Lipid necks, large curvature bridges, are shown to be metastable.
problem Understanding the energetically prohibitive yet ubiquitous lipid necks in cell membranes.
method Geometric triality approach to demonstrate metastability.
result Lipid necks can exist for finite but potentially long times without stabilizing mechanisms.
Triality connects three polynomial bases in Lie algebra studies.
problem Understanding relationships between invariant polynomials.
method Triality of mSO(8,C) and Hitchin fibrations. result Explicit automorphisms linking polynomial bases identified.
Following an idea of Dadok, Harvey and Lawson, we apply the triality property of SO(8) to study the comass of certain self-dual 4-forms on R^8. In particular, we prove that the Cayley 4-form has comass 1 and that any self-dual 4-form realizing the maximal Wirtinger ratio is SO(8)-conjugate to the Cayley 4-form. We also…
The purpose of this article is to give a geometric interpretation to the so-called "twelve surfaces of Darboux", or "Darboux wreath", which appear by applying repeatedly certain simple transformations to a given infinitesimal isometric deformation of a surface in euclidean three space. This interpretation is a differen…
Formula derived for Lie algebra E8's bracket.
problem Calculating the bracket of the exceptional Lie algebra E8.
method Based on triality and oct-octonions, following Barton-Sudbery description.
result Explicit formula for E8's bracket.
Using octonions and the triality property of Spin(8), we find explicit formulae for the Lie brackets of the exceptional simple real Lie algebras f4 and f4∗, i.e. the Lie algebras of the isometry groups of the Cayley projective plane and the Cayley hyperbolic plane. As an application, we cla…
Researchers found a maximal antipodal set of three elements in a 7x7 sphere space.
problem Determining the maximal antipodal set in the outer 3-symmetric space S7imesS7. method Investigated the polar and maximal antipodal set P for the given 3-symmetric space S7imesS7. result The maximal antipodal set P has three elements. We give a computer free proof of the Deligne, Cohen and deMan formulas for the dimensions of the irreducible g-modules appearing in the tensor powers of g, where g ranges over the exceptional complex simple Lie algebras. We give additional dimension formulas for the exceptional series, as well as uniform dimensio…
We present a binary code for spinors and Clifford multiplication using non-negative integers and their binary expressions, which can be easily implemented in computer programs for explicit calculations. As applications, we present explicit descriptions of the triality automorphism of Spin(8), explicit representations…
New tensor framework connects Fisher information, hypergraphs, and multi-observable correlations.
problem Missing structure in pairwise Fisher graphs for multi-observable radiation patterns.
method Higher-order Fisher tensors and natural exponential-family coordinates.
result Exact triality of Fisher tensors, cumulants, and hypergraphs.
This article is an exposition of a body of existing results, together with an announcement of recent results. We discuss a theory of polytopes associated to bipartite graphs and trinities, developed by Kálmán, Postnikov and others. This theory exhibits a variety of interesting duality and triality relations, and extend…
The article consists of the Russian and English variants of Ph.D. Thesis in which the answers is given on the following questions: 1. how to construct the spinor formalism for n=6; 2. how to construct the spinor formalism for n=8; 3. how to prolong the Riemannian connection from the tangent bundle into the spinor one w…
New method produces positive colored superpolynomials from four-point functions.
problem Constructing positive superpolynomials for colored link invariants.
method Four-point functions from topological vertices.
result New relation between super- and hyperpolynomials.
We consider a triality between the Zermelo navigation problem, the geodesic flow on a Finslerian geometry of Randers type, and spacetimes in one dimension higher admitting a timelike conformal Killing vector field. From the latter viewpoint, the data of the Zermelo problem are encoded in a (conformally) Painleve-Gullst…
We propose that a simple, Lagrangian 2d N=(0,2) duality interface between the 3d N=2 XYZ model and 3d N=2 SQED can be associated to the simplest triangulated 4-manifold: the 4-simplex. We then begin to flesh out a dictionary between more general triangulated 4-manifolds with boundar…
We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of this structure in terms of invariant spinor-valued 1-forms, which are harmonic …
Geometric GNNs improve graph discrimination through GWL.
problem Discriminating geometric graphs embedded in Euclidean space.
method Proposed a geometric version of the Weisfeiler-Leman test (GWL) for geometric graphs.
result Characterized the expressive power of geometric GNNs based on physical symmetries.
Geometric Algebra Transformer (GATr) handles various geometric data types efficiently.
problem Lack of a single architecture for diverse geometric data types.
method GATr uses projective geometric algebra, equivariant to E(3), and is a Transformer architecture.
result GATr outperforms non-geometric and equivariant baselines in various geometric tasks.
The differential geometric aspects of Geometric Phases are reviewed.
A geometric triangulation of a Riemannian manifold is a triangulation where the interior of each simplex is totally geodesic. Bistellar moves are local changes to the triangulation which are higher dimensional versions of the flip operation of triangulations in a plane. We show that geometric triangulations of a compac…
Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
tf_geometric simplifies graph deep learning in TensorFlow.
problem Efficient graph deep learning in TensorFlow.
method Kernel libraries and infrastructures for GNNs.
result tf_geometric supports various graph tasks and provides efficient GNN models.
Geometric Bass martingales linked to Brownian motion and geometric Brownian motion.
problem Modeling continuous martingales with prescribed initial and terminal distributions.
method Developed geometric Bass martingales and established their properties.
result Explicit bijection and representation of geometric Bass martingales.
Study geometric bounds on generalized Ricci flow.
problem No specific problem stated; focuses on bounds.
method Analogous geometric quantities and bounds proven.
result Geometric and analytic bounds established.
Researchers geometrically define asymptotic coordinates in General Relativity.
problem Understanding the asymptotic behavior of relativistic initial data sets.
method Geometrization of asymptotic flatness and analysis of geometric invariants.
result Geometrically defined asymptotic coordinates for mass, energy, momentum, and angular momentum.
GDB bridges geometric states with improved accuracy and generality.
problem Challenges in predicting geometric state evolution in complex systems.
method Geometric Diffusion Bridge (GDB) framework using equivariant diffusion bridges.
result GDB surpasses existing methods in accurately bridging geometric states.
Estimates small eigenvalues for geometrically finite manifolds.
problem Estimating small eigenvalues of Schrödinger operators.
method Geometrically finite manifolds, Riemannian vector bundles.
result Estimates the number of small eigenvalues.
We show that for a strongly convergent sequence of geometrically finite Kleinian groups with geometrically finite limit, the Cannon-Thurston maps of limit sets converge uniformly. If however the algebraic and geometric limits differ, as in the well known examples due to Kerckhoff and Thurston, then provided the geometr…
In this paper we study a collection of jet geometrical concepts, we refer to d-tensors, relativistic time dependent semisprays, harmonic curves and nonlinear connections on the 1-jet space J1(R;M), necessary to the construction of a Miron's-like geometrization for Lagrangians depending on a relativistic time. The geome…
Survey on conservation laws for geometric PDEs.
problem Modeling polyharmonic maps.
method Conservation law approach.
result Overview of conservation laws in geometric PDEs.
Sharp geometric inequalities for free boundary hypersurfaces in balls.
problem Understanding geometric properties of free boundary hypersurfaces in balls.
method Proving a family of sharp geometric inequalities.
result Family of sharp geometric inequalities for free boundary hypersurfaces in balls.
Equivariant networks improve geometric prediction without scalar approximations.
problem Efficiently predicting geometric tensors in real-world scenarios.
method Equivariant networks for geometric prediction.
result Equivariant networks can generalize to unseen systems for geometric prediction.
Combines topological and geometric approaches to data analysis.
problem Understanding when and how geometric objects intersect.
method Connects topological and geometric concepts of curvature.
result Reconceptualizes curvature and links it to hyperconvexity.
Geometric Occam's Razor shapes deep learning solutions.
problem Understanding the regularization in over-parameterized neural networks.
method Analyzing the geometric model complexity and Dirichlet energy in neural networks.
result Over-parameterized neural networks are implicitly regularized by geometric model complexity.
Geometric structures modeled on rational homogeneous manifolds are studied to characterize rational homogeneous manifolds and to prove their deformation rigidity. To generalize these characterizations and deformation rigidity results to quasihomogeneous varieties, we first study horospherical varieties and geometric st…
It is well-known that the LIE(Locally Induction Equation) admit soliton-type solutions and same soliton solutions arise from different and apparently irrelevant physical models. By comparing the solitons of LIE and Killing magnetic geodesics, we observe that these solitons are essentially decided by two families of iso…
Extends geometric structures to manifolds with new operators.
problem No specific problem stated; extending geometric structures.
method Defines new gradient and Laplace operators on manifolds with geometric structures.
result Provides properties of the new operators.
Enhanced spectral clustering for geometric graphs improves clustering accuracy.
problem Ineffective standard spectral clustering for geometric graphs.
method Higher-order spectral clustering using higher-order eigenvectors.
result Established weak and strong consistency for Soft Geometric Block Model.
Study on geometrically formal metrics on complex manifolds.
problem Existence and properties of geometrically formal metrics on complex manifolds.
method Topological and cohomological obstructions, detailed analysis for specific manifolds, and metric constructions.
result Existence and non-existence conditions for geometrically formal metrics on various complex manifolds.
Geometric techniques reveal new insights into Gromov-Witten invariants.
problem Formulating Gromov-Witten invariants for complete intersections in projective space.
method Combining geometric group theory and geometric topology, focusing on geodesic laminations.
result Primitive cohomologies unify mathematical formulations of Gromov-Witten invariants.
The paper proves geometric inequalities in sphere using locally constrained flows.
problem Deriving geometric inequalities in sphere.
method Established the longtime existence and convergence of a locally constrained flow.
result Proved new families of three-term geometric inequalities in sphere.
GeoAdaLer enhances geometric understanding of Adam for stochastic optimization.
problem Understanding geometric principles behind Adam's success in stochastic optimization.
method Introduces GeoAdaLer, an adaptive learning method based on geometric properties.
result Extends interpretability and effectiveness in complex optimization scenarios.
CP2 uses geometric information to improve conformal prediction robustness.
problem CP fails under geometric data shifts, losing coverage guarantees.
method Integrates geometric pose information into CP via canonicalization.
result Integrating geometric information with CP ensures robustness under geometric shifts.
Maps between non-compact surfaces can have geometric kernels under certain conditions.
problem Understanding when maps between non-compact surfaces have geometric kernels.
method Using Brown's proper fundamental group to establish sufficient conditions for geometric kernels.
result Characterization of conjugacy classes in the proper fundamental group and sufficient conditions for geometric kernels.
Paper connects geometric structures to algebra in high dimensions.
problem Understanding geometric structures in high dimensions.
method Relating minimal left ideals on Clifford algebras to geometric structures.
result Established a connection between algebraic and geometric properties.
New proof of Willmore inequality using geometric divergence inequality.
problem Proving the Willmore inequality for bounded domains.
method Using a parametric geometric inequality derived from a divergence form geometric differential inequality.
result New proofs of quantitative Willmore-type and weighted Minkowski inequalities.
Unified approach to geometric structure equivalence problem.
problem Equivalence problem of geometric structures.
method Unified framework, step prolongation, structure function γ. result Unified scheme for equivalence problem of geometric structures.
New geometric SDEs and discretizations on Riemannian manifolds with error bounds.
problem Modeling diffusion processes on Riemannian manifolds with geometric SDEs.
method Introduced a new construction of geometric SDEs and provided non-asymptotic error bounds.
result First non-asymptotic error bound for geometric Euler-Murayama discretization.