Rotation Of Group Research Articles

Dihedral beams

Abstract In this work, a group theory-based formulation that introduces new classes of dihedral-symmetric beams is presented. Our framework leverages the algebraic properties of the dihedral group of rotations and reflections to transform input beams into closed-form families of dihedral-invariant wavefields, which will be referred to as dihedral beams. Each transformation is associated with a specific dihedral group in such a way that each family of dihedral beams exhibits the symmetries of its corresponding group. Our approach is inspired by one of the outcomes of this work: elegant Hermite–Gauss beams can be described as a dihedral interference pattern of elegant traveling waves, a new set of solutions to the paraxial equation also developed in this paper. Particularly, when taking elegant traveling waves as input beams, they transform into elegant dihedral beams possessing quasi-crystalline properties and including features like phase singularities, self-healing, and pseudo-nondiffracting propagation, as well as containing elegant Hermite and Laguerre–Gauss beams as special cases. Our approach can be applied to arbitrary scalar and vector input beams and constitutes a general group-theory formulation that can be extended beyond the dihedral group.

The Partially-Matched-Sample Correction in Pseudo Panel Minimum Distance Estimation

Abstract Certain repeated cross-sectional data sets, such as the Current Population Survey (CPS), use special sampling designs by which samples from different times periods are partially-matched. This paper proposes a correction to the optimal weighting matrix in minimum distance (MD) estimation of pseudo panel models to account for such partially-matched samples. This partially-matched-sample correction may be needed if the sample matching rate is nontrivial and, at the same time, there is a fixed effect, a serially correlated idiosyncratic error, or both in the underlying linear panel data model data generating process, all of which lead to a block diagonal structure of the optimal weighting matrix. Using the correction can result in considerable efficiency gains both in finite sample and asymptotically. Furthermore, it is shown that this correction is needed not only for the optimal MD estimator but for any MD estimator to make the inference right. As an illustration, the correction is applied to the classical question of estimating the monetary return to education using the yearly Merged Outgoing Rotation Group (MORG) files from CPS.

Symmetry invariants and classes of quasiparticles in magnetically ordered systems having weak spin-orbit coupling

Symmetry invariants of a group specify the classes of quasiparticles, namely the classes of projective irreducible co-representations in systems having that symmetry. More symmetry invariants exist in discrete point groups than the full rotation group O(3), leading to new quasiparticles restricted to lattices that do not have any counterpart in a vacuum. We focus on the fermionic quasiparticle excitations under “spin-space group” symmetries, applicable to materials where long-range magnetic order and itinerant electrons coexist. We provide a list of 218 classes of new quasiparticles that can only be realized in the spin-space groups. These quasiparticles have at least one of the following properties that are qualitatively distinct from those discovered in magnetic space group(MSG)s, and distinct from each other:(i) degree of degeneracy,(ii) dispersion as function of momentum, and(iii) rules of coupling to external probe fields. We rigorously prove this result as a theorem that directly relates these properties to the symmetry invariants, and then illustrate this theorem with a concrete example, by comparing three 12-fold fermions having different sets of symmetry invariants including one discovered in MSG. Our approach can be generalized to realize more quasiparticles whose little co-groups are beyond those considered in our work.

Differences in spinal axial rotation angles during the lumbar-locked rotation test between hyper and normal thoracic rotation groups: A cross-sectional study.

Trunk rotation is important in many sporting activities The thoracic spine has reciprocal relationships with the lumbar and pelvic spines, such that reduced flexibility in the lumbar or thoracic spine can lead to abnormal patterns of trunk movement and pain. However, few studies have investigated the relative trunk rotation mobilities of the thorax, lumbar, and pelvis. To compare thoracic, lumbar, and pelvic rotation angles during the lumbar-locked rotation test between hyper and normal thoracic rotation groups. Thirty-two young, active participants were enrolled in this study. After the attachment of inertial measurement units at the T1, T7, T12, L3, and S2 levels, the participants were required to stand in a comfortable upright posture for 5s to allow postural measurements before performing the lumbar-locked rotation test. The participants were then divided into hyper thoracic rotation and normal thoracic rotation groups based on T1 angle measurements obtained during the lumbar-locked rotation test. The hyper thoracic rotation group had significantly higher thoracic rotation angles on both the right (p< 0.05) and left (p< 0.05) sides compared with the normal thoracic rotation group. Furthermore, we observed flat lumbar lordosis in the hyper thoracic rotation group compared with the normal thoracic rotation group, particularly in the lower lumbar region in standing posture. Our data suggest that evaluations of thoracic mobility should consider relative thoracic, lumbar, and pelvic motions, rather than the T1 angle alone. This study provides a basis for health professionals to evaluate movement dysfunctions associated with thoracic hypermobility.

Influence of the rotation of the diverting loop ileostomy in rectal cancer surgery on small-bowel obstruction: A multicenter prospective study conducted by the Clinical Study Group of Osaka University, Colorectal Group

AimsWhether rotation of a diverting loop ileostomy during rectal cancer surgery, for reducing the catastrophic effect of an anastomotic leakage, affects the incidence of small-bowel obstruction has not been fully investigated. The purpose of this study is to explore whether technical maneuvers in diverting loop ileostomy creation, including its rotation, are associated with increased incidence of small-bowel obstruction in rectal tumor surgery. MethodsThis multicenter prospective study was conducted by the Clinical Study Group of Osaka University, which comprises 24 major institutions. Patients with rectal adenocarcinoma scheduled for laparoscopic/robotic low anterior resection or intersphincteric resection with a diverting loop ileostomy were included. A total of 451 patients were prospectively enrolled between July 2015 and April 2021. The primary endpoint was the relevance of loop ileostomy rotation to the incidence of small-bowel obstruction; the secondary endpoints included the origin of the small-bowel obstruction and length of hospital stay. ResultsSmall-bowel obstruction was observed in 10.8% in the nonrotated group and 12.3% in the rotated group, with no significant difference (P > .99). The only risk factor identified for small-bowel obstruction was distance from the ileocecal valve, with a significant difference in 16 patients (7.3%) with a distance of ≤30 cm and 16 patients (15.4%) in a distance of >30 cm (P = .028). ConclusionRotation of the diverting loop ileostomy had no significant effect on the incidence of small-bowel obstruction.

Modeling and Optimization Algorithm for Rotational Irrigation Group Based on Connectivity Weighted Index

Abstract Manual calculations for rotational irrigation groups are inefficient, and a single objective cannot provide farmers with diverse options. Based on considering both engineering standards and practical needs, this study designs a connectivity index and proposes a new model with the optimization objectives of balanced flow and minimum connectivity. To address infeasible solutions and constraint issues, A hybrid genetic algorithm based on variable neighborhoods is adopted to enhance the search capability of the classical non-dominated sorting genetic algorithm. The effectiveness and universality of the model and algorithm are verified through multiple real-world cases. The research shows that the proposed multi-objective model is more practical than a single-objective model, and farmers can be provided with diverse Pareto solutions by adjusting the weight index ω. The variable neighborhood search algorithm enhances the search capability within the radius threshold, improves population diversity, and analyzes the radius threshold and branch pipe flow parameters to explore factors that affect algorithm performance and indicators.

Relationship between the course of postoperative pelvic axis rotation and shoulder balance in patients with Lenke types 1 and 2 adolescent idiopathic scoliosis

BackgroundPosteroanterior radiographs of patients with adolescent idiopathic scoliosis (AIS) show bilateral differences in the iliac wings. This is due to pelvic axis rotation (PAR) associated with scoliosis. We often encounter cases wherein the PAR changes with surgery and postoperatively. We investigated the course of preoperative PAR and the relationship between PAR and shoulder balance in patients with Lenke 1,2 AIS. MethodsIn total, 28 patients with Lenke 1,2 AIS undergoing scoliosis correction were included. The PAR and shoulder parameters were measured on posteroanterior radiographs. The correlation between the measured parameters and the extent of changes in each parameter was also examined. ResultsEleven patients (39.3 %) underwent preoperative PAR. Six patients (21.4 %) showed a greater change in PAR from 1 week to 2 years postoperatively. The rotation did not change significantly from preoperatively to immediately postoperatively but changed during the first three months postoperatively. The rotation group had significantly more balanced shoulders at 2 years (P = 0.025). The rotation group had a greater change in shoulder balance in the postoperative course significantly (P < 0.05). The extent of change in pelvic rotation from 1 week to 2 years postoperatively correlated with the extent of change in shoulder balance. ConclusionThe PAR in patients with Lenke 1,2 AIS significantly changed during 3 months postoperatively. Patients with Lenke 1,2 AIS with preoperative PAR can be expected to have postoperative shoulder rebalancing.

Magneto-viscoelastic rod model for hard-magnetic soft rods under 3D large deformation: Theory and numerical implementation

The main purpose of this work is to develop a three-dimensional (3D) viscoelastic rod model for hard-magnetic soft (HMS) rods under large deformation which are widely used active structures in soft robotics. To do so, the Simo’s viscoelasticity theory has been rationally incorporated into the geometrically exact 3D curved rod model. The proposed model includes the deformation modes of axial tension, shear, bending, and torsion, which is applicable to the HMS rods with arbitrarily initial curved and twisted geometries under 3D large deformation. The viscoelastic constitutive equations of the HMS rod in the present formulation are formulated, which include the general relaxation functions. To obtain the expression for the magnetic load, the rotation-based magnetic free energy density is introduced, and the governing equations of the HMS rod with magnetic load and body force are presented. To obtain the numerical implementation, an implicit time integration algorithm that simply extends the generalized-α method for the rotation group, and the corresponding variational formulation and its linearization of the rod model are derived. To validate the model, five numerical examples, including 2D dynamic buckling, 3D static, and 3D dynamic problem are presented. The dynamic problems include the dynamic snap-through behavior of a bistable HMS arch and damped oscillation of a quarter arc cantilever under 3D deformation. The simulation results show good agreement with the results reported in the literature.

Ternary Associativity and Ternary Lie Algebras at Cube Roots of Unity

We propose a new approach to extend the notion of commutator and Lie algebra to algebras with ternary multiplication laws. Our approach is based on the ternary associativity of the first and second kinds. We propose a ternary commutator, which is a linear combination of six triple products (all permutations of three elements). The coefficients of this linear combination are the cube roots of unity. We find an identity for the ternary commutator that holds due to the ternary associativity of either the first or second kind. The form of this identity is determined by the permutations of the general affine group GA(1,5)⊂S5. We consider this identity as a ternary analog of the Jacobi identity. Based on the results obtained, we introduce the concept of a ternary Lie algebra at cube roots of unity and provide examples of such algebras constructed using ternary multiplications of rectangular and three-dimensional matrices. We also highlight the connection between the structure constants of a ternary Lie algebra with three generators and an irreducible representation of the rotation group. The classification of two-dimensional ternary Lie algebras at cube roots of unity is proposed.

Impact of the Aortomitral Positional Anatomy on Atrioventricular Conduction Disorder Following Mitral Valve Surgery.

Variations in the aortomitral positional anatomy, including aortic root rotation appear to be related to variations in the location of the conduction system, including the bundle of His. However, little is known about their clinical significance. This study included 147 patients with normal ECGs who underwent mitral valve surgery. The aortomitral anatomy was classified using preoperative 3-dimensional transesophageal echocardiography, and postoperative conduction disorders, including atrioventricular block and bundle branch block, were analyzed. Variations classified as aortomitral appearance were designated as having a center appearance (85.7%, n=126/147) or lateral appearance (14.3%, n=21/147) on the basis of whether the aortic root was located at the center or was shifted to the left fibrous trigone side. Subsequently, those with a center appearance, aortic root rotation was classified as having a center rotation (83.3% [n=105/126]), in which the commissure of the left and noncoronary aortic leaflet was located at the center, lateral rotation (14.3% [n=18/126]), rotated to the left trigone side, or medial rotation (2.4% [n=3/126]), rotated to the right. The incidence of 3-month persistent new-onset conduction disorder was higher in the lateral appearance than the center appearance group (21.1% versus 5.0%; P=0.031) and higher in the lateral rotation than in the center or medial rotation groups (29.4% versus 1.0% versus 0.0%, respectively; P<0.001). Aortomitral variations can be classified using 3-dimensional transesophageal echocardiography. Lateral appearance and lateral rotation are risk factors for conduction disorders in mitral valve surgery.

Comparative study on the effects of role changes in simulation training among Korean nursing students

BackgroundSimulation education, based on experiential learning, helps nursing students develop coping strategies through reflective problem-solving in a safe environment. This enhances their ability to respond to similar new experiences and improves their caring abilities. Most nursing schools use simulation education, although the applications vary. This study identified a more effective simulation teaching method for nursing students by comparing the effects between two groups: one that performs one role per scenario and another that rotates through multiple roles per scenario. MethodsUsing a randomized control group pre-post design, we equally divided 62 students aged ≥19 years enrolled at a university in Changwon city, South Korea into a single-scenario role rotation group (SRRG) and a multiscenario role fixation group (MRFG). Data were analyzed by performing χ2 tests, independent t-tests, and paired t-tests. ResultsLearning confidence and critical thinking significantly changed in the SRRG; clinical competency significantly changed in the MRFG. ConclusionsExposure to multiple scenario simulations improves clinical competency. Changing roles in a single scenario boost learning confidence and critical thinking disposition.

Parallelly Sliced Optimal Transport on Spheres and on the Rotation Group

Sliced optimal transport, which is basically a Radon transform followed by one-dimensional optimal transport, became popular in various applications due to its efficient computation. In this paper, we deal with sliced optimal transport on the sphere Sd-1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {S}^{d-1}$$\\end{document} and on the rotation group SO(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SO}(3)$$\\end{document}. We propose a parallel slicing procedure of the sphere which requires again only optimal transforms on the line. We analyze the properties of the corresponding parallelly sliced optimal transport, which provides in particular a rotationally invariant metric on the spherical probability measures. For SO(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SO}(3)$$\\end{document}, we introduce a new two-dimensional Radon transform and develop its singular value decomposition. Based on this, we propose a sliced optimal transport on SO(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{SO}(3)$$\\end{document}. As Wasserstein distances were extensively used in barycenter computations, we derive algorithms to compute the barycenters with respect to our new sliced Wasserstein distances and provide synthetic numerical examples on the 2-sphere that demonstrate their behavior for both the free- and fixed-support setting of discrete spherical measures. In terms of computational speed, they outperform the existing methods for semicircular slicing as well as the regularized Wasserstein barycenters.

Impact of lateral cortical notching on biomechanical performance in cephalomedullary nailing for unstable pertrochanteric fractures

PurposeIn pertrochanteric femur fractures the risk for fracture healing complications increases with the complexity of the fracture. In addition to dynamization along the lag screw, successful fracture healing may also be facilitated by further dynamization along the shaft axis. The aim of this study was to investigate the mechanical stability of additional axial notch dynamization compared to the standard treatment in an unstable pertrochanteric femur fracture treated with cephalomedullary nailing.MethodsIn 14 human cadaver femora, an unstable pertrochanteric fracture was stabilized with a cephalomedullary nail. Additional axial notch dynamization was enabled in half of the samples and compared against the standard treatment (n = 7). Interfragmentary motion, axial construct stiffness and load to failure were investigated in a stepwise increasing cyclic load protocol.ResultsMean load to failure (1414 ± 234 N vs. 1428 ± 149 N, p = 0.89) and mean cycles to failure (197,129 ± 45,087 vs. 191,708 ± 30,490, p = 0.81) were equivalent for axial notch dynamization and standard treatment, respectively. Initial construct stiffness was comparable for both groups (axial notch dynamization 684 [593–775] N/mm, standard treatment 618 [497–740] N/mm, p = 0.44). In six out of seven specimens the additional axial dynamization facilitated interfragmentary compression, while maintaining its mechanical stability. After initial settling of the constructs, there were no statistically significant differences between the groups for either subsidence or rotation of the femoral head fragment (p ≤ 0.30).ConclusionAxial notch dynamization provided equivalent mechanical stability compared to standard treatment in an unstable pertrochanteric fracture. Whether the interfragmentary compression generated by axial notch dynamization will promote fracture healing through improved fracture reduction needs to be evaluated clinically.

Characterising the Haar measure on the [formula omitted]-adic rotation groups via inverse limits of measure spaces

We determine the Haar measure on the compact p-adic special orthogonal groups of rotations SO(d)p in dimension d=2,3, by exploiting the machinery of inverse limits of measure spaces, for every prime p>2. We characterise the groups SO(d)p as inverse limits of finite groups, of which we provide parametrisations and orders, together with an equivalent description through a multivariable Hensel lifting. Supplying these finite groups with their normalised counting measures, we get an inverse family of Haar measure spaces for each SO(d)p. Finally, we constructively prove the existence of the so-called inverse limit measure of these inverse families, which is explicitly computable, and prove that it gives the Haar measure on SO(d)p. Our results pave the way towards the study of the irreducible projective unitary representations of the p-adic rotation groups, with potential applications to the recently proposed p-adic quantum information theory.

Classification of rotation-invariant biomedical images using equivariant neural networks

Transmission electron microscopy (TEM) is an imaging technique used to visualize and analyze nano-sized structures and objects such as virus particles. Light microscopy can be used to diagnose diseases or characterize e.g. blood cells. Since samples under microscopes exhibit certain symmetries, such as global rotation invariance, equivariant neural networks are presumed to be useful. In this study, a baseline convolutional neural network is constructed in the form of the commonly used VGG16 classifier. Thereafter, it is modified to be equivariant to the p4 symmetry group of rotations of multiples of 90° using group convolutions. This yields a number of benefits on a TEM virus dataset, including higher top validation set accuracy by on average 7.6% and faster convergence during training by on average 23.1% of that of the baseline. Similarly, when training and testing on images of blood cells, the convergence time for the equivariant neural network is 7.9% of that of the baseline. From this it is concluded that augmentation strategies for rotation can be skipped. Furthermore, when modelling the accuracy versus amount of TEM virus training data with a power law, the equivariant network has a slope of − 0.43 compared to − 0.26 of the baseline. Thus the equivariant network learns faster than the baseline when more training data is added. This study extends previous research on equivariant neural networks applied to images which exhibit symmetries to isometric transformations.

Minimizing under relaxed symmetry constraints: Triple and N-junctions

We consider a nonnegative potential $W:\mathbb{R}^2\rightarrow\mathbb{R}$ invariant under the action of the rotation group $C_N$ of the regular polygon with $N$ sides, $N\geq 3$. We assume that $W$ has $N$ nondegenerate zeros and prove the existence of a $N$-junction solution to the vector Allen-Cahn equation. The proof is variational and is based on sharp lower and upper bounds for the energy and on a new pointwise estimate for vector minimizers.

Quench dynamics in lattices above one dimension: The free fermionic case

We begin a systematic investigation of quench dynamics in higher-dimensional lattice systems considering the case of noninteracting fermions with conserved particle number. We prepare the system in a translational-invariant nonequilibrium initial state, the simplest example being a classical configuration with fermions at fixed positions on the lattice, and let it evolve in time. We characterize the system's dynamics by measuring the entanglement between a finite connected region and its complement. We observe the transmutation of entanglement entropy into thermodynamic entropy and investigate how this process depends on the shape and orientation of the region with respect to the underlying lattice. Interestingly, we find that irregular regions display a distinctive multislope entanglement growth, while the dependence on the orientation angle is generically fairly weak. This is particularly true for regions with a large (discrete) rotational symmetry group. The main tool of our analysis is the celebrated quasiparticle picture of Calabrese and Cardy, which we generalize to describe the case at hand. Specifically, we show that for generic initial configurations (even when restricting to classical ones) one has to allow for the production of multiplets involving n&gt;2 quasiparticles and carrying nondiagonal correlations. We obtain quantitatively accurate predictions, tested against exact numerics, and propose an efficient Monte Carlo based scheme to evaluate them for arbitrary connected regions of generic higher-dimensional lattices. Published by the American Physical Society 2024

Time budgets differ in horses during continuous and space-restricted rotational grazing

Horses can become obese and develop related health issues such as laminitis from excessive grazing on high-quality pasture grass; limiting pasture intake can allow weight loss to occur. The objective of this study was to determine the effect of space-restricted rotational grazing on body weight (BW) and time budgets in horses. Eight mature geldings and mares with maintenance-only requirements were randomly assigned to either a space-restricted rotational grazing group (SRG; BW 512 ± 6 kg; n = 4) or a continuous grazing group (CG; BW 517 ± 49 kg; n = 4) for 42 d SRG horses grazed an area with dimensions to provide 80–90 % of mean digestible energy requirement for the 4 horses over a 7-d grazing period; whereas, the CG horses continuously grazed similar non-toxic endophyte-infected tall fescue pasture providing greater than maintenance requirements for the 42 d Horses in the SRG group were moved to a new area every 7 d for 6 weeks. On d 7 at 1600 h of each week, horses were brought inside, and feed was withheld overnight. At 0700 h the next day, BWs were recorded prior to turnout. Observers recorded behaviors simultaneously on SRG and CG horses every six minutes throughout the day three days per week according to an ethogram. This included 30 s scans of all horses. Proportion of grazing and standing had an inverse relationship. Proportion of grazing was affected by the treatment by time interaction, which grazing was displayed more in SRG than CG during weeks 2 and 3, and then reversed weeks 4, 5 and 6.

Invariant measures on p-adic Lie groups: the p-adic quaternion algebra and the Haar integral on the p-adic rotation groups

We provide a general expression of the Haar measure—that is, the essentially unique translation-invariant measure—on a p-adic Lie group. We then argue that this measure can be regarded as the measure naturally induced by the invariant volume form on the group, as it happens for a standard Lie group over the reals. As an important application, we next consider the problem of determining the Haar measure on the p-adic special orthogonal groups in dimension two, three and four (for every prime number p). In particular, the Haar measure on SO(2,Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SO}(2,\\mathbb {Q}_p)$$\\end{document} is obtained by a direct application of our general formula. As for SO(3,Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SO}(3,\\mathbb {Q}_p)$$\\end{document} and SO(4,Qp)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext {SO}(4,\\mathbb {Q}_p)$$\\end{document}, instead, we show that Haar integrals on these two groups can conveniently be lifted to Haar integrals on certain p-adic Lie groups from which the special orthogonal groups are obtained as quotients. This construction involves a suitable quaternion algebra over the field Qp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {Q}_p$$\\end{document} and is reminiscent of the quaternionic realization of the real rotation groups. Our results should pave the way to the development of harmonic analysis on the p-adic special orthogonal groups, with potential applications in p-adic quantum mechanics and in the recently proposed p-adic quantum information theory.

Comparative Analysis of Thoracic Rotation Exercises: Range of Motion Improvement in Standing and Quadruped Variants.

There have been few investigations into the effectiveness of thoracic spine exercises for improving thoracic range of motion (ROM) in any plane. This study assessed the effectiveness of two thoracic spine exercises: one in the quadruped position and one in the thoracic standing position. We determined how these exercises affect thoracic spine mobility ROM over a 2-week intervention period. Thirty-nine healthy participants were enrolled and assigned to a Quadruped Thoracic Rotation group (n=17 participants: 9 females and 8 males) or Flamenco Thoracic Spine Rotation group (n=22: 14 females and 8 males). All participants were administered a KOJI AWARENESSTM screening test, and the initial thoracic spine ROM before intervention exercise was measured in a laboratory setting. Quadruped Thoracic Rotation was performed as the quadruped exercise and Flamenco Thoracic Spine Rotation as the standing exercise. The KOJI AWARENESSTM thoracic spine test and ROM were evaluated on the day after the first exercise session and again after the program. Despite their different approaches to thoracic mobility, the quadruped exercise and standing exercise achieved equivalent improvement in thoracic ROM after 2 weeks. Practitioners have a range of exercise options for enhancing thoracic mobility based on their environmental or task-specific needs.