In this paper, we consider one-parameter families of new extrinsic differential geometries on spacelike hypersurfaces in the lightcone. These geometries are constructed by applying two of one-parameter families of Legendrian dualities between pseudo-spheres in Lorentz-Minkowski space. These Legendrian dualities have been recently given as a part of extensions of the mandala of Legendrian dualities in the previous research of the authors.

We show that Hardy's inequalities for Laguerre expansions hold on the space $L^1(0,\infty)$ when the Laguerre parameters $\alpha$ are positive, and we prove that although the inequality holds on the real Hardy space $H^1(0,\infty)$ if $\alpha= 0$, it does not hold on $L^1(0,\infty)$. Further, Hardy's inequality for Hermite expansion is established on $L^1(0,\infty)$.

We strengthen previous results on the fundamental groups of the Hawaiian earring and wild Peano continua. Let $X$ be a path-connected, locally path-connected, first countable space which is not locally semi-simply connected at any point. If the fundamental group $\pi_1(X)$ is a subgroup of a free product $*_{j \in J}H_j$, then it is contained in a conjugate subgroup to some $H_j$.

D. W. Henderson established the metric topology vertion of J. H. C. Whitehead's Theorem on small subdivisions of simplicial complexes. However, his proof is valid only for locally finite-dimensional simplicial complexes. In this note, we give a complete proof of Henderson's Theorem for arbitrary simplicial complexes.

We calculate the group of homotopy classes of homotopy self-equivalences of 4-manifolds with $\pi_1$-free second homotopy.

Let $k$ be an algebraically closed field of positive characteristic, $G$ a reductive group over $k$, and $V$ a finite dimensional $G$-module. Let $B$ be a Borel subgroup of $G$, and $U$ its unipotent radical. We prove that if $S = \mathrm{Sym} \: V$ has a good filtration, then $S^U$ is $F$-pure.

Shioji and Takahashi proved that for every bounded sequence $\{a_n\}^{\infty}_{n=0}$ of real numbers, $$\{\phi(\{a_n\}^{\infty}_{n=0}) \: | \: \phi \mathrm{\: is \: a \: Banach \: limit} \}$$ $$\: \: = \displaystyle\bigcap\limits_{j=1}^{\infty} \overline{\mathrm{co}} \{(n+1)^-1 \displaystyle\sum\limits_{k=0}^n a_{k+m} \: | \: n \geq j, m \geq 0 \}.$$ We generalize this result to bounded sequences of vectors and also apply it to bounded measurable functions.

Let $(G,K)$ be one of the following Hermitian symmetric pair: $SU(p,q),S(U(p) \: \times \: U(q)))$, $(Sp(n,$\mathbf{R}$ ,U(n))$, or $(SO^*(2n),U(n))$. Let $G_C$ and $K_C$ be the complexifications of $G$ and $K$, respectively, $Q$ the maximal parabolic subgroup of $G_C$ whose Levi part is $K_C$, and $V$ the holomorphic tangent space at the origin of $G/K$. It is known that the ring of $K_C$ -invariant differential operators on $V$ has a generating system $\{\mathit{\Gamma_k}\}$ given in terms of determinant or Pfaffian that plays an essential role in the Capelli identities. Our main result is that determinant or Pfaffian of a deformation of the twisted moment map on the holomorphic cotangent bundle of $G_C / Q$ provides a generating function for the principal symbols of $\mathit{\Gamma_k}$'s.

We consider geometric decompositions of aspherical 4-manifolds which fibre over 2-orbifolds. We show that no such manifold admits infinitely many fibrations over hyperbolic base orbifolds and that “most” Seifert fibred 4-manifolds over hyperbolic bases have a decomposition induced from a decomposition of the base.

We present an Itô type formula for a Gaussian process, in which only the one-marginals of the Gaussian process are involved. Thus, this formula is well adapted to the study of processes increasing in the convex order, in a Gaussian framework. In particular, we give conditions ensuring that processes defined as integrals, with respect to one parameter, of exponentials of two-parameter Gaussian processes, are increasing in the convex order with respect to the other parameter. Finally, we construct Gaussian sheets allowing to exhibit martingales with the same one-marginals as the previously defined processes.

In 1932, Pontrjagin and Schnirelmann [15] proved the classical theorem which characterizes topological dimension by use of box-counting dimensions. They proved their theorem by use of geometric arguments in some Euclidean spaces. In this paper, by use of dimensional theoretical techniques in an abstract topological space, we investigate strong relations between metrics of spaces and box-counting dimensions. First, by use of the numerical information of normal sequences of finite open covers of a space $X$, we prove directly the following theorem characterizing topological dimension dim $X$.

THEOREM 0.1. Let $X$ be a nonempty separable metric space. Then $$\mathrm{dim}X = \min \left\{ \displaystyle\liminf\limits_{i \to \infty} \frac{\log_3 | \: \mathcal{U}_i|}{i} \left.\right| \{\mathcal{U}\}^\infty_{i=1} \text{is a normal star-sequence of finite open covers of} X \text{and a development of} X \right\}$$ $$ = \min \left\{ \displaystyle\liminf\limits_{i \to \infty} \frac{\log_2 | \: \mathcal{U}_i|} {i} \left.\right| \{\mathcal{U}_i\}^\infty_{i = 1} \text{is a normal delta-sequence of finite open covers of} X \text{and a development of} X \right\}.$$ Next, we study box-counting dimensions $\dim_B(X,d)$ by use of Alexandroff-Urysohn metrics $d$ induced by normal sequences. We show that the above theorem implies Pontrjagin-Schnirelmann theorem. The proof is different from the one of Pontrjagin and Schnirelmann (see [15]). By use of normal sequences, we can construct freely metrics $d$ which control the values of $\log N (\epsilon,d) / | \log \epsilon|$. In particular, we can construct chaotic metrics with respect to the determination of the box-counting dimensions as follows.

In our recent paper [5] in this journal, we have studied strong relations between metrics of spaces and box-counting dimensions by use of Alexandroff-Urysohn metrics d induced by normal sequences. In this addendum, we intend to improve the main theorems given in [5, Theorem 0.1 and 0.2] and give the complete solution for a problem of metrics and two box-counting dimensions.

We extend the one dimensional Denjoy-Sacksteder theorems to some diffeomorphism groups of smooth compact closed n-dimensional manifolds. More precisely, we show the existence of a non trivial stabilizer for actions of “quasi-conformal groups” admitting an “exceptional” minimal set which is of “strongly decreasing type”; our results include the classical Denjoy-Sacksteder theorems.

A positive definite Hermitian lattice is called regular if it represents all integers which can be represented locally by the lattice. We investigate binary regular Hermitian lattices over imaginary quadratic fields Q$$(\sqrt{-m})$$ and provide a complete list of the normal binary regular Hermitian lattices.

Consider the Navier-Stokes flow past a rotating obstacle with a general time-dependent angular velocity and a time-dependent outflow condition at infinity. After rewriting the problem on a fixed domain, one obtains a non-autonomous system of equations with unbounded drift terms. It is shown that the solution to a model problem in the whole space case R^d is governed by a strongly continuous evolution system on L^p_σ(R^d) for 1 < p < ∞. The strategy is to derive a representation formula, similar to the one known in the case of non-autonomous Ornstein-Uhlenbeck equations. This explicit formula allows to prove L^p-L^q estimates and gradient estimates for the evolution system. These results are key ingredients to obtain (local) mild solutions to the full nonlinear problem by a version of Kato's iteration scheme.

Let $(G,H) = (U(p,q),U(p-1,q) \times U(1))$ and $\{\Gamma_n\}$ a tower of congruence uniform lattices in $G$. By the period integrals of automorphic forms on $\Gamma \backslash G$ along $\Gamma_n \cap H\backslash H$ , we introduce a certain discrete measure $d \mu^H_{\Gamma_n}$ on the $H$-spherical unitary dual of $G$. It is shown that the sequence of measures $d \mu^H_{\Gamma_n}$ with growing $n$ converges in a weak sense to the Plancherel measure $d \mu^H$ for the symmetric space $H\backslash G$.