Harmonic Spheres Research Articles

Torus-like solutions for the Landau-de Gennes model. Part II: Topology of [formula omitted]-equivariant minimizers

We study energy minimization of a continuum Landau-de Gennes energy functional for nematic liquid crystals, in a three-dimensional axisymmetric domain and in a restricted class of S1-equivariant (i.e., axially symmetric) configurations. We assume smooth and nonvanishing S1-equivariant (e.g. homeotropic) Dirichlet boundary condition and a physically relevant norm constraint (the Lyuksyutov constraint) in the interior. Relying on our previous results in the nonsymmetric setting [16], we prove partial regularity of minimizers away from a possible finite set of interior singularities located on the symmetry axis. For a suitable class of domains and boundary data, we show that for smooth minimizers (torus solutions) the level sets of the signed biaxiality are generically finite unions of tori of revolution. Concerning nonsmooth minimizers (split solutions), we characterize their asymptotic behavior around any singular point in terms of explicit S1-equivariant harmonic maps into S4, whence the generic level sets of the signed biaxiality contains invariant topological spheres. Finally, in the model case of a nematic droplet, we provide existence of torus solutions, at least when the boundary data are suitable uniaxial deformations of the radial anchoring, and existence of split solutions for boundary data which are suitable linearly full harmonic spheres.

Fragility in glassy liquids: A structural approach based on machine learning.

The rapid rise of viscosity or relaxation time upon supercooling is a universal hallmark of glassy liquids. The temperature dependence of viscosity, however, is quite nonuniversal for glassy liquids and is characterized by the system's "fragility," with liquids with nearly Arrhenius temperature-dependent relaxation times referred to as strong liquids and those with super-Arrhenius behavior referred to as fragile liquids. What makes some liquids strong and others fragile is still not well understood. Here, we explore this question in a family of harmonic spheres that range from extremely strong to extremely fragile, using "softness," a structural order parameter identified by machine learning to be highly correlated with dynamical rearrangements. We use a support vector machine to identify softness as the same linear combination of structural quantities across the entire family of liquids studied. We then use softness to identify the factors controlling fragility.

Vibrational density of states of jammed packing at high dimensions: Mean-field theory.

Several mean-field theories predict that the Hessian matrix of amorphous solids converges the Wishart matrix in the limit of the large spatial dimensions d→∞. Motivated by these results, we calculate here the density of states of random packing of harmonic spheres by mapping the Hessian of the original system to the Wishart matrix. We compare our result with that of previous numerical simulations of harmonic spheres in several spatial dimensions d=3, 5, and 9. For small pressure p≪1 (near jamming), we find a good agreement even in d=3, and obtain better agreements in larger d, suggesting that the approximation becomes exact in the limit d→∞.

Nonexistence of the Quasi-harmonic Spheres and Harmonic Spheres into Certain Manifold

Nonexistence of the Quasi-harmonic Spheres and Harmonic Spheres into Certain Manifold

Morse Index Bound for Minimal Two Spheres

Given a closed manifold of dimension at least three, with non-trivial homotopy group pi _3(M) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bounded by one, with sum of their energies realizing a geometric invariant width.

Особливості трактовки сонатної форми у фортепіанній творчості Л. Бетховена: до проблеми побудови виконавської концепції (на прикладі сонат op. 101, 106, 109, 111)

The purpose of the article is to consider the peculiarities of the interpretation of the sonata form in the piano work of L. Beethoven in the light of the problem of constructing a performing mixture. The methodology of the work is based on a comprehensive study of historiographic, genre-compositional, artistic-semantic and potential-performing approaches. The scientific novelty of this article is due to the identification in the piano work of L. Beethoven and consideration of the features of the interpretation of the sonata form in these periods, the identification of figurative and dramatic prerequisites for stylistic and stylistic innovations in the last four sonatas (op. 101, 106, 109, 111). Conclusions. In the piano cycles of the late period of L. Beethoven’s work, significant changes are observed in the interpretation of the sonata form, the ratio of intra-conflict and harmonic spheres is modified, which has a significant impact on the general understanding of musical dramaturgy. In the cycles of past periods, the conflict state of thought is wider, which coincides – since the latter does not always really become a reality and exists only in human thinking as an ideal (hence the disproportion in its development). In sonata cycles op. 101, 106, 109, 111 and conflict, and the visible ideal in address rightness, their inverse relationship: the conflict is interpreted in its “narrow” meaning – as an intra-subjective state, and harmony – as an objective one – as a union of man and being. In the sonatas of the early and middle periods of L. Beethoven’s work, he limits the intra-conflict and harmonic sphere to no higher than the general level of quality; in the last sonatas, the harmonies, affirmed as the conclusion of the solution of the conflict, the character of the maximum level of generality, since they combine the ideal and effectiveness (the ideal is open in an objective meeting, and it becomes a reality for the individual), emotional and perceived, objective and perceived in content and is assumed in terms of complete merging with him personality).

Morse index bound for minimal two spheres

Given a closed manifold of dimension at least three, with non trivial homotopy group \pi_3(M) and a generic metric, we prove that there is a finite collection of harmonic spheres with Morse index bound one, with sum of their energies realizes a geometric invariant width.

The principles of re-intonation of multinational folklore in the work of Ludwig van Beethoven (on the example of the collection of arrangements “Songs of Different Nations”)

The principles of re-intonation of multinational folklore in the work of Ludwig van Beethoven (on the example of the collection of arrangements “Songs of Different Nations”)

Adiabatic Limit in Yang–Mills Equations in R⁴

Our goal is to present an approach to the proof of the harmonic spheres conjecture based on the adiabatic limit construction. This construction allows to associate with an arbitrary Yang–Mills G-field on the Euclidean 4-dimensional space a harmonic map of the Riemann sphere to the loop space of the group G

Temperature dependence of the transition packing fraction of thermal jamming in a harmonic soft sphere system

The glassy dynamics of soft harmonic spheres are often mapped onto the dynamics of hard spheres by considering an effective diameter for the soft particles and therefore an effective packing fraction. While in this approach the thermal fluctuations within valleys of the energy landscape are covered, the crossing of energy barriers from one valley into another usually is neglected. Here we argue—motivated by studies of the glass transition based on explorations of the energy landscape—that the crossing of energy barriers can be attributed by an effective decrease of the glass transition packing fraction with increasing temperature T according to T0.2. Furthermore, we reanalyzing data of soft sphere simulations. Since fitting scaling laws to simulation data always allows for some arbitrariness, we cannot prove based on the simulation data that our idea of a shift of the glass transition packing fraction due to barrier crossings is the only possible way to explain the discrepancies that have been observed previously. However, we show that a possible explanation of the simulation data is given by our approach to characterize the dynamics of soft spheres by both, the previously-considered temperature-dependent effective packing fraction due to the increase of the mean overlap between neighboring particles with stronger thermal fluctuations and the newly introduced increase of the glass transition packing with an increasing probability of barrier crossings.

Liu-Nagel phase diagrams in infinite dimension

We study Harmonic Soft Spheres as a model of thermal structural glasses in the limit of infinite dimension. We show that cooling, compressing and shearing a glass lead to a Gardner transition and, hence, to a marginally stable amorphous solid as found for Hard Spheres systems. A general outcome of our results is that a reduced stability of the glass favors the appearance of the Gardner transition. Therefore using strong perturbations, e.g. shear and compression, on standard glasses or using weak perturbations on weakly stable glasses, e.g. the ones prepared close to the jamming point, are the generic ways to induce a Gardner transition. The formalism that we discuss allows to study general perturbations, including strain deformations that are important to study soft glassy rheology at the mean field level.

Harmonic spheres in outer symmetric spaces, their canonical elements and Weierstrass-type representations

Making use of Murakami’s classification of outer involutions in a Lie algebra and following the Morse-theoretic approach to harmonic two-spheres in Lie groups introduced by Burstall and Guest, we obtain a new classification of harmonic two-spheres in outer symmetric spaces and a Weierstrass-type representation for such maps. Several examples of harmonic maps into classical outer symmetric spaces are given in terms of meromorphic functions on \(S^2\).

Convergence of Yang–Mills–Higgs fields

In this paper, we study the convergence of Yang-Mills-Higgs fields defined on fiber bundles over Riemann surfaces where the fiber is a compact symplectic manifold and the conformal structure of the Riemann surface is allowed to vary. We show that away from the nodes, the YMH fields converges, up to gauge, to a smooth YMH field modulo finitely many harmonic spheres, while near the nodes where the conformal structure degenerates, the YMH fields converges to a pair consisting of a flat connection and a twisted geodesic (with potential) after finitely many times of blowing-up's. In particular, we prove a generalized energy identity and give a refined analysis of the neck.

Twistor Interpretation of Harmonic Spheres and Yang–Mills Fields

We consider the twistor descriptions of harmonic maps of the Riemann sphere into Kähler manifolds and Yang–Mills fields on four-dimensional Euclidean space. The motivation to study twistor interpretations of these objects comes from the harmonic spheres conjecture stating the existence of the bijective correspondence between based harmonic spheres in the loop space \(\Omega G\) of a compact Lie group \(G\) and the moduli space of Yang–Mills \(G\)-fields on \(\mathbb R^4\).

Harmonic maps of finite uniton number and their canonical elements

We classify all harmonic maps with finite uniton number from a Riemann surface into an arbitrary compact simple Lie group $$G$$ , whether $$G$$ has trivial centre or not, in terms of certain pieces of the Bruhat decomposition of the group $$\Omega _\mathrm {alg}{G}$$ of algebraic loops in $$G$$ and corresponding canonical elements. This will allow us to give estimations for the minimal uniton number of the corresponding harmonic maps with respect to different representations and to make more explicit the relation between previous work by different authors on harmonic two-spheres in classical compact Lie groups and their inner symmetric spaces and the Morse theoretic approach to the study of such harmonic two-spheres introduced by Burstall and Guest. As an application, we will also give some explicit descriptions of harmonic spheres in low-dimensional spin groups making use of spinor representations.

Quantitative stratification and the regularity of harmonic map flow

In this paper, we prove estimates and quantitative regularity results for the harmonic map flow. First, we consider \(H^1_\mathrm{loc }\)-maps \(u\) defined on a parabolic ball \(P\subset M^m\times \mathbb {R}\) and with target manifold \(N\), that have bounded Dirichlet-energy and Struwe-energy. We define a quantitative stratification, which groups together points in the domain into quantitative weakly singular strata \(\mathcal {S}^j_{\eta ,r}(u)\) according to the number of approximate symmetries of \(u\) at certain scales. We prove that their tubular neighborhoods have small volume, namely \(\mathrm{Vol}\left( T_r(\mathcal {S}^j_{\eta ,r}(u))\right) \le Cr^{m+2-j-\varepsilon }\), where \(C\) depends on \(\eta , \epsilon \) and some additional parameters; for the precise statement see Theorem 1.5. In particular, this generalizes the known Hausdorff estimate \(\dim \mathcal {S}^j(u)\le j\) for the weakly singular strata of suitable weak solutions of the harmonic map flow. As an application, specializing to Chen-Struwe solutions with target manifolds that do not admit certain harmonic and quasi-harmonic spheres, we obtain refined Minkowski estimates for the singular set, which generalize a result of Lin-Wang (Anal Geom 7(2):397–429, 1999). We also obtain \(L^p\)-estimates for the reciprocal of the regularity scale. Our results for harmonic map flow are analogous to results for mean curvature flow we proved in Cheeger et al. (Geom Funct Anal 23(3):828–847, 2013).

Crossovers in the dynamics of supercooled liquids probed by an amorphous wall.

We study the relaxation dynamics of a binary Lennard-Jones liquid in the presence of an amorphous wall generated from equilibrium particle configurations. In qualitative agreement with the results presented by Kob et al. [Nat. Phys. 8, 164 (2012).] for a liquid of harmonic spheres, we find that our binary mixture shows a saturation of the dynamical length scale close to the mode-coupling temperature T(c). Furthermore we show that, due to the broken symmetry imposed by the wall, signatures of an additional change in dynamics become apparent at a temperature well above T(c). We provide evidence that this modification in the relaxation dynamics occurs at a recently proposed dynamical crossover temperature T(s) > T(c), which is related to the breakdown of the Stokes-Einstein relation. We find that this dynamical crossover at T(s) is also observed for the harmonic spheres as well as a WCA liquid, showing that it may be a general feature of glass-forming systems.

Extended solutions in a noncommutative sigma model

The notion of an extended solution plays a key role in Uhlenbeck’s approach to the description of harmonic maps to the unitary group. In the present paper, we extend this notion to the case of a noncommutative sigma model (a quantum analog of harmonic spheres in the unitary group) and use it to establish that the minimum uniton number of any diagonal solution is equal to the canonical rank of this solution.

Finite-size effects in the dynamics of glass-forming liquids

We present a comprehensive theoretical study of finite-size effects in the relaxation dynamics of glass-forming liquids. Our analysis is motivated by recent theoretical progress regarding the understanding of relevant correlation length scales in liquids approaching the glass transition. We obtain predictions both from general theoretical arguments and from a variety of specific perspectives: mode-coupling theory, kinetically constrained and defect models, and random first-order transition theory. In the last approach, we predict in particular a nonmonotonic evolution of finite-size effects across the mode-coupling crossover due to the competition between mode-coupling and activated relaxation. We study the role of competing relaxation mechanisms in giving rise to nonmonotonic finite-size effects by devising a kinetically constrained model where the proximity to the mode-coupling singularity can be continuously tuned by changing the lattice topology. We use our theoretical findings to interpret the results of extensive molecular dynamics studies of four model liquids with distinct structures and kinetic fragilities. While the less fragile model only displays modest finite-size effects, we find a more significant size dependence evolving with temperature for more fragile models, such as Lennard-Jones particles and soft spheres. Finally, for a binary mixture of harmonic spheres we observe the predicted nonmonotonic temperature evolution of finite-size effects near the fitted mode-coupling singularity, suggesting that the crossover from mode-coupling to activated dynamics is more pronounced for this model. Finally, we discuss the close connection between our results and the recent report of a nonmonotonic temperature evolution of a dynamic length scale near the mode-coupling crossover in harmonic spheres.

Microscopic theory of the jamming transition of harmonic spheres

We develop a microscopic theory to analyze the phase behavior and compute correlation functions of dense assemblies of soft repulsive particles both at finite temperature, as in colloidal materials, and at vanishing temperature, a situation relevant for granular materials and emulsions. We use a mean-field statistical mechanical approach which combines elements of liquid state theory to replica calculations to obtain quantitative predictions for the location of phase boundaries, macroscopic thermodynamic properties, and microstructure of the system. We focus, in particular, on the derivation of scaling properties emerging in the vicinity of the jamming transition occurring at large density and zero temperature. The new predictions we obtain for pair correlation functions near contact are tested using computer simulations. Our work also clarifies the conceptual nature of the jamming transition and its relation to the phenomenon of the glass transition observed in atomic liquids.