In this work we present a general introduction to the Signorini problem (or thin obstacle problem). It is a self-contained survey that aims to cover the main currently known results regarding the thin obstacle problem. We present the theory with some proofs, from the optimal regularity of solutions and classification of free boundary points to more recent results on the non-regular part of the free boundary and generic regularity.

If $S$ is the degree of a permutation group and $s$ is the maximum degree of its elements, then $S\le 2s-2$. We show that this inequality is sharp for some permutation group if and only if $s$ is a power of $2$, and then there is exactly one such permutation group up to isomorphism. The unique example is an elementary Abelian $2$-group that arises from a punctured Hadamard code. Then we discuss the solutions of $S=2s-3$ and $S=2s-4$.

This paper aims to study solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup of the general skew linear group ${\text{GL}}_n(D)$ over a division ring $D$. It turns out that in the case where $D$ is non-commutative, if such maximal subgroups exist, then either it is abelian or . Also, if $F$ is an infinite field and $n\ge 5$, then every locally solvable maximal subgroup of a normal subgroup of ${\text{GL}}_n(F)$ is abelian.

The regular subgroup determining an induced Hopf Galois structure for a Galois extension $L/K$ is obtained as the direct product of the corresponding regular groups of the inducing subextensions. We describe here the associated Hopf algebra and Hopf action of an induced structure and we prove that they are obtained by tensoring the corresponding inducing objects. In order to deal with their associated orders we develop a general method to compute bases and free generators in terms of matrices coming from representation theory of Hopf modules. In the case of an induced Hopf Galois structure this allows us to decompose the associated order, assuming that inducing subextensions are arithmetically disjoint.

In this paper, we study the existence and uniqueness of bounded viscosity solutions for parabolic Hamilton–Jacobi monotone systems in which the diffusion term is driven by variable-order nonlocal operators whose kernels depend on the space-time variable. We prove the existence of solutions via Perron’s method, and considering Hamiltonians with linear and superlinear nonlinearities related to their gradient growth we state a comparison principle for bounded sub and supersolutions. Moreover, we present steady-state large time behavior with an exponential rate of convergence.

In this paper we introduce the notion of evolution rank and give a decomposition of an evolution algebra into its annihilator and extending evolution subspaces having evolution rank one. This decomposition can be used to prove that in nondegenerate evolution algebras any family of natural and orthogonal vectors can be extended to a natural basis. The central results are the characterization of those families of orthogonal linearly independent vectors which can be extended to a natural basis.

We also consider ideals in perfect evolution algebras and prove that they coincide with basic ideals.

Nilpotent elements of order three can be localized (in a perfect evolution algebra over a field in which every element is a square) by merely looking at the structure matrix: any vanishing principal minor provides one. Conversely, if a perfect evolution algebra over an arbitrary field has a nilpotent element of order three, then its structure matrix has a vanishing principal minor.

We finish by considering the adjoint evolution algebra and relating its properties to the corresponding ones in the initial evolution algebra.

In this article we prove both norm and modular Hardy inequalities for class functions in one-dimensional fractional Orlicz–Sobolev spaces.

We give an improved polynomial bound on the complexity of the equation solvability problem, or more generally, of finding the value sets of polynomials over finite nilpotent rings. Our proof depends on a result in additive combinatorics, which may be of independent interest.

We study the group-invariant continuous polynomials on a Banach space $X$ that separate a given set $K$ in $X$ and a point $z$ outside $K$. We show that if $X$ is a real Banach space, $G$ is a compact group of $\mathcal{L}(X)$, $K$ is a $G$-invariant set in $X$, and $z$ is a point outside $K$ that can be separated from $K$ by a continuous polynomial $Q$, then $z$ can also be separated from $K$ by a $G$-invariant continuous polynomial $P$. It turns out that this result does not hold when $X$ is a complex Banach space, so we present some additional conditions to get analogous results for the complex case. We also obtain separation theorems under the assumption that $X$ has a Schauder basis which give applications to several classical groups. In this case, we obtain characterizations of points which can be separated by a group-invariant polynomial from the closed unit ball.

Let $\ell$ and $p$ be prime numbers and $K_{n,m}=\mathbb Q(p^{\frac{1}{\ell^n}},\zeta_{2\ell^{m}})$. We study the $\ell$-class group of $K_{n,m}$ in this paper. When $\ell=2$, we determine the structure of the $2$-class group of $K_{n,m}$ for all $(n,m)\in \mathbb Z_{\geq 0}^2$ in the case $p\equiv 3, 5\bmod{8}$, and for $(n,m)=(n,0)$, $(n,1)$, or $(1,m)$ in the case $p\equiv 7\bmod{16}$, generalizing the results of Parry about the $2$-divisibility of the class number of $K_{2,0}$. We also obtain results about the $\ell$-class group of $K_{n,m}$ when $\ell$ is odd and in particular when $\ell=3$. The main tools we use are class field theory, including Chevalley’s ambiguous class number formula and its generalization by Gras, and a stationary result about the $\ell$-class groups in the $2$-dimensional Kummer tower $\{K_{n,m}\}$.

In this paper, the theory of Gelfand problems is adapted to the 1-Laplacian setting. Concretely, we deal with the following problem:

where $\mathrm{\Omega }\subset {ℝ}^N$ ($N\ge 1$) is a domain, $\lambda \ge 0$, and $f:[0,+\infty [\to ]0,+\infty [$ is any continuous increasing and unbounded function with .

We prove the existence of a threshold ${\lambda }^{*}=\frac{h(\mathrm{\Omega })}{f(0)}$ ($h(\mathrm{\Omega })$ being the Cheeger constant of $\mathrm{\Omega }$) such that there exists no solution when and the trivial function is always a solution when $\lambda \le {\lambda }^{*}$. The radial case is analyzed in more detail, showing the existence of multiple (even singular) solutions as well as the behavior of solutions to problems involving the $p$-Laplacian as $p$ tends to $1$, which allows us to identify proper solutions through an extra condition.

For $S_g(x,y)=x-g(y)$, $x,y\in {ℝ}^n$, $g\in O(n)$, we investigate the Lebesgue measure and Hausdorff dimension of $S_g(A)$ given the dimension of $A$, both for general Borel subsets of ${ℝ}^{2n}$ and for product sets.

A well-known conjecture by Lovász and Plummer from the 1970s asserted that a bridgeless cubic graph has exponentially many perfect matchings. It was solved in the affirmative by Esperet et al. ([13]). On the other hand, Chudnovsky and Seymour ([8]) proved the conjecture in the special case of cubic planar graphs. In our work we consider random bridgeless cubic planar graphs with the uniform distribution on graphs with $n$ vertices. Under this model we show that the expected number of perfect matchings in labeled bridgeless cubic planar graphs is asymptotically $c{\gamma }^n$, where and $\gamma ~1.14196$ is an explicit algebraic number. We also compute the expected number of perfect matchings in (not necessarily bridgeless) cubic planar graphs and provide lower bounds for unlabeled graphs. Our starting point is a correspondence between counting perfect matchings in rooted cubic planar maps and the partition function of the Ising model in rooted triangulations.

This paper is devoted to proving the differential invariance of the multiplicity of real and complex analytic sets. In particular, we prove the real version of the Gau–Lipman theorem, i.e., it is proved that the multiplicity mod $2$ of real analytic sets is a differential invariant. We also prove a generalization of the Gau–Lipman theorem.

In order to obtain a closed orientable convex projective $4$-manifold with small positive Euler characteristic, we build an explicit example of convex projective Dehn filling of a cusped hyperbolic $4$-manifold through a continuous path of projective cone-manifolds.

We study nearly holomorphic Siegel Eisenstein series of general levels and characters on ${ℍ}_{2n}$, the Siegel upper half space of degree $2n$. We prove that the Fourier coefficients of these Eisenstein series (once suitably normalized) lie in the ring of integers of ${\mathbb{Q}}_p$ for all sufficiently large primes $p$. We also prove that the pullbacks of these Eisenstein series to ${ℍ}_n\times {ℍ}_n$ are cuspidal under certain assumptions.

The purpose of this note is to supply a correct proof of a proposition in the article quoted above.