Wavelet coefficients of a process have arguments shift and scale. It can thus be viewed as a time series along shift for each scale. We have considered in the previous study general wavelet coefficient domain estimators and revealed a localization property with respect to shift. In this paper, we formulate the localization property with respect to scale, which is more difficult than that of shift. Two factors that govern the decay rate of cross-scale covariance are indicated. The factors are both functions of vanishing moments and scale-lags. The localization property is then successfully applied to formulate limiting variance in the central limit theorem associated with Hurst index estimation problem of fractional Brownian motion. Especially, we can find the optimal upper bound $J$ of scales $1, \ldots, J$ used in the estimation to be $J = 5$ by an evaluation of the diagonal component of the limiting variance, in virtue of the scale localization property.

In this paper we consider invariant Matsumoto metrics which are induced by invariant Riemannian metrics and invariant vector fields on homogeneous spaces, and then we give the flag curvature formula of them. Also we study the special cases of naturally reductive spaces and bi-invariant metrics. We end the article by giving some examples of geodesically complete Matsumoto spaces.

In this paper we show that stochastic differential equations with jumps and non-Lipschitz coefficients have $(\xi,W,N_{p})$-pathwise unique strong solutions by the Euler--Maruyama approximation. Moreover, the Euler--Maruyama discretisation has an optimal strong convergence rate.

We shall introduce a notion of $S^{1}$-fibred nilBott tower. It is an iterated $S^{1}$-bundle whose top space is called an $S^{1}$-fibred nilBott manifold and the $S^{1}$-bundle of each stage realizes a Seifert construction. The $S^{1}$-fibred nilBott tower is a generalization of real Bott tower from the viewpoint of fibration. In this note we shall prove that any $S^{1}$-fibred nilBott manifold is diffeomorphic to an infranilmanifold. According to the group extension of each stage, there are two classes of $S^{1}$-fibred nilBott manifolds which is defined as finite type or infinite type. We discuss their properties.

For any weight vector $\chi$ of positive integers, the weighted projective space $\mathbb{P}(\chi)$ is a projective toric variety, and has orbifold singularities in every case other than standard projective space. Our principal aim is to study the algebraic topology of $\mathbb{P}(\chi)$, paying particular attention to its localisation at individual primes $p$. We identify certain $p$-primary weight vectors $\pi$ for which $\mathbb{P}(\pi)$ is homeomorphic to an iterated Thom space, and discuss how any weighted projective space may be reassembled from its $p$-primary parts. The resulting Thom isomorphisms provide an alternative to Kawasaki's calculation of the cohomology ring of $\mathbb{P}(\chi)$, and allow us to recover Al Amrani's extension to complex $K$-theory. Our methods generalise to arbitrary complex oriented cohomology algebras and their dual homology coalgebras, as we demonstrate for complex cobordism theory, the universal example. In particular, we describe a fundamental class that belongs to the complex bordism coalgebra of $\mathbb{P}(\chi)$, and may be interpreted as a resolution of singularities.

In this paper we study involutions on minimal surfaces of general type with $p_{g} = q = 0$ and $K^{2} = 7$. We focus on the classification of the birational models of the quotient surfaces and their branch divisors induced by an involution.

Poincaré proved existence of meromorphic function solution of a certain kind of system of multiplication formulae and claimed that the class of those functions is new. In this paper we exemplify Poincaré's meromorphic functions being truly new in a rigorous sense. We prove that those functions cannot be expressed rationally (nor algebraically) by solutions of linear difference equations, the exponential function $\mathrm{e}^{x}$, the trigonometric functions $\cos x$ and $\sin x$, the Weierstrass function $\wp(x)$ and any other functions satisfying first order algebraic difference equations, where the transforming operator of the difference equations is one sending $y(x)$ to $y(2x)$, not to $y(x+1)$.

On the basis of a result of Barrett [2], we show that members of certain classes of abstract Levi flat manifolds with boundary, whose Levi foliation contains a compact leaf with contracting, flat holonomy, admit no $\mathit{CR}$ embedding as a hypersurface of a complex manifold. In particular, it follows that the foliation constructed in [6] is not embeddable.

M. Kiyota, T. Okuyama and T. Wada recently proved that each $2$-block of a symmetric group $\Sigma_{n}$ contains a unique irreducible Brauer character of height $0$. We present a more conceptual proof of this result.

We consider a parametric nonlinear elliptic equation driven by the Dirichlet $p$-Laplacian. We study the existence, nonexistence and multiplicity of positive solutions as the parameter $\lambda$ varies in $\mathbb{R}^{+}_{0}$ and the potential exhibits a $p$-superlinear growth, without satisfying the usual in such cases Ambrosetti--Rabinowitz condition. We prove a bifurcation-type result when the reaction has ($p-1$)-sublinear terms near zero (problem with concave and convex nonlinearities). We show that a similar bifurcation-type result is also true, if near zero the right hand side is ($p-1$)-linear.

Let $X$ be a Calabi--Yau threefold and $\mu$ the symmetric trilinear form on the second cohomology group $H^{2}(X,\mathbb{Z})$ defined by the cup product. We investigate the interplay between the Chern classes $c_{2}(X)$, $c_{3}(X)$ and the trilinear form $\mu$, and demonstrate some numerical relations between them. When the cubic form $\mu(x,x,x)$ has a linear factor over $\mathbb{R}$, some properties of the linear form and the residual quadratic form are also obtained.

We are interested in periodic points on the boundaries of rotation domains of rational functions. R. Pérez-Marco showed that there are no periodic points on the boundaries of Siegel disks having Jordan neighborhoods with certain properties [12]. In this paper, we consider periodic points on the boundaries of rotation domains under more weakly conditions.

In this paper we obtain the $L^{p}$-boundedness for the maximal functions and the singular integrals associated to surfaces $(y,\phi(\lvert y\rvert))$ with rough kernels, $1 < p < \infty$. The analogue estimate is also established for the corresponding maximal singular integrals.

In this paper, we study the gradient estimates for positive solutions to the following nonlinear parabolic equation \begin{equation*} \frac{\partial u}{\partial t} = \triangle_{f}u+cu^{-\alpha} \end{equation*} on complete noncompact manifolds with Bakry--Émery Ricci curvature bounded below, where $\alpha$, $c$ are two real constants and $\alpha > 0$.

The degenerate affine and affine BMW algebras arise naturally in the context of Schur--Weyl duality for orthogonal and symplectic Lie algebras and quantum groups, respectively. Cyclotomic BMW algebras, affine Hecke algebras, cyclotomic Hecke algebras, and their degenerate versions are quotients. In this paper the theory is unified by treating the orthogonal and symplectic cases simultaneously; we make an exact parallel between the degenerate affine and affine cases via a new algebra which takes the role of the affine braid group for the degenerate setting. A main result of this paper is an identification of the centers of the affine and degenerate affine BMW algebras in terms of rings of symmetric functions which satisfy a ``cancellation property'' or ``wheel condition'' (in the degenerate case, a reformulation of a result of Nazarov). Miraculously, these same rings also arise in Schubert calculus, as the cohomology and K-theory of isotropic Grassmannians and symplectic loop Grassmannians. We also establish new intertwiner-like identities which, when projected to the center, produce the recursions for central elements given previously by Nazarov for degenerate affine BMW algebras, and by Beliakova--Blanchet for affine BMW algebras.