We show that in a metric space $X$ with annular convexity, uniform domains are precisely those Gromov hyperbolic domains whose quasiconformal structure on the Gromov boundary agrees with that on the boundary in $X$. As an application, we show that quasimöbius maps between geodesic spaces with annular convexity preserve uniform domains. These results are quantitative.

Let $\Omega\subset\mathbb{R}^N$ be a bounded domain. We study the best constant of the Sobolev trace embedding $W^{1,\infty}(\Omega) \hookrightarrow L^\infty (\partial\Omega)$ for functions that vanish in a subset $A\subset \Omega$, which we call the hole. That is we deal with the minimization problem $S_A^T =\inf\|u\|_{W^{1,\infty}(\Omega)}/\|u\|_{L^\infty(\partial\Omega)}$ for functions that verify $u|_A = 0$. We find that there exists an optimal hole that minimizes the best constant $S_A^T$ among subsets of $\Omega$ of prescribed volume and we give a geometrical characterization of this optimal hole. In fact, minimizers associated to these holes are cones centered at some points $x_0^*$ on $\partial\Omega$ with respect to the arc-length metric in $\Omega$ and the best holes are of the form $A^*=\Omega\setminus B_d(x_0^*,r^*)$ where the ball is taken again with respect of the arc-length metric.

A similar analysis can be performed for the best constant of the embedding $W^{1,\infty}(\Omega) \hookrightarrow L^\infty( \Omega)$ with holes. In this case, we also find that minimizers associated to optimal holes are cones centered at some points $x_0^*$ on $\partial\Omega$ and the best holes are of the form $A^*=\Omega\setminus B_d(x_0^*,r^*)$.

We introduce a geometric motivated method to construct immersions into $\mathbb{H}^2\times\mathbb{R}$ from a smooth function $\varphi$ defined on an open set of the unit disc, and study the relation between the geometry of the immersion in terms of partial differential equations for $\varphi$. We give two applications of the method. First, we introduce the class of surfaces generated by harmonic functions and show that they have properties analogous to minimal surfaces in $\mathbb {R}^3$. We also exhibit an explicit local relation between CMC 1/2 and flat surfaces in $\mathbb{H}^2\times\mathbb{R}$.

Let $S$ be a standard $\mathbb{N}^r$-graded algebra over a local ring $A$, and let $M$ be a finitely generated $\mathbb{Z}^r$-graded module over $S$. We characterize the Cohen–Macaulayness of $M$ in terms of the vanishing of certain sheaf cohomology modules. As a consequence, we apply our result to study the Cohen–Macaulayness of multi-Rees modules. Our work extends previous studies on the Cohen–Macaulayness of multi-Rees algebras.

In this article, we study the distribution of zeros of the derivative of the Selberg zeta function for compact three-dimensional hyperbolic spaces. We obtain an asymptotic formula for the counting function of its zeros. This is a three-dimensional version of the celebrated work of Wenzhi Luo. We also deduce other asymptotic formulas relating to its zeros from the above formula.

Let $E$ be an $L_1$-predual and $B\subset B_{E^*}$ be a boundary. We show that any bounded $\sigma(E,B)$-compact subset of $E$ is weakly compact. We also present an example of an $L_1$-predual $E$ that is not angelic in the $\sigma(E,\operatorname{ext} B_{E^*})$-topology.

This paper considers sufficient conditions for a Young function of type $t^{p} \varphi(t)$, with $p$ greater than one, so that certain Sobolev–Orlicz capacities have the same null sets. Examples of such Young functions are given too.

Let $R$ be a Noetherian commutative ring, $\langle a_1,\dots, a_n\rangle$ a sequence of elements of $R$, $I=(a_1,\dots,a_n)$ the ideal generated by the elements $a_i$ and $I_i=(a_1,\dots,a_i)$, $i=0,1,\dots, n$, the ideal generated by the first $i$ elements of the sequence. A c-sequence is a sequence $\langle a_1,\dots, a_n\rangle$ which satisfies the condition \[ [I_{i-1}I^k : a_i] \cap I^k=I_{i-1}I^{k-1} \] for every $i \in\{1, \dots, n\}$ and every $k\geq1$. It generates an ideal of linear type. We characterize c-sequences in terms of the corresponding sequences in the Rees algebra of the ideal generated by the elements of the sequence. We then characterize monomial c-sequences of three terms.

In this paper, we show that the tilting modules over a cluster-tilted algebra $A$ lift to tilting objects in the associated cluster category $\mathcal{C}_H$. As a first application, we describe the induced exchange relation for tilting $A$-modules arising from the exchange relation for tilting object in $\mathcal{C}_H$. As a second application, we exhibit tilting $A$-modules having cluster-tilted endomorphism algebras.

We characterize convergence and divergence types for Fuchsian groups in terms of the critical exponent of convergence and modified functions of the Poincaré series for certain subgroups associated with ends of the quotient Riemann surfaces. As an application of this result, we prove that convergence and divergence type are not invariant under a quasiconformal automorphism of the unit disk.

We study the instability of standing-wave solutions $e^{i\omega t}\phi_{\omega}(x)$ to the inhomogeneous nonlinear Schrödinger equation \[ i\varphi_t=-\triangle\varphi+|x|^2\varphi-|x|^b|\varphi |^{p-1}\varphi, \quad x\in\mathbb{R}^N, \] where $b \gt 0$ and $\phi_{\omega}$ is a ground-state solution. The results of the instability of standing-wave solutions reveal a balance between the frequency $\omega$ of wave and the power of nonlinearity $p$ for any fixed $b \gt 0$.

Fabry’s theorem on the singularities of power series is improved: the maximum density in the assumptions of this theorem is replaced by an interior density of Beurling–Malliavin type.

Let $E$ be a closed set in $\mathbb{R}^n$ and $\mathcal{W}$ a Whitney decomposition of $\mathbb{R}^n\setminus E$. Choosing one point from the interior of each cube in $\mathcal{W}$ we obtain a set $F$ and then we say that the set $E\cup F$ is a Whitney modification of $E$. The Whitney modification of a measure $\mu$ on $\mathbb{R}^n$ to $E\cup F$ is a measure $\nu$ defined on $E\cup F$ by $\nu\equiv\mu$ on $E$ and by $\nu(\{x\})=\mu(I_x)$ for every $x\in F$, where $I_x\in\mathcal{W}$ is the cube containing the point $x$. We prove that a measure on $E\cup F$ is doubling if and only if it is the Whitney modification of a doubling measure on $\mathbb{R}^n$. As its application, we show that there are metric spaces $X,Y$ and a nonquasisymmetric homeomorphism $f$ of $X$ onto $Y$ such that a measure $\mu$ on $X$ is doubling if and only if its image $\mu\circ f^{-1}$ is doubling on $Y$.

Given a positive function $F$ on $S^n$ which satisfies a convexity condition, we define the $r$-th anisotropic mean curvature function $H^F_r$ for hypersurfaces in $\mathbb{R}^{n+1}$ which is a generalization of the usual $r$-th mean curvature function. Let $X : M\to\mathbb{R}^{n+1}$ be an $n$-dimensional closed hypersurface with $H^F_{r+1}=\mathrm{constant}$, for some $r$ with $0\leq r\leq n-1$, which is a critical point for a variational problem. We show that $X(M)$ is stable if and only if $X(M)$ is the Wulff shape.

We establish an uncertainty principle over arbitrary compact groups, generalizing several previous results. Specifically, we show that if $\mathrm{P}$ and $\mathrm{R}$ are operators on $L^2(G)$ such that $\mathrm{P}$ commutes with projection onto every measurable subset of $G$ and $\mathrm{R}$ commutes with left-multiplication by elements of $G$, then $\|\operatorname{PR}\| \leq\|\mathrm{P} \cdot\chi_G \|_2 \|\mathrm {R}\|_2$, where $\chi_G : g \mapsto1$ is the characteristic function of $G$. As a consequence, we show that every nonzero function $f$ in $L^2(G)$ satisfies $\mu(\operatorname{\mathbf{supp}} f)\cdot \sum_{\rho\in\hat G} d_\rho\operatorname{\mathbf{rank}} \hat f(\rho) \geq1$.

The paper is devoted to the generalization of the theory of Hoelder Complexes, i.e., Lipschitz classification of germs of semialgebraic surfaces, for the definable surfaces in o-minimal structures. The theory is based on the Rosenlicht valuations on the corresponding Hardy fields. We obtain a complete answer for the case of polynomially bounded o-minimal structures and for the case of isolated singularities for general o-minimal structures.

The $n$-type vectors introduced by Geramita, Harima, and Shin are in 1—1 correspondence with the Hilbert functions of Artinian lex ideals. Letting $\mathbb{A} =\{ a_1,\ldots,a_n\}$ define the degrees of a regular sequence, we construct $\mathrm{lpp}_{\le }(\mathbb{A})$-vectors which are in 1—1 correspondence with the Hilbert functions of certain lex plus powers ideals (depending on $\mathbb{A}$). This construction enables us to show that the residual of a lex plus powers ideal in an appropriate regular sequence is again a lex plus powers ideal. We then use this result to show that the Eisenbud–Green–Harris conjecture is equivalent to showing that lex plus powers ideals have the largest last graded Betti numbers (it is well known that the Eisenbud–Green–Harris conjecture is equivalent to showing that lex plus powers ideals have the largest first graded Betti numbers).

We give conditions for the shadow boundary of a Riemannian submanifold $M$ to be regular. We prove that a helix hypersurface is ruled. By studying some relations between these natural submanifolds, we show that a minimal helix shadow boundary hypersurface of $M$ is totally geodesic in $M$.

We give an explicit formula relating the infinitesimal generators of the Loewner differential equation and the Hadamard variation. This is applied to establish an extension of the Hadamard variation to the case of arbitrary simply-connected domains and to prove the existence of Loewner chains with arbitrary smooth initial generator starting at an arbitrary univalent function which is sufficiently smooth up to the boundary. As another application of this method, we show that every subordination chain $f_t$ is differentiable almost everywhere and satisfies a Loewner equation, without assuming that $f_t'(0)$ is continuous.

We study a number of conditions on the Hilbert function of a level Artinian algebra which imply the Weak Lefschetz Property (WLP). Possibly the most important open case is whether a codimension 3 SI-sequence forces the WLP for level algebras. In other words, does every codimension 3 Gorenstein algebra have the WLP? We give some new partial answers to this old question: we prove an affirmative answer when the initial degree is 2, or when the Hilbert function is relatively small. Then we give a complete answer to the question of what is the largest socle degree forcing the WLP.