We characterize complete intersection matrix Schubert varieties, generalizing the classical result on one-sided ladder determinantal varieties. We also give a new proof of the $F$-rationality of matrix Schubert varieties. Although it is known that such varieties are $F$-regular (hence $F$-rational) by the global $F$-regularity of Schubert varieties, our proof is of independent interest since it does not require the Bott–Samelson resolution of Schubert varieties. As a consequence, this provides an alternative proof of the classical fact that Schubert varieties in flag varieties are normal and have rational singularities.

In this paper, we obtain some results about the local homology modules of Artinian modules, and by Matlis duality we obtain some results about the local cohomology modules of finitely generated modules.

The present paper is a sequel to our paper “Metric characterization of isometries and of unital operator spaces and systems.” We characterize certain common objects in the theory of operator spaces (unitaries, unital operator spaces, operator systems, operator algebras, and so on), in terms which are purely linear-metric, by which we mean that they only use the vector space structure of the space and its matrix norms. In the last part, we give some characterizations of operator algebras (which are not linear-metric in our strict sense described in the paper).

We study the random sampling of band-limited functions of several variables. If a band-limited function with bandwidth has its essential support on a cube of volume $R^{d}$, then $\mathcal{O}(R^{d}\log R^{d})$ random samples suffice to approximate the function up to a given error with high probability.

We determine the homotopy types of gauge groups of principle $G$-bundles over closed, connected nonorientable surfaces for $G=U(n)$ and $G$ a simply-connected, compact Lie group. Applications are made to moduli spaces of stable vector bundles.

In 1841, Delaunay (J. Math. Pures Appl. 6 (1841) 309–320) showed that the intersection of a constant mean curvature surface of revolution in $\mathbf{R}^{3}$ and a plane $\Pi$ that contains its axis of symmetry $l$ can be described as the trace of the focus of a conic when this conic rolls without slipping in the plane $\Pi$ along the line $l$. In the same way surfaces of revolution are foliated by circles perpendicular to the axis of symmetry, helicoidal surfaces are foliated by helices, all of them symmetric to a line $l$. Roughly speaking, helicoidal surfaces are surfaces invariant under a screw-motion. In this paper, we show that the intersection of a helicoidal minimal surface $S$ in $\mathbf{R}^{3}$ and a plane $\pi$ perpendicular to line $l$—where $l$ is the axis of symmetry of the screw motion—is characterized by the property that if we roll the curve $C=S\cap\pi$ on a flat treadmill located on another plane $\Pi$, then, the point $P=\pi\cap l$ describes a hyperbola on the plane $\Pi$ centered at the fixed point of contact of the treadmill with the curve $C$. This way of generating a curve using another curve, similar to the well known “Roulette,” was introduced by the author in (Pacific J. Math. 258 (2012) 459–485) and it was called the “TreadmillSled.” We will also prove several properties of the TreadmillSled, in particular we will classify all curves that are the TreadmillSled of another curve.

We prove variation-norm estimates for the Walsh model of the truncated bilinear Hilbert transform, extending related results of Lacey, Thiele, and Demeter. The proof uses analysis on the Walsh phase plane and two new ingredients: (i) a variational extension of a lemma of Bourgain by Nazarov–Oberlin–Thiele, and (ii) a variation-norm Rademacher–Menshov theorem of Lewko–Lewko.

We give an abstract version of the hard Lefschetz theorem, the Lefschetz decomposition and the Hodge–Riemann theorem for compact Kähler manifolds. Some examples are studied for compact symplectic Kähler manifolds.

This paper offers a sufficient condition, based on Maharam (Proc. Natl. Acad. Sci. USA 28 (1942) 108–111) and re-emphasized by Hoover and Keisler (Trans. Amer. Math. Soc. 286 (1984) 159–201), for the validity of Lyapunov’s theorem on the range of a nonatomic vector measure taking values in an infinite-dimensional Banach space that is not necessarily separable nor has the Radon–Nikodym property (RNP). In particular, we obtain an extension of a corresponding result due to Uhl (Proc. Amer. Math. Soc. 23 (1969) 158–163). The proposed condition is also shown to be necessary in the sense formalized by Keisler and Sun (Adv. Math. 221 (2009) 1584–1607), and thereby closes a question of long-standing as regards an infinite-dimensional generalization of the theorem. The result is applied to obtain short simple proofs of recent results on the convexity of the integral of a set-valued function, and on the characterization of restricted cores of a saturated economy.

For a complete Riemannian manifold without boundary which a unimodular locally compact group properly cocompact acts on it, under some conditions, we define and study the analytic torsion on it by using the $G$-trace defined in ($L^{2}$-index formula for proper cocompact group actions, preprint). For a fiber bundle $\pi:M\to B$, if there is a unimodular locally compact group acts fiberwisely properly and cocompact on it, we define the torsion form for it, and show that the zero degree part of the torsion form is the analytic torsion. This can be viewed as an extension of the $L^{2}$-analytic torsion.

On the Bergman space in the unit disk, we study the intertwining relation for Volterra type operators, whose intertwining operator is a composition operator. We also investigate the “compact” intertwining relations for Volterra type operators. As obvious consequences, the essential commutativity of Volterra type and composition operators are characterized. At the end of the paper, we find a new connection between the Bergman space and little Bloch space through this essential commutativity.

We improve the exponent in (Int. Math. Res. Not. IMRN 1993 (1993) 61–66) for the discrete restriction to the $n$ dimensional sphere, from $p=\frac{2(n+1)}{n-3}$ to $p=\frac{2n}{n-3}$, when $n\ge4$.

It is a folk conjecture that for $\alpha>1/2$ there is no $\alpha$-Hölder surface in the subRiemannian Heisenberg group. Namely, it is expected that there is no embedding from an open subset of $\mathbb{R}^{2}$ into the Heisenberg group that is Hölder continuous of order strictly greater than $1/2$. The Heisenberg group here is equipped with its Carnot–Carathéodory distance. We show that, in the case that such a surface exists, it cannot be of essential bounded variation and it intersects some vertical line in at least a topological Cantor set.

Let $k$ be an arbitrary field (of arbitrary characteristic) and let $X=[x_{i,j}]$ be a generic $m\times n$ matrix of variables. Denote by $I_{2}(X)$ the ideal in $k[X]=k[x_{i,j}:i=1,\ldots,m;j=1,\ldots,n]$ generated by the $2\times2$ minors of $X$. Using Gröbner basis, we give a recursive formulation for the lengths of the $k[X]$-module $k[X]/(I_{2}(X)+(x_{1,1}^{q},\ldots,x_{m,n}^{q}))$ as $q$ varies over all positive integers. This is a generalized Hilbert–Kunz function, and our formulation proves that it is a polynomial function in $q$. We apply our method to give closed forms for these Hilbert–Kunz functions for cases $m\le2$.

Let $A$ be a ring and $R$ be a polynomial or a power series ring over $A$. When $A$ has dimension zero, we show that the Bass numbers and the associated primes of the local cohomology modules over $R$ are finite. Moreover, if $A$ has dimension one and $\pi$ is an nonzero divisor, then the same properties hold for prime ideals that contain $\pi$. These results do not require that $A$ contains a field. As a consequence, we give a different proof for the finiteness properties of local cohomology over unramified regular local rings. In addition, we extend previous results on the injective dimension of local cohomology modules over certain regular rings of mixed characteristic.

We consider weighted norm inequalities for the Riesz potentials $I_{\alpha}$, also referred to as fractional integral operators. First, we prove mixed $A_{p}\mbox{-}A_{\infty}$ type estimates in the spirit of (Indiana Univ. Math. J. 61 (2012) 2041–2052, Anal. PDE 6 (2013) 777–818, Houston J. Math. 38 (2012) 799–814). Then we prove strong and weak type inequalities in the case $p<q$ using the so-called log bump conditions. These results complement the strong type inequalities of Pérez (Indiana Univ. Math. J. 43 (1994) 663–683) and answer a conjecture from (Weights, extrapolation and the theory of Rubio de Francia (2011) Birkhäuser). For both sets of results, our main tool is a corona decomposition adapted to fractional averages.