^{\mathrm{3}},-\mathrm{\Delta }\phi =u^{\mathrm{2}},\underset{\left|x\right|\to +\infty }{\lim }\phi (x)=\mathrm{0.}

Under certain hypotheses on V and a general spectral assumption, the existence and multiplicity of solutions are obtained via variational methods.

We state that the ag-transitive automorphism group of a 2-\upsilon ,\mathrm{5},\lambda design \mathcal{D} is primitive of affine type or almost simple type. We also find that there are up to isomorphism 20 2-\upsilon ,\mathrm{5},\lambda designs admitting flag-transitive automorphism groups with socle of a sporadic simple group.

A supertree is a connected and acyclic hypergraph. We investigate the supertrees with the extremal spectral radii among several kinds of r-uniform supertrees. First, by using the matching polynomials of supertrees, a new and useful grafting operation is proposed for comparing the spectral radii of supertrees, and its applications are shown to obtain the supertrees with the extremal spectral radii among some kinds of r-uniform supertrees. Second, the supertree with the third smallest spectral radius among the r-uniform supertrees is deduced. Third, among the r-uniform supertrees with a given maximum degree, the supertree with the smallest spectral radius is derived. At last, among the r-uniform starlike supertrees, the supertrees with the smallest and the largest spectral radii are characterized.

Let {{\displaystyle \mathcal{D}}}_N be the multicomponent twisted Heisenberg-Virasoro algebra. We compute the second continuous cohomology group with coeficients in ℂ and study the bihamiltonian Euler equations associated to {{\displaystyle \mathcal{D}}}_N and its central extensions.

We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators. Then we give the equivalent norms via the Peetre type maximal functions and atomic decomposition of these spaces.

Let {\mathrm{\Lambda }}_{(0,0)}=\left(\begin{array}{cc}A& {}_A{N}_B\\ {}_B{N}_A& B\end{array}\right) be a Morita ring, where the bimodule homomorphisms \varphiand \psi are zero. We study the finite presentedness, locally coherence, pure projectivity, pure injectivity, and FP-injectivity of modules over {\mathrm{\Lambda }}_{(\mathrm{0},\mathrm{0})}. Some applications are then given.

We mainly study the super-biderivations of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras. In particular, we prove that all super-biderivations of not-finitely graded Lie superalgebras related to generalized super-Virasoro algebras are inner.

We study a kind of partial information non-zero sum differential games of mean-field backward doubly stochastic differential equations, in which the coefficient contains not only the state process but also its marginal distribution, and the cost functional is also of mean-field type. It is required that the control is adapted to a sub-filtration of the filtration generated by the underlying Brownian motions. We establish a necessary condition in the form of maximum principle and a verification theorem, which is a sufficient condition for Nash equilibrium point. We use the theoretical results to deal with a partial information linear-quadratic (LQ) game, and obtain the unique Nash equilibrium point for our LQ game problem by virtue of the unique solvability of mean-field forward-backward doubly stochastic differential equation.