In a topological connection theory, we give a general way of constructing flat slicing functions in locally trivial principal bundles with a discrete group as structure group. Slicing functions play a role of connections in smooth category. By applying this construction to the universal principal bundle over a classifying space which comes from the Milnor construction, we obtain an explicit description of the universal flat slicing function. Using this explicit nature, we show that flat slicing functions given to respective contexts are pulled back from the universal one.

This paper is concerned with a diffusive Lotka-Volterra prey-predator model with finitely many protection zones for the prey species. We discuss the stability of trivial and semi-trivial steady-state solutions, and we also study the existence and non-existence of positive steady-state solutions. It is proved that there exists a certain critical growth rate of the prey for survival. Moreover, it is shown that when cross-diffusion is present, under certain conditions, the critical value decreases as the number of protection zones increases. On the other hand, it is also shown that when cross-diffusion is absent, the critical value does not always decrease even if the number of protection zones increases.

In this paper we introduce the concept of quadratic operator perspective for a continuous function $\text{Φ}$ defined on the positive semi-axis of real numbers, the invertible operator $T$ and operator $V$ on a Hilbert space by

$${◯}_{\Phi }(V,T):=T^{*}\Phi (|V{T}^{-1}{|}^2)T.$$

This generalize the quadratic weighted operator geometric mean of $(T,V)$ defined by

$$T{\odot }_{v}V:=||V{T}^{-1}{|}^{v}T{|}^2$$

for $v\in [0,1]$ and the quadratic relative operator entropy defined by

$$\odot \left(T\text{|}V\right):=T^{*}\ln \left({\left|V{T}^{-1}\right|}^2\right)T.$$

Some inequalities for this perspective of convex functions are established. Applications for quadratic weighted operator geometric mean and quadratic relative operator entropy are also provided.

This paper is concerned with consistency properties of the dimensionality estimation criteria AIC, BIC, and ${\text{C}}_p$ in CCA (Canonical Correlation Analysis) between $p$ variables and $q(\le p)$ variables, based on a sample of size $N=n+1$. The consistency properties of the criteria are studied under a high-dimensional asymptotic framework such that $p$ and $n$ tend to infinity satisfying $p/n\to c\in [0,1)$, and under two types of assumptions on the order of the population canonical correlations, where $q$ is fixed. We note that there are cases that the criteria based on AIC and ${\text{C}}_p$ are consistent, but the criterion based on BIC is not consistent. Through a Monte Carlo simulation experiment, our results are checked numerically, and the estimation criteria are compared.

The isomorphism classes of the images of irreducible two-dimensional Galois representations with squarefree conductor are determined.