We show that the sum of Betti numbers of the intersections of two real flag manifolds in a complex flag manifold consisting of sequences of complex subspaces in a complex vector space are invariants of such real flag manifolds. We also show that similar claims hold for the intersections of two quaternionic flag manifolds and the intersections of real and quaternionic flag manifolds.
We study the $L^\infty$-decay estimate and the asymptotic behavior of solutions to dissipative nonlinear Schrödinger equations. In some known results, Kita-Nakamura established an $L^{\infty}$-decay result under a weak dissipative condition for nonlinear power $p \gt 2.686$. In this paper, we extend the lower bound of the admissible range to $p \gt 5/2$, still under weak dissipation and with small initial data in $H^2$ satisfying $x^2 u_0 \in L^2$. Our result also contrasts with the work of Jin-Jin-Li, Chinese mathematicians, who treated the same problem under a strong dissipative condition and removed the size restriction on the initial data. Notably, we have solved the problem of extending the nonlinear power $p$ range for the modified scattering and decay estimates without the strong dissipative condition.
We consider the Cauchy problem for the derivative fractional cubic nonlinear Schrödinger equation \[ \begin{cases} \partial_{t}u-\dfrac{1}{\alpha}\left\vert \partial_{x}\right\vert ^{\alpha}u=i\lambda\partial_{x}(\left\vert u\right\vert ^{2}u), & t\gt 0,~ x\in\mathbb{R}, \\ u\left( 0,x\right) =u_{0}\left( x\right) , & x\in\mathbb{R}, \end{cases} \] where $\alpha\in\left( 4,5\right) ,$ and $\lambda\in\mathbb{R}.$ We show that the large time asymptotics of solutions has a logarithmic correction in the phase comparing with the corresponding linear case. We develop the method based on the factorization techniques. Also we apply the known results on the $\mathbf{L}^{2}$ - boundedness of pseudodifferential operators.
In this paper, we establish the truncated non-integrated defect relation for meromorphic maps from complete Kähler manifolds into the projective space intersecting hyperlpanes with the distributive constant of these hyperplanes. As an application, the uniqueness theorem is also given.
It is shown that the growth of the hyperbolic derivative of an analytic self-map phi of the unit disc naturally restricts the behaviour of its modulus yielding new integral growth estimates for z bar right arrow log 1-vertical bar phi(0)vertical bar(2)/1-vertical bar phi(z)vertical bar(2). function is used to derive a kind of embedding theorem involving hyperbolic derivative. This result is further used to characterize weighted hyperbolic Dirichlet classes in terms of hyperbolic oscillation. Certain inclusion relations related to hyperbolic Besov classes are also briefly discussed.
