1 13 4. Free rational curves 14 5. Uniruledness 16 5.1. Definitions equivalent to uniruledness 16 5.2. Examples and consequences of uniruledness 18 6. Rational connectivity 20 6.1. Definitions equivalent to rational connectivity 21 6.2. Rational chain-connectivity implies rational connectivity 22 6.3. Examples and consequences of rational connectivity 26 7. The MRC quotient 27 7.1. Quotients by...

Let \(M= \Gamma \setminus \mathbb {H}_d\) be a compact quotient of the d-dimensional Heisenberg group \(\mathbb by lattice subgroup \(\Gamma \). We show that eigenvalue counting function \(N^\alpha \left( \lambda \right) \) for any fixed element family second order differential operators \(\left\{ \mathcal {L}_\alpha \right\} on M has asymptotic behavior \sim C_{d,\alpha } {\text {vol}}\left( M...

This study aims to contribute improving mental health services and establishing a direction for disaster survivors by verifying the effectiveness of TLS (Training Life Skills) app, management mobile application. Altogether, 22 received access app (with guidance) eight weeks; we analyzed its examining each participant’s electroencephalography data, which were collected while they utilizing app. ...

We describe hypergeometric functions of Deligne-Mostow type for open subsets of the configuration space of six points on P, induced from those for seven points on P. The seven point ball quotient example DM(2, 1) does not appear on Mostow’s original list [9], but does appear on Thurston’s corrected version [12]. We show that DM(2, 1) is a finite cover of the moduli space of cubic surfaces MC en...

We study the notion of a strongly normal sequence in the dual E∗ of a Banach space E. In particular, we prove that the following three conditions are equivalent: (1) E∗ has a strongly normal sequence, (2) (E∗, σ(E∗, E)) has a Schauder basic sequence, (3) E has an infinite-dimensional separable quotient.

The quotient correlation is defined here as an alternative to Pearson’s correlation that is more intuitive and flexible in cases where the tail behavior of data is important. It measures nonlinear dependence where the regular correlation coefficient is generally not applicable. One of its most useful features is a test statistic that has high power when testing nonlinear dependence in cases whe...

The quotient correlation is defined here as an alternative to Pearson’s correlation that is more intuitive and flexible in cases where the tail behavior of data is important. It measures nonlinear dependence where the regular correlation coefficient is generally not applicable. One of its most useful features is a test statistic that has high power when testing nonlinear dependence in cases whe...

The computation of a right eigenvector and corresponding finite eigenvalue of a large sparse generalised eigenproblem Ax = λMx using preconditioned Rayleigh quotient iteration and the simplified JacobiDavidson method is considered. Both methods are inner-outer iterative methods and we consider GMRES and FOM as iterative algorithms for the (inexact) solution of the inner systems that arise. The ...

In this paper, we first define the notion of $\mathcal{F}$-cosmall quotient for an additive exact substructure $\mathcal{F}$ structure $\mathcal{E}$ in category $\mathcal{A}$. We show that every is right minimal some cases. also give definition $\mathcal{F}$-superfluous and relate it approximation modules. As application, investigate our results a pure-exact $\mathcal{F}$.