Multiplicatively badly approximable matrices up to logarithmic factors

Let \[||x||\] denote the distance from \[x \in \mathbb{R}\] to the nearest integer. In this paper, we prove a new existence and density result for matrices \[A \in {\mathbb{R}^{m \times n}}\] satisfying the inequality \[\mathop {\lim \inf }\limits_{|q{|_\infty } \to + \infty } \prod\limits_{j = 1}^n {\max } \{ 1,|{q_j}|\} \log {\left( {\prod\limits_{j = 1}^n {\max } \{ 1,|{q_j}|\} } \right)^{m + n - 1}}\prod\limits_{i = 1}^m {{A_i}q} > 0,\] where q ranges in \[{\mathbb{Z}^n}\] and A i denote the rows of the matrix A . This result extends previous work of Moshchevitin both to arbitrary dimension and to the inhomogeneous setting. The estimates needed to apply Moshchevitin’s method to the case m > 2 are not currently available. We therefore develop a substantially different method, based on Cantor-like set constructions of Badziahin and Velani. Matrices with the above property also appear to have very small sums of reciprocals of fractional parts. This fact helps us to shed light on a question raised by Lê and Vaaler on such sums, thereby proving some new estimates in higher dimension.

ONE-LEVEL DENSITY OF LOW-LYING ZEROS OF QUADRATIC HECKE L-FUNCTIONS OF IMAGINARY QUADRATIC NUMBER FIELDS

In this paper, we prove a one level density result for the low-lying zeros of quadratic Hecke L-functions of imaginary quadratic number fields of class number 1. As a corollary, we deduce, essentially, that at least $(19-\cot (1/4))/16 = 94.27\ldots \%$ of the L-functions under consideration do not vanish at 1/2.

Relative Density of SLM-Produced Aluminum Alloy Parts: Interpretation of Results

Micrographic image analysis, tomography and the Archimedes method are commonly used to analyze the porosity of Selective Laser Melting (SLM)-produced parts and then to estimate the relative density. This article deals with the limitation of the relative density results to conclude on the quality of a part manufactured by additive manufacturing and focuses on the interpretation of the relative density result. To achieve this aim, two experimental methods are used: the image analysis method, which provides local information on the distribution of porosity, and the Archimedes method, which provides access to global information. To investigate this, two different grades of aluminum alloy, AlSi7Mg0.6 and AM205, were used in this study. The study concludes that an analysis of the metallographic images to calculate the relative density of the part depends on the areas chosen for the analysis. In addition, the results show that the Archimedes method has limitations, particularly related to the choice of reference materials for calculating relative density. It can be observed, for example, that, depending on the experimental conditions, the calculation can lead to relative densities higher than 100%, which is inconsistent. This article shows that it is essential that a result of relative density obtained from Archimedes measurements be supplemented by an indication of the reference density used.

Effective Density for Inhomogeneous Quadratic Forms I: Generic Forms and Fixed Shifts

We establish effective versions of Oppenheim’s conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed shift vectors and generic quadratic forms. When the shift is rational we prove a counting result, which implies the optimal density for values of generic inhomogeneous forms. We also obtain a similar density result for fixed irrational shifts satisfying an explicit Diophantine condition. The main technical tool is a formula for the 2nd moment of Siegel transforms on certain congruence quotients of $SL_n(\mathbb{R}),$ which we believe to be of independent interest. In a sequel, we use different techniques to treat the companion problem concerning generic shifts and fixed quadratic forms.

On the regularity of Cauchy hypersurfaces and temporal functions in closed cone structures

We complement our work on the causality of upper semi-continuous distributions of cones with some results on Cauchy hypersurfaces. We prove that every locally stably acausal Cauchy hypersurface is stable. Then we prove that the signed distance [Formula: see text] from a spacelike hypersurface [Formula: see text] is, in a neighborhood of it, as regular as the hypersurface, and by using this fact we give a proof that every Cauchy hypersurface is the level set of a Cauchy temporal (and steep) function of the same regularity as the hypersurface. We also show that in a globally hyperbolic closed cone structure, compact spacelike hypersurfaces with boundary can be extended to Cauchy spacelike hypersurfaces of the same regularity. We end the work with a separation result and a density result.

Skyrmion phase and competing magnetic orders on a breathing kagomé lattice

Magnetic skyrmion textures are realized mainly in non-centrosymmetric, e.g. chiral or polar, magnets. Extending the field to centrosymmetric bulk materials is a rewarding challenge, where the released helicity/vorticity degree of freedom and higher skyrmion density result in intriguing new properties and enhanced functionality. We report here on the experimental observation of a skyrmion lattice (SkL) phase with large topological Hall effect and an incommensurate helical pitch as small as 2.8 nm in metallic Gd3Ru4Al12, which materializes a breathing kagomé lattice of Gadolinium moments. The magnetic structure of several ordered phases, including the SkL, is determined by resonant x-ray diffraction as well as small angle neutron scattering. The SkL and helical phases are also observed directly using Lorentz-transmission electron microscopy. Among several competing phases, the SkL is promoted over a low-temperature transverse conical state by thermal fluctuations in an intermediate range of magnetic fields.