Unbounded operators having self-adjoint,subnormal, or hyponormal powers

We show that if a densely defined closable operator A is such that the resolvent set of A² is nonempty, then A is necessarily closed. This result is then extended to the case of a polynomial $p(A)$. We also generalize a recent result by Sebestyén–Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given. One of them is a proof that if T is a quasinormal (unbounded) operator such that $T^n$ is normal for some $n\ge 2$, then T is normal. Hence a closed subnormal operator T such that $T^n$ is normal is itself normal. We also show that if a hyponormal (nonnecessarily bounded) operator A is such that $A^p$ and $A^q$ are self-adjoint for some coprime numbers p and q, then A must be self-adjoint.     

Thom property and Milnor–Lê fibration for analytic maps

Let (X, 0) be the germ of either a subanalytic set $X\subset {\mathbb{R}}^n$ or a complex analytic space $X\subset {\mathbb{C}}^n$, and let $f:(X,0)\to ({\mathbb{K}}^k,0)$ be a $\mathbb{K}$-analytic map-germ, with $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, respectively. When $k=1$, there is a well-known topological locally trivial fibration associated with f, called the Milnor–Lê fibration of f, which is one of the main pillars in the study of singularities of maps and spaces. However, when $k>1$ that is not always the case. In this paper, we give conditions which guarantee that the image of f is well-defined as a set-germ, and that f admits a Milnor–Lê fibration. We also give conditions for f to have the Thom property. Finally, we apply our results to mixed function-germs of type $f\overline{g}:(X,0)\to (\mathbb{C},0)$ on a complex analytic surface $X\subset {\mathbb{C}}^n$ with arbitrary singularity.     

Estimates for the entropy numbers of the Nikol'skii–Besov classes of functions with mixed smoothness in the space of quasi-continuous functions

We obtained order estimates for the entropy numbers of the Nikol'skii–Besov classes of functions $B_{p,\theta }^{\mathit{r}}({\mathbb{T}}^d)$ with mixed smoothness in the metric of the space of quasi-continuous functions $QC({\mathbb{T}}^d)$. We also showed that for $2\le p\le \infty$, , $r_1>\frac{1}{2}$, $d\ge 2$, the estimate of the corresponding asymptotic characteristic is exact in order.     

A singular Liouville equation on planar domains

We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, $-\mathrm{\Delta }u=(\log u+f(u)){\chi }_{\{u>0\}}$ in $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with Dirichlet boundary condition. The function f has exponential growth, which can be subcritical or critical with respect to the Trudinger–Moser inequality. We study the energy functional $I_{\epsilon }$ corresponding to the perturbed equation $-\mathrm{\Delta }u+g_{\epsilon }(u)=f(u)$, where $g_{\epsilon }$ is well defined at 0 and approximates $-\log u$. We show that $I_{\epsilon }$ has a critical point $u_{\epsilon }$ in $H_0^1(\mathrm{\Omega })$, which converges to a legitimate nontrivial nonnegative solution of the original problem as $\epsilon \to 0$. We also investigate the problem with $f(u)$ replaced by $\lambda f(u)$, when the parameter $\lambda >0$ is sufficiently large.     

Invariant hypercomplex structures and algebraic curves

We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{g}\mathfrak{l}(k,\mathbb{C})$ correspond to algebraic curves C of genus ${(k-1)}^2$, equipped with a flat projection $\pi :C\to {\mathbb{P}}^1$ of degree k, and an antiholomorphic involution $\sigma :C\to C$ covering the antipodal map on ${\mathbb{P}}^1$.     

Quantifying properties (K) and (μs$\mu ^{s}$)

A Banach space X has property (K), whenever every weak* null sequence in the dual space admits a convex block subsequence ${(f_n)}_{n=1}^{\infty }$ so that $⟨f_n,x_n⟩\to 0$ as $n\to \infty$ for every weakly null sequence ${(x_n)}_{n=1}^{\infty }$ in X; X has property $({\mu }^s)$ if every weak* null sequence in $X^{\ast }$ admits a subsequence so that all of its subsequences are Cesàro convergent to 0 with respect to the Mackey topology. Both property $({\mu }^s)$ and reflexivity (or even the Grothendieck property) imply property (K). In this paper, we propose natural ways for quantifying the aforementioned properties in the spirit of recent results concerning other familiar properties of Banach spaces.     

On the (p,q)$(p,q)$-type strong law of large numbers for sequences of independent random variables

Li, Qi, and Rosalsky (Trans. Amer. Math. Soc., 368 (2016), no. 1, 539–561) introduced a refinement of the Marcinkiewicz–Zygmund strong law of large numbers (SLLN), the so-called $(p,q)$-type SLLN, where and $q>0$. They obtained sets of necessary and sufficient conditions for this new type SLLN for two cases: , $q>p$, and . Results for the case where and remain open problems. This paper gives a complete solution to these problems. We consider random variables taking values in a real separable Banach space $\mathbf{B}$, but the results are new even when $\mathbf{B}$ is the real line. Furthermore, the conditions for a sequence of random variables $\left\{X_n,n\ge 1\right\}$ satisfying the $(p,q)$-type SLLN are shown to provide an exact characterization of stable type p Banach spaces.     

Isomorphisms of Galois groups of number fields with restricted ramification

Let K be a number field and S a set of primes of K. We write $K_S/K$ for the maximal extension of K unramified outside S and $G_{K,S}$ for its Galois group. In this paper, we answer the following question under some assumptions: “For $i=1,2$, let $K_i$ be a number field, $S_i$ a (sufficiently large) set of primes of $K_i$ and $\sigma :G_{K_1,S_1}\stackrel{\sim }{\to }{G}_{K_2,S_2}$ an isomorphism. Is σ induced by a unique isomorphism between $K_{1,S_1}/K_1$ and $K_{2,S_2}/K_2$?” Here, the main assumption is about the Dirichlet density of $S_i$.     

Cones of lines having high contact with general hypersurfaces and applications

Given a smooth hypersurface $X\subset {\mathbb{P}}^{n+1}$ of degree $d⩾2$, we study the cones $V_p^h\subset {\mathbb{P}}^{n+1}$ swept out by lines having contact order $h⩾2$ at a point $p\in X$. In particular, we prove that if X is general, then for any $p\in X$ and $2⩽h⩽\min \{n+1,d\}$, the cone $V_p^h$ has dimension exactly $n+2-h$. Moreover, when X is a very general hypersurface of degree $d⩾2n+2$, we describe the relation between the cones $V_p^h$ and the degree of irrationality of k-dimensional subvarieties of X passing through a general point of X. As an application, we give some bounds on the least degree of irrationality of k-dimensional subvarieties of X passing through a general point of X, and we prove that the connecting gonality of X satisfies $d-⌊\frac{\sqrt{16n+25}-3}{2}⌋⩽conn.gon(X)⩽d-⌊\frac{\sqrt{8n+1}+1}{2}⌋$.     

Exponential decay for semilinear wave equation with localized damping in the hyperbolic space

In this paper, we study the exponential decay of the energy associated to an initial value problem involving the wave equation on the hyperbolic space ${\mathbb{B}}^N$. The space ${\mathbb{B}}^N$ is the unit disc of ${\mathbb{R}}^N$ endowed with the Riemannian metric g given by $g_{ij}=p^2{\delta }_{ij}$, where $p(x)=\frac{2}{1-{|x|}^2}$ and ${\delta }_{ij}=1$, if $i=j$ and ${\delta }_{ij}=0$, if $i\ne j$. Making an appropriate change, the problem can be seen as a singular problem on the boundary of the open ball endowed with the euclidean metric. The proof is based on the multiplier techniques combined with the use of Hardy's inequality, in a version due to the Brezis–Marcus, which allows us to overcome the difficulty involving the singularities.     

Spectral heat content on a class of fractal sets for subordinate killed Brownian motions

We study the spectral heat content for a class of open sets with fractal boundaries determined by similitudes in ${\mathbb{R}}^d$, $d\ge 1$, with respect to subordinate killed Brownian motions via $\alpha /2$-stable subordinators and establish the asymptotic behavior of the spectral heat content as $t\to 0$ for the full range of $\alpha \in (0,2)$. Our main theorems show that these asymptotic behaviors depend on whether the sequence of logarithms of the coefficients of the similitudes is arithmetic when $\alpha \in [d-\mathfrak{b},2)$, where $\mathfrak{b}$ is the interior Minkowski dimension of the boundary of the open set. The main tools for proving the theorems are the previous results on the spectral heat content for Brownian motions and the renewal theorem.     

Totally real flat minimal surfaces in the hyperquadric

In this paper, we study geometry of totally real minimal surfaces in the complex hyperquadric $Q_{N-2}$, and obtain some characterizations of the harmonic sequence generated by these minimal immersions. For totally real flat surfaces that are minimal immersed in both $Q_{N-2}$ and $\mathbb{C}{P}^{N-1}$, we determine them for $N=4,5,6$, and give a classification theorem when they are Clifford solutions.     

Theorems of Wolff–Denjoy type for semigroups of nonexpansive mappings in geodesic spaces

We show that if $S={\{f_t:Y\to Y\}}_{t\ge 0}$ is a one-parameter continuous semigroup of nonexpansive mappings acting on a complete locally compact geodesic space $(Y,d)$ that satisfies some geometric properties, then there exists $\xi \in \partial Y$ such that S converge uniformly on bounded sets of Y to ξ. In particular, our result applies to strictly convex bounded domains in ${\mathbb{R}}^n$ or ${\mathbb{C}}^n$ with respect to a large class of metrics including Hilbert's and Kobayashi's metrics.     

Vanishing of higher order Alexander-type invariants of plane curves

The higher order degrees are Alexander-type invariants of complements to an affine plane curve. In this paper, we characterize the vanishing of such invariants for a curve C given as a transversal union of plane curves $C^{\prime }$ and $C^{\prime \prime }$ in terms of the finiteness and the vanishing properties of the invariants of $C^{\prime }$ and $C^{\prime \prime }$, and whether or not they are irreducible. As a consequence, we prove that the multivariable Alexander polynomial ${\mathrm{\Delta }}_C^{multi}$ is a power of $(t-1)$, and we characterize when ${\mathrm{\Delta }}_C^{multi}=1$ in terms of the defining equations of $C^{\prime }$ and $C^{\prime \prime }$. Our results impose obstructions on the class of groups that can be realized as fundamental groups of complements of a transversal union of curves.