arXiv subjects

Andrea Pedrini

Publications and source records attributed to Andrea Pedrini.

13 recordsLinked to original sources

arXiv2025

Surface Nematic Quasi-Uniformity

Line fields on surfaces are a means to describe the nematic order that may pattern them. The least distorted nematic fields are called uniform, but they can only exist on surfaces with negative constant Gaussian curvature. To identify the least distorted nematic fields on a generic surface, we relax the notion of uniformity into that of quasi-uniformity and prove that all such fields are parallel transported (in Levi-Civita's sense) by the geodesics of the surface. Both global and local constructions of quasi-uniform fields are presented to illustrate both richness and significance of the proposed notion.

cond-mat.soft↗

arXiv2025

Surface Nematic Uniformity

An ant-like observer confined to a two-dimensional surface traversed by stripes would wonder whether this striped landscape could be devised in such a way as to appear to be the same wherever they go. Differently stated, this is the problem studied in this paper. In a more technical jargon, we determine all possible uniform nematic fields on a smooth surface. It was already known that for such a field to exist, the surface must have constant negative Gaussian curvature. Here, we show that all uniform nematic fields on such a surface are parallel transported (in Levi-Civita's sense) by special systems of geodesics, which (with scant inventiveness) are termed uniform. We prove that, for every geodesic on the surface, there are two systems of uniform geodesics that include it; they are conventionally called right and left, to allude at a possible intrinsic definition of handedness. We found explicitly all uniform fields for Beltrami's pseudosphere. Since both geodesics and uniformity are preserved under isometries, by a classical theorem of Minding, the solution for the pseudosphere carries over all other admissible surfaces, thus providing a general solution to the problem (at least in principle).

cond-mat.soft↗

arXiv2023

Relieving nematic geometric frustration in the plane

Frustration in nematic-ordered media (endowed with a director field) is treated in a purely geometric fashion in a flat, two-dimensional space. We recall the definition of quasi-uniform distortions and envision these as viable ways to relieve director fields prescribed on either a straight line or the unit circle. We prove that using a planar spiral is the only way to fill the whole plane with a quasi-uniform distortion. Apart from that, all relieving quasi-uniform distortions can at most be defined in a half-plane; however, in a generic sense, they are all asymptotically spirals.

math-ph↗

arXiv2021

Ridge approximation for thin nematic polymer networks

Nematic polymer networks (NPNs) are nematic elastomers within which the nematic director is enslaved to the elastic deformation. The elastic free energy of a NPN sheet of thickness $h$ has both stretching and bending components (the former scaling like h, the latter scaling like $h^3$). NPN sheets bear a director field $\mathbf{m}$ imprinted in them (usually, uniformly throughout their thickness); they can be activated by changing the nematic order (e.g. by illumination or heating). This paper illustrates an attempt to compute the bending energy of a NPN sheet and to show which role it can play in determining the activated shape. Our approach is approximate: the activated surface consists of flat sectors connected by ridges, where the unit normal jumps and the bending energy is concentrated. By increasing the number of ridges, we should get closer to the real situation, where the activated surface is smooth and the bending energy is distributed on it. The method is applied to a disk with imprinted a spiraling hedgehog. It is shown that upon activation the disk, like a tiny hand, is able to grab a rigid lamina.

cond-mat.soft↗

arXiv2020

Ridge energy for thin nematic polymer networks

Minimizing the elastic free energy of a thin sheet of nematic polymer network among smooth isometric immersions is the strategy purported by the mainstream theory. In this paper, we broaden the class of admissible spontaneous deformations: we consider ridged isometric immersions, which can cause a sharp ridge in the immersed surfaces. We propose a model to compute the extra energy distributed along such ridges. This energy comes from bending; it is shown under what circumstances it scales quadratically with the sheet's thickness, falling just in between stretching and bending energies. We put our theory to the test by studying the spontaneous deformation of a disk on which a radial hedgehog was imprinted at the time of crosslinking. We predict the number of folds that develop in terms of the degree of order induced in the material by external agents (such as heat and illumination).

cond-mat.soft↗

arXiv2019

Liquid Crystal Distortions Revealed by an Octupolar Tensor

The classical theory of liquid crystal elasticity as formulated by Oseen and Frank describes the (orientable) optic axis of these soft materials by a director $\mathbf{n}$. The ground state is attained when $\mathbf{n}$ is uniform in space; all other states, which have a non-vanishing gradient $\nabla\mathbf{n}$, are distorted. This paper proposes an algebraic (and geometric) way to describe the local distortion of a liquid crystal by constructing from $\mathbf{n}$ and $\nabla\mathbf{n}$ a third-rank, symmetric and traceless tensor $\mathbf{A}$ (the octupolar tensor). The (nonlinear) eigenvectors of $\mathbf{A}$ associated with the local maxima of its cubic form $Φ$ on the unit sphere (its octupolar potential) designate the directions of distortion concentration. The octupolar potential is illustrated geometrically and its symmetries are charted in the space of distortion characteristics, so as to educate the eye to capture the dominating elastic modes. Special distortions are studied, which have everywhere either the same octupolar potential or one with the same shape, but differently inflated.

cond-mat.soft↗

arXiv2018

Lily-like twist distribution in toroidal nematics

Toroidal nematics are droplets of nematic liquid crystals in the form of a circular torus. When the nematic director is subject to planar degenerate boundary conditions, the bend-only director field with vector lines along the parallels of all internal torodial shells is an equilibrium solution for all values of the elastic constants. Local stability analyses have shown that an instability is expected to occur for sufficiently small values of the twist elastic constant. It is natural to conjecture that in this regime the global equilibrium would be characterized by a maximum twist deflection on the boundary of the torus, with a twist distribution over the torus' cross-section represented by a fennel-like surface. We prove that surprisingly the stable twist distribution is instead represented by a lily-like surface. Thus the overall maximum twist deflection falls well within the torus. To cope with the complexity of the elastic free-energy functional in the fully non-linear setting, we developed an ad hoc deep-learning optimization method, which here is also duly validated and documented for it promises to be applicable to other similar problems, equally intractable analytically.

cond-mat.soft↗

arXiv2018

Instability of toroidal nematics

Toroidal nematics are nematic liquid crystals confined within a circular torus and subject to planar degenerate anchoring on the boundary of the torus. They may be droplets floating in an isotropic environment or cavities carved out of a solid substrate. A universal solution of Frank's elastic free energy is an equilibrium configuration for the nematic director field, irrespective of the values of the elastic constants, whose vector lines are the coaxial parallels of the torus. We explore the local stability of this configuration and identify a range of parameters where the main drive towards instability does not come from the surface-like elastic constant $K_{24}$ being large, but from the the ratio $K_2/K_3$ of the twist to bend elastic constants being small, which also makes our study relevant to chromonic liquid crystals.

cond-mat.soft↗

arXiv2017

Tarski's Theorem on Intuitionistic logic, for polyhedra

In 1938, Tarski proved that a formula is not intuitionistically valid if, and only if, it has a counter-model in the Heyting algebra of open sets of some topological space. In fact, Tarski showed that any Euclidean space R^n with n >= 1 suffices, as does e.g. the Cantor space. In particular, intuitionistic logic cannot detect topological dimension in the frame of all open sets of a Euclidean space. By contrast, we consider the lattice of open subpolyhedra of a given compact polyhedron P \subseteq R^n, prove that it is a locally finite Heyting subalgebra of the (non-locally-finite) algebra of all open sets of R^n, and show that intuitionistic logic is able to capture the topological dimension of P through the bounded-depth axiom schemata. Further, we show that intuitionistic logic is precisely the logic of formulæ valid in all Heyting algebras arising from polyhedra in this manner. Thus, our main theorem reconciles through polyhedral geometry two classical results: topological completeness in the style of Tarski, and Jaskowski's theorem that intuitionistic logic enjoys the finite model property. Several questions of interest remain open. E.g., what is the intermediate logic of all closed triangulable manifolds?

math.LO↗

arXiv2015

From Freudenthal's Spectral Theorem to projectable hulls of unital Archimedean lattice-groups, through compactifications of minimal spectra

We use a landmark result in the theory of Riesz spaces - Freudenthal's 1936 Spectral Theorem - to canonically represent any Archimedean lattice-ordered group $G$ with a strong unit as a (non-separating) lattice-group of real valued continuous functions on an appropriate $G$-indexed zero-dimensional compactification $w_GZ_G$ of its space $Z_G$ of \emph{minimal} prime ideals. The two further ingredients needed to establish this representation are the Yosida representation of $G$ on its space $X_G$ of \emph{maximal} ideals, and the well-known continuous surjection of $Z_G$ onto $X_G$. We then establish our main result by showing that the inclusion-minimal extension of this representation of $G$ that separates the points of $Z_G$ - namely, the sublattice subgroup of ${\rm C}\,(Z_G)$ generated by the image of $G$ along with all characteristic functions of clopen (closed and open) subsets of $Z_G$ which are determined by elements of $G$ - is precisely the classical projectable hull of $G$. Our main result thus reveals a fundamental relationship between projectable hulls and minimal spectra, and provides the most direct and explicit construction of projectable hulls to date. Our techniques do require the presence of a strong unit.

math.FA↗

arXiv2015

Some Further Evidence about Magnification and Shape in Neural Gas

Neural gas (NG) is a robust vector quantization algorithm with a well-known mathematical model. According to this, the neural gas samples the underlying data distribution following a power law with a magnification exponent that depends on data dimensionality only. The effects of shape in the input data distribution, however, are not entirely covered by the NG model above, due to the technical difficulties involved. The experimental work described here shows that shape is indeed relevant in determining the overall NG behavior; in particular, some experiments reveal richer and complex behaviors induced by shape that cannot be explained by the power law alone. Although a more comprehensive analytical model remains to be defined, the evidence collected in these experiments suggests that the NG algorithm has an interesting potential for detecting complex shapes in noisy datasets.

cs.NE↗

arXiv2014

The Euler valuation on MV-algebras

Every finitely presented MV-algebra A has a unique idempotent valuation E assigning value 1 to every basic element of A. For each element a of A, E(a) turns out to coincide with the Euler characteristic of the open set of maximal ideals m of A such that a/m is nonzero.

math.RA↗

arXiv2012

The Euler characteristic of a polyhedron as a valuation on its coordinate vector lattice

A celebrated theorem of Hadwiger states that the Euler-Poincaré characteristic is the the unique invariant and continuous valuation on the distributive lattice of compact polyhedra in R^n that assigns value one to each convex non-empty such polyhedron. This paper provides an analogue of Hadwiger's result for finitely presented unital vector lattices (i.e. real vector spaces with a compatible lattice order, also known as Riesz spaces). The vector lattice of continuous and piecewise (affine) linear real-valued functions on a compact polyhedron, with operations defined pointwise from the vector lattice R, is a finitely presented unital vector lattice; and it is a non-trivial fact that all such vector lattices arise in this manner, to within an isomorphism. Each function in such a vector lattice can be written as a linear combination of a subset of distinguished elements that we call vl-Schauder hats. We prove here that the functional that assigns to each non-negative piecewise linear function on the polyhedron the Euler-Poincaré characteristic of its support is the unique vl-valuation (a special class of valuations on vector lattices) that assigns one to each vl-Schauder hat of the vector lattice.

math.MG↗