A pseudo-Kähler manifold is a natural generalization of a Kähler manifold. It is well-known that any generalized flag manifold has pseudo-Kähler metrics. Moreover, there exists a $T$-root system corresponding to a generalized flag manifold. In this paper, we investigate the signatures of invariant pseudo-Kähler metrics on a generalized flag manifold of which the $T$-root system becomes one of the irreducible reduced root systems (in general, a $T$-root system is not an irreducible reduced root system).

Fairness and centredness of ideals in commutative rings, i.e., the relations between assassins and weak assassins of a module, its small or large torsion submodule, and the corresponding quotients, are studied. General criteria as well as more specific results about idempotent or nil ideals are given, and several examples are presented.

We give a definition of scattering matrices based on the asymptotic behaviors of generalized eigenfunctions proving existence of radial limits of the functions and show that these scattering matrices are equivalent to the ones defined by wave operator approach in long-range $N$-body problems including the problems of Coulomb interaction potentials. Equivalence of stationary and time-dependent definitions of the generalized Fourier transforms is also shown.

In this article, the testing equality of the standardized generalized variance (SGV) of $k$ multivariate normal distributions with possibly unequal dimensions is studied. The conventional likelihood-ratio test statistic reveals a serious bias with an increase in dimensions. Therefore, we present a new test statistic that eliminates this bias, and propose an asymptotic approximation-based test. Our proposed test is valid not only in high-dimensional settings but also in large sample settings. Additionally, we obtain the asymptotic non-null distribution of the proposed test and the approximate confidence interval of the SGV under high-dimensional and large sample settings. Finally, we investigate the finite sample and dimension behavior of this test using Monte Carlo simulations.

We discuss orthogonal Chebyshev-Frolov lattices, their generating matrices and their use in Frolov cubature formula. We give a detailed account on coordinate-permuted systems that lead to fast computation and enumeration of such lattices. In particular, we explain the recurrences identified in (K. Suzuki and T. Yoshiki, Hiroshima Math. J., 49(1):139–159, 2019) via a plain constructive approach exhibiting a new hierarchical basis of polynomials. Dual Chebyshev-Frolov lattices and their generating matrices are also studied. Lattices enumeration in axis-parallel boxes is discussed.