In this paper the generic bifurcations of the Minkowski symmetry set for 1-parameter families of plane curves are classified and the necessary and sufficient geometric criteria for each type are given. The Minkowski symmetry set is an analogue of the standard Euclidean symmetry set, and is defined to be the locus of ce…
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48 results for “symmetry sets”
The investigation of 3D euclidean symmetry sets (SS) and medial axis is an important area, due in particular to their various important applications. The pre-symmetry set of a surface M in 3-space (resp. smooth closed curve in 2D) is the set of pairs of points which contribute to the symmetry set, that is, the closure …
Study on geodesics in a specific sub-Riemannian structure with two types of behavior.
problem Symmetries and geodesics in a sub-Riemannian structure with growth vector (4,7).
method Symmetry reduction to analyze geodesics.
result Existence of two distinct types of geodesics: those that avoid the fixed point set and those that are contained within it.
The paper examines the geometry of a curve's centre symmetry set.
problem Global geometrical properties of a curve's centre symmetry set.
method Study of the envelope of affine chords.
result Number of singularities and asymptotes of the centre symmetry set.
Study shows instability of certain MOTSs with continuous symmetry.
problem Stability of MOTSs with continuous symmetry.
method Analysis of initial data sets with continuous symmetry and non-preserved MOTSs.
result Exotic MOTSs are unstable except in exceptional cases.
Develops non-parametric tests for group symmetry in data.
problem Lack of statistical tests for group symmetry in data.
method Formulates and implements non-parametric tests for distributional symmetry under specified groups.
result Develops tests for conditional invariance/equivariance and applies them to real-world data.
Origami patterns are classified based on their symmetry groups.
problem Classifying the symmetry groups of origami patterns.
method Iteratively compute intersection points and lines to construct mathematical origami sets, then classify them based on wallpaper groups.
result Determine which wallpaper groups can be constructed from given origami patterns.
New findings on hidden symmetries in ReLU networks.
problem Understanding the redundancy and symmetries in ReLU network parameter space.
method Analyzing parameter settings and function classes for various network architectures.
result For certain network architectures, there are no hidden symmetries.
Invariants found for tau-symmetric bihamiltonian systems.
problem Understanding symmetries in bihamiltonian systems.
method Proof of infinite Virasoro symmetries for tau-symmetric bihamiltonian deformations.
result Infinite set of Virasoro symmetries for tau-symmetric bihamiltonian systems.
The generalized Feix--Kaledin construction shows that c-projective -manifolds with curvature of type are precisely the submanifolds of quaternionic -manifolds which are fixed points set of a special type of quaternionic action . In this paper, we consider this construction in the presence of in…
Invariant kernels reduce rank and improve generalization across dimensions.
problem Symmetry in high-dimensional data impacts kernel matrix rank and learning algorithms.
method Compute invariant polynomial kernel ranks under various groups acting on data.
result Symmetry decreases kernel rank, making it independent of data dimension.
Hidden symmetries and the generic spectral setting of generalized laplacians on homogeneous spacesmath.DG
Study spectral settings of generalized Laplacians on homogeneous spaces.
problem Understanding the spectral properties of generalized Laplacians on compact homogeneous spaces.
method Investigates the generic spectral configuration of operators on -invariant metrics on .
result The spectral setting depends on -isometries and hidden symmetries.
Study connects symmetries in dynamical systems to phase plane representations.
problem Understanding symmetries in dynamical systems and their phase plane realizations.
method Analysis of symmetries in differential equations and phase plane representations, establishing correspondence and lifting conditions.
result Every symmetry generator in one formulation corresponds uniquely to a generator in the other, with a lifting condition to solve.
Study on symmetry defects of complete intersections in complex space.
problem Characterizing symmetry defects of complete intersections.
method Analyzing midpoints of chords connecting points in complete intersections.
result Symmetry defect of complete intersections is an algebraic variety.
We fully develop the concept of causal symmetry introduced in Class. Quant. Grav. 20 (2003) L139. A causal symmetry is a transformation of a Lorentzian manifold (V,g) which maps every future-directed vector onto a future-directed vector. We prove that the set of all causal symmetries is not a group under the usual comp…
Unified framework for enforcing, discovering, and promoting symmetry in machine learning.
problem Symmetry in machine learning models and data.
method Unified mathematical framework using Lie derivatives, convex regularization, and nuclear norm relaxation.
result Unified approach to symmetry in machine learning tasks.
Non-compact inaudibility of weak symmetry and commutativity via generalized Heisenberg groupsmath.DG
Paper proves non-compact inaudibility of symmetry and commutativity.
problem Proving inaudibility of symmetry and commutativity in non-compact settings.
method Using isospectral pairs of generalized Heisenberg groups.
result Proved inaudibility of weak symmetry and commutativity.
Study of intrinsic symmetry groups of links, finding counterexamples.
problem Whether every subgroup of the symmetric group is an intrinsic symmetry group of some link.
method Defined intrinsic symmetry groups and provided counterexamples.
result For n > 5, there does not exist an n-component link L for which S(L) is the alternating group.
The Regge symmetry is a set of remarkable relations between two tetrahedra whose edge lengths are related in a simple fashion. It was first discovered as a consequence of an asymptotic formula in mathematical physics. Here we give a simple geometric proof of Regge symmetries in Euclidean, spherical, and hyperbolic geom…
This work extends PAC-Bayesian learning guarantees to non-compact symmetries and non-invariant data.
problem Lack of theoretical guarantees explaining the benefits of symmetries in machine learning models.
method Adapting and tightening PAC-Bayes bounds for non-compact symmetries and non-invariant data distributions.
result Theoretical evidence that symmetric models are preferable for symmetric data, beyond compact groups and invariant distributions.
Part I. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method…
Study symmetries in smoothed polygonal links.
problem Computing Khovanov homology and link invariants.
method Cube of resolutions with combinatorial structure.
result New group-theoretic invariants of links.
Abstract: Generalized reduction methods for symmetries in graded geometry.
problem Generalized reduction of symmetries in graded geometry.
method Graded symplectic reduction for Courant, Dirac, and generalized complex structures.
result Systematic recovery of reduction schemes for exact cases.
We show that the distribution of symmetry of a naturally reductive nilpotent Lie group coincides with the invariant distribution induced by the set of fixed vectors of the isotropy. This extends a known result on compact naturally reductive spaces. We also address the study of the quotient by the foliation of symmetry.
Study of symmetries in 2D Yang-Mills theory, including orbifolds and higher forms.
problem Understanding symmetries and anomalies in 2D Yang-Mills theory.
method Combining continuum methods, topological defects, and higher gauge theory.
result Unified description of higher and lower form gauge fields, identifying spontaneous symmetry breaking.
No-Arbitrage Symmetriesq-fin.MF
The paper defines symmetries in no-arbitrage markets.
problem Characterizing transformations preserving no-arbitrage.
method Geometric formalization in discrete time models.
result Local characterization of no-arbitrage symmetries.
The paper simplifies symmetries in complex geometric structures.
problem Redundancy in conditions for symmetry reduction in polysymplectic and polycosymplectic structures.
method Exploring and proving necessary and sufficient conditions for polycosymplectic reduction.
result A one-to-one relationship between polycosymplectic reduction and the reduction of a larger polysymplectic manifold.
Study invariant Lipschitz bandits, improving regret bounds.
problem Optimizing decisions under symmetry in online settings.
method Integrates side observations using group orbits into UniformMesh algorithm.
result Improved regret bound for invariant Lipschitz bandit class.
In this paper, we present the point symmetry group of three-dimensional homogeneous Helmholtz equation, when we consider the cylindrical coordinate system. In continuation, we present a complete set of functionally independent invariants of the equation along with the form of the general solution provided by these inva…
Study symmetric linear bandits with hidden symmetry, achieving improved regret bounds.
problem High-dimensional linear bandits with hidden symmetry.
method Model selection within low-dimensional subspaces to learn hidden symmetry.
result Achieved improved regret bounds of and .
This work relaxes GNN symmetries to approximate automorphisms, improving model performance.
problem Improving graph neural network performance on asymmetric graphs.
method Formalizing approximate symmetries via graph coarsening, introducing a bias-variance formula.
result Best generalization performance achieved by choosing a larger symmetry group than automorphisms but smaller than permutations.
Causal symmetriesgr-qc
Based on the recent work \cite{PII} we put forward a new type of transformation for Lorentzian manifolds characterized by mapping every causal future-directed vector onto a causal future-directed vector. The set of all such transformations, which we call causal symmetries, has the structure of a submonoid which contain…
Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.
problem Understanding 4D generalized Ricci flows with nilpotent symmetry.
method Immortal solutions, type III curvature and diameter estimates, new energy monotonicity.
result Blowdown limits lie in a finite-dimensional family of solutions.
Bayesian framework detects symmetries in chaotic dynamical systems.
problem Detecting symmetries in chaotic attractors for insights into dynamical system structure.
method Bayesian framework using Gibbs posterior constructed from Wasserstein distances.
result Bayesian framework accurately recovers symmetries under high noise and small sample sizes.
Let P and Q be convex polyhedra in E3 with face lattices F(P) and F(Q) and symmetry groups G(P) and G(Q), respectively. Then, P and Q are called face equivalent if there is a lattice isomorphism between F(P) and F(Q); P and Q are called symmetry equivalent if the action of G(P) on F(P) is equivalent to the action of G(…
Causal symmetriesmath-ph
We define a new type of transformation for Lorentzian manifolds characterized by mapping every causal future-directed vector onto a causal future-directed vector. The set of all such transformations, which we call causal symmetries, has the structure of a submonoid. Some of their properties are investigated and we give…
New method produces reflections with nonseparating fixed points.
problem Constructing hyperbolic manifolds with reflective symmetries.
method Standard method for constructing closed hyperbolic manifolds.
result Fixed point sets of reflections are nonseparating.
We show that the space of algebraic covariant derivative curvature tensors R' is generated by Young symmetrized tensor products W*U or U*W, where W and U are covariant tensors of order 2 and 3 whose symmetry classes are irreducible and characterized by the following pairs of partitions: {(2),(3)}, {(2),(2 1)} or {(1 1)…
Study on MHD equilibria on curved spaces without symmetries.
problem Analyzing MHD equilibria on curved spaces without symmetries.
method Examined MHD equilibria on Riemannian 3-manifolds with various adapted metrics.
result Found that for an open and dense set of adapted metrics, MHD equilibria on compact 3-manifolds without boundary admit no continuous Killing symmetries.
Hidden symmetry of a G'-space X is defined by an extension of the G'-action on X to that of a group G containing G' as a subgroup. In this setting, we study the relationship between the three objects: (A) global analysis on X by using representations of G (hidden symmetry); (B) global analysis on X by using representat…
Study of sub-Riemannian problem on specific Lie groups, revealing symmetries and bounds.
problem Sub-Riemannian problem on solvable, non-nilpotent Lie groups.
method Qualitative phase-space analysis of Hamiltonian system, focusing on vertical component.
result Explicit upper bound for cut time in terms of pendulum period.
Data analysis and data mining are concerned with unsupervised pattern finding and structure determination in data sets. "Structure" can be understood as symmetry and a range of symmetries are expressed by hierarchy. Such symmetries directly point to invariants, that pinpoint intrinsic properties of the data and of the …
Since the foundational work of Chenciner and Montgomery in 2000 there has been a great deal of interest in choreographic solutions of the n-body problem: periodic motions where the n bodies all follow one another at regular intervals along a closed path. The principal approach combines variational methods with symmetry…
Given a pair of planar curves, one can define its generalized area distance, a concept that generalizes the area distance of a single curve. In this paper, we show that the generalized area distance of a pair of planar curves is an improper indefinite affine spheres with singularities, and, reciprocally, every indefini…
New triangulations of octonionic projective plane found with restricted symmetry groups.
problem Finding symmetry groups of 27-vertex triangulations of manifolds like the octonionic projective plane.
method Using Smith and Bredon's results on transformation groups to restrict possible symmetry groups.
result List of 26 subgroups of S27 containing all possible symmetry groups of 27-vertex triangulations of manifolds like the octonionic project plane.
Study examines stability of MOTS in symmetric initial data sets.
problem Stability of marginally outer trapped surfaces in symmetric initial data sets.
method Characterizes instability through vector decomposition and properties of normal and tangent components.
result Characterizes instability of MOTS by the nature of zero sets and divergences.
Symmetries of Analytic Pathsmath-ph
The symmetries of paths in a manifold are classified with respect to a given pointwise proper action of a Lie group on . Here, paths are embeddings of a compact interval into . There are at least two types of symmetries: Firstly, paths that are parts of an integral curve of a fundamental vector field on $…
Study nearly parallel G2-structures with torus symmetry using multi-moment maps.
problem Characterize and construct nearly parallel G2-structures with torus symmetry.
method Use multi-moment map techniques and analyze the geometry of the base spaces.
result Locally, the construction may produce examples with four-torus symmetry.