How Neighborly Can a Centrally Symmetric Polytope Be?

We show that there exist k-neighborly centrally symmetric d-dimensional polytopes with 2(n + d) vertices, where

Exact formulae and matrix-less eigensolvers for block banded symmetric Toeplitz matrices

Precise asymptotic expansions for the eigenvalues of a Toeplitz matrix \(T_n(f)\), as the matrix size n tends to infinity, have recently been obtained, under suitable assumptions on the associated generating function f. A restriction is that f has to be polynomial, monotone, and scalar-valued. In this paper we focus on the case where \(\mathbf {f}\) is an \(s\times s\) matrix-valued trigonometric polynomial with \(s\ge 1\), and \(T_n(\mathbf {f})\) is the block Toeplitz matrix generated by \(\mathbf {f}\), whose size is \(N(n,s)=sn\). The case \(s=1\) corresponds to that already treated in the literature. We numerically derive conditions which ensure the existence of an asymptotic expansion for the eigenvalues. Such conditions generalize those known for the scalar-valued setting. Furthermore, following a proposal in the scalar-valued case by the first author, Garoni, and the third author, we devise an extrapolation algorithm for computing the eigenvalues of banded symmetric block Toeplitz matrices with a high level of accuracy and a low computational cost. The resulting algorithm is an eigensolver that does not need to store the original matrix, does not need to perform matrix-vector products, and for this reason is called matrix-less. We use the asymptotic expansion for the efficient computation of the spectrum of special block Toeplitz structures and we provide exact formulae for the eigenvalues of the matrices coming from the \(\mathbb {Q}_p\) Lagrangian Finite Element approximation of a second order elliptic differential problem. Numerical results are presented and critically discussed.     

Local and global solutions for some parabolic nonlocal problems

We study local and global existence of solutions for some semilinear parabolic initial boundary value problems with autonomous nonlinearities having a “Newtonian” nonlocal term.     

Joint Modeling of Multiple Network Views

Latent space models (LSM) for network data rely on the basic assumption that each node of the network has an unknown position in a D-dimensional Euclidean latent space: generally the smaller the distance between two nodes in the latent space, the greater their probability of being connected. In this article, we propose a variational inference approach to estimate the intractable posterior of the LSM. In many cases, different network views on the same set of nodes are available. It can therefore be useful to build a model able to jointly summarize the information given by all the network views. For this purpose, we introduce the latent space joint model (LSJM) that merges the information given by multiple network views assuming that the probability of a node being connected with other nodes in each network view is explained by a unique latent variable. This model is demonstrated on the analysis of two datasets: an excerpt of 50 girls from “Teenage Friends and Lifestyle Study” data at three time points and the Saccharomyces cerevisiae genetic and physical protein–protein interactions. Supplementary materials for this article are available online.     

Synthesis and characterization of high-molecular-weight syndiotactic amorphous polypropylene

Heterocycle-fused titanium indenyl silylamido dimethyl complexes produce very high molecular weight polypropylene having a prevailingly syndiotactic microstructure with syndiotactic pentad contents rrrr up to 40-55% (sam-PP). The samples are basically amorphous and may slowly develop a low level of crystallinity (16-20%) at room temperature. A structural characterization has shown that sam-PP samples crystallize in disordered modifications of the helical form I of syndiotactic polypropylene (s-PP). The stretching of compression-molded films of sam-PP samples produce oriented crystalline fibers in the trans-planar mesomorphic form of s-PP. The low stereoregularity prevents the formation of the ordered trans-planar form III of s-PP, which instead is obtained in stretched fibers of the highly stereoregular and crystalline s-PP. The trans-planar mesomorphic form, obtained in stretched fibers, in turn transforms into the helical form I upon releasing the tension. The analysis of the mechanical properties has shown that sam-PP samples show good elastic behavior in a large range of deformation with remarkable strength, due to the presence of crystallinity. A comparison with the mechanical properties of less syndiotactic and fully amorphous samples is reported. These fully amorphous samples present lower strength and experience rapid viscous flow of the chains at high deformations and/or by application of stresses for long times. The higher strength in the semicrystalline sam-PP samples makes these materials interesting thermoplastic elastomers showing high toughness and ductility.     

A maximal atmospheric mixing from a maximal CP violating phase

We point out an elegant mechanism to predict a maximal atmospheric angle, which is based on a maximal CP violating phase difference between second and third lepton families in the flavour symmetry basis. In this framework, a discussion of the general formulas for θ₁₂${\theta }_{12}$, |U_e3|$|U_{e3}|$, δ   and their possible correlations in some limiting cases is provided. We also present an explicit realisation in terms of an SO(3)$\mathit{SO}(3)$ flavour symmetry model.