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 ${\mathbb{P}}^1$.     

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.     

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.     

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.     

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.     

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 topological and combinatorial structures of pointed stable curves over algebraically closed fields of positive characteristic

In this paper, we study anabelian geometry of curves over algebraically closed fields of positive characteristic. Let $X^{•}=(X,D_X)$ be a pointed stable curve over an algebraically closed field of characteristic $p>0$ and ${\mathrm{\Pi }}_{X^{•}}$ the admissible fundamental group of $X^{•}$. We prove that there exists a group-theoretical algorithm, whose input datum is the admissible fundamental group ${\mathrm{\Pi }}_{X^{•}}$, and whose output data are the topological and the combinatorial structures associated with $X^{•}$. This result can be regarded as a mono-anabelian version of the combinatorial Grothendieck conjecture in the positive characteristic. Moreover, by applying this result, we construct clutching maps for moduli spaces of admissible fundamental groups.     

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.     

Elliptic Weingarten hypersurfaces of Riemannian products

Let $M^n$ be either a simply connected space form or a rank-one symmetric space of the noncompact type. We consider Weingarten hypersurfaces of $M\times \mathbb{R}$, which are those whose principal curvatures $k_1,\cdots ,k_n$ and angle function $𝛩$ satisfy a relation $W(k_1,\cdots ,k_n,{𝛩}^2)=0$, being W a differentiable function which is symmetric with respect to $k_1,\cdots ,k_n$. When $\partial W/\partial k_i>0$ on the positive cone of ${\mathbb{R}}^n$, a strictly convex Weingarten hypersurface determined by W is said to be elliptic. We show that, for a certain class of Weingarten functions W, there exist rotational strictly convex Weingarten hypersurfaces of $M\times \mathbb{R}$ which are either topological spheres or entire graphs over M. We establish a Jellett–Liebmann-type theorem by showing that a compact, connected and elliptic Weingarten hypersurface of either ${\mathbb{S}}^n\times \mathbb{R}$ or ${\mathbb{H}}^n\times \mathbb{R}$ is a rotational embedded sphere. Other uniqueness results for complete elliptic Weingarten hypersurfaces of these ambient spaces are obtained. We also obtain existence results for constant scalar curvature hypersurfaces of ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$ which are either rotational or invariant by translations (parabolic or hyperbolic). We apply our methods to give new proofs of the main results by Manfio and Tojeiro on the classification of constant sectional curvature hypersurfaces of ${\mathbb{S}}^n\times \mathbb{R}$ and ${\mathbb{H}}^n\times \mathbb{R}$.     

Generalization of Hardy–Littlewood maximal inequality with variable exponent

Let $p(·)$ be a measurable function defined on ${\mathbb{R}}^d$ and $p_{-}:={inf}_{x\in {\mathbb{R}}^d}p(x)$. In this paper, we generalize the Hardy–Littlewood maximal operator. In the definition, instead of cubes or balls, we take the supremum over all rectangles the side lengths of which are in a cone-like set defined by a given function ψ. Moreover, instead of the integral means, we consider the $L_{q(·)}$-means. Let $p(·)$ and $q(·)$ satisfy the log-Hülder condition and $p(·)=q(·)r(·)$. Then, we prove that the maximal operator is bounded on $L_{p(·)}$ if and is bounded from $L_{p(·)}$ to the weak $L_{p(·)}$ if $1\le r_{-}\le \infty$. We generalize also the theorem about the Lebesgue points.     

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

For a positive integer N, let $X_0(N)$ be the modular curve over $\mathbf{Q}$ and $J_0(N)$ its Jacobian variety. We prove that the rational cuspidal subgroup of $J_0(N)$ is equal to the rational cuspidal divisor class group of $X_0(N)$ when $N=p^{2}M$ for any prime p and any squarefree integer M. To achieve this, we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m,h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on $X_0(N)$ under a mild assumption.     

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.     

Improved Bohr inequality for harmonic mappings

In order to improve the classical Bohr inequality, we explain some refined versions for a quasi-subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk $\mathbb{D}$. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form $f=h+\overline{g}$, where $g(0)=0$, the analytic part h is bounded by 1 and that $|g^{\prime }(z)|\le k|h^{\prime }(z)|$ in $\mathbb{D}$ and for some $k\in [0,1]$.     

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.     

Weyl families of transformed boundary pairs

Let $(\mathfrak{L},\mathrm{\Gamma })$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space $\mathfrak{H}$. Let $M_{\mathrm{\Gamma }}$ be the Weyl family corresponding to $(\mathfrak{L},\mathrm{\Gamma })$. We cope with two main topics. First, since $M_{\mathrm{\Gamma }}$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_{\mathrm{\Gamma }}(z)$, for some $z\in \mathbb{C}\setminus \mathbb{R}$, becomes a nontrivial task. Regarding $M_{\mathrm{\Gamma }}(z)$ as the (Shmul'yan) transform of $zI$ induced by Γ, we give conditions for the equality in $\overline{M_{\mathrm{\Gamma }}(z)}\subseteq \overline{M_{\overline{\mathrm{\Gamma }}}(z)}$ to hold and we compute the adjoint $M_{\overline{\mathrm{\Gamma }}}{(z)}^{\ast }$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^{+}$ is nonempty. Based on the criterion for the closeness of $M_{\mathrm{\Gamma }}(z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family $M_{\mathrm{\Gamma }}$ corresponding to a boundary relation Γ for $T^{+}$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$ with its Weyl family $M_{{\mathrm{\Gamma }}^{\prime }}$. The transformation scheme is either ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }{V}^{-1}$ or ${\mathrm{\Gamma }}^{\prime }=V\mathrm{\Gamma }$ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},\mathrm{\Gamma })$ and $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$; the formula for $M_{{\mathrm{\Gamma }}^{\prime }}-M_{\mathrm{\Gamma }}$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},{\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1)$ with $ker\mathrm{\Gamma }=T$ and $T_0=T_0^{\ast }$ (second scheme, Hilbert space case).