Action Of Symmetry Group Research Articles

Алгоритмы перечисления симметричных хордовых диаграмм

Numerous applications require exhaustive lists of strings, which may be subject to various requirements, for example, such as their non-equivalence under the action of symmetry groups on them. The necklace is the equivalence class of r-ary strings under rotation. An unlabeled r-ary necklace is an equivalence class of r-ary strings under rotation and permutation of alphabetic characters. Let G be a symmetry group acting on a given set of necklaces T. A necklace x  T is called symmetric if there exists an element g  G, g  e such that gx = x. Other definitions of the symmetry of necklaces are possible, equivalent to the above. All of them, in one way or another, are involved in determining the symmetry of the figure compared to the unlabelled necklace. A flat figure is called symmetrical if it is self-aligned during movements of space R2 i. e. during its isometric transformations. A special case of figures and, accordingly, necklaces are chord diagrams. Chord diagrams represent an object of research that is interesting from different aspects (knot theory, Feynman diagrams, representations of Lie algebras), and has been studied by many authors. The article presents algorithms for enumerating symmetric chord diagrams and shows their connection with the knot theory using simple examples. The list of symmetrical chord diagrams may, in particular, be of interest in the field of mathematical poetry, which studies the dynamics of poetic stanzas horizontally (rhythm, syllabic volume of verses) and vertically (rhyme schemes, refrains, and other stylistic devices).

Lagrangian reduction of forced discrete mechanical systems

In this paper we propose a process of Lagrangian reduction and reconstruction for symmetric discrete-time mechanical systems acted on by external forces, where the symmetry group action on the configuration manifold turns it into a principal bundle. We analyze the evolution of momentum maps and Poisson structures under different conditions.

Conditions for symmetry reduction of polysymplectic and polycosymplectic structures

For Hamiltonian field theories on polysymplectic manifolds with a symmetry group action and a momentum map, we explore the redundancy in a set of necessary conditions that has appeared in the literature, for a generalized version of the Marsden–Weinstein symmetry reduction theorem. Next, we prove a necessary and sufficient condition for polycosymplectic reduction. We relate polycosymplectic reduction in a one-to-one way to the reduction of an associated larger polysymplectic manifold. Throughout the paper, we provide examples and discuss special cases.

The Structure of Differential Invariants for a Free Symmetry Group Action

The Structure of Differential Invariants for a Free Symmetry Group Action

Hamiltonian neural networks with automatic symmetry detection.

Recently, Hamiltonian neural networks (HNNs) have been introduced to incorporate prior physical knowledge when learning the dynamical equations of Hamiltonian systems. Hereby, the symplectic system structure is preserved despite the data-driven modeling approach. However, preserving symmetries requires additional attention. In this research, we enhance HNN with a Lie algebra framework to detect and embed symmetries in the neural network. This approach allows us to simultaneously learn the symmetry group action and the total energy of the system. As illustrating examples, a pendulum on a cart and a two-body problem from astrodynamics are considered.

A Group-Theoretic Approach to the Bifurcation Analysis of Spatial Cosserat-Rod Frameworks with Symmetry

We present a general approach to the bifurcation analysis of spatial framework structures with symmetry. While group-theoretic methods for bifurcation problems with symmetry are well known, their actual implementation in the context of geometrically exact frameworks is not straightforward. We consider spatial structures comprising assemblages of Cosserat rods; the main difficulty arises from the nonlinear configuration space, due to the presence of (cross-sectional) rotation fields. We avoid this via a single-rod formulation, developed earlier by one of the authors, whereby the governing equations are embedded in a slightly enlarged linear space. The field equations for a framework then comprise the assembly of all such rod equations, supplemented by compatibility and equilibrium conditions at the joints. We demonstrate their equivariance under the natural symmetry group action, and the implementation of group-theoretic methods is now clear within the enlarged linear space. All potential generic, symmetry-breaking bifurcations are predicted a-priori. We then employ an open-source path-following code, which can detect and compute simple, one-dimensional bifurcations; multiple bifurcation points are beyond its capabilities. For the latter, we construct symmetry-reduced problems implemented by appropriate substructures. Multiple bifurcations are rendered simple, and the path-following code is again applicable. We first analyze a simple tripod framework, providing all details of our methodology. We then treat a more complex hexagonal space frame via the same approach. Both structures exhibit simple and double bifurcation points.

Quotient maps and configuration spaces of hard disks

Hard disks systems are often considered as prototypes for simple fluids. In a statistical mechanics context, the hard disk configuration space is generally quotiented by the action of various symmetry groups. The changes in the topological and geometric properties of the configuration spaces effected by such quotient maps are studied for small numbers of disks on a square and hexagonal torus. A metric is defined on the configuration space and the various quotient spaces that respects the desired symmetries. This is used to construct explicit triangulations of the configuration spaces as alpha-complexes. Critical points of the hard disk potential on a configuration space are associated with changes in the topology of the accessible part of the configuration space as a function of disk radius, are conjectured to be related to the configurational entropy of glassy systems, and could reveal the origins of phase transitions in other systems. The number of critical points and their topological and geometric properties are found to depend on the symmetries by which the configuration space is quotiented.Graphic abstract

A different look at the optimal control of the Brockett integrator

We propose a method to analyze the three-dimensional nonholonomic system known as the Brockett integrator and to derive the (energy) optimal trajectories between two given points. For systems with nonholonomic constraint, it is well-known that the energy optimal trajectories corresponds to sub-Riemannian geodesics under a proper sub-Riemannian metric. Our method uses symmetry reduction and an analysis of the quotient space associated with the action of a (symmetry) group on . By lifting the Riemannian geodesics with respect to an appropriate metric from the quotient space back to the original space , we derive the optimal trajectories of the original problem.

Programmability of covariant quantum channels

A programmable quantum processor uses the states of a program register to specify one element of a set of quantum channels which is applied to an input register. It is well-known that such a device is impossible with a finite-dimensional program register for any set that contains infinitely many unitary quantum channels (Nielsen and Chuang's No-Programming Theorem), meaning that a universal programmable quantum processor does not exist. The situation changes if the system has symmetries. Indeed, here we consider group-covariant channels. If the group acts irreducibly on the channel input, these channels can be implemented exactly by a programmable quantum processor with finite program dimension (via teleportation simulation, which uses the Choi-Jamiolkowski state of the channel as a program). Moreover, by leveraging the representation theory of the symmetry group action, we show how to remove redundancy in the program and prove that the resulting program register has minimum Hilbert space dimension. Furthermore, we provide upper and lower bounds on the program register dimension of a processor implementing all group-covariant channels approximately.

Complete classification of rational solutions of A2n-Painlevé systems

We provide a complete classification and an explicit representation of rational solutions to the fourth Painlevé equation PIV and its higher order generalizations known as the A2n-Painlevé or Noumi-Yamada systems. The construction of solutions makes use of the theory of cyclic dressing chains of Schrödinger operators. Studying the local expansions of the solutions around their singularities we find that some coefficients in their Laurent expansion must vanish, which express precisely the conditions of trivial monodromy of the associated potentials. The characterization of trivial monodromy potentials with quadratic growth implies that all rational solutions can be expressed as Wronskian determinants of suitably chosen sequences of Hermite polynomials. The main classification result states that every rational solution to the A2n-Painlevé system corresponds to a cycle of Maya diagrams, which can be indexed by an oddly coloured integer sequence. Finally, we establish the link with the standard approach to building rational solutions, based on applying Bäcklund transformations on seed solutions, by providing a representation for the symmetry group action on coloured sequences and Maya cycles.

The Minimal Dimension of a Sphere with an Equivariant Embedding of the Bouquet of g Circles is $$2g-1$$

To embed the bouquet of $g$ circles $B_g$ into the $n$-sphere $S^n$ so that its full symmetry group action extends to an orthogonal actions on $S^n$, the minimal $n$ is $2g-1$. This answers a question raised by B. Zimmermann.

Calabi–Yau generalized complete intersections and aspects of cohomology of sheaves

We consider generalized complete intersection manifolds in the product space of projective spaces and work out useful aspects pertaining to the cohomology of sheaves over them. First, we present and prove a vanishing theorem on the cohomology groups of sheaves for subvarieties of the ambient product space of projective spaces. We then prove a birational equivalence between configuration matrices of complete intersection Calabi–Yau manifolds. We also present a formula of the genus of curves in generalized complete intersection manifolds. Some of these curves arise as the fixed point locus of certain symmetry group action on the generalized complete intersection Calabi–Yau manifolds. We also make a blowing-up along curves by which one can generate new Calabi–Yau manifolds. Moreover, an approach on spectral sequences is used to compute Hodge numbers of generalized complete intersection Calabi–Yau manifolds and the genus of curves therein.

On the development of symmetry-preserving finite element schemes for ordinary differential equations

In this paper we introduce a procedure, based on the method of equivariant moving frames, for formulating continuous Galerkin finite element schemes that preserve the Lie point symmetries of initial value problems for ordinary differential equations. Our methodology applies to projectable and non-projectable symmetry group actions, to ordinary differential equations of arbitrary order, and finite element approximations of arbitrary polynomial degree. Several examples are included to illustrate various features of the symmetry-preserving process. We summarise extensive numerical experiments showing that symmetry-preserving finite element schemes may provide better long term accuracy than their non-invariant counterparts and can be implemented on larger elements.

Clebsch-Lagrange variational principle and geometric constraint analysis of relativistic field theories

Inspired by the Clebsch optimal control problem, we introduce a new variational principle that is suitable for capturing the geometry of relativistic field theories with constraints related to a gauge symmetry. Its special feature is that the Lagrange multipliers couple to the configuration variables via the symmetry group action. The resulting constraints are formulated as a condition on the momentum map of the gauge group action on the phase space of the system. We discuss the Hamiltonian picture and the reduction in the gauge symmetry by stages in this geometric setting. We show that the Yang–Mills–Higgs action and the Einstein–Hilbert action fit into this new framework after a (1 + 3)-splitting. Moreover, we recover the Gauß constraint of Yang–Mills–Higgs theory and the diffeomorphism constraint of general relativity as momentum map constraints.

Symmetries of the refined D1/D5 BPS spectrum

We examine the large N 1/4-BPS spectrum of the symmetric orbifold CFT SymN (M ) deformed to the supergravity point in moduli space for M = K3 and T4. We consider refinement under both left- and right-moving SU(2)R symmetries of the superconformal algebra, and decompose the spectrum into characters of the algebra. We find that at large N the character decomposition satisfies an unusual property, in which the degeneracy only depends on a certain linear combination of left- and right-moving quantum numbers, suggesting deeper symmetry structure. Furthermore, we consider the action of discrete symmetry groups on these degeneracies, where certain subgroups of the Conway group are known to play a role. We also comment on the potential for larger discrete symmetry groups to appear in the large N limit.

Landau-Ginzburg orbifolds and symmetries of K3 CFTs

Recent developments in the study of the moonshine phenomenon, including umbral and Conway moonshine, suggest that it may play an important role in encoding the action of finite symmetry groups on the BPS spectrum of K3 string theory. To test and clarify these proposed K3-moonshine connections, we study Landau-Ginzburg orbifolds that flow to conformal field theories in the moduli space of K3 sigma models. We compute K3ellipticgeneratwinedbydiscretesymmetriesthataremanifestintheUVdescription, though often inaccessible in the IR. We obtain various twining functions coinciding with moonshine predictions that have not been observed in physical theories before. These include twining functions arising from Mathieu moonshine, other cases of umbral moonshine, and Conway moonshine. For instance, all functions arising from M11 ⊂ 2.M12 moonshine appear as explicit twining genera in the LG models, which moreover admit a uniform description in terms of its natural 12-dimensional representation. Our results provide strong evidence for the relevance of umbral moonshine for K3 symmetries, as well as new hints for its eventual explanation.

Symmetry and conservation laws in semiclassical wave packet dynamics

We formulate symmetries in semiclassical Gaussian wave packet dynamics and find the corresponding conserved quantities, particularly the semiclassical angular momentum, via Noether’s theorem. We consider two slightly different formulations of Gaussian wave packet dynamics; one is based on earlier works of Heller and Hagedorn and the other based on the symplectic-geometric approach by Lubich and others. In either case, we reveal the symplectic and Hamiltonian nature of the dynamics and formulate natural symmetry group actions in the setting to derive the corresponding conserved quantities (momentum maps). The semiclassical angular momentum inherits the essential properties of the classical angular momentum as well as naturally corresponds to the quantum picture.

The Lie–Poisson structure of the symmetry reduced regularized n-body problem

This paper investigates the symmetry reduction of the regularized n-body problem. The three body problem, regularized through quaternions, is examined in detail. We show that for a suitably chosen symmetry group action the space of quadratic invariants is closed and the Hamiltonian can be written in terms of the quadratic invariants. The corresponding Lie–Poisson structure is isomorphic to the Lie algebra . Finally, we generalize this result to the n-body problem for .

Symmetry orbits of supergravity black holes

Black-hole solutions of supergravity theories form families that realize the deep nonlinear “duality” symmetries of these theories. They form orbits under the action of these symmetry groups, with extremal (i.e., BPS) solutions at the limits of such orbits. An important technique for analyzing such solution families uses timelike dimensional reduction and exchanges the stationary black-hole problem for a nonlinear sigma-model problem. We characterize families of extremal or BPS solutions by nilpotent orbits under the duality symmetries, based on a trigraded or pentagraded decomposition of the corresponding duality-group algebra.

Replacing ordinary derivatives by gauge derivatives in the continuum theory of dislocations to compensate the action of the symmetry group

When the symmetry group of a body is continuous it plays a fundamental role on the nonlinear continuum theory of dislocations: it induces a non-uniqueness to the field that describes the defects – the uniform reference – and affects also other fundamental ingredients of the theory. The purpose of the present paper is to examine how certain important quantities of the dislocation theory are affected from symmetry's group action. Apart from the uniform reference we study how the deformation gradient, the first and second Piola–Kirchhoff stress tensors, the elasticities of the material and the momentum equation are affected from the action of the symmetry group. This action is inhomogeneous, namely, differs from point to point. A similar inhomogeneous action of a group may be found in gauge theories. Prompt by the gauge approach, we propose the use of the gauge covariant exterior derivative to compensate for the action of the symmetry group on the uniform reference. The main advantage of using this derivative is that the momentum equation for the static case retains its divergence form. It remains an open question how the Yang–Mills potentials may be determined for the present theory.