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The relaxation dynamics of granular materials is more like that of complex fluids than that of thermal glass-forming systems, owing to the absence of the ‘cage effect’. Against the grain We can all claim familiarity with granular materials (think of sand, for example), so it might be surprising to find that there is still much to learn about their properties. When left undisturbed, a granular system will eventually settle into a stable structure; but perturb the system and the grains will move about, in a manner thought to resemble that of atoms in a slowly flowing glassy system. New results by Binquan Kou et al . show that this glassy analogy doesn't hold. The team used X-ray tomography to follow the three-dimensional motion of grains in a perturbed granular medium, and see dynamic behaviour that is distinct from that encountered in a glassy system. They trace this deviant behaviour to the presence of friction between the grains. Granular materials such as sand, powders and foams are ubiquitous in daily life and in industrial and geotechnical applications 1 , 2 , 3 , 4 . These disordered systems form stable structures when unperturbed, but in the presence of external influences such as tapping or shear they ‘relax’, becoming fluid in nature. It is often assumed that the relaxation dynamics of granular systems is similar to that of thermal glass-forming systems 3 , 5 . However, so far it has not been possible to determine experimentally the dynamic properties of three-dimensional granular systems at the particle level. This lack of experimental data, combined with the fact that the motion of granular particles involves friction (whereas the motion of particles in thermal glass-forming systems does not), means that an accurate description of the relaxation dynamics of granular materials is lacking. Here we use X-ray tomography to determine the microscale relaxation dynamics of hard granular ellipsoids subject to an oscillatory shear. We find that the distribution of the displacements of the ellipsoids is well described by a Gumbel law 6 (which is similar to a Gaussian distribution for small displacements but has a heavier tail for larger displacements), with a shape parameter that is independent of the amplitude of the shear strain and of the time. Despite this universality, the mean squared displacement of an individual ellipsoid follows a power law as a function of time, with an exponent that does depend on the strain amplitude and time. We argue that these results are related to microscale relaxation mechanisms that involve friction and memory effects (whereby the motion of an ellipsoid at a given point in time depends on its previous motion). Our observations demonstrate that, at the particle level, the dynamic behaviour of granular systems is qualitatively different from that of thermal glass-forming systems, and is instead more similar to that of complex fluids. We conclude that granular materials can relax even when the driving strain is weak.

Motivated by our study of the complex Banach conjecture, we characterize a complex ellipsoids \\(\\mathcal E\\) as compact subsets of \\(\\mathbb C^n\\), with the property that every complex line intersect \\(\\mathcal E\\) either in a single point or in the complex affine image of the unit disk. This characterization leads to the main interest of this paper. We study the topological behavior of compact subsets of \\(\\mathbb{CP}^n\\) with the property that any complex line that intersects them does either at a single point, at the boundary of a complex disk, or along the entire line. In particular, we are interested in quadratic \\(\\R\\)-algebraic subsets of \\(\\mathbb{CP}^n\\).

Any two-qubit state can be faithfully represented by a steering ellipsoid inside the Bloch sphere, but not every ellipsoid inside the Bloch sphere corresponds to a two-qubit state. We give necessary and sufficient conditions for when the geometric data describe a physical state and investigate maximal volume ellipsoids lying on the physical-unphysical boundary. We derive monogamy relations for steering that are strictly stronger than the Coffman-Kundu-Wootters (CKW) inequality for monogamy of concurrence. The CKW result is thus found to follow from the simple perspective of steering ellipsoid geometry. Remarkably, we can also use steering ellipsoids to derive non-trivial results in classical Euclidean geometry, extending Eulerʼs inequality for the circumradius and inradius of a triangle.

Given a set of points C = {x1, x2, ..., xm} ⊆ Rn, what is the minimum volume ellipsoid that encloses it? Equally interestingly, one may ask: What is the maximum volume ellipsoid that can be embedded in the set of points without containing any? These problems have a number of applications beside being interesting in their own right. In this paper we review the important results concerning these and suggest an evolutionary-type approach for their solution. We will also highlight computational results.

In 1970, Schneider introduced the \\(m\\)th-order extension of the difference body \\(DK\\) of a convex body \\(K\\subset\\mathbb R^n\\), the convex body \\(D^m(K)\\) in \\(\\mathbb R^{nm}\\). He conjectured that its volume is minimized for ellipsoids when the volume of \\(K\\) is fixed. In this work, we solve a dual version of this problem: we show that the volume of the polar body of \\(D^m(K)\\) is maximized precisely by ellipsoids. For \\(m=1\\) this recovers the symmetric case of the celebrated Blaschke-Santaló inequality. We also show that Schneider's conjecture cannot be tackled using standard symmetrization techniques, contrary to this new inequality. As an application for our results, we prove Schneider's conjecture asymptotically á la Bourgain-Milman. We also consider a functional version.

We investigate strictly convex hypersurfaces in Euclidean space that are parameterized by their support function. We obtain a differential equation for the support function restricted to curves on the sphere, and we give explicit parameterizations of ellipsoids in R[sup.n+1] as the inverse of their Gauss map, where symmetry plays an important role.

We study the \\(L^p\\) mapping properties of the strong spherical maximal function, which is a multiparameter generalisation of Stein's spherical maximal function. We show that this operator is bounded on \\(L^p\\) for \\(p > 2\\) in all dimensions \\(n \\geq 3\\). This matches the conjectured sharp range \\(p>(n+1)/(n-1)\\) when \\(n=3\\). For \\(n=2\\) the analogous estimate was recently proved by Chen, Guo and Yang. Our result builds upon and improves an earlier bound of Lee, Lee and Oh. The main novelty is an estimate in discretised incidence geometry that bounds the volume of the intersection of thin neighbourhoods of axis-parallel ellipsoids. This estimate is then interpolated with the Fourier analytic \\(L^p\\)-Sobolev estimates of Lee, Lee and Oh.

This article presents a robust control law based on the Attractive Ellipsoids Method applied to perform attitude control of a satellite that has inertial wheels and relatively small dimensions such as microsatellites or other similar architectures. The novelty lies in the fact that under certain characterized external and parametric perturbations it is possible to track the desired angular trajectory with an optimization method to determine the computed torque. Additionally, a demonstration of this methodology is carried out with a numerical example.

It is shown that Alesker's solution of McMullen's conjecture implies the following stronger version of the conjecture: Every continuous, translation invariant, \\(k\\)-homogeneous valuation on convex bodies in \\(\\mathbb{R}^n\\) can be approximated uniformly on compact subsets by finite linear combinations of mixed volumes involving at most \\(N_{n,k}\\) summands, where \\(N_{n,k}\\) is a constant depending on \\(n\\) and \\(k\\) only. Moreover, \\(n-k-1\\) of the arguments of the mixed volumes can be chosen to be ellipsoids that do not depend on the valuation. The result is based on a corresponding description of smooth valuations in terms of finite linear combinations of mixed volumes.

A bstract We analyze the charge-to-mass structure of BPS states in general infinite-distance limits of N = 2 compactifications of Type IIB string theory on Calabi-Yau three-folds, and use the results to sharpen the formulation of the Swampland Conjectures in the presence of multiple gauge and scalar fields. We show that the BPS bound coincides with the black hole extremality bound in these infinite distance limits, and that the charge-to-mass vectors of the BPS states lie on degenerate ellipsoids with only two non-degenerate directions, regardless of the number of moduli or gauge fields. We provide the numerical value of the principal radii of the ellipsoid in terms of the classification of the singularity that is being approached. We use these findings to inform the Swampland Distance Conjecture, which states that a tower of states becomes exponentially light along geodesic trajectories towards infinite field distance. We place general bounds on the mass decay rate λ of this tower in terms of the black hole extremality bound, which in our setup implies λ ≥ 1 / 6 . We expect this framework to persist beyond N = 2 as long as a gauge coupling becomes small in the infinite field distance limit.