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In this paper, Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are investigated. The focus is on the convergence of successive symmetrals with respect to a sequence of i-dimensional subspaces of ℝ n . Such a sequence is called universal for a family of sets if the successive symmetrals of any set in the family converge to a ball with center at the origin. New universal sequences for the main symmetrizations, for all valid dimensions i of the subspaces, are found by combining two groups of results. The first, published separately, provides finite sets ℱ of subspaces such that reflection symmetry (or rotational symmetry) with respect to each subspace in ℱ implies full rotational symmetry. In the second, proved here, a theorem of Klain for Steiner symmetrization is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber symmetrizations, showing that if a sequence of subspaces is drawn from a finite set ℱ of subspaces, the successive symmetrals of any compact convex set converge to a compact convex set that is symmetric with respect to any subspace in ℱ appearing infinitely often in the sequence. It is also proved that for Steiner, Schwarz, and Minkowski symmetrizations, a sequence of i-dimensional subspaces is universal for the class of compact sets if and only if it is universal for the class of compact convex sets, and Klain’s theorem is shown to hold for Schwarz symmetrization of compact sets.

We shall prove a convergence result relative to sequences of Minkowski symmetrals of general compact sets. In particular, we investigate the case when this process is induced by sequences of subspaces whose elements belong to a finite family, following the path marked by Klain in Klain (2012, Advances in Applied Mathematics 48, 340–353), and the generalizations in Bianchi et al. (2019, Convergence of symmetrization processes, preprint) and Bianchi et al. (2012, Indiana University Mathematics Journal 61, 1695–1710). We prove an analogous result for fiber symmetrization of a specific class of compact sets. The idempotency for symmetrizations of this family of sets is investigated, leading to a simple generalization of a result from Klartag (2004, Geometric and Functional Analysis 14, 1322–1338) regarding the approximation of a ball through a finite number of symmetrizations, and generalizing an approximation result in Fradelizi, Làngi and Zvavitch (2020, Volume of the Minkowski sums of star-shaped sets, preprint).

Many real data sets contain numerical features (variables) whose distribution is far from normal (Gaussian). Instead, their distribution is often skewed. In order to handle such data it is customary to preprocess the variables to make them more normal. The Box–Cox and Yeo–Johnson transformations are well-known tools for this. However, the standard maximum likelihood estimator of their transformation parameter is highly sensitive to outliers, and will often try to move outliers inward at the expense of the normality of the central part of the data. We propose a modification of these transformations as well as an estimator of the transformation parameter that is robust to outliers, so the transformed data can be approximately normal in the center and a few outliers may deviate from it. It compares favorably to existing techniques in an extensive simulation study and on real data.

We investigate necessary and sufficient conditions on the weights for the Hardy-Rellich inequalities to hold, and propose a new way to use the notion of Bessel pair to establish the optimal Hardy-Rellich type inequalities. Our results sharpened earlier Hardy-Rellich and Rellich type inequalities in the literature. We also study several results about the symmetry and symmetry breaking properties of the Rellich type and Hardy-Rellich type inequalities, and then partially answered an open question raised by Ghoussoub and Moradifam. Namely, we will present conditions on the weights such that the Rellich type and Hardy-Rellich type inequalities hold for all functions if and only if the same inequalities hold for all radial functions.

We provide symmetrization results as mass concentration comparisons for solutions to singular parabolic equations in the cylinder$$\\Omega \\times (0,T)$$Ω × ( 0 , T ) ,$$T>0$$T > 0 . Here,$$\\Omega \\subset {\\mathbb {R}}^N$$Ω ⊂ R N ($$N \\ge 2$$N ≥ 2 ) is a bounded open set, featuring a lower order term that is singular in the solution variable s .

This chapter provides an introduction to the theory of decoherence‐free subspaces (DFSs), noiseless subsystems (NSs), and dynamical decoupling (DD) – the key tools for decoherence mitigation strategies. The second and third sections of the chapter discuss decoherence‐free subspaces, and define and analyze the collective dephasing model and explain how to combine the corresponding DFS encoding with universal quantum computation. The next section considers the same problem in the context of the more general collective decoherence model. The fifth section analyzes NSs, a key generalization of DFSs. The subsequent section defines DD by analyzing the protection of a single qubit against pure dephasing and against general decoherence. The seventh section discusses DD as a symmetrization procedure; the combining of DD with DFS is discussed in the eighth section. The penultimate section addresses concatenated dynamical decoupling (CDD). In the final section, DD is linked to the representation theory ideas underlying NSs theory.

In this paper, we give necessary and sufficient conditions for the rigidity of the perimeter inequality under Schwarz symmetrization. The term rigidity refers to the situation in which the equality cases are only obtained by translations of the symmetric set. In particular, we prove that the sufficient conditions for rigidity provided in M. Barchiesi, F. Cagnetti and N. Fusco [Stability of the Steiner symmetrization of convex sets. J. Eur. Math. Soc. 15 (2013), 1245-1278.] are also necessary.

Using first-principles calculations, this study evaluates the structure, equation of state, and elasticity of three compositions of phase D up to 75 GPa: (1) the magnesium endmember [MgSi2O4(OH)2], (2) the aluminum endmember [Al2SiO4(OH)2], and (3) phase D with 50% Al-substitution [AlMg0.5Si1.5O4(OH)2]. We find that the Mg-endmember undergoes hydrogen-bond symmetrization and that this symmetrization is linked to a 22% increase in the bulk modulus of phase D, in agreement with previous studies. Al2SiO4(OH)2 also undergoes hydrogen-bond symmetrization, but the concomitant increase in bulk modulus is only 13%—a significant departure from the 22% increase of the Mg-endmember. Additionally, Al-endmember phase D is denser (2%–6%), less compressible (6%–25%), and has faster compressional (6%–12%) and shear velocities (12%–15%) relative to its Mg-endmember counterpart. Finally, we investigated the properties of phase D with 50% Al-substitution [AlMg0.5Si1.5O4(OH)2], and found that the hydrogen-bond symmetrization, equation of state parameters, and elastic constants of this tie-line composition cannot be accurately modeled by interpolating the properties of the Mg- and Al-endmembers.

The richness of the phase diagram of water reduces drastically at very high pressures where only two molecular phases, proton-disordered ice VII and proton-ordered ice VIII, are known. Both phases transform to the centered hydrogen bond atomic phase ice X above about 60 GPa, i.e., at pressures experienced in the interior of large ice bodies in the universe, such as Saturn and Neptune, where nonmolecular ice is thought to be the most abundant phase of water. In this work, we investigate, by Raman spectroscopy up to megabar pressures and ab initio simulations, how the transformation of ice VII in ice X is affected by the presence of salt inclusions in the ice lattice. Considerable amounts of salt can be included in ice VII structure under pressure via rock–ice interaction at depth and processes occurring during planetary accretion. Our study reveals that the presence of salt hinders proton order and hydrogen bond symmetrization, and pushes ice VII to ice X transformation to higher and higher pressures as the concentration of salt is increased.