Let $\mathrm{SL}(2,\mathbb{H})$ be the group of $2 × 2$ quaternionic matrices with Dieudonné determinant one. The group $\mathrm{SL}(2,\mathbb{H})$ acts on the five dimensional hyperbolic space by isometries. We investigate extremality of Jørgensen type inequalities in $\mathrm{SL}(2,\mathbb{H})$. Along the way, we derive Jørgensen type inequalities for quaternionic Möbius transformations which extend earlier inequalities obtained by Waterman and Kellerhals.

We define a generalized fixed contact angle condition for $n$-varifold and establish a boundary monotonicity formula. The results are natural generalizations of those for the Neumann boundary condition considered by Grüter-Jost [7].

In this paper we give a new Koksma-Hlawka type inequality for Quasi-Monte Carlo (QMC) integration. QMC integration of a function $f\colon[0,1)^s\rightarrow\mathbb{R}$ by a finite point set $\mathcal{P}\subset[0,1)^s$ is the approximation of the integral $I(f):=\int_{[0,1)^s}f(\mathbf{x})\,d\mathbf{x}$ by the average $I_{\mathcal{P}}(f):=\frac{1}{|\mathcal{P}|}\sum_{\mathbf{x} \in \mathcal{P}}f(\mathbf{x})$. We treat a certain class of point sets $\mathcal{P}$ called digital nets. A Koksma-Hlawka type inequality is an inequality providing an upper bound on the integration error $\text{Err}(f;\mathcal{P}):=I(f)-I_{\mathcal{P}}(f)$ of the form $|\text{Err}(f;\mathcal{P})|\le C\cdot \|f\|\cdot D(\mathcal{P})$. We can obtain a Koksma-Hlawka type inequality by estimating bounds on $|\hat{f}(\mathbf{k})|$, where $\hat{f}(\mathbf{k})$ is a generalized Fourier coefficient with respect to the Walsh system. In this paper we prove bounds on the Walsh coefficients $\hat{f}(\mathbf{k})$ by introducing an operator called ‘dyadic difference’ $\partial_{i,n}$. By converting dyadic differences $\partial_{i,n}$ to derivatives $\frac{\partial }{\partial x_i}$, we get a new bound on $|\hat{f}(\mathbf{k})|$ for a function $f$ whose mixed partial derivatives up to order $\alpha$ in each variable are continuous. This new bound is smaller than the known bound on $|\hat{f}(\mathbf{k})|$ under some instances. The new Koksma-Hlawka type inequality is derived using this new bound on the Walsh coefficients.

In this paper, we consider a $C_{p}$ type criterion for ANOVA model with a tree ordering ($\mathrm{TO}$) $\theta_{1}\leq\theta_{j}, (j=2,\ldots,l)$ where $\theta_{1},\ldots\theta_{l}$ are population means. In general, under ANOVA model with the $\mathrm{TO}$, the usual $C_{p}$ criterion has a bias to a risk function, and the bias depends on unknown parameters. In order to solve this problem, we calculate a value of the bias, and we derive its unbiased estimator. By using this estimator, we provide an unbiased $C_{p}$ type criterion for ANOVA model with the $\mathrm{TO}$, called $\mathrm{TO}C_{p}$. A penalty term of the $\mathrm{TO}C_{p}$ is simply defined as a function of an indicator function and maximum likelihood estimators. Furthermore, we show that the $\mathrm{TO}C_{p}$ is the uniformly minimum-variance unbiased estimator (UMVUE) of a risk function.

This paper is devoted to the existence and (in)stability of nonconstant steady-states in a system of a semilinear parabolic equation coupled to an ODE, which is a simplified version of a receptor-ligand model of pattern formation. In the neighborhood of a constant steady-state, we construct spatially heterogeneous steady-states by applying the bifurcation theory. We also study the structure of the spectrum of the linearized operator and show that bifurcating steady-states are unstable against high wave number disturbances. In addition, we consider the global behavior of the bifurcating branches of nonconstant steady-states. These are quite different from classical reaction-diffusion systems where all species diffuse.