Orthogonal Group Research Articles

Improved Metal-Free Approach for the Synthesis of Protected Thiol Containing Thymidine Nucleoside Phosphoramidite and Its Application for the Synthesis of Ligatable Oligonucleotide Conjugates.

Oligonucleotide conjugates are versatile scaffolds that can be applied in DNA-based screening platforms and ligand display or as therapeutics. Several different chemical approaches are available for functionalizing oligonucleotides, which are often carried out on the 5' or 3' end. Modifying oligonucleotides in the middle of the sequence opens the possibility to ligate the conjugates and create DNA strands bearing multiple different ligands. Our goal was to establish a complete workflow that can be applied for such purposes from monomer synthesis to templated ligation. To achieve this, a monomer is required with an orthogonal functional group that can be incorporated internally into the oligonucleotide sequence. This is followed by conjugation with different molecules and ligation with the help of a complementary template. Here, we show the synthesis and the application of a thiol-modified thymidine nucleoside phosphoramidite to prepare ligatable oligonucleotide conjugates. The conjugations were performed both in solution and on solid phase, resulting in conjugates that can be assembled into multivalent oligonucleotides decorated with tissue-targeting peptides using templated ligation.

Neighboring extremal nonlinear model predictive control of a rigid body on SO(3)

AbstractThe issue of implementing nonlinear model predictive control (NMPC) on mechanical systems evolving on special orthogonal group (SO(3)) is taken into consideration in the first place. Necessary conditions of optimality are extracted based on Lie group variational integrators, leading to a two-point boundary value problem (TPBVP) which is solved using sensitivity derivatives and indirect shooting methods. Fast Newton-like methods referred to as fast solvers which are commonly used to solve the TPBVP are established based on the repetition of a nonlinear process. The numerical schemes employed to alleviate the computation burden consist of eliminating some constraint-related but non-essential terms in the trend of sensitivity derivatives calculation and for solving the TPBVP equations. As another claim, assuming that a first attempt to resolve the NMPC problem is accessible, the problem subjected to some changes in its initial conditions (due to some re-planning schemes) can be resolved cost-effectively based on it. Instead of solving the whole optimization process from the scratch, the optimal control inputs and states of the system are updated based on the neighboring extremal (NE) method. For this purpose, two approaches are considered: applying NE method on the first solution that leads to a neighboring optimal solution, or assisting this latter by updating the NMPC-related optimization using exact TPBVP equations at some predefined intermediate steps. It is shown through an example that the first method is not accurate enough due to error accumulations. In contrast, the second method preserves the accuracy while reducing the computation time significantly.

The essential p-dimension of the split finite quasi-simple groups of classical Lie type

In this paper, we compute the essential p-dimension of the split finite quasi-simple groups of classical Lie type at the defining prime, specifically the quasi-simple groups arising from the general linear and special linear groups, the symplectic groups, and the orthogonal groups.

Approximate and discrete Euclidean vector bundles

Abstract We introduce $\varepsilon $ -approximate versions of the notion of a Euclidean vector bundle for $\varepsilon \geq 0$ , which recover the classical notion of a Euclidean vector bundle when $\varepsilon = 0$ . In particular, we study Čech cochains with coefficients in the orthogonal group that satisfy an approximate cocycle condition. We show that $\varepsilon $ -approximate vector bundles can be used to represent classical vector bundles when $\varepsilon> 0$ is sufficiently small. We also introduce distances between approximate vector bundles and use them to prove that sufficiently similar approximate vector bundles represent the same classical vector bundle. This gives a way of specifying vector bundles over finite simplicial complexes using a finite amount of data and also allows for some tolerance to noise when working with vector bundles in an applied setting. As an example, we prove a reconstruction theorem for vector bundles from finite samples. We give algorithms for the effective computation of low-dimensional characteristic classes of vector bundles directly from discrete and approximate representations and illustrate the usage of these algorithms with computational examples.

When are the natural embeddings of classical invariant rings pure?

Abstract Consider a reductive linear algebraic group G acting linearly on a polynomial ring S over an infinite field; key examples are the general linear group, the symplectic group, the orthogonal group, and the special linear group, with the classical representations as in Weyl’s book: For the general linear group, consider a direct sum of copies of the standard representation and copies of the dual; in the other cases, take copies of the standard representation. The invariant rings in the respective cases are determinantal rings, rings defined by Pfaffians of alternating matrices, symmetric determinantal rings and the Plücker coordinate rings of Grassmannians; these are the classical invariant rings of the title, with $S^G\subseteq S$ being the natural embedding. Over a field of characteristic zero, a reductive group is linearly reductive, and it follows that the invariant ring $S^G$ is a pure subring of S, equivalently, $S^G$ is a direct summand of S as an $S^G$ -module. Over fields of positive characteristic, reductive groups are typically no longer linearly reductive. We determine, in the positive characteristic case, precisely when the inclusion $S^G\subseteq S$ is pure. It turns out that if $S^G\subseteq S$ is pure, then either the invariant ring $S^G$ is regular or the group G is linearly reductive.

Improvement of RSS-Based Measurement Based on Adaptive Kalman Filter Considering the Anisotropy on Antenna in Dynamic Environment

The classic propagation model of received signal strength (RSS) measurement is significantly affected by the environment. This paper divides the error sources of RSS ranging into three parts and deals with them in different methods. First, considering the influence of UAV pose on RSS, the compensation model is established through Gaussian Process Regression (GPR) for the first time by mapping the quaternion of the UAV and the attenuation caused by the influence of anisotropy on the antenna in the corresponding the special orthogonal group <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">SO</i> (3). On this basis, the distribution of random compensation variables is derived by unscented transformation (UT) and covariance propagation. Besides, a polynomial function is employed for fitting the distance-dependent environment noise. Then, to deal with the RSS measurement instability caused by the environment noise and the uncertainty of compensation, the adaptive Kalman filter (AKF) is introduced. Finally, an UWB-based environmental adaptive algorithm (UEAA) is proposed to improve ranging precision by using the UWB device in a dynamic environment. By setting the threshold parameter, the UEAA enables the classic RSS ranging model to have a certain adaptive ability for the dynamic environment of UAV, which significantly reduces the error caused by model parameter mismatch. Several experiments are carried out on the UAV platform, and the results show that the improvement of the RSS propagation model is adequate.

Finite subgroups of automorphisms of K3 surfaces

AbstractWe give a complete classification of finite subgroups of automorphisms of K3 surfaces up to deformation. The classification is in terms of Hodge theoretic data associated to certain conjugacy classes of finite subgroups of the orthogonal group of the K3 lattice. The moduli theory of K3 surfaces, in particular the surjectivity of the period map and the strong Torelli theorem allow us to interpret this datum geometrically. Our approach is computer aided and involves Hermitian lattices over number fields.

Constrained Attitude Control of Uncertain Spacecraft With Appointed-Time Control Performance

This article studies the appointed-time attitude tracking control of the spacecraft on the special orthogonal group, with the attitude forbidden zone, the parameter uncertainties, and the external disturbances. A novel projection function is proposed, such that the normalized boresight vector of the sensitive instrument is mapped to a reduced dimensional vector in the Euclidean space. If the reduced dimensional vector is uniformly bounded, the constraint on the attitude forbidden zone will be satisfied at all the time. By virtue of the designed reduced dimensional vector and the associated auxiliary vectors, a set of vector-based error functions, the appointed-time performance constraints, and the according switching law is carefully constructed. The proposed vector-based adaptive control scheme ensures that the spacecraft attitude can satisfy the attitude constraint and appointed-time control performance simultaneously, in the presence of parameter uncertainties and external disturbances. Simulation results show the effectiveness of the designed control scheme.

Anti-disturbance Trajectory Tracking Control for a Quadrotor UAV with Input Constraints⋆

This paper presents the trajectory tracking control for a quadrotor unmanned aerial vehicle(UAV) with disturbances and input constraints. The UAV's system is decomposed into an outer position loop and an inner attitude loop. The position tracking control is achieved by a class of saturated proportional-derivative(PD) control laws. Assuming that the unknown disturbances mainly effect on the translational dynamics, an adaptive control term is investigated to compensate the effects of the disturbances. The position tracking control force implies a coupled-attitude system that connects the inner attitude loop with the outer position loop. An intrinsic PD control law, which is directly designed on the special orthogonal group SO(3), is proposed to track the desired attitude. In order to reduce the calculational difficulty of the attitude tracking control law, an observer-based control law is further developed while the control law and observer can be designed separately. Simulation results complete this work.

Second-order sliding-mode on SO(3) and fault-tolerant spacecraft attitude control

As one of attitude representation methods for a rigid body, the rotation matrices defined on the special orthogonal matrix group SO(3) can give the unique, global and nonsingular solution. For the attitude control systems evolving on Lie group SO(3)×R3, a second-order sliding surface is constructed and proved to be a Lie subgroup, along which the reduced-order dynamics is almost globally asymptotically stable at the equilibrium state. Based on the new sliding-mode surface, a fault-tolerant controller is designed to deal with the actuator faults and external disturbances in attitude control systems. The new method is numerically simulated and further implemented in a semi-physical simulation system of an air-bearing platform to verify its effectiveness in spacecraft practical engineering.

3d [formula omitted] SO/USp adjoint SQCD: s-confinement and exact identities

We study 3d N=2 SQCD with symplectic and orthogonal gauge groups and adjoint matter. For USp(2n) with two fundamentals and SO(N) with one vector these models have been recently shown to s-confine. Here we corroborate the validity of this proposal by relating it to the confinement of USp(2n) with four fundamentals and an antisymmetric tensor, using exact mathematical results coming from the analysis of the partition function on the squashed three-sphere. Our analysis allows us to conjecture new s-confining theories for a higher number of fundamentals and vectors, in presence of linear monopole superpotentials. We then prove the new dualities through a chain of adjoint deconfinements and s-confining dualities.

Simple lattices and free algebras of modular forms

We study the algebras of modular forms on type IV symmetric domains for simple lattices; that is, lattices for which every Heegner divisor occurs as the divisor of a Borcherds product. For every simple lattice L of signature (n,2) with 3≤n≤10, we prove that the graded algebra of modular forms for the maximal reflection subgroup of the orthogonal group of L is freely generated. We also show that, with five exceptions, the graded algebra of modular forms for the maximal reflection subgroup of the discriminant kernel of L is also freely generated.

Stretched Newell–Littlewood coefficients

Newell-Littlewood coefficients $n_{\mu,\nu}^{\lambda}$ are the multiplicities occurring in the decomposition of products of universal characters of the orthogonal and symplectic groups. They may also be expressed, or even defined directly in terms of Littlewood-Richardson coefficients, $c_{\mu,\nu}^{\lambda}$. Both sets of coefficients have stretched forms $c_{t\mu,t\nu}^{t\lambda}$ and $n_{t\mu,t\nu}^{t\lambda}$, where $t\kappa$ is the partition obtained by multiplying each part of the partition $\kappa$ by the integer $t$. It is known that $c_{t\mu,t\nu}^{t\lambda}$ is a polynomial in $t$ and here it is shown that $n_{t\mu,t\nu}^{t\lambda}$ is an Ehrhart quasi-polynomial in $t$ with minimum quasi-period at most $2$. The evaluation of $n_{t\mu,t\nu}^{t\lambda}$ is effected both by deriving their generating function and by establishing a hive model analogous to that used for the calculation of $c_{t\mu,t\nu}^{t\lambda}$. These two approaches lead to a whole battery of conjectures about the nature of the quasi-polynomials $n_{t\mu,t\nu}^{t\lambda}$. These include both positivity, stability and saturation conjectures that are supported by a significant amount of data from a range of examples.

On the 𝔸1-Invariance of K2 Modeled on Linear and Even Orthogonal Groups

Abstract Let $k$ be an arbitrary field. In this paper, we show that in the linear case ($\Phi =\mathsf {A}_{\ell }$, $\ell \geq 4$) and even orthogonal case ($\Phi = \mathsf {D}_{\ell }$, $\ell \geq 7$, $\textrm {char}(k)\neq 2$), the unstable functor $\textrm {K}_{2}(\Phi , -)$ possesses the $\mathbb {A}^{1}$-invariance property in the geometric case, that is, ${\textrm {K}}_{2}(\Phi , R[t]) = {\textrm {K}}_{2}(\Phi , R)$ for a regular ring $R$ containing $k$. As a consequence, the unstable ${\textrm {K}}_{2}$ groups can be represented in the unstable $\mathbb {A}^{1}$-homotopy category $\mathscr {H}_{\bullet }(k)$ as fundamental groups of the simply-connected Chevalley–Demazure group schemes $\textrm {G}(\Phi ,-)$. Our invariance result can be considered as the ${\textrm {K}}_{2}$-analogue of the geometric case of Bass–Quillen conjecture. We also show for a semilocal regular $k$-algebra $A$ that ${\textrm {K}}_{2}(\Phi , A)$ embeds as a subgroup into ${\textrm {K}}^{\textrm {M}}_{2}(\textrm {Frac}\,A)$.

On the attitude estimation of nonholonomic wheeled mobile robots

This paper addresses the attitude estimation problem of wheeled mobile robots (WMRs) with nonholonomic constraints moving in a plane. We consider the case that only the robot’s Cartesian position and linear velocity are available from sensor measurements. Based on the kinematic model, three attitude observers are proposed. The first observer is designed directly on the Special Orthogonal group SO(2) and guarantees that for almost all initial conditions, the estimated attitude converges to the actual one with an exponential convergence rate. The second observer estimates the heading direction of the mobile robot, i.e., its dynamics evolves on the unit circle S1. The first and second observers rely on linear velocity measurements. On the other hand, the third observer presents a globally exponentially stable error dynamics and requires only Cartesian position measurements but does not evolve on S1. Additionally, a control-observer algorithm is proposed to solve the trajectory-tracking problem of WMRs. Finally, the performances of the proposed attitude observers and control law were evaluated by a set of experiments on a commercial WMR.

Schwarzschild Spacetimes: Topology

This paper is devoted to the geometric theory of a Schwarzschild spacetime, a basic objective in applications of the classical general relativity theory. In a broader sense, a Schwarzschild spacetime is a smooth manifold, endowed with an action of the special orthogonal group SO(3) and a Schwarzschild metric, an SO(3)-invariant metric field, satisfying the Einstein equations. We prove the existence of and find all Schwarzschild metrics on two topologically non-equivalent manifolds, R×(R3∖{(0,0,0)}) and S1×(R3∖{(0,0,0)}). The method includes a classification of SO(3)-invariant, time-translation invariant and time-reflection invariant metrics on R×(R3∖{(0,0,0)}) and a winding mapping of the real line R onto the circle S1. The resulting family of Schwarzschild metrics is parametrized by an arbitrary function and two real parameters, the integration constants. For any Schwarzschild metric, one of the parameters determines a submanifold, where the metric is not defined, the Schwarzschild sphere. In particular, the family admits a global metric whose Schwarzschild sphere is empty. These results transfer to S1×(R3∖{(0,0,0)}) by the winding mapping. All our assertions are derived independently of the signature of the Schwarzschild metric; the signature can be chosen as an independent axiom.

Earth Velocity and Rigid-Body Attitude Estimation on SO(3) Using Biased Measurements

This article addresses the problem of simultaneously estimating: 1) the attitude of a rigid body on the special orthogonal group of order 3 ( <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {SO}(3)$</tex-math></inline-formula> ); 2) the Earth’s spin vector expressed in body-fixed coordinates; and 3) a pair of time-varying sensor measurement biases. The available sensor readings include biased angular velocity measurements collected from a set of high-grade rate gyros sensitive to the rotation of the planet and biased vector observations of some known inertial reference field. The estimation solution consists of a Lagrangian-based pseudo Kalman filtering routine, which is partially constrained to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\mathsf {SO}(3)$</tex-math></inline-formula> vis-á-vis the attitude. Most noticeably, the geometric relationship between the Earth’s angular velocity and the inertial field is leveraged to relax common practical, as well as theoretical, demands closely linked to conventional attitude estimation solutions, namely, the need for explicit body-fixed measurements of two inertial vector fields. The proposed algorithm is validated through a set of realisticsimulations. Experimental results are also included to further demonstrate the performance of this strategy in a real-world application.

On the expected number of real roots of polynomials and exponential sums

The expected number of real projective roots of orthogonally invariant random homogeneous real polynomial systems is known to be equal to the square root of the Bézout number. A similar result is known for random multi-homogeneous systems, invariant through a product of orthogonal groups. In this note, those results are generalized to certain families of sparse polynomial systems, with no orthogonal invariance assumed.

Novel Special Orthogonal Group Optimization for Coarse Alignment Method of SINS on a Rocking Base

In order to solve the coarse alignment problem of the strapdown inertial navigation system on a rocking base, a fast coarse alignment method using the Special Orthogonal Group optimization has been proposed in this paper. In this method, based on the alignment idea of tracing gravitational apparent motion in inertial frame, the model of coarse alignment on a rocking base has been established using the Special Orthogonal Group directly. A new attitude error function has been proposed on the basis of the cosines between the measurement vector and predictive vector to describe the error between the estimated attitude and the true one. In order to directly reflect the change in the attitude error in the new innovation term and enable the attitude error to converge to zero as fast as possible, the gradient of the new attitude error function has been selected as the new innovation term to compensate for the attitude in the estimation process. Finally, the stability of the proposed optimization estimation method has been proved by employing the Lyapunov stability theory. Simulation and experiment results show that the method presented in this paper exhibits good performance in terms of alignment accuracy and time and can be applied to coarse alignment under a rocking base under different environments.

Commutators of random matrices from the unitary and orthogonal groups

We investigate the statistical properties of C = uvu−1v−1, when u and v are independent random matrices, uniformly distributed with respect to the Haar measure of the groups U(N) and O(N). An exact formula is derived for the average value of power sum symmetric functions of C, and also for products of the matrix elements of C, similar to Weingarten functions. The density of eigenvalues of C is shown to become constant in the large-N limit, and the first N−1 correction is found.