In this article we treat two closely related problems: 1) the upper semi-continuity property for Almgren minimal sets in regions with regular boundary; and 2) the uniqueness property for all the $2$-dimensional minimal cones in $\mathbb R^3$.

Given an open set $\Omega\subset\mathbb R^n$, a closed set $E\subset \Omega$ is said to be Almgren minimal of dimension $d$ in $\Omega$ if it minimizes the $d$-Hausdorff measure among all its Lipschitz deformations in $\Omega$. We say that a $d$-dimensional minimal set $E$ in an open set $\Omega$ admits upper semi-continuity if, whenever $\{f_n(E)\}_n$ is a sequence of deformations of $E$ in $\Omega$ that converges to a set $F$, then we have ${\mathcal H}^d(F)\ge \limsup_n {\mathcal H}^d(f_n(E))$. This guarantees in particular that $E$ minimizes the $d$-Hausdorff measure, not only among all its deformations, but also among limits of its deformations.

As proved in [19], when several $2$-dimensional minimal cones are all translational and sliding stable, and admit the uniqueness property, then their almost orthogonal union stays minimal. As a consequence, the uniqueness property obtained in the present paper, together with the translational and sliding stability properties proved in [18] and [20] permit us to use all known $2$-dimensional minimal cones in $\mathbb R^n$ to generate new families of minimal cones by taking their almost orthogonal unions.

The upper semi-continuity property is also helpful in various circumstances: when we have to carry on arguments using Hausdorff limits and some properties do not pass to the limit, the upper semi-continuity can serve as a link. As an example, it plays a very important role throughout [19].

In this paper we consider a partial overdetermined mixed boundary value problem in domains inside a cone as in [18]. We show that, in cones having an isoperimetric property, the only domains which admit a solution and which minimize a torsional energy functional are spherical sectors centered at the vertex of the cone. We also show that cones close in the $C^{1,1}$-metric to an isoperimetric one are also isoperimetric, generalizing so a result of [1]. This is achieved by using a characterization of constant mean curvature polar graphs in cones which improves a result of [18].

The aim of this paper is to continue the study of asymptotic expansions and summability in a monomial in any number of variables, as introduced in [3, 15]. In particular, we characterize these expansions in terms of bounded derivatives and we develop Tauberian theorems for the summability processes involved. Furthermore, we develop and apply the Borel-Laplace analysis in this framework to prove the monomial summability of solutions of a specific class of singularly perturbed PDEs.

Let $G$ be a finite group and $\mathcal{A}_p(G)$ be the poset of nontrivial elementary abelian $p$-subgroups of $G$. Quillen conjectured that $O_p(G)$ is nontrivial if $\mathcal{A}_p(G)$ is contractible. We prove that $O_p(G)\neq 1$ for any group $G$ admitting a $G$-invariant acyclic $p$-subgroup complex of dimension $2$. In particular, it follows that Quillen's conjecture holds for groups of $p$-rank $3$. We also apply this result to establish Quillen's conjecture for some particular groups not considered in the seminal work of Aschbacher-Smith.

Let $L/K$ be a $G$-Galois extension of fields with an $H$-Hopf Galois structure of type $N$. We study the Galois correspondence ratio $GC(G, N)$, which is the proportion of intermediate fields $E$ with $K \subseteq E \subseteq L$ that are in the image of the Galois correspondence for the $H$-Hopf Galois structure on $L/K$. The Galois correspondence ratio for a Hopf Galois structure can be found by translating the problem into counting certain subgroups of the corresponding skew brace. We look at skew braces arising from finite radical algebras $A$ and from Zappa–Szép products of finite groups, and in particular when $A^3 = 0$ or the Zappa–Szép product is a semidirect product, in which cases the corresponding skew brace is a bi-skew brace, that is, a set $G$ with two group operations $\circ$ and $\star$ in such a way that $G$ is a skew brace with either group structure acting as the additive group of the skew brace. We obtain the Galois correspondence ratio for several examples. In particular, if $(G, \circ, \star)$ is a bi-skew brace of squarefree order $2m$ where $(G, \circ) \cong Z_{2m}$ is cyclic and $(G, \star) \cong D_m$ is dihedral, then for large $m$, $GC(Z_{2m},D_m)$ is close to $1/2$ while $GC(D_m, Z_{2m})$ is near $0$.

In the setting of the unbounded derived category $\mathbf{D}(R)$ of a ring $R$ of weak global dimension at most one we consider t-structures with a definable coaisle. The t-structures among these which are stable (that is, the t-structures which consist of a pair of triangulated subcategories) are precisely the ones associated to a smashing localization of the derived category. In this way, our present results generalize those of [8] to the non-stable case. As in the stable case [8], we confine for the most part to the commutative setting, and give a full classification of definable coaisles in the local case, that is, over valuation domains. It turns out that, unlike in the stable case of smashing subcategories, the definable coaisles do not always arise from homological ring epimorphisms. We also consider a non-stable version of the Telescope Conjecture for t-structures and give a ring-theoretic characterization of the commutative rings of weak global dimension at most one for which it is satisfied.

It is well known that the set of values of a lower central word in a group $G$ need not be a subgroup. For a fixed lower central word $\gamma_r$ and for $p\ge 5$, Guralnick showed that if $G$ is a finite $p$-group such that the verbal subgroup $\gamma_r(G)$ is abelian and $2$-generator, then $\gamma_r(G)$ consists only of $\gamma_r$-values. In this paper we extend this result, showing that the assumption that $\gamma_r(G)$ is abelian can be dropped. Moreover, we show that the result remains true even if $p\!=\!3$. Finally, we prove that the analogous result for pro-$p$ groups is true.

We are interested in the set of normal sequences in the space $\{0,1\}^{\mathbb N}$ with a given frequency of the pattern $11$ in the positions $k$, $2k$. The topological entropy of such sets is determined.

A foliation is of toric type when it has a combinatorial reduction of singularities. We show that every toric type foliation on $(\mathbb{C}^3,0)$ without saddle-nodes has invariant surface. We extend the argument of Cano-Cerveau for the nondicritical case to the compact dicritical components of the exceptional divisor. These components are projective toric surfaces and the isolated invariant branches of the induced foliation extend to closed irreducible curves. We build the invariant surface as a germ along the singular locus and those closed irreducible invariant curves. The result of Ortiz-Bobadilla-Rosales-Gonzalez-Voronin about the distribution of invariant branches in dimension two is a key argument in our proof.

We show that there can be no algorithm to decide whether infinite recursively described acyclic aspherical $2$-complexes are contractible. We construct such a complex that is contractible if and only if the Collatz conjecture holds.

In this paper we study the existence of uniform a priori estimates for positive solutions to Navier problems of higher order Lane–Emden equations \begin{equation}\label{0-0} (-\Delta)^{m}u(x)=u^{p}(x), \quad x\in\Omega, \end{equation} for all large exponents $p$, where $\Omega\subset\mathbb{R}^{n}$ is a star-shaped or strictly convex bounded domain with $C^{2m-2}$ boundary, $n\geq4$, and $2\leq m\leq\frac{n}{2}$. Our results extend those of previous authors for second order $m=1$ to general higher order cases $m\geq2$.

A classical result of harmonic analysis asserts that if a weight $w$ satisfies Muckenhoupt's condition $A_\infty$, then the unweighted class $\mathit{BMO}$ is contained in the weighted space $\mathit{BMO}(w)$. The paper identifies the norm of this embedding in the one-dimensional setting. Specifically, for any function $f\in \mathit{BMO}(\mathbb{R})$ and any weight $w\in A_\infty(\mathbb{R})$ of characteristic $[w]_{A_\infty}$, we have the estimate $$ \|f\|_{\mathit{BMO}(w)}\leq e\sqrt{2}[w]_{A_\infty}\|f\|_{\mathit{BMO}}. $$ The constant $e\sqrt{2}=3.8442\dots$ is the best possible. We also prove a sharp version of this result in which the characteristic $[w]_{A_\infty}$ of the weight is fixed. Further extensions to the theory of martingales are obtained.

In a weighted Killing warped product $M^n_f\times_\rho\mathbb R$ with war\-ping me\-tric $\langle\,,\rangle_{\!M}+{\rho}^2\,dt$, where the warping function $\rho$ is a real positive function defined on $M^n$ and the weighted function $f$ does not depend on the parameter $t\in\mathbb R$, we use equivariant bifurcation theory in order to establish sufficient conditions that allow us to guarantee the existence of bifurcation instants, or the local rigidity for a family of open sets $\{\Omega_{\gamma}\}_{\gamma\in I}$ whose boundaries $\partial\Omega_{\gamma}$ are hypersurfaces with constant weighted mean curvature. For this, we analyze the number of negative eigenvalues of a certain Schrödinger operator and study its evolution. Furthermore, we obtain a characterization of a stable closed hypersurface $x\colon \Sigma^n\hookrightarrow M^n_f\times_\rho\mathbb R$ with constant weighted mean curvature in terms of the first eigenvalue of the $f$-Laplacian of $\Sigma^n$.

We use normal sections to relate the curvature locus of regular (resp. singular corank $1$) $3$-manifolds in $\mathbb{R}^6$ (resp.\ $\mathbb R^5$) with regular (resp.\ singular corank $1$) surfaces in $\mathbb R^5$ (resp. $\mathbb R^4$). For example, we show how to generate a Roman surface by a family of ellipses different to Steiner's way. We also study the relations between the regular and singular cases through projections. We show that there is a commutative diagram of projections and normal sections which relates the curvature loci of the different types of manifolds, and therefore, that the second order geometry of all of them is related. In particular, we define asymptotic directions for singular corank $1$ $3$-manifolds in $\mathbb R^5$ and relate them to asymptotic directions of regular $3$-manifolds in $\mathbb R^6$ and singular corank $1$ surfaces in $\mathbb R^4$.