Generalized Regularity and the Symmetry of Branches of ''Botanological'' Networks

We derive the generalized regularity of convex quadrilaterals in R^2, which gives a new evolutionary class of convex quadrilaterals that we call generalized regular quadrilaterals in R^2. The property of generalized regularity states that the Simpson line defined by the two Steiner points passes through the corresponding Fermat-Torricelli point of the same convex quadrilateral. We prove that a class of generalized regular convex quadrilaterals consists of convex quadrilaterals, such that their two opposite sides are parallel. We solve the problem of vertical evolution of a ''botanological'' thumb (a two way communication weighted network) w.r to a boundary rectangle in R^2 having two roots,two branches and without having a main branch, by applying the property of generalized regularity of weighted rectangles. We show that the two branches have equal weights and the two roots have equal weights, if the thumb inherits a symmetry w.r to the midperpendicular line of the two opposite sides of the rectangle, which is perpendicular to the ground (equal branches and equal roots). The geometric, rotational and dynamic plasticity of weighted networks for boundary generalized regular tetrahedra and weighted regular tetrahedra lead to the creation of ''botanological'' thumbs and ''botanological'' networks (with a main branch) having symmetrical branches

Conditional Expectation, Entropy, and Transport for Convex Gibbs Laws in Free Probability

Let $(X_1,\dots ,X_m)$ be self-adjoint noncommutative random variables distributed according to the free Gibbs law given by a sufficiently regular convex and semi-concave potential $V$, and let $(S_1,\dots ,S_m)$ be a free semicircular family. For $k < m$, we show that conditional expectations and conditional non-microstates free entropy given $X_1$, …, $X_k$ arise as the large $N$ limit of the corresponding conditional expectations and entropy for the $N \times N$ random matrix models associated to $V$. Then, by studying conditional transport of measure for the matrix models, we construct an isomorphism $\mathrm{W}^*(X_1,\dots ,X_m) \to \mathrm{W}^*(S_1,\dots ,S_m)$ that maps $\mathrm{W}^*(X_1,\dots ,X_k)$ to $\mathrm{W}^*(S_1,\dots ,S_k)$ for each $k = 1, \dots , m$ and that also witnesses the Talagrand inequality for the law of $(X_1,\dots ,X_m)$ relative to the law of $(S_1,\dots ,S_m)$.

The Platonic solids and fundamental tests of quantum mechanics

The Platonic solids is the name traditionally given to the five regular convex polyhedra, namely the tetrahedron, the octahedron, the cube, the icosahedron and the dodecahedron. Perhaps strongly boosted by the towering historical influence of their namesake, these beautiful solids have, in well over two millennia, transcended traditional boundaries and entered the stage in a range of disciplines. Examples include natural philosophy and mathematics from classical antiquity, scientific modeling during the days of the European scientific revolution and visual arts ranging from the renaissance to modernity. Motivated by mathematical beauty and a rich history, we consider the Platonic solids in the context of modern quantum mechanics. Specifically, we construct Bell inequalities whose maximal violations are achieved with measurements pointing to the vertices of the Platonic solids. These Platonic Bell inequalities are constructed only by inspecting the visible symmetries of the Platonic solids. We also construct Bell inequalities for more general polyhedra and find a Bell inequality that is more robust to noise than the celebrated Clauser-Horne-Shimony-Holt Bell inequality. Finally, we elaborate on the tension between mathematical beauty, which was our initial motivation, and experimental friendliness, which is necessary in all empirical sciences.

Domains of elliptic operators on sets in Wiener space

Let [Formula: see text] be a separable Banach space endowed with a non-degenerate centered Gaussian measure [Formula: see text]. The associated Cameron–Martin space is denoted by [Formula: see text]. Consider two sufficiently regular convex functions [Formula: see text] and [Formula: see text]. We let [Formula: see text] and [Formula: see text]. In this paper, we study the domain of the self-adjoint operator associated with the quadratic form [Formula: see text] and we give sharp embedding results for it. In particular, we obtain a characterization of the domain of the Ornstein–Uhlenbeck operator in Hilbert space with [Formula: see text] and on half-spaces, namely if [Formula: see text] and [Formula: see text] is an affine function, then the domain of the operator defined via (0.1) is the space [Formula: see text] where [Formula: see text] is the Feyel–de La Pradelle Hausdorff–Gauss surface measure.

The Electronic Structure of Graphene Nanoislands: A CAS-SCF and NEVPT2 Study

This paper presents a tight binding and ab initio study of finite graphene nanostructures. The attention is focused on three types of regular convex polygons: triangles, rhombuses, and hexagons, which are the most simple high-symmetry convex structures that can be ideally cut out of a graphene layer. Three different behaviors are evidenced for these three classes of compounds: closed-shells for hexagons; low-spin open-shells for rhombuses; high-spin open-shells for triangles.