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For every given $\beta<0$, we study the problem of maximizing the first Robin eigenvalue of the Laplacian $\lambda_\beta(\Omega)$ among convex (not necessarily smooth) sets $\Omega\subset\mathbb{S}^{n}$ with fixed perimeter. In particular, denoting by $\sigma_n$ the perimeter of the $n$-dimensional hemisphere, we show that for fixed perimeters $P<\sigma_n$, geodesic balls maximize the eigenvalue. Moreover, we prove a quantitative stability result for this isoperimetric inequality in terms of volume difference between $\Omega$ and the ball $D$ of the same perimeter.

The current standard model of cosmology successfully describes a variety of measurements, but the nature of its main ingredients, dark matter and dark energy, remains unknown. Euclid is a medium-class mission in the Cosmic Vision 2015-2025 programme of the European Space Agency (ESA) that will provide high-resolution optical imaging, as well as near-infrared imaging and spectroscopy, over about 14,000 deg^2 of extragalactic sky. In addition to accurate weak lensing and clustering measurements that probe structure formation over half of the age of the Universe, its primary probes for cosmology, these exquisite data will enable a wide range of science. This paper provides a high-level overview of the mission, summarising the survey characteristics, the various data-processing steps, and data products. We also highlight the main science objectives and expected performance., Comment: Paper submitted as part of the A&A special issue`Euclid on Sky'

In this paper, we establish a comparison principle in terms of Lorentz norms and point-wise inequalities between a positive solution $u$ to the Poisson equation with non-homogeneous Neumann boundary conditions and a specific positive solution $v$ to the Schwartz symmetrized problem, which is related to $u$ through an additional boundary condition.

An $\mathbb{F}_q$- linear set $L=L_U$ of $\Lambda=\mathrm{PG}(V, \mathbb{F}_{q^n}) \cong \mathrm{PG}(r-1,q^n)$ is a set of points defined by non-zero vectors of an $\mathbb{F}_q$-subspace $U$ of $V$. The integer $\dim_{\mathbb{F}_q} U$ is called the rank of $L$. In [G. Lunardon, O. Polverino: Translation ovoids of orthogonal polar spaces. Forum Math. 16 (2004)], it was proven that any $\mathbb{F}_q$-linear set $L$ of $\Lambda$ of rank $u$ such that $\langle L \rangle=\Lambda$ is either a canonical subgeometry of $\Lambda$ or there are a $(u-r-1)$-dimensional subspace $\Gamma$ of $\mathrm{PG}(u-1,q^n) \supset \Lambda$ disjoint from $\Lambda$ and a canonical subgeometry $\Sigma \cong \mathrm{PG}(u-1,q)$ disjoint from $\Gamma$ such that $L$ is the projection of $\Sigma$ from $\Gamma$ onto $\Lambda$. The subspace $\Gamma$ is called the vertex of the projection. In this article, we will show a method to reconstruct the vertex $\Gamma$ for a peculiar class of linear sets of rank $u = n(r - 1)$ in $\mathrm{PG}(r - 1, q^n)$ called evasive linear sets. Also, we will use this result to characterize some families of linear sets of the projective line $\mathrm{PG}(1,q^n)$ introduced from 2018 onward, by means of certain properties of their projection vertices, as done in [B. Csajb\'{o}k, C. Zanella: On scattered linear sets of pseudoregulus type in $\mathrm{PG}(1, q^t)$, Finite Fields Appl. 41 (2016)] and in [C. Zanella, F. Zullo: Vertex properties of maximum scattered linear sets of $\mathrm{PG}(1, q^n)$. Discrete Math. 343(5) (2020)].

Quantum computing represents a cutting-edge frontier in artificial intelligence. It makes use of hybrid quantum-classical computation which tries to leverage quantum mechanic principles that allow us to use a different approach to deep learning classification problems. The work presented here falls within the context of the AGILE space mission, launched in 2007 by the Italian Space Agency. We implement different Quantum Convolutional Neural Networks (QCNN) that analyze data acquired by the instruments onboard AGILE to detect Gamma-Ray Bursts from sky maps or light curves. We use several frameworks such as TensorFlow-Quantum, Qiskit and PennyLane to simulate a quantum computer. We achieved an accuracy of 95.1% on sky maps with QCNNs, while the classical counterpart achieved 98.8% on the same data, using however hundreds of thousands more parameters., Comment: 4 pages, 2 figures, proceedings of the ADASS XXXIII (2023) conference, to appear in ASP Conference Serie

The aim of this paper is to study supersoluble skew braces, a class of skew braces that encompasses all finite skew braces of square-free order. It turns out that finite supersoluble skew braces have Sylow towers, and that in an arbitrary supersoluble skew brace $B$ many relevant skew brace-theoretical properties are easier to identify: for example, a centrally nilpotent ideal of $B$ is $B$-centrally nilpotent, a fact that simplifies the computational search for the Fitting ideal; also, $B$ has finite multipermutational level if and only if $(B,+)$ is nilpotent. Given a finite presentation of the structure skew brace $G(X,r)$ associated with a finite non-degenerate solution of the Yang--Baxter Equation (YBE), there is an algorithm that decides if $G(X,r)$ is supersoluble or not. Moreover, supersoluble skew braces are examples of almost polycyclic skew braces, so they give rise to solutions of the YBE on which one can algorithmically work on., Comment: 37 pages; to be published in Forum of Mathematics, Sigma

In this paper, we prove a Serrin-type result for an elliptic system of equations, overdetermined with both Dirichlet and a generalized Neumann conditions. With this tool, we characterize the critical shapes under volume constraint of some domain functionals.

The subgroup lattice of a group is a great source of information about the structure of the group itself. The aim of this paper is to use a similar tool for studying profinite groups. In more detail, we study the lattices of closed or open subgroups of a profinite group and its relation with the whole group. We show, for example, that procyclic groups are the only profinite groups with a distributive lattice of closed or open subgroups, and we give a sharp characterization of profinite groups whose lattice of closed (or open) subgroups satisfies the Dedekind modular law; we actually give a precise description of the behaviour of modular elements of the lattice of closed subgroups. We also deal with the problem of carrying some structural information from a profinite group to another one having an isomorphic lattice of closed (or open) subgroups. Some interesting consequences and related results concerning decomposability and the number of profinite groups with a given lattice of closed (or open) subgroups are also obtained., Comment: 23 pages

Methods are studied to compute the boosting effects produced by the observer motion that modifies and transfers to higher l the isotropic monopole frequency spectrum of the cosmic background. Explicit analytical solutions for spherical harmonic coefficients are presented and applied to various background spectra, alleviating computational effort. High l frequency spectra are led by higher order derivatives of the spectrum. Tabulated frequency spectra are computed with a relatively poor frequency resolution in comparison with the Doppler shift, calling for interpolation. They are affected by uncertainties due to intrinsic inaccuracies in modelling, observational data or limited computation accuracy, propagate and increase with the derivative order, possibly preventing a trustworthy computation to higher l and of the observed monopole. We filter the original function and the multipole spectra to derive reliable predictions of the harmonic coefficients. For spectra expressed in Taylor series, we derive explicit solutions for the harmonic coefficients up to l=6 in terms of spectra derivatives. We consider filters and study the quality of these methods on suitable analytical approximations, polluted with simulated noise. We consider the extragalactic sources microwave background from radio loud AGN and the 21cm line superimposed to the CMB. Gaussian pre-filtering coupled to a real space filtering of derivatives allows accurate predictions up to l=6, while log-log polynomial representation gives accurate solutions at any l. Describing the 21 cm model variety is difficult, so it is relevant to relax assumptions. Pre-filtering gives accurate predictions up to l=3-4, while further filtering or boosting amplification/deamplification method improves the results allowing reasonable estimations. The methods can extend the range of realistic background models manageable with a fast computation., Comment: 21 pages, 20 figures, revised version accepted on 09/01/2024 for pubblication on Astronomy and Astrophysics. Original manuscript submitted on 20/10/2023

The aim of this short note is to completely answer Questions 2.34 and 2.35 of arXiv:1806.01127. In particular, we show that a finite strong-nil skew brace $B$ of abelian type need not be right-nilpotent, but that this is the case if~$B$ is of nilpotent type and $b\ast b=0$ for all $b\in B$ (our examples show that this is the best possible result)., Comment: arXiv admin note: text overlap with arXiv:2310.07474. A typo is corrected in proof of Lemma 2.1 and a step of the proof is explained with more detail

Nipotency of skew braces is related to certain types of solutions of the Yang-Baxter equation. This paper delves into the study of centrally nilpotent skew braces. In particular, we study their torsion theory (Section 4.1) and we introduce an "index" for subbraces (Section 4.2), but we also show that the product of centrally nilpotent ideals need not be centrally nilpotent (Example B), a rather peculiar fact. To cope with these examples, we introduce a special type of nilpotent ideal, using which, we define a {\it good} Fitting ideal. Also, a Frattini ideal is defined and its relationship with the Fitting ideal is investigated. A key ingredient in our work is the characterisation of the commutator of ideals in terms of absorbing polynomials (Section 3); this solves Problem 3.4 of arXiv:2109.04389. Moreover, we provide an example (Example A) showing that the idealiser of a subbrace (as defined in arXiv:2205.01572v2) does not exist in general., Comment: 44 pages, definition of centralizer is modified

Let $\sigma=\{\sigma_j\,:\, j\in J\}$ be a partition of the set $\mathbb{P}$ of all prime numbers. A subgroup $X$ of a finite group $G$ is~\textit{$\sigma$-subnormal} in $G$ if there exists a chain of subgroups $$X=X_0\leq X_1\leq\ldots\leq X_n=G$$ such that, for each $1\leq i\leq n-1$, $X_{i-1}\trianglelefteq X_i$ or $X_i/(X_{i-1})_{X_i}$ is a $\sigma_{j_i}$-group for some $j_i\in J$. Skiba~[12] studied the main properties of $\sigma$-subnormal subgroups in finite groups and showed that the set of all $\sigma$-subnormal subgroups plays a relevant role in the structure of a finite soluble group. In [5], we laid the foundation of a general theory of $\sigma$-subnormal subgroups (and $\sigma$-series) in locally finite groups. It turns out that the main difference between the finite and the locally finite case concerns the behaviour of the join of $\sigma$-subnormal subgroups: in finite groups, $\sigma$-subnormal subgroups form a sublattice of the lattice of all subgroups [3], but this is no longer true for arbitrary locally finite groups. This is similar to what happens with subnormal subgroups, so it makes sense to study the class $\mathfrak{S}_\sigma^\infty$ (resp. $\mathfrak{S}_\sigma$) of locally finite groups in which the join of (resp. of finitely many) $\sigma$-subnormal subgroups is $\sigma$-subnormal. Our aim is to study how much one can extend a group in one of these classes before going outside the same class (see for example Theorems~3.6, 3.8, 5.5 and 5.7). Also, $\sigma$-subnormality criteria for the join of $\sigma$-subnormal subgroups are obtained: similarly to a celebrated theorem of Williams (see [15]), we give a necessary and sufficient conditions for a join of two $\sigma$-subnormal subgroups to always be $\sigma$-subnormal; consequently, we show that the join of two orthogonal $\sigma$-subnormal subgroups is $\sigma$-subnormal (extending a result of Roseblade [11])., Comment: 26pp

The aim of this paper is to show that the structure skew brace associated with a finite non-degenerate solution of the Yang-Baxter equation is finitely presented., Comment: arXiv admin note: text overlap with arXiv:2210.08598

By exploiting the connection between scattered $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^3$ and minimal non degenerate $3$-dimensional rank metric codes of $\mathbb{F}_{q^m}^{n}$, $n \geq m+2$, described in [2], we will exhibit a new class of codes with parameters $[m+2,3,m-2]_{q^m/q}$ for infinite values of $q$ and $m \geq 5$ odd. Moreover, by studying the geometric structures of these scattered subspaces, we determine the rank weight distribution of the associated codes.

Seafloor deformation monitoring is now performed in the marine sector of the Campi Flegrei volcanic area. MEDUSA infrastructure consists of 4 buoys at depths of 40-96m equipped with cGPS receivers, accelerometers and magnetic compasses to monitor buoy status and a seafloor module with a bottom pressure recorder. We study the seafloor deformation in the caldera. Previously we show that cGPS onland network and MEDUSA timeseries for the years 2017-2020 are in agreement with the deformation predicted by a Mogi model describing the observed deformation of an active volcano. Only for buoy A data differ significantly from model, at 6.9sigma and 23.7sigma for the horizontal speed (v) and direction. We devised a new method to reconstruct the sea bottom displacement including cGPS and compass data. The method, applied to buoy A and validated also on C, uses compass data to correct cGPS positions accounting for pole inclination. Including systematic errors, the internal consistency, within 3sigma (2sigma) for the speed (angle), between the results derived for different maximum inclinations of the buoy pole up to 3.5deg shows that the method allows to significantly reduce the impact of the pole inclination which can alter the estimation. We find good convergence of the velocity and deformation angle for increasing values of the buoy pole inclination. We found v=3.521+-0.039(stat)+-0.352(syst)cm/yr and an angle -115.159+-0.670(stat)+-7.630(syst)deg. The relative impact of potential systematics (statistical) effects increases (decreases) with cutoff. Our analysis gives v consistent with Mogi at 5.2sigma(stat) or 0.5sigma(stat and syst), and a deformation angle consistent at 4.3sigma(stat) or at 0.3sigma. The module of the vectorial difference between v from the data and Mogi diminishes by a factor 7.65+-1.23(stat) or +-5.78(stat+syst) compared with previous work. Potential improvements are discussed., Comment: 26 pages, 10 figures, accepted for pubblication on Journal of Geodesy

We are interested in the thermal insulation of a bounded open set $\Omega$ surrounded by a set whose thickness is locally described by $\varepsilon h$, where $h$ is a non-negative function defined on the boundary $\partial\Omega$. We study the problem in the limit for $\varepsilon$ going to zero using a first-order asymptotic development by $\Gamma$-convergence.

In this paper, we introduce a symmetrization technique for the gradient of a $\BV$ function, which separates its absolutely continuous part from its singular part (sum of the jump and the Cantorian part). In particular, we prove an $\text{\emph{L}}^{\text{1}}$ comparison between the function and its symmetrized. Furthermore, we apply this result to obtain Saint-Venant type inequalities for some geometric functionals.

If $(X,r)$ is a finite non-degenerate set-theoretic solution of the Yang--Baxter equation, the additive group of the structure skew brace $G(X,r)$ is an $FC$-group, i.e. a group whose elements have finitely many conjugates. Moreover, its multiplicative group is virtually abelian, so it is also close to an $FC$-group itself. If one additionally assumes that the derived solution of $(X,r)$ is indecomposable, then for every element $b$ of $G(X,r)$ there are finitely many elements of the form $b*c$ and $c*b$, with $c\in G(X,r)$. This naturally leads to the study of a brace-theoretic analogue of the class of $FC$-groups. For this class of skew braces, the fundamental results and their connections with the solutions of the YBE are described: we prove that they have good torsion and radical theories and they behave well with respect to certain nilpotency concepts and finite generation., Comment: 24 pages. Accepted for publication in Publicacions Matem\`atiques

In [A. Neri, P. Santonastaso, F. Zullo. Extending two families of maximum rank distance codes], the authors extended the family of $2$-dimensional $\mathbb{F}_{q^{2t}}$-linear MRD codes recently found in [G. Longobardi, G. Marino, R. Trombetti, Y. Zhou. A large family of maximum scattered linear sets of $\mathrm{PG}(1,q^n)$ and their associated MRD codes]. Also, for $t \geq 5$ they determined equivalence classes of the elements in this new family and provided the exact number of inequivalent codes in it. In this article, we complete the study of the equivalence issue removing the restriction $t \geq 5$. Moreover, we prove that in the case when $t=4$, the linear sets of the projective line $\mathrm{PG}(1,q^{8})$ ensuing from codes in the relevant family, are not equivalent to any one known so far.

By means of a suitable weighted rearrangement, we obtain various apriori bounds for the solutions to a Robin problem. Among other things, we derive a family of Faber-Krahn type inequalities.

We study the thermal insulation of a bounded body $\Omega\subset\mathbb{R}^n$, under a prescribed heat source $f>0$, via a bulk layer of insulating material. We consider a model of heat transfer between the insulated body and the environment determined by convection; this corresponds to Robin boundary conditions on the free boundary of the layer. We show that a minimal configuration exists and that it satisfies uniform density estimates.

We revisit and extend the connections between $\mathbb{F}_{q^m}$-linear rank-metric codes and evasive $\mathbb{F}_q$-subspaces of $\mathbb{F}_{q^m}^k$. We give a unifying framework in which we prove in an elementary way how the parameters of a rank-metric code are related to special geometric properties of the associated evasive subspace, with a particular focus on the generalized rank weights. In this way, we can also provide alternative and very short proofs of known results on scattered subspaces. We then use this simplified point of view in order to get a geometric characterization of near MRD codes and a clear bound on their maximal length. Finally we connect the theory of quasi-MRD codes with $h$-scattered subspaces of maximum dimension, extending to all the parameters sets the already known results on MRD codes., Comment: 20 pages

The aim of this short note is to provide a proof to a statement of Sierpi\'nski concerning the number of possible sums of a series (of type $\lambda<\aleph_1$) of arbitrary ordinal numbers.

We study a shape optimization problem involving a solid $K\subset\mathbb{R}^n$ that is maintained at constant temperature and is enveloped by a layer of insulating material $\Omega$ which obeys a generalized boundary heat transfer law. We minimize the energy of such configurations among all $(K,\Omega)$ with prescribed measure for $K$ and $\Omega$, and no topological or geometrical constraints. In the convection case (corresponding to Robin boundary conditions on $\partial\Omega$) we obtain a full description of minimizers, while for general heat transfer conditions, we prove the existence and regularity of solutions and give a partial description of minimizers.

We study the behaviour, when $p \to +\infty$, of the first $p$-Laplacian eigenvalues with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that the limit of the eigenfunctions is a viscosity solution to an eigenvalue problem for the so-called $\infty$-Laplacian.