Showing papers on "Symmetry (geometry) published in 2018"

PCN: Point Completion Network

Abstract: Shape completion, the problem of estimating the complete geometry of objects from partial observations, lies at the core of many vision and robotics applications. In this work, we propose Point Completion Network (PCN), a novel learning-based approach for shape completion. Unlike existing shape completion methods, PCN directly operates on raw point clouds without any structural assumption (e.g. symmetry) or annotation (e.g. semantic class) about the underlying shape. It features a decoder design that enables the generation of fine-grained completions while maintaining a small number of parameters. Our experiments show that PCN produces dense, complete point clouds with realistic structures in the missing regions on inputs with various levels of incompleteness and noise, including cars from LiDAR scans in the KITTI dataset.

On finite symmetries and their gauging in two dimensions

Abstract: It is well-known that if we gauge a ℤn symmetry in two dimensions, a dual ℤn symmetry appears, such that re-gauging this dual ℤn symmetry leads back to the original theory. We describe how this can be generalized to non-Abelian groups, by enlarging the concept of symmetries from those defined by groups to those defined by unitary fusion categories. We will see that this generalization is also useful when studying what happens when a non-anomalous subgroup of an anomalous finite group is gauged: for example, the gauged theory can have non-Abelian group symmetry even when the original symmetry is an Abelian group. We then discuss the axiomatization of two-dimensional topological quantum field theories whose symmetry is given by a category. We see explicitly that the gauged version is a topological quantum field theory with a new symmetry given by a dual category.

Quantitative mappings between symmetry and topology in solids

Abstract: The study of spatial symmetries was accomplished during the last century and had greatly improved our understanding of the properties of solids. Nowadays, the symmetry data of any crystal can be readily extracted from standard first-principles calculation. On the other hand, the topological data (topological invariants), the defining quantities of nontrivial topological states, are in general considerably difficult to obtain, and this difficulty has critically slowed down the search for topological materials. Here we provide explicit and exhaustive mappings from symmetry data to topological data for arbitrary gapped band structure in the presence of time-reversal symmetry and any one of the 230 space groups. The mappings are completed using the theoretical tools of layer construction and symmetry-based indicators. With these results, finding topological invariants in any given gapped band structure reduces to a simple search in the mapping tables provided.

Conversion Rules for Weyl Points and Nodal Lines in Topological Media.

Abstract: According to a widely held paradigm, a pair of Weyl points with opposite chirality mutually annihilate when brought together. In contrast, we show that such a process is strictly forbidden for Weyl points related by a mirror symmetry, provided that an effective two-band description exists in terms of orbitals with opposite mirror eigenvalue. Instead, such a pair of Weyl points convert into a nodal loop inside a symmetric plane upon the collision. Similar constraints are identified for systems with multiple mirrors, facilitating previously unreported nodal-line and nodal-chain semimetals that exhibit both Fermi-arc and drumhead surface states. We further find that Weyl points in systems symmetric under a $\ensuremath{\pi}$ rotation composed with time reversal are characterized by an additional integer charge that we call helicity. A pair of Weyl points with opposite chirality can annihilate only if their helicities also cancel out. We base our predictions on topological crystalline invariants derived from relative homotopy theory, and we test our predictions on simple tight-binding models. The outlined homotopy description can be directly generalized to systems with multiple bands and other choices of symmetry.

Minimalist approach to the classification of symmetry protected topological phases

Abstract: A number of proposals with differing predictions (e.g. Borel group cohomology, oriented cobordism, group supercohomology, spin cobordism, etc.) have been made for the classification of symmetry protected topological (SPT) phases. Here we treat various proposals on an equal footing and present rigorous, general results that are independent of which proposal is correct. We do so by formulating a minimalist Generalized Cohomology Hypothesis, which is satisfied by existing proposals and captures essential aspects of SPT classification. From this Hypothesis alone, formulas relating classifications in different dimensions and/or protected by different symmetry groups are derived. Our formalism is expected to work for fermionic as well as bosonic phases, Floquet as well as stationary phases, and spatial as well as on-site symmetries. As an application, we predict that the complete classification of 3-dimensional bosonic SPT phases with space group symmetry $G$ is $H^4_{\rm Borel}\left(G;U(1)\right) \oplus H^1_{\rm group}\left(G;\mathbb Z\right)$, where the $H^1$ term classifies phases beyond the Borel group cohomology proposal.

A Step-by-Step Guide for Geometric Morphometrics of Floral Symmetry.

Abstract: This paper provides a step-by-step guide for the morphological analysis of corolla and the decomposition of corolla shape variation into its symmetric and asymmetric components The shape and symmetric organization of corolla are key traits in the developmental and evolutionary biology of flowering plants The various spatial layout of petals can exhibit bilateral symmetry, rotational symmetry or more complex combination of symmetry types Here, I describe a general landmark-based geometric morphometric framework for the full statistical shape analysis of corolla and exemplify its use with four fully worked out case studies including tissue treatment, imaging, landmark data collection, file formatting, and statistical analyses: (i) bilateral symmetry (Fedia graciliflora), (ii) two perpendicular axes of bilateral symmetry (Erysimum mediohispanicum), (iii) rotational symmetry (Vinca minor), and (iv) Combined bilateral and rotational symmetry (Trillium undulatum) The necessary tools for such analyses are not implemented in standard morphometric software and they are therefore provided here as functions running in the R environment Principal component analysis is used to separate symmetric and asymmetric components of variation, respectively quantifying variation among and within individuals For bilaterally symmetric flowers, only one component of left-right asymmetric variation is extracted, while flowers with more complex symmetric layout have components of asymmetric variation associated with each symmetry operator implied (eg left-right asymmetry and adaxial-abaxial asymmetry) Fundamental information on the genetic, developmental, and environmental determinants of shape variation can be inferred from this decomposition (eg directional asymmetry, fluctuating asymmetry) and further exploited to document patterns of canalization, developmental stability, developmental modularity and morphological integration Even if symmetry and asymmetry are not the primary interest of a study on corolla shape variation, statistical and anatomical arguments support the use of the framework advocated This didactic protocol will help both morphometricians and non-morphometricians to further understand the role of symmetry in the development, variation and adaptive evolution of flowers

Mean curvature self-shrinkers of high genus: Non-compact examples

Abstract: We give the first rigorous construction of complete, embedded self-shrinking hypersurfaces under mean curvature flow, since Angenent's torus in 1989. The surfaces exist for any sufficiently large prescribed genus $g$, and are non-compact with one end. Each has $4g+4$ symmetries and comes from desingularizing the intersection of the plane and sphere through a great circle, a configuration with very high symmetry. Each is at infinity asymptotic to the cone in $\mathbb{R}^3$ over a $2\pi/(g+1)$-periodic graph on an equator of the unit sphere $\mathbb{S}^2\subseteq\mathbb{R}^3$, with the shape of a periodically "wobbling sheet". This is a dramatic instability phenomenon, with changes of asymptotics that break much more symmetry than seen in minimal surface constructions. The core of the proof is a detailed understanding of the linearized problem in a setting with severely unbounded geometry, leading to special PDEs of Ornstein-Uhlenbeck type with fast growth on coefficients of the gradient terms. This involves identifying new, adequate weighted H\"older spaces of asymptotically conical functions in which the operators invert, via a Liouville-type result with precise asymptotics.

Symmetry and Dirac points in graphene spectrum

Abstract: Existence and stability of Dirac points in the dispersion relation of operators periodic with respect to the hexagonal lattice is investigated for different sets of additional symmetries. The following symmetries are considered: rotation by $2\pi/3$ and inversion, rotation by $2\pi/3$ and horizontal reflection, inversion or reflection with weakly broken rotation symmetry, and the case where no Dirac points arise: rotation by $2\pi/3$ and vertical reflection. All proofs are based on symmetry considerations. In particular, existence of degeneracies in the spectrum is deduced from the (co)representation of the relevant symmetry group. The conical shape of the dispersion relation is obtained from its invariance under rotation by $2\pi/3$. Persistence of conical points when the rotation symmetry is weakly broken is proved using a geometric phase in one case and parity of the eigenfunctions in the other.

Higher-order bulk-boundary correspondence for topological crystalline phases

Abstract: We study the bulk-boundary correspondence for topological crystalline phases, where the crystalline symmetry is an order-two (anti)symmetry, unitary or antiunitary. We obtain a formulation of the bulk-boundary correspondence in terms of a subgroup sequence of the bulk classifying groups, which uniquely determines the topological classification of the boundary states. This formulation naturally includes higher-order topological phases as well as topologically nontrivial bulk systems without topologically protected boundary states. The complete bulk and boundary classification of higher-order topological phases with an additional order-two symmetry or antisymmetry is contained in this work.

Space-Time Crystal and Space-Time Group.

Abstract: Crystal structures and the Bloch theorem play a fundamental role in condensed matter physics. We extend the static crystal to the dynamic ``space-time'' crystal characterized by the general intertwined space-time periodicities in $D+1$ dimensions, which include both the static crystal and the Floquet crystal as special cases. A new group structure dubbed a ``space-time'' group is constructed to describe the discrete symmetries of a space-time crystal. Compared to space and magnetic groups, the space-time group is augmented by ``time-screw'' rotations and ``time-glide'' reflections involving fractional translations along the time direction. A complete classification of the 13 space-time groups in one-plus-one dimensions ($1+1\mathrm{D}$) is performed. The Kramers-type degeneracy can arise from the glide time-reversal symmetry without the half-integer spinor structure, which constrains the winding number patterns of spectral dispersions. In $2+1\mathrm{D}$, nonsymmorphic space-time symmetries enforce spectral degeneracies, leading to protected Floquet semimetal states. We provide a general framework for further studying topological properties of the ($D+1$)-dimensional space-time crystal.

A PDE construction of the Euclidean $\Phi^4_3$ quantum field theory

Abstract: We present a new construction of the Euclidean $\Phi^4$ quantum field theory on $\mathbb{R}^3$ based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on $\mathbb{R}^3$ defined on a periodic lattice of mesh size $\varepsilon$ and side length $M$. We introduce a new renormalized energy method in weighted spaces and prove tightness of the corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow \infty$. Every limit point is non-Gaussian and satisfies reflection positivity, translation invariance and stretched exponential integrability. These properties allow to verify the Osterwalder--Schrader axioms for a Euclidean QFT apart from rotation invariance and clustering. Our argument applies to arbitrary positive coupling constant, to multicomponent models with $O(N)$ symmetry and to some long-range variants. Moreover, we establish an integration by parts formula leading to the hierarchy of Dyson--Schwinger equations for the Euclidean correlation functions. To this end, we identify the renormalized cubic term as a \emph{distribution} on the space of Euclidean fields.

Free-space remote sensing of rotation at photon-counting level

Abstract: The rotational Doppler effect associated with light's orbital angular momentum (OAM) has been found as a powerful tool to detect rotating bodies. However, this method was only demonstrated experimentally on the laboratory scale under well controlled conditions so far. And its real potential lies at the practical applications in the field of remote sensing. We have established a 120-meter long free-space link between the rooftops of two buildings and show that both the rotation speed and the rotational symmetry of objects can be identified from the detected rotational Doppler frequency shift signal at photon count level. Effects of possible slight misalignments and atmospheric turbulences are quantitatively analyzed in terms of mode power spreading to the adjacent modes as well as the transfer of rotational frequency shifts. Moreover, our results demonstrate that with the preknowledge of the object's rotational symmetry one may always deduce the rotation speed no matter how strong the coupling to neighboring modes is. Without any information of the rotating object, the deduction of the object's symmetry and rotational speed may still be obtained as long as the mode spreading efficiency does not exceed 50 %. Our work supports the feasibility of a practical sensor to remotely detect both the speed and symmetry of rotating bodies.

Emergence of symmetry selectivity in the visual areas of the human brain: fMRI responses to symmetry presented in both frontoparallel and slanted planes.

Abstract: Symmetry is effortlessly perceived by humans across changes in viewing geometry. Here, we re-examined the network subserving symmetry processing in the context of up-to-date retinotopic definitions of visual areas. Responses in object selective cortex, as defined by functional localizers, were also examined. We further examined responses to both frontoparallel and slanted symmetry while manipulating attention both toward and away from symmetry. Symmetry-specific responses first emerge in V3 and continue across all downstream areas examined. Of the retinotopic areas, ventral occipital VO1 showed the strongest symmetry response, which was similar in magnitude to the responses observed in object selective cortex. Neural responses were found to increase with both the coherence and folds of symmetry. Compared to passive viewing, drawing attention to symmetry generally increased neural responses and the correspondence of these neural responses with psychophysical performance. Examining symmetry on the slanted plane found responses to again emerge in V3, continue through downstream visual cortex, and be strongest in VO1 and LOB. Both slanted and frontoparallel symmetry evoked similar activity when participants performed a symmetry-related task. However, when a symmetry-unrelated task was performed, fMRI responses to slanted symmetry were reduced relative to their frontoparallel counterparts. These task-related changes provide a neural signature that suggests slant has to be computed ahead of symmetry being appropriately extracted, known as the "normalization" account of symmetry processing. Specifically, our results suggest that normalization occurs naturally when attention is directed toward symmetry and orientation, but becomes interrupted when attention is directed away from these features.

AFLOW-SYM: Platform for the complete, automatic and self-consistent symmetry analysis of crystals

Abstract: Determination of the symmetry profile of structures is a persistent challenge in materials science. Results often vary amongst standard packages, hindering autonomous materials development by requiring continuous user attention and educated guesses. Here, we present a robust procedure for evaluating the complete suite of symmetry properties, featuring various representations for the point-, factor-, space groups, site symmetries, and Wyckoff positions. The protocol determines a system-specific mapping tolerance that yields symmetry operations entirely commensurate with fundamental crystallographic principles. The self consistent tolerance characterizes the effective spatial resolution of the reported atomic positions. The approach is compared with the most used programs and is successfully validated against the space group information provided for over 54,000 entries in the Inorganic Crystal Structure Database. Subsequently, a complete symmetry analysis is applied to all 1.7$+$ million entries of the AFLOW data repository. The AFLOW-SYM package has been implemented in, and made available for, public use through the automated, $\textit{ab-initio}$ framework AFLOW.

Rigorous use of symmetry within the construction of multidimensional potential energy surfaces.

Abstract: A method is presented, which allows for the rigorous use of symmetry within the construction of multidimensional potential energy surfaces (PESs). This approach is based on a crude but very fast energy estimate, which retains the symmetry of a molecule. This enables the efficient use of coordinate systems, which mix molecular and permutational symmetry, as, for example, in the case of normal coordinates with subsets of localized normal coordinates. The impact of symmetry within the individual terms of an expansion of the PES is studied together with a symmetry consideration within the individual electronic structure calculations. A trade between symmetry within the surface and the electronic structure calculations has been observed and has been investigated in dependence on different coordinate systems. Differences occur between molecules belonging to Abelian point groups in contrast to non-Abelian groups, in which further benefits can be achieved by rotating normal coordinates belonging to degenerate vibrational frequencies. In general, the exploitation of surface symmetry was found to be very important within the construction of PESs of small and medium-sized molecules—irrespective of the coordinate system. Benchmark calculations are provided for formaldehyde, ethene, chloromethane, and cubane.

A Survey of Closed Self-Shrinkers with Symmetry

Abstract: In this paper, we survey known results on closed self-shrinkers for mean curvature flow and discuss techniques used in recent constructions of closed self-shrinkers with classical rotational symmetry. We also propose new existence and uniqueness problems for closed self-shrinkers with bi-rotational symmetry and provide numerical evidence for the existence of new examples.

Chiral 3d SU(3) SQCD and N=2 mirror duality.

Abstract: Recently a very interesting three-dimensional $\mathcal{N}=2$ supersymmetric theory with $SU(3)$ global symmetry was discussed by several authors. We denote this model by $T_x$. This was conjectured to have two dual descriptions, one with explicit supersymmetry and emergent flavor symmetry and the other with explicit flavor symmetry and emergent supersymmetry. We discuss a third description of the model which has both flavor symmetry and supersymmetry manifest. We then investigate models which can be constructed by using $T_x$ as a building block gauging the global symmetry and paying special attention to the global structure of the gauge group. We conjecture several cases of $\mathcal{N}=2$ mirror dualities involving such constructions with the dual being either a simple $\mathcal{N}=2$ Wess-Zumino model or a discrete gauging thereof.

On the algebra of symmetries of Laplace and Dirac operators

Abstract: We consider a generalization of the classical Laplace operator, which includes the Laplace–Dunkl operator defined in terms of the differential-difference operators associated with finite reflection groups called Dunkl operators. For this Laplace-like operator, we determine a set of symmetries commuting with it, in the form of generalized angular momentum operators, and we present the algebraic relations for the symmetry algebra. In this context, the generalized Dirac operator is then defined as a square root of our Laplace-like operator. We explicitly determine a family of graded operators which commute or anticommute with our Dirac-like operator depending on their degree. The algebra generated by these symmetry operators is shown to be a generalization of the standard angular momentum algebra and the recently defined higher-rank Bannai–Ito algebra.

Linking Entanglement and Discrete Anomaly

Abstract: In 3d Chern-Simons theory, there is a discrete one-form symmetry, whose symmetry group is isomorphic to the center of the gauge group. We study the ‘t Hooft anomaly associated to this discrete one-form symmetry in theories with generic gauge groups, A, B, C, D-types. We propose to detect the discrete anomaly by computing the Hopf state entanglement in the subspace spanned by the symmetry generators and develop a systematical way based on the truncated modular S matrix. We check our proposal for many examples.

Chiral 3 d SU(3) SQCD and N = 2 $$ \mathcal{N}=2 $$ mirror duality

Abstract: Recently a very interesting three-dimensional $$ \mathcal{N}=2 $$ supersymmetric theory with SU(3) global symmetry was discussed by several authors. We denote this model by Tx. This was conjectured to have two dual descriptions, one with explicit supersymmetry and emergent flavor symmetry and the other with explicit flavor symmetry and emergent supersymmetry. We discuss a third description of the model which has both flavor symmetry and supersymmetry manifest. We then investigate models which can be constructed by using Tx as a building block gauging the global symmetry and paying special attention to the global structure of the gauge group. We conjecture several cases of $$ \mathcal{N}=2 $$ mirror dualities involving such constructions with the dual being either a simple $$ \mathcal{N}=2 $$ Wess-Zumino model or a discrete gauging thereof.

Data depth and floating body

Abstract: Little known relations of the renown concept of the halfspace depth for multivariate data with notions from convex and affine geometry are discussed. Halfspace depth may be regarded as a measure of symmetry for random vectors. As such, the depth stands as a generalization of a measure of symmetry for convex sets, well studied in geometry. Under a mild assumption, the upper level sets of the halfspace depth coincide with the convex floating bodies used in the definition of the affine surface area for convex bodies in Euclidean spaces. These connections enable us to partially resolve some persistent open problems regarding theoretical properties of the depth.

Information Content in Inverse Source with Symmetry and Support Priors

Abstract: This paper illustrates how inverse source problems are affected by certain symmetry and support priors concerning the source space. The study is developed for a prototype configuration where the field radiated by square integrable strip sources is observed in far-zone. Three symmetry priors are considered: the source is a priori known to be a real or Hermitian or even (resp. odd) function. Instead, as spatial priors we assume that the source support consists of a single or multiple disjoint domains. The role of the aforementioned priors is assessed against some metrics commonly used to characterise inverse source problems such as the number of degrees of freedom, the point-spread function and the “information content” measured through the Kolmogorov entropy.

An identity of symmetry for the degenerate Frobenius-Euler Polynomials

Abstract: Abstract In this paper, we prove an identity of symmetry for the higher-order degenerate Frobenius-Euler polynomials and derive the recurrence relations and multiplication theorem type result for the degenerate Frobenius-Euler polynomials.

Determination of Patterson group symmetry from sparse multi‐crystal data sets in the presence of an indexing ambiguity

Abstract: Combining X-ray diffraction data from multiple samples requires determination of the symmetry and resolution of any indexing ambiguity. For the partial data sets typical of in situ room-temperature experiments, determination of the correct symmetry is often not straightforward. The potential for indexing ambiguity in polar space groups is also an issue, although methods to resolve this are available if the true symmetry is known. Here, a method is presented to simultaneously resolve the determination of the Patterson symmetry and the indexing ambiguity for partial data sets.

A Symmetry-Based Method for LiDAR Point Registration

Abstract: LiDAR point registration is a key procedure for the acquisition of complete point cloud datasets. It has great significance for the fusion of multisource LiDAR data. In general, the widely used methods for LiDAR point registration can be categorized into three types: auxiliary methods, direct methods, and feature methods. However, for the registration of complex objects (e.g., stadium and tower), such methods may face varying degrees of technical problems owing to the unavailability of auxiliary data or targets, requirement of sufficient overlapping areas, and difficulty in feature extraction and matching. In the real world, numerous objects with extremely complicated geometric shapes have the characteristic of symmetry. This study focuses on complex objects with symmetry and tries to exploit their intrinsic symmetry characteristic in order to facilitate their point cloud registration. A symmetry-based method for LiDAR point registration is proposed, in which the general idea is to derive 3-D central axes from multisource point clouds, based on the symmetry of objects. The proposed method consists of six main steps: detection of rotational symmetry, adaptive point cloud slicing, central point extraction, central axis fitting, central axis matching, and orientation and positioning. Comparative experiments and quantitative evaluations are conducted. The experimental results indicate that the proposed framework can achieve satisfactory registration of objects with rotational symmetry.

Symmetric Objects Become Special in Perception Because of Generic Computations in Neurons.

Abstract: Symmetry is a salient visual property: It is easy to detect and influences perceptual phenomena from segmentation to recognition. Yet researchers know little about its neural basis. Using recordings from single neurons in monkey IT cortex, we asked whether symmetry-being an emergent property-induces nonlinear interactions between object parts. Remarkably, we found no such deviation: Whole-object responses were always the sum of responses to the object's parts, regardless of symmetry. The only defining characteristic of symmetric objects was that they were more distinctive compared with asymmetric objects. This was a consequence of neurons preferring the same part across locations within an object. Just as mixing diverse paints produces a homogeneous overall color, adding heterogeneous parts within an asymmetric object renders it indistinct. In contrast, adding identical parts within a symmetric object renders it distinct. This distinctiveness systematically predicted human symmetry judgments, and it explains many previous observations about symmetry perception. Thus, symmetry becomes special in perception despite being driven by generic computations at the level of single neurons.

On skew-symmetric games

Abstract: By resorting to the vector space structure of finite games, skew-symmetric games (SSGs) are proposed and investigated as a natural subspace of finite games. First of all, for two player games, it is shown that the skew-symmetric games form an orthogonal complement of the symmetric games. Then for a general SSG its linear representation is given, which can be used to verify whether a finite game is skew-symmetric. Furthermore, some properties of SSGs are also obtained in the light of its vector subspace structure. Finally, a symmetry-based decomposition of finite games is proposed, which consists of three mutually orthogonal subspaces: symmetric subspace, skew-symmetric subspace and asymmetric subspace. An illustrative example is presented to demonstrate this decomposition.