Let $E(1)$ be the first Johnson-Wilson spectrum at the prime two. In this paper, we calculate the homotopy groups of the $E(1)$-localized sphere spectrum with a grading over the Picard group of the stable homotopy category of $E(1)$-local spectra.

Abstract. Let $\varphi : \mathbb R^n \times [0,\infty) \rightarrow [0,\infty)$ satisfy that $\varphi(x,\cdot)$ is an Orlicz function for any given $x \in \mathbb R^n$, and $\varphi(\cdot,t)$ is a Muckenhoupt $A_\infty$ weight uniformly in $t \in (0,\infty)$. The weak Musielak-Orlicz Hardy space $WH^\varphi(\mathbb R^n)$ is defined to be the set of all tempered distributions such that their grand maximal functions belong to the weak Musielak-Orlicz space $WL^\varphi(\mathbb R^n)$. In this paper, we discuss the boundedness of the Calderón-Zygmund operator with variable kernel from $WH^\varphi(\mathbb R^n)$ to $WL^\varphi(\mathbb R^n)$. These results are new even for the classical weighted weak Hardy space and probably new for the classical weak Hardy space.

It is well known that the set of all functions $g$ such that “$f \in L^{p_1}\Rightarrow fg \in L^{p_2}$” is $L^{p_3}$, if $1/p_2 = 1/p_1 + 1/p_3$ with $p_i \in (0,\infty], i = 1, 2, 3$. In this paper we characterize the set of all functions $g$ such that “$f \in \mathrm w L^{\phi_1} \Rightarrow fg \in \mathrm w L^{\phi_2}$”, where $\mathrm w L^{\phi_i}, i = 1, 2$, are weak Orlicz spaces.

For a given angle $\theta$, consider the configuration space $C_n$ of equilateral $n$-gons in $\mathbf R^3$ whose bond angles are equal to $\theta$ except for two successive ones. We show that when $n \ge 8$ and $\theta$ is sufficiently close to the inner angle $\frac{n-2}{n}\pi$ of the regular $n$-gon, $C_n$ is homeomorphic to the $(n-4)$-dimensional sphere $S^{n-4}$.

Let $N_g$ be the non-orientable surface of genus $g$, $\mathrm{MCG}(N_g)$ the mapping class group of $N_g$, $\mathscr{T} (N_g)$ the index $2$ subgroup generated by all Dehn twists of $\mathrm{MCG}(N_g)$. We prove that for odd genus, $(1)$ if $g = 4k + 3 (k\ge1), \mathrm{MCG}(N_g)$ can be generated by three elements of finite order; $(2)$ if $g = 4k + 1 (k\ge2)$, $\mathscr{T} (N_g)$ can be generated by three elements of finite order.

Given a representation of a link group, we introduce a trilinear form as a topological invariant. We show that, if the link is either hyperbolic or a knot with the malnormal peripheral subgroup, then the trilinear form is equal to the pairing of the (twisted) triple cup product and the fundamental relative 3-class. We give some examples illustrating the main results.

In this paper, we evaluate the asymptotic bias of $C_p$ type criterion for model selection in the GEE (generalized estimating equation) method when the sample and cluster sizes are large. We present the asymptotic properties of GEE estimator and the model selection criterion. Then, we present the order of the asymptotic bias of PMSEG (the prediction mean squared error in the GEE).

An oscillatory hexagonal solution in a two component reaction-diffusion system with a non-local term is studied. By applying the center manifold theory, we obtain a four-dimensional dynamical system that informs us about the bifurcation structure around the trivial solution. Our results suggest that the oscillatory hexagonal solution can bifurcate from a stationary hexagonal solution via the Hopf bifurcation. This provides a reasonable explanation for the existence of the oscillatory hexagon.