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Capping protein, a heterodimeric protein composed of α and β subunits, is a key cellular component regulating actin filament assembly and organization. It binds to the barbed ends of the filaments and works as a ‘cap’ by preventing the addition and loss of actin monomers at the end. Here we describe the crystal structure of the chicken sarcomeric capping protein CapZ at 2.1 Å resolution. The structure shows a striking resemblance between the α and β subunits, so that the entire molecule has a pseudo 2‐fold rotational symmetry. CapZ has a pair of mobile extensions for actin binding, one of which also provides concomitant binding to another protein for the actin filament targeting. The mobile extensions probably form flexible links to the end of the actin filament with a pseudo 2 1 helical symmetry, enabling the docking of the two in a symmetry mismatch.

The authors study noncompact surfaces evolving by mean curvature flow (mcf). For an open set of initial data that are C^3-close to round, but without assuming rotational symmetry or positive mean curvature, the authors show that mcf solutions become singular in finite time by forming neckpinches, and they obtain detailed asymptotics of that singularity formation. The results show in a precise way that mcf solutions become asymptotically rotationally symmetric near a neckpinch singularity.

With the development of 3D measurement technology, 3D vision sensors and object pose estimation methods have been developed for robotic loading and unloading. In this work, an end-to-end deep learning method on point clouds, PointNetRGPE, is proposed to estimating the grasping pose of SCARA robot. In PointNetRGPE model, the point cloud and class number are fused into a point-class vector, and several PointNet-like networks are used to estimate the robot grasping pose, containing 3D translation and 1D rotation. Considering that rotational symmetry is very common in man-made and industrial environments, a novel architecture is introduced into PointNetRGPE to solve the pose estimation problem with rotational symmetry in the z -axis direction. Additionally, an experimental platform is built containing an industrial robot and a binocular stereo vision system, and a dataset with three subsets is set up. Finally, the PointNetRGPE is tested on the dataset, and the success rates of three subsets are 98.89%, 98.89%, and 94.44% respectively.

Objective and MethodsThe reliable estimation of phase transition physicochemical properties such as boiling and melting points can be valuable when designing compounds with desired physicochemical properties. This study explores the role of external rotational symmetry in determining boiling and melting points of select organic compounds. Using experimental data from the literature, the entropies of boiling and fusion were obtained for 541 compounds. The statistical significance of external rotational symmetry number on entropies of phase change was determined by using multiple linear regression. In addition, a series of aliphatic hydrocarbons, polysubstituted benzenes, and di-substituted napthalenes are used as examples to demonstrate the role of external symmetry on transition temperature.ResultsThe results reveal that symmetry is not well correlated with boiling point but is statistically significant in melting point.ConclusionThe lack of correlation between the boiling point and the symmetry number reflects the fact that molecules have a high degree of rotational freedom in both the liquid and the vapor. On the other hand, the strong relationship between symmetry and melting point reflects the fact that molecules are rotationally restricted in the crystal but not in the liquid. Since the symmetry number is equal to the number of ways that the molecule can be properly oriented for incorporation into the crystal lattice, it is a significant determinant of the melting point.

Two related questions are discussed. The first is when reflection symmetry in a finite set of i-dimensional subspaces, i ∈ {1, ..., n − 1}, implies full rotational symmetry, that is, the closure of the group generated by the reflections equals O(n). For i = n−1, this has essentially been solved by Burchard, Chambers, and Dranovski, but new results are obtained for i ∈ {1, ..., n−2}. The second question, to which an essentially complete answer is given, is when (full) rotational symmetry with respect to a finite set of i-dimensional subspaces, i ∈ {1, ..., n−2}, implies full rotational symmetry, that is, the closure of the group generated by all the rotations about each of the subspaces equals SO(n). The latter result also shows that a closed set in ℝ n that is invariant under rotations about more than one axis must be a union of spheres with their centers at the origin.

Using the strong-field approximation we systematically investigate the selection rules for high-order harmonic generation and the symmetry properties of the angle-resolved photoelectron spectra for various atomic and molecular targets exposed to one-component and two-component laser fields. These include bicircular fields and orthogonally polarized two-color fields. The selection rules are derived directly from the dynamical symmetries of the driving field. Alternatively, we demonstrate that they can be obtained using the conservation of the projection of the total angular momentum on the quantization axis. We discuss how the harmonic spectra of atomic targets depend on the type of the ground state or, for molecular targets, on the pertinent molecular orbital. In addition, we briefly discuss some properties of the high-order harmonic spectra generated by a few-cycle laser field. The symmetry properties of the angle-resolved photoelectron momentum distribution are also determined by the dynamical symmetry of the driving field. We consider the first two terms in a Born series expansion of the T matrix, which describe the direct and the rescattered electrons. Dynamical symmetries involving time translation generate rotational symmetries obeyed by both terms. However, those that involve time reversal generate reflection symmetries that are only observed by the direct electrons. Finally, we explain how the symmetry properties, imposed by the dynamical symmetry of the driving field, are altered for molecular targets.

In this paper, Steiner and Schwarz symmetrizations, and their most important relatives, the Minkowski, Minkowski-Blaschke, fiber, inner rotational, and outer rotational symmetrizations, are investigated. The focus is on the convergence of successive symmetrals with respect to a sequence of i-dimensional subspaces of ℝ n . Such a sequence is called universal for a family of sets if the successive symmetrals of any set in the family converge to a ball with center at the origin. New universal sequences for the main symmetrizations, for all valid dimensions i of the subspaces, are found by combining two groups of results. The first, published separately, provides finite sets ℱ of subspaces such that reflection symmetry (or rotational symmetry) with respect to each subspace in ℱ implies full rotational symmetry. In the second, proved here, a theorem of Klain for Steiner symmetrization is extended to Schwarz, Minkowski, Minkowski-Blaschke, and fiber symmetrizations, showing that if a sequence of subspaces is drawn from a finite set ℱ of subspaces, the successive symmetrals of any compact convex set converge to a compact convex set that is symmetric with respect to any subspace in ℱ appearing infinitely often in the sequence. It is also proved that for Steiner, Schwarz, and Minkowski symmetrizations, a sequence of i-dimensional subspaces is universal for the class of compact sets if and only if it is universal for the class of compact convex sets, and Klain’s theorem is shown to hold for Schwarz symmetrization of compact sets.

Lattice structures exhibit periodicity along three orthogonal axes, which limits their mechanical isotropy and design flexibility. In this study, the lattice structures with multifold rotational symmetries were proposed and designed based on body-centered cubic (BCC) structures. The specimens of the proposed structures were fabricated through laser powder bed fusion (LPBF) additive manufacturing technology. A finite element model was constructed to analyze the quasi-static compression behavior of multifold rotationally symmetric BCC lattice structures. The lightweight potential of the additive manufactured structures was evaluated through experiments. The effect of the minimum rotation angle on the manufacturability and uniaxial compressive mechanical behavior of the multifold rotationally symmetric BCC lattice structures was investigated through experimental analysis. The results show that the reduction of the minimum rotation angle improves the dimensional accuracy of LPBF-fabricated multifold rotationally symmetric structures and enhances their overall elastic modulus and yield strength. The lattice structure with higher loading capacity can be achieved by reducing the minimum rotation angle. Experimental and finite element simulation results reveal that multifold rotationally symmetric BCC lattice structures exhibit controllable anisotropy in stress distribution and failure modes compared to conventional lattice structures. This study provides a reference for the application of lattice structures in the design of rotationally moving components such as circular toolholders, shafts, pumps, and impellers.

The Lamé curve is an extension of an ellipse, the latter being a special case. Dr. Johan Gielis further extended the Lamé curve in the polar coordinate system by introducing additional parameters (n1, n2, n3; m): rφ=1Acosm4φn2+1Bsinm4φn3−1/n1, which can be applied to model natural geometries. Here, r is the polar radius corresponding to the polar angle φ; A, B, n1, n2 and n3 are parameters to be estimated; m is the positive real number that determines the number of angles of the Gielis curve. Most prior studies on the Gielis equation focused mainly on its applications. However, the Gielis equation can also generate a large number of shapes that are rotationally symmetric and axisymmetric when A = B and n2 = n3, interrelated with the parameter m, with the parameters n1 and n2 determining the shapes of the curves. In this paper, we prove the relationship between m and the rotational symmetry and axial symmetry of the Gielis curve from a theoretical point of view with the condition A = B, n2 = n3. We also set n1 and n2 to take negative real numbers rather than only taking positive real numbers, then classify the curves based on extremal properties of r(φ) at φ = 0, π/m when n1 and n2 are in different intervals, and analyze how n1, n2 precisely affect the shapes of Gielis curves.