Orthogonal Group Research Articles

A new tableau model for representations of the special orthogonal group

A new tableau model for representations of the special orthogonal group

A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization

A Spectral Method for Joint Community Detection and Orthogonal Group Synchronization

Effect of different orthogonal double frequency ultrasonic assisted freezing on the quality of sea bass.

The ice crystals formed in the body of the fish after freezing will cause irreversible damage to the fish's tissues, resulting in a decline in the fish quality. Therefore, based on the single frequency and double frequency ultrasonic freezing technology, the influence of orthogonal ultrasonic on the sea bass quality was studied. The results showed that the orthogonal ultrasonic wave could effectively improve the utilization rate of ultrasonic. In addition, SEM images showed that the muscle tissue in the dual frequency orthogonal ultrasonic assisted freezing group (DOUAF-40 (H) 20 (V)) was more uniform and dense. DOUAF-40 (H) 20 (V) group did not cause excessive oxidation of myofibrin on the one hand, and on the other hand reduced the duration of lipid oxidation in fish. The results showed that the orthogonal ultrasonic freezing technology inhibited the impact on fish quality during the freezing process, which provided a reference for the food freezing industry to improve aquatic products.

On the Geometry of the Flag Varieties of Orthogonal group and Symplectic group

The aim of this paper is to discuss an important discourse, as most literatures highlighted the dimension of partial flag in a certain manner, according to a fixed definition that does not include all possible cases. In the present paper the existence of two different cases of flags in the partial flag varieties of the groups and is explained wholly. In addition to this we will proceed by calculating the dimensions for all the given cases after pertaining the signature of these flags that are studied.

Visualizing the Dickson–Siegel–Eichler–Roy elementary orthogonal group

In this paper, we define a set of elementary orthogonal matrices similar to Vaserstein’s definition of elementary symplectic matrices. We prove that the elementary orthogonal group generated by these matrices is a conjugate of the Dickson–Siegel–Eichler–Roy (DSER) elementary orthogonal group defined on a free quadratic module with a hyperbolic summand.

A higher-dimensional Chevalley restriction theorem for orthogonal groups

We prove a higher-dimensional Chevalley restriction theorem for orthogonal groups, which was conjectured by Chen and Ngô for reductive groups. In characteristic p>2, we also prove a weaker statement. In characteristic 0, the theorem implies that the categorical quotient of a commuting scheme by the diagonal adjoint action of the group is integral and normal. As applications, we deduce some trace identities and a certain multiplicative property of the Pfaffian over an arbitrary commutative algebra.

Synthesis of N,N′-Unsymmetrical 9-Amino-5,7-dimethyl-bispidines

A new approach to the preparation of N,N'-unsymmetrically substituted 9-aminobispidines through aminal bridge removal reaction has been developed, the principal feature of which is the ability to selectively functionalize all three nitrogen atoms. The intermediates of 1,3-diazaadamantane's aminal bridge removal reaction are characterized, and the mechanism of this reaction is proposed based on the analysis of their structures. Representatives of the previously unknown saturated heterocyclic 1,5,9-triazatricyclo[5.3.1.03,8]undecane system were obtained and structurally characterized. Thus, it was possible for the first time to obtain 3,7,9-trisubstituted bispidines containing acetyl, Boc, and benzyl groups at nitrogen atoms, which can be independently removed (orthogonal protective groups).

Attitude Synchronization of a Group of Rigid Bodies Using Exponential Coordinates.

Currently, managing a group of satellites or robot manipulators requires coordinating their motion and work in a cooperative way to complete complex tasks. The attitude motion coordination and synchronization problems are challenging since attitude motion evolves in non-Euclidean spaces. Moreover, the equation of motions of the rigid body are highly nonlinear. This paper studies the attitude synchronization problem of a group of fully actuated rigid bodies over a directed communication topology. To design the synchronization control law, we exploit the cascade structure of the rigid body's kinematic and dynamic models. First, we propose a kinematic control law that induces attitude synchronization. As a second step, an angular velocity-tracking control law is designed for the dynamic subsystem. We use the exponential coordinates of rotation to describe the body's attitude. Such coordinates are a natural and minimal parametrization of rotation matrices which almost describe every rotation on the Special Orthogonal group SO(3). We provide simulation results to show the performance of the proposed synchronization controller.

The proportion of non-degenerate complementary subspaces in classical spaces

Given positive integers $$e_1,e_2$$ , let $$X_i$$ denote the set of $$e_i$$ -dimensional subspaces of a fixed finite vector space $$V=({\mathbb F}_q)^{e_1+e_2}$$ . Let $$Y_i$$ be a non-empty subset of $$X_i$$ and let $$\alpha _i = |Y_i|/|X_i|$$ . We give a positive lower bound, depending only on $$\alpha _1,\alpha _2,e_1,e_2,q$$ , for the proportion of pairs $$(S_1,S_2)\in Y_1\times Y_2$$ which intersect trivially. As an application, we bound the proportion of pairs of non-degenerate subspaces of complementary dimensions in a finite classical space that intersect trivially. This problem is motivated by an algorithm for recognizing classical groups. By using techniques from algebraic graph theory, we are able to handle orthogonal groups over the field of order 2, a case which had eluded Niemeyer, Praeger, and the first author.

Non-invertible Condensation, Duality, and Triality Defects in 3+1 Dimensions

We discuss a variety of codimension-one, non-invertible topological defects in general 3+1d QFTs with a discrete one-form global symmetry. These include condensation defects from higher gauging of the one-form symmetries on a codimension-one manifold, each labeled by a discrete torsion class, and duality and triality defects from gauging in half of spacetime. The universal fusion rules between these non-invertible topological defects and the one-form symmetry surface defects are determined. Interestingly, the fusion coefficients are generally not numbers, but 2+1d TQFTs, such as invertible SPT phases, $${\mathbb {Z}}_N$$ gauge theories, and $$U(1)_N$$ Chern-Simons theories. The associativity of these algebras over TQFT coefficients relies on nontrivial facts about 2+1d TQFTs. We further prove that some of these non-invertible symmetries are intrinsically incompatible with a trivially gapped phase, leading to nontrivial constraints on renormalization group flows. Duality and triality defects are realized in many familiar gauge theories, including free Maxwell theory, non-abelian gauge theories with orthogonal gauge groups, $${{{\mathcal {N}}}}=1,$$ and $${{{\mathcal {N}}}}=4$$ super Yang-Mills theories.

Refinement of von Neumann-type inequalities on product Eaton triples

In this paper, a von Neumann-type inequality is studied on an Eaton triple $ (V,G,D) $, where $ V $ is a real inner product space, $ G $ is a compact subgroup of the orthogonal group $ O (V) $, and $ D \subset V $ is a closed convex cone. By using an inner structure of an Eaton triple, a refinement of this inequality is shown. In the special case $ G = O ( V ) $, a refinement of the Cauchy-Schwarz inequality is obtained.

SYK Model with global symmetries in the double scaling limit

We discuss the double scaling limit of the SYK model with global symmetries. We develop the chord diagram techniques to compute the moments of the Hamiltonian and the two point function in the presence of arbitrary chemical potential. We also derive a transfer matrix acting on an auxiliary hilbert space which can capture the chord diagram contributions. We present explicit results for the case of classical group symmetries namely orthogonal, unitary and symplectic groups. We also find the partition functions at fixed charges.

Spaces of states of the two-dimensional $O(n)$ and Potts models

We determine the spaces of states of the two-dimensional O(n)O(n) and QQ-state Potts models with generic parameters n,Q\in \mathbb{C}n,Q∈ℂ as representations of their known symmetry algebras. While the relevant representations of the conformal algebra were recently worked out, it remained to determine the action of the global symmetry groups: the orthogonal group for the O(n)O(n) model, and the symmetric group S_QSQ for the QQ-state Potts model. We do this by two independent methods. First we compute the twisted torus partition functions of the models at criticality. The twist in question is the insertion of a group element along one cycle of the torus: this breaks modular invariance, but allows the partition function to have a unique decomposition into characters of irreducible representations of the global symmetry group. Our second method reduces the problem to determining branching rules of certain diagram algebras. For the O(n)O(n) model, we decompose representations of the Brauer algebra into representations of its unoriented Jones-Temperley-Lieb subalgebra. For the QQ-state Potts model, we decompose representations of the partition algebra into representations of the appropriate subalgebra. We find explicit expressions for these decompositions as sums over certain sets of diagrams, and over standard Young tableaux. We check that both methods agree in many cases. Moreover, our spaces of states are consistent with recent bootstrap results on four-point functions of the corresponding CFTs.

The Heat Equation on Submanifolds in Lie Groups and Random Motions on Spheres

We studied the random variable Vt=volS2(gtB∩B), where B is a disc on the sphere S2 centered at the north pole and (gt)t≥0 is the Brownian motion on the special orthogonal group SO(3) starting at the identity. We applied the results of the theory of compact Lie groups to evaluate the expectation of Vt for 0≤t≤τ, where τ is the first time when Vt vanishes. We obtained an integral formula using the heat equation on some Riemannian submanifold ΓB seen as the support of the function f(g)=volS2(gB∩B) immersed in SO(3). The integral formula depends on the mean curvature of ΓB and the diameter of B.

Invariants for the Weil representation and modular units for orthogonal groups of signature (2,2)

We show that the space of invariants for the Weil representation for discriminant groups which contain self-dual isotropic subgroups is spanned by the characteristic functions of the self-dual isotropic subgroups. As an application, we construct modular units for certain orthogonal groups in signature (2,2) using Borcherds products.

In-motion initial alignment method using Cayley–Kalman filter on special orthogonal group

Aiming at the in-motion initial alignment problem of strapdown inertial navigation system (SINS), this paper proposed a precise direct in-motion alignment method based on attitude matrix representation and Cayley–Kalman filter on the special orthogonal group. There are two innovations in this paper. First, as an element of the special orthogonal group [Formula: see text], the error attitude matrix is used to construct an in-motion alignment model, and it is transformed into the vector space to represent the attitude error state. The error matrix was used as a state to decrease the influence of the state-dependent problem. Second, this method uses the Cayley transform to replace the Taylor expansion to derive the update equation, which is a no approximation mapping relation to assure accuracy when deriving the filtering algorithm. Simulation and experimental results demonstrate the proposed alignment method has advantages over the existing methods in alignment accuracy and alignment time and performs well in the in-motion alignment process of SINS.

Security of medical images based on special orthogonal group and Galois field

Security of medical images based on special orthogonal group and Galois field

Relative Haagerup property for arbitrary von Neumann algebras

We introduce the relative Haagerup approximation property for a unital, expected inclusion of arbitrary von Neumann algebras and show that if the smaller algebra is finite then the notion only depends on the inclusion itself, and not on the choice of the conditional expectation. Several variations of the definition are shown to be equivalent in this case, and in particular the approximating maps can be chosen to be unital and preserving the reference state. The concept is then applied to amalgamated free products of von Neumann algebras and used to deduce that the standard Haagerup property for a von Neumann algebra is stable under taking free products with amalgamation over finite-dimensional subalgebras. The general results are illustrated by examples coming from q-deformed Hecke-von Neumann algebras and von Neumann algebras of quantum orthogonal groups.

Aritmethic lattices of SO(1,n) and units of group rings

We establish that standard arithmetic subgroups of a special orthogonal group SO(1,n) are conjugacy separable. As an application we deduce this property for unit groups of certain integer group rings. We also prove that finite quotients of group of units of any of these group rings determines the original group ring.

Spectral Decomposition of Discrepancy Kernels on the Euclidean Ball, the Special Orthogonal Group, and the Grassmannian Manifold

To numerically approximate Borel probability measures by finite atomic measures, we study the spectral decomposition of discrepancy kernels when restricted to compact subsets of mathbb {R}^d. For restrictions to the Euclidean ball in odd dimensions, to the rotation group textrm{SO}(3), and to the Grassmannian manifold mathcal {G}_{2,4}, we compute the kernels’ Fourier coefficients and determine their asymptotics. The L_2-discrepancy is then expressed in the Fourier domain that enables efficient numerical minimization based on the nonequispaced fast Fourier transform. For textrm{SO}(3), the nonequispaced fast Fourier transform is publicly available, and, for mathcal {G}_{2,4}, the transform is derived here. We also provide numerical experiments for textrm{SO}(3) and mathcal {G}_{2,4}.