The homological property of the associated graded ring of an ideal is an important problem in commutative algebra and algebraic geometry. In this paper we explore the almost Cohen–Macaulayness of the associated graded ring of stretched $\mathfrak{m}$-primary ideals in the case where the reduction number attains almost minimal value in a Cohen–Macaulay local ring $(A,\mathfrak{m})$. As an application, we present complete descriptions of the associated graded ring of stretched $\mathfrak{m}$-primary ideals with small reduction number.

For positive integers $d, m, n \geq 1$ with $(m, n) \neq (1, 1)$ and $\mathbb{K} = \mathbb{R}$ or $\mathbb{C}$, let $\mathbb{Q}^{d,m}_{n}(\mathbb{K})$ denote the space of $m$-tuples $(f_{1}(z), \ldots, f_m(z)) \in \mathbb{K} [z]^{m}$ of $\mathbb{K}$-coefficients monic polynomials of the same degree $d$ such that polynomials $\{f_{k}(z)\}_{k=1}^{m}$ have no common real root of multiplicity $\geq n$ (but may have complex common root of any multiplicity). These spaces can be regarded as one of generalizations of the spaces defined and studied by Arnold and Vassiliev, and they may be also considered as the real analogues of the spaces studied by Farb–Wolfson. In this paper, we shall determine their homotopy types explicitly and generalize our previous results.

We show that a system of unstable higher Toda brackets can be defined inductively.

Our purpose is to show the existence of a Calabi–Yau structure on the punctured cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ of the Cayley projective plane $P^{2}\mathbb{O}$ and to construct a Bargmann type transformation from a space of holomorphic functions on $T^{*}_{0}(P^2\mathbb{O})$ to $L_{2}$-space on $P^{2}\mathbb{O}$. The space of holomorphic functions corresponds to the Fock space in the case of the original Bargmann transformation. A Kähler structure on $T^{*}_{0}(P^{2}\mathbb{O})$ was given by identifying it with a quadric in the complex space $\mathbb{C}^{27} \backslash \{0\}$ and the natural symplectic form of the cotangent bundle $T^{*}_{0}(P^2\mathbb{O})$ is expressed as a Kähler form. Our construction of the transformation is the pairing of polarizations, one is the natural Lagrangian foliation given by the projection map $\boldsymbol{q}:T^{*}_{0}(P^2\mathbb{O}) \longrightarrow P^{2}\mathbb{O}$ and the other is the polarization given by the Kähler structure.

The transformation gives a quantization of the geodesic flow in terms of one parameter group of elliptic Fourier integral operators whose canonical relations are defined by the graph of the geodesic flow action at each time. It turns out that for the Cayley projective plane the results are not same with other cases of the original Bargmann transformation for Euclidean space, spheres and other projective spaces.

We generalize the recent result of Erdoğan, Goldberg and Green on the $L^{p}$-boundedness of wave operators for two dimensional Schrödinger operators and prove that they are bounded in $L^{p}(\mathbb{R}^2)$ for all $1 < p < \infty$ if and only if the Schrödinger operator possesses no $p$-wave threshold resonances, viz. Schrödinger equation $(-\Delta + V(x))u(x) = 0$ possesses no solutions which satisfy $u(x) = (a_1 x_1 + a_2 x_2) |x|^{-2} + o(|x|^{-1})$ as $|x| \to \infty$ for an $(a_1, a_2) \in \mathbb{R}^2 \setminus \{(0,0)\}$ and, otherwise, they are bounded in $L^{p}(\mathbb{R}^2)$ for $1 < p \leq 2$ and unbounded for $2 < p < \infty$. We present also a new proof for the known part of the result.

In this article we show a Cartan decomposition for reductive Riemannian Gelfand pairs and an induction of spherical functions for Riemannian Gelfand pairs. With the induction we find that the property of the symmetry of spherical functions, which is known for Riemannian symmetric pairs, can also be induced from the corresponding property of smaller dimension. A Fourier transform of a positive function for a Riemannian Gelfand pair with abelian unipotent radical is also given under some condition on its support by using the symmetry of spherical function.

We define regular points of an extremal subset in an Alexandrov space and study their basic properties. We show that a neighborhood of a regular point in an extremal subset is almost isometric to an open subset in Euclidean space and that the set of regular points in an extremal subset has full measure and is dense in it. These results actually hold for strained points in an extremal subset. Applications include the volume convergence of extremal subsets under a noncollapsing convergence of Alexandrov spaces, and the existence of a cone fibration structure of a metric neighborhood of the regular part of an extremal subset. In an appendix, a deformation retraction of a metric neighborhood of a general extremal subset is constructed.

In the present article, we obtain an optimal support function of weighted $L^{2}$ integrations on superlevel sets of psh weights, which implies the strong openness property of multiplier ideal sheaves.

In this paper, we analyze ends of zero mean curvature surfaces of mixed (or non-mixed) type in the Lorentzian 3-space $\mathbf{R}^{2,1}$. Among these, we show that spacelike or timelike planar ends are $C^{\infty}$ in the compactification $\hat{L}$ of $\mathbf{R}^{2,1}$ as in the case of minimal surfaces in the Euclidean 3-space $\mathbf{R}^3$. On the other hand, lightlike planar ends are not $C^{\infty}$. Each lightlike planar end of a mixed type surface has the following additional parts: the converging part (a lightlike line in $\mathbf{R}^{2,1}$), the diverging part (the set of the points in $\hat{L} \setminus \mathbf{R}^{2,1}$ corresponding to zero-divisors), and the border of these two parts. We show that such an end is $C^{\infty}$ on the first two parts almost everywhere, while there appears an isolated singularity in the form of $(x^3, x\tau + \mbox{``higher order terms''}, \tau)$ on the border. We also show that conelike singularities of mixed type appear on the lightlike lines in special cases.

The main purpose of this work is to generalize the $S^{3}_{\boldsymbol{w}}$ Sasaki join construction $M \star_{\boldsymbol{l}} S^{3}_{\boldsymbol{w}}$ described in the authors' 2016 paper when the Sasakian structure on $M$ is regular, to the general case where the Sasakian structure is only quasi-regular. This gives one of the main results, Theorem 3.2, which describes an inductive procedure for constructing Sasakian metrics of constant scalar curvature. In the Gorenstein case ($c_{1}(\mathscr{D}) = 0$) we construct a polynomial whose coeffients are linear in the components of $\boldsymbol{w}$ and whose unique root in the interval $(1, \infty)$ completely determines the Sasaki–Einstein metric. In the more general case we apply our results to prove that there exists infinitely many smooth 7-manifolds each of which admit infinitely many inequivalent contact structures of Sasaki type admitting constant scalar curvature Sasaki metrics (see Corollary 6.15). We also discuss the relationship with a recent paper of Apostolov and Calderbank as well as the relation with K-stability.