Presence Of Symmetry Research Articles

Symmetry facilitates shape constancy for smoothly curved 3d objects.

We tested whether the presence of symmetry improves shape discrimination across changes in viewpoint and lighting for smoothly curved 3D objects. We constructed symmetric and asymmetric versions of random 3D shapes by manipulating their spherical harmonic representations. Matched objects had the same power spectra and appear highly similar except for the presence of symmetry. Observers discriminated sequentially presented pairs of either symmetric or asymmetric objects. Objects were presented in conditions that provided different 3D cues: shading only, stereo only, and combined shading and stereo. To control for 2D cues, standard and test objects had matched boundary contours and were rendered with different light sources. Test objects were also rotated in depth by variable amounts (0° to 60°). Across all viewpoint and 3D cue conditions, we found that shape discrimination for symmetric objects was better than for asymmetric objects. The symmetry benefit was not limited to monocular viewing or to conditions with large rotations in depth. In a second experiment, we blocked trials by viewpoint rotation to eliminate uncertainty in object orientation. This improved performance for asymmetric objects relative to symmetric objects, suggesting that symmetry contributes by providing a cue to object orientation. However, a symmetry advantage was still observed in all shape cue conditions, so this was not the sole source of benefit. Our results demonstrate that symmetry improves shape constancy for smooth 3D objects and suggest that one role of symmetry is to provide a reference orientation for an object.

Capturing Structural Complexity of Innovation Diffusion through System Dynamics

Several shortcomings of traditional innovation diffusion research create major impediments to the ‘holistic’ understanding of diffusion of innovations as well as to the development of effective strategies of policy intervention to facilitate diffusion process. Among the major criticisms of diffusion research are the ‘selection biases’ of many diffusion studies and the futility of curve fitting as an adequate test of theoretical relevance. These limitations can be avoided by substantive and methodological changes in diffusion research; incorporating and integrating into selective critical structural variables to map their interaction, and explaining the inherent dynamism. There are several research articles that discuss different approaches to modeling innovation diffusion over time. Conventionally, the approaches in explaining the innovation diffusion process assume that the process takes place in a stable and homogeneous system in which the innovation diffuses or spreads without being affected by the system’s structural variables even under external influences. Many studies have established that the presence of symmetry is not the general rule in innovation diffusion process. Also examined are these models, and the need to further modify in order to improve the holistic understanding of the dynamic structural complexities and forces driving the processes of innovation and diffusion is recognized.

PIECEWISE SMOOTH REVERSIBLE DYNAMICAL SYSTEMS AT A TWO-FOLD SINGULARITY

This paper focuses on the existence of closed orbits around a two-fold singularity of 3D discontinuous systems of the Filippov type in the presence of symmetries.

Gauge Transformations, Twisted Poisson Brackets and Hamiltonization of Nonholonomic Systems

In this paper we study the problem of Hamiltonization of nonholonomic systems from a geometric point of view. We use gauge transformations by 2-forms (in the sense of Severa and Weinstein in Progr Theoret Phys Suppl 144:145 154 2001) to construct different almost Poisson structures describing the same nonholonomic system. In the presence of symmetries, we observe that these almost Poisson structures, although gauge related, may have fundamentally different properties after reduction, and that brackets that Hamiltonize the problem may be found within this family. We illustrate this framework with the example of rigid bodies with generalized rolling constraints, including the Chaplygin sphere rolling problem. We also see through these examples how twisted Poisson brackets appear naturally in nonholonomic mechanics.

Partial symmetry breaking by local search in the group

The presence of symmetry in constraint satisfaction problems can cause a great deal of wasted search effort, and several methods for breaking symmetries have been reported. In this paper we describe a new method called Symmetry Breaking by Nonstationary Optimisation, which interleaves local search in the symmetry group with backtrack search on the constraint problem. It can be tuned to break each symmetry with an arbitrarily high probability with high runtime overhead, or as a lightweight but still powerful method with low runtime overhead. It has negligible memory requirement, it combines well with static lex-leader constraints, and its benefit increases with problem hardness.

The Mathematical Structure of Information Bottleneck Methods

Information Bottleneck-based methods use mutual information as a distortion function in order to extract relevant details about the structure of a complex system by compression. One of the approaches used to generate optimal compressed representations is by annealing a parameter. In this manuscript we present a common framework for the study of annealing in information distortion problems. We identify features that should be common to any annealing optimization problem. The main mathematical tools that we use come from the analysis of dynamical systems in the presence of symmetry (equivariant bifurcation theory). Through the compression problem, we make connections to the world of combinatorial optimization and pattern recognition. The two approaches use very different vocabularies and consider different problems to be “interesting”. We provide an initial link, through the Normalized Cut Problem, where the two disciplines can exchange tools and ideas.

Best constants in second-order Sobolev inequalities on compact Riemannian manifolds in the presence of symmetries

Let (M,g) be a smooth compact 3\leq n-dimensional Riemannian manifold, and G a subgroup of the isometry group of (M,g). We establish the best constants in second-order for a Sobolev inequality when the functions are G-invariant.

Equivariant normal forms for parameterized delay differential equations with applications to bifurcation theory

In this paper, we develop an efficient approach to compute the equivariant normal form of delay differential equations with parameters in the presence of symmetry. We present and justify a process that involves center manifold reduction and normalization preserving the symmetry, and that yields normal forms explicitly in terms of the coefficients of the original system. We observe that the form of the reduced vector field relies only on the information of the linearized system at the critical point and on the inherent symmetry, and the normal forms give critical information about not only the existence but also the stability and direction of bifurcated spatiotemporal patterns. We illustrate our general results by some applications to fold bifurcation, equivariant Hopf bifurcation and Hopf-Hopf interaction, with a detailed case study of additive neurons with delayed feedback.

Classifying quantum phases using matrix product states and projected entangled pair states

We give a classification of gapped quantum phases of one-dimensional systems in the framework of Matrix Product States (MPS) and their associated parent Hamiltonians, for systems with unique as well as degenerate ground states, and both in the absence and presence of symmetries. We find that without symmetries, all systems are in the same phase, up to accidental ground state degeneracies. If symmetries are imposed, phases without symmetry breaking (i.e., with unique ground states) are classified by the cohomology classes of the symmetry group, this is, the equivalence classes of its projective representations, a result first derived in [X. Chen, Z.-C. Gu, and X.-G. Wen, Phys. Rev. B 83, 035107 (2011); arXiv:1008.3745]. For phases with symmetry breaking (i.e., degenerate ground states), we find that the symmetry consists of two parts, one of which acts by permuting the ground states, while the other acts on individual ground states, and phases are labelled by both the permutation action of the former and the cohomology class of the latter. Using Projected Entangled Pair States (PEPS), we subsequently extend our framework to the classification of two-dimensional phases in the neighborhood of a number of important cases, in particular systems with unique ground states, degenerate ground states with a local order parameter, and topological order. We also show that in two dimensions, imposing symmetries does not constrain the phase diagram in the same way it does in one dimension. As a central tool, we introduce the isometric form, a normal form for MPS and PEPS which is a renormalization fixed point. Transforming a state to its isometric form does not change the phase, and thus, we can focus on to the classification of isometric forms.

The σ-isotypic decomposition and the σ-index of reversible-equivariant systems

This work is concerned with dynamical systems in presence of symmetries and reversing symmetries. We describe a construction process of subspaces that are invariant by linear Γ-reversible-equivariant mappings, where Γ is the compact Lie group of all the symmetries and reversing symmetries of such systems. These subspaces are the σ-isotypic components, first introduced by Lamb and Roberts in (1999) [10] and that correspond to the isotypic components for purely equivariant systems. In addition, by representation theory methods derived from the topological structure of the group Γ, two algebraic formulae are established for the computation of the σ-index of a closed subgroup of Γ. The results obtained here are to be applied to general reversible-equivariant systems, but are of particular interest for the more subtle of the two possible cases, namely the non-self-dual case. Some examples are presented.

The filling law: A general framework for leaf folding and its consequences on leaf shape diversity

Leaves are packed in a bud in different ways, being flat, rolled, or folded, but always filling the whole bud volume. This “filling law” has many consequences, in particular on the shapes of growing folded leaves. This is shown here for different types of folding and packing. The folded volume is roughly a part of an ellipsoid, with the veins on the outside rounded face and the lamina margin on the adaxial plane. The veins on the abaxial side protect the fragile lamina inside. The first general consequence of the folds and the space limitation of the lamina growth is the presence of symmetries on the leaf shape, and the second is the quantitative relationships between the sizes of the lobes and sinuses. For particular geometries, the leaf lamina can be limited by lateral veins, creating spoon-like lobes, or tangent cuts, creating asymmetrical wavy perimeters. Changes in the packing between different cultivars correspond to changes in the mature leaf shapes. Each particular case shows how pervasive the geometrical consequences of the filling law are.

Energy bands: Chern numbers and symmetry

Energy bands formed by rotation–vibrational states of molecules in the presence of symmetry and their qualitative modifications under variation of some control parameters are studied within the semi-quantum model. Rotational variables are treated as classical whereas a finite set of vibrational states is considered as quantum. In the two-state approximation the system is described in terms of a fiber bundle with the base space being a two-dimensional sphere, the classical phase space for rotational variables. Generically this rank 2 complex vector bundle can be decomposed into two complex line bundles characterized by a topological invariant, the first Chern class. A general method of explicit calculation of Chern classes and of their possible modifications under variation of control parameters in the presence of symmetry is suggested. The construction of iso-Chern diagrams which split the space of control parameters into connected domains with fixed Chern numbers is suggested. A detailed analysis of the rovibrational model Hamiltonian for a D 3 invariant molecule possessing two vibrational states transforming according to the two-dimensional irreducible representation is done to illustrate non-trivial restrictions imposed by symmetry on possible values of Chern classes.

Symmetry facilitates 3D shape discrimination across changes in viewpoint

Perceiving 3D shape constancy across changes in viewpoint is a challenging problem. We tested whether the presence of symmetry can facilitate shape discrimination for smoothly curved, solid 3D objects. Symmetric and asymmetric random shapes were generated to have the same angular power spectra, expressed as spherical harmonics. Objects were presented in three viewing conditions: shading-only, stereo-only, and combined shading and stereo. Shaded objects were illuminated by a diffused point light source, and a mirror haploscope was used to present binocular images. Objects were presented sequentially and observers judged whether the two shapes were same or different. The test objects were either the same orientation or differed by a rotation in depth of ±15°, ±30°, or ±60° relative to the standard object. In all conditions, we found that shape discrimination performance decreased with rotation in depth. Overall, shape discrimination for symmetric objects was better than for asymmetric objects. The presence of stereo information improved performance for both symmetric and asymmetric objects. The results suggest that structural constraints like symmetry are important for viewpoint invariant shape perception, and that stereo information improves shape perception even for structured 3D objects.

Synchrony-Breaking Bifurcation at a Simple Real Eigenvalue for Regular Networks 1: 1-Dimensional Cells

We study synchrony-breaking local steady-state bifurcation in networks of dynamical systems when the critical eigenvalue is real and simple, using singularity theory to transform the bifurcation into normal form. In a general dynamical system, a generic steady-state local bifurcation from a trivial state is transcritical. In the presence of symmetry, a pitchfork is also possible generically. Network structure introduces constraints that may change the generic behavior. We consider regular networks, in which all cells have the same type and all arrows have the same type, and every cell receives inputs from the same number of arrows. A characterization of all smooth admissible maps permits a singularity-theoretic analysis based on Liapunov–Schmidt reduction. Assuming that the cells have 1-dimensional internal dynamics, we give conditions on the critical eigenvectors of the linearization and its transpose that determine when a generic bifurcation is transcritical, pitchfork, or more degenerate. We prove that for all regular n-cell networks, such bifurcations are generically n-determined. In the path-connected case, this is improved to $(n-1)$-determined. In bidirectional networks, generic bifurcation is transcritical or pitchfork, but the role of symmetry is minor. In the general case, degenerate cases can occur: the network must have at least 4 cells (5 in the path-connected case). We give examples of networks for which generic bifurcations are degenerate, including a 6-cell network with a normal form that is determined only at degree 6 and a path-connected 5-cell network with a normal form that is determined only at degree 4.

Set-based parameter estimation for symmetric network motifs

AbstractDeriving a predictive model in systems biology is a complex task. One major problem is the typically large network size, which renders the analysis with standard methods difficult. Symmetry, as omnipresent in nature, was used in many applications to encounter this problem. In this work, we investigate the influence of symmetry on set-based parameter estimation. We show that the presence of symmetry in a model can be used to significantly simplify the parameter estimation problem. This is done by determining a symmetry-adapted basis, corresponding to a linear representation of a finite group, in which the problem size is of smaller dimension. We demonstrate the applicability of this approach for several common network motifs, as e.g. the Michaelis-Menten reaction and the feedforward motif.

Semi‐isometric Registration of Line Features for Flexible Fitting of Protein Structures

In this paper, we study a registration problem that is motivated by a practical biology problem - fitting protein structures to low-resolution density maps. We consider registration between two sets of lines features (e.g., helices in the proteins) that have undergone not a single, but multiple isometric transformations (e.g., hinge-motions). The problem is further complicated by the presence of symmetry in each set. We formulate the problem as a clique-finding problem in a product graph, and propose a heuristic solution that includes a fast clique-finding algorithm unique to the structure of this graph. When tested on a suite of real protein structures, the algorithm achieved high accuracy even for very large inputs containing hundreds of helices.

Sharp Nash inequalities on manifolds with boundary in the presence of symmetries

In this paper we establish the best constant A ˜ o p t ( M ¯ ) for the trace Nash inequality on a n -dimensional compact Riemannian manifold in the presence of symmetries, which is an improvement over the classical case due to the symmetries which arise and reflect the geometry of manifold. This is particularly true when the data of the problem is invariant under the action of an arbitrary compact subgroup G of the isometry group I s ( M , g ) , where all the orbits have infinite cardinal.

Perturbation of Yamabe equation on Iwasawa N groups in presence of symmetry

Let G be a simple Lie group of real rank one and N be in the Iwasawa decomposition of G. Under the assumption of some symmetries, we obtain an existent result for the nonlinear equation ΔNu + (1 + ɛK(x, z))u2*−1 = 0 on N, which generalizes the result of Malchiodi and Uguzzoni to the Kohn’s subelliptic context on N in presence of symmetry.

Symmetry in Boolean Satisfiability

This paper reviews recent approaches on how to accelerate Boolean Satisfiability (SAT) search by exploiting symmetries in the problem space. SAT search algorithms traverse an exponentially large search space looking for an assignment that satisfies a set of constraints. The presence of symmetries in the search space induces equivalence classes on the set of truth assignments. The goal is to use symmetries to avoid traversing all assignments by constraining the search to visit a few representative assignments in each equivalence class. This can lead to a significant reduction in search runtime without affecting the completeness of the search.

Dependence of magnetic field generation by thermal convection on the rotation rate: A case study

Dependence of magnetic field generation on the rotation rate is explored by direct numerical simulation of magnetohydrodynamic convective attractors in a plane layer of conducting fluid with square periodicity cells for the Taylor number varied from zero to 2000, for which the convective fluid motion halts (other parameters of the system are fixed). We observe 5 types of hydrodynamic (amagnetic) attractors: two families of two-dimensional (i.e. depending on two spatial variables) rolls parallel to sides of periodicity boxes of different widths and parallel to the diagonal, travelling waves and three-dimensional “wavy” rolls. All types of attractors, except for one family of rolls, are capable of kinematic magnetic field generation. We have found 21 distinct nonlinear convective MHD attractors (13 steady states and 8 periodic regimes) and identified bifurcations in which they emerge. In addition, we have observed a family of periodic, two-frequency quasiperiodic and chaotic regimes, as well as an incomplete Feigenbaum period doubling sequence of bifurcations of a torus followed by a chaotic regime and subsequently by a torus with 1/3 of the cascade frequency. The system is highly symmetric. We have found two novel global bifurcations reminiscent of the SNIC bifurcation, which are only possible in the presence of symmetries. The universally accepted paradigm, whereby an increase of the rotation rate below a certain level is beneficial for magnetic field generation, while a further increase inhibits it (and halts the motion of fluid on continuing the increase), remains unaltered, but we demonstrate that this “large-scale” picture lacks many significant details.