Orthogonal Group Research Articles

The inverse Galois problem for orthogonal groups

We prove many cases of the Inverse Galois Problem for those simple groups arising from orthogonal groups over finite fields. For example, we show that the finite simple groups Ω 2 n + 1 ( p ) \Omega _{2n+1}(p) and P Ω 4 n + ( p ) \operatorname {P}\!\Omega _{4n}^+(p) both occur as the Galois group of a Galois extension of the rationals for all integers n ≥ 2 n\geq 2 and all primes p ≥ 5 p\geq 5 . We obtain our representations by studying families of twists of elliptic curves and using some known cases of the Birch and Swinnerton-Dyer conjecture along with a big monodromy result of Hall.

An intrinsic Bayesian bound for estimators on the Lie groups [formula omitted] and [formula omitted

In the past decade, estimation on manifolds has been paid an increased attention for making it possible to intrinsically account for constraints on the unknown parameters. Applications are widespread from computer vision to target tracking. To evaluate the performance of the algorithms, it is of interest to derive bounds on the estimation error. This issue has been mostly investigated for parametric estimation on Riemannian manifolds. This paper addresses the problem of computing a Bayesian lower bound on the norm of the estimation error when both the unknown variables and the observations belong to specific manifolds called Lie groups (LGs). For that purpose, a metric is considered that preserves the geometric properties of the latter. We firstly derive an inequality on the correlation matrix of this intrinsic error under the mild assumption of unimodular matrix LGs. Then, we obtain a closed-form expression of the bound for the special orthogonal group SO(3) and the special Euclidean group SE(3). Finally, we detail its derivation in the case when both the prior distribution and the likelihood are concentrated Gaussian distributions. The proposed Bayesian bound is implemented and tested on two estimation problems involving both unknown parameters and observations on SE(3): the inference of the pose of a camera and that of the centroid of a cluster of space debris.

Infinite dimensional representations of orthogonal groups of quadratic forms with finite index

Infinite dimensional representations of orthogonal groups of quadratic forms with finite index

Multi-Objective Optimization in Ultrasonic Polishing of Silicon Carbide via Taguchi Method and Grey Relational Analysis.

As high-level equipment and advanced technologies continue toward sophistication, ultrasonic technology is extensively used in the polishing process of difficult-to-process materials to achieve efficiently smooth surfaces with nanometer roughness. The polishing of silicon carbide, an indispensable difficult-to-machine optical material, is extremely challenging due to its high hardness and good wear resistance. To overcome the current silicon carbide (SiC) ultrasonic polishing (UP) process deficiencies and strengthen the competitiveness of the UP industry, the multi-objective optimization based on the Taguchi-GRA method for the UP process with SiC ceramic to obtain the optimal process parameter combination is a vital and urgently demanded task. The orthogonal experiment, analysis of variance, grey relational analysis (GRA), and validation were performed to optimize the UP schemes. For a single objective of roughness and removal rate, the influence degree is abrasive size > preloading force > abrasive content > spindle speed > feed rate, and spindle speed > abrasive size > feed rate > preloading force > abrasive content, respectively. Moreover, the optimal process combination integrating these two objectives is an abrasive content of 14 wt%, abrasive size of 2.5 μm, preloading force of 80 N, spindle speed of 8000 rpm, and feed rate of 1 mm/s. The optimized workpiece surface morphology is better, and the roughness and removal rate are increased by 7.14% and 28.34%, respectively, compared to the best orthogonal group. The Taguchi-GRA method provides a more scientific approach for evaluating the comprehensive performance of polishing. The optimized process parameters have essential relevance for the ultrasonic polishing of SiC materials.

Effective Density for Inhomogeneous Quadratic Forms II: Fixed Forms and Generic Shifts

Abstract We establish effective versions of Oppenheim’s conjecture for generic inhomogeneous quadratic forms. We prove such results for fixed quadratic forms and generic shifts. Our results complement our previous paper [13] where we considered generic forms and fixed shifts. In this paper, we use ergodic theorems and in particular we establish a strong spectral gap with effective bounds for some representations of orthogonal groups, which do not possess Kazhdan’s property $(T)$.

Fundamental Fears in Greece (I)

Abstract: Introduction: The present study examined the psychometric properties of a Greek adaptation of the Anxiety Sensitivity Index-3 (ASI-3; Taylor et al., 2007 ). Method: We translated the ASI-3 following a forward-backward method and then, in addition to measures of anxiety and depression (DASS-21; Lovibond & Lovibond, 1995 ; Lyrakos et al., 2011 ), we administered it to a nonclinical general population sample ( N = 611) recruited online. Results: Confirmatory factor analysis revealed that a bifactor model with three orthogonal group factors best fit the data, followed by a correlated three-factor model. An examination of the dimensionality of the ASI-3 and the reliability of its dimensions suggested the presence of a reliable, strong AS general factor and comparatively weaker group factors. The ASI-3 appears to measure AS invariantly across gender. We report preliminary evidence for its convergent, discriminant, and divergent validity. Conclusion: The Greek adaption of the ASI-3 revealed adequate psychometric properties. Future studies should explore its criterion-related validity by administering the Greek adaptation of the ASI-3 to clinical samples and explore its relationship to other key constructs of anxiety sensitivity’s nomological network.

Trifunctional Dibromomaleimide Reagents Built Around A Lysine Scaffold Deliver Site-selective Dual-modality Antibody Conjugation.

We describe the synthesis and application of a selection of trifunctional reagents for the dual-modality modification of native, solvent accessible disulfide bonds in trastuzumab. The reagents were developed from the dibromomaleimide (DBM) platform with two orthogonal clickable functional groups built around a lysine core. We also describe the development of an aryl diselenide additive which enables antibody disulfide reduction in 4 minutes and a rapid overall reduction-bridging-double click sequence.

Kleinian sphere packings, reflection groups, and arithmeticity

In this paper we study crystallographic sphere packings and Kleinian sphere packings, introduced first by Kontorovich and Nakamura in 2017 and then studied further by Kapovich and Kontorovich in 2021. In particular, we solve the problem of existence of crystallographic sphere packings in certain higher dimensions posed by Kontorovich and Nakamura. In addition, we present a geometric doubling procedure allowing to obtain sphere packings from some Coxeter polyhedra without isolated roots, and study “properly integral” packings (that is, ones which are integral but not superintegral). Our techniques rely extensively on computations with Lorentzian quadratic forms, their orthogonal groups, and associated higher–dimensional hyperbolic polyhedra.

Exact first moments of the RV coefficient by invariant orthogonal integration

The RV coefficient measures the similarity between two multivariate configurations, and its significance testing has attracted various proposals in the last decades. We present a new approach, the invariant orthogonal integration, permitting to obtain the exact first four moments of the RV coefficient under the null hypothesis.Our proposal can be applied to any multivariate setting endowed with Euclidean distances between the observations. It also covers the weighted setting of observations of unequal importance, where the exchangeability assumption, justifying the usual permutation tests, breaks down.The proposed RV moments express as simple functions of the kernel eigenvalues occurring in the weighted multidimensional scaling of the two configurations (spectral effective dimensionality, spectral skewness and spectral excess kurtosis). The expressions for the third and fourth moments seem original, and explain the marked asymmetry and kurtosis of the RV coefficient. They permit to test the significance of the RV coefficient by Cornish–Fisher cumulant expansion, beyond the normal approximation, as illustrated on a small dataset.The first three moments can be obtained by elementary means, but computing the fourth moment requires a more sophisticated apparatus, the Weingarten calculus for orthogonal groups.

Local Plücker formulas for orthogonal groups

Local Plücker formulas for orthogonal groups

Classification of all constant solutions of SU(2) Yang–Mills equations with arbitrary current in pseudo-Euclidean space ℝp,q

We present a classification and an explicit form of all constant solutions of the Yang–Mills equations with [Formula: see text] gauge symmetry for an arbitrary constant non-Abelian current in pseudo-Euclidean space [Formula: see text] of arbitrary finite dimension [Formula: see text]. Using hyperbolic singular value decomposition and two-sheeted covering of orthogonal group by spin group, we solve the nontrivial system for constant solutions of the Yang–Mills equations of [Formula: see text] cubic equations with [Formula: see text] unknowns and [Formula: see text] parameters in the general case. We present a new symmetry of this system of equations. All solutions in terms of the potential, strength and invariant of the Yang–Mills field are presented. Nonconstant solutions of the Yang–Mills equations can be considered in the form of series of perturbation theory using all constant solutions as a zeroth approximation.

Envelopes and classifying spaces

AbstractFor a split semisimple algebraic group H with its split maximal torus S, let be the restriction homomorphism of the Chow rings of the classifying spaces of H and S, where W is the Weyl group. A constraint on the image of f, given by the Steenrod operations, has been applied to the spin groups in a previous paper. Here, we describe and apply to the spin groups another constraint, which is given by the reductive envelopes of H. We also recover in this way some older results on orthogonal groups.

2-Reflective Lattices of Signature (n, 2) with n ≥ 8

Abstract An even lattice $M$ of signature $(n,2)$ is called $2$-reflective if there is a non-constant modular form for the orthogonal group of $M$, which vanishes only on quadratic divisors orthogonal to $2$-roots of $M$. In 2017, Ma [ 25] proved that there are only finitely many $2$-reflective lattices of signature $(n,2)$ with $n\geq 7$. In this paper, we extend the finiteness result of Ma to $n\geq 5$ and show that there are exactly forty-two $2$-reflective lattices of signature $(n,2)$ with $n\geq 8$.

Versatile Production of Multivariate, Hyperdimensional End Group and Main Chain Functionalized Polyolefins.

The (stereoselective) living coordinative copolymerization of 1-alkenes with 4-aryl-1,6-heptadienes, in both the absence and presence of multiple equivalents of a reversible chain transfer agent, is established as a highly versatile strategy for production of multivariate hyperdimensional functionalized semi-crystalline or amorphous polyolefins that optionally possess either mono- or difunctionalized (telechelic) end-groups in combination with a programmable level of incorporation of orthogonal functional groups within the main-chain. The non-conjugated diene comonomers are readily obtained from a diverse range of aryl carboxaldehyde precursors through a one-step bis-allylation process. These results serve to provide a new platform for exploring the science and technology of a vast new landscape of functionalized classes of polyolefins that are now accessible in practical and scalable quantities.

Generalized Quaternions and Matrix Algebra

In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained real and complex matrices corresponding to the real and complex basis of the generalized quaternions. Also, we investigated the basis features of real and complex matrices. We get Pauli matrices corresponding to generalized quaternions. Then, we have shown that the algebra produced by these matrices is isomorphic to the Clifford algebra Cl(E_αβ^3) produced by generalized space E_αβ^3.
 Finally, we studied the relations among the symplectic matrices group corresponding to generalized unit quaternions, generalized unitary matrices group, and generalized orthogonal matrices group.

Versatile Production of Multivariate, Hyperdimensional End Group and Main Chain Functionalized Polyolefins

AbstractThe (stereoselective) living coordinative copolymerization of 1‐alkenes with 4‐aryl‐1,6‐heptadienes, in both the absence and presence of multiple equivalents of a reversible chain transfer agent, is established as a highly versatile strategy for production of multivariate hyperdimensional functionalized semi‐crystalline or amorphous polyolefins that optionally possess either mono‐ or difunctionalized (telechelic) end‐groups in combination with a programmable level of incorporation of orthogonal functional groups within the main‐chain. The non‐conjugated diene comonomers are readily obtained from a diverse range of aryl carboxaldehyde precursors through a one‐step bis‐allylation process. These results serve to provide a new platform for exploring the science and technology of a vast new landscape of functionalized classes of polyolefins that are now accessible in practical and scalable quantities.

Identification of potential quality markers of Zishen Yutai pill based on spectrum-effect relationship analysis.

Introduction: The current quality evaluation of traditional Chinese medicine (TCM) is difficult to attribute to clinical efficacy due to the complexity of TCM. Zishen Yutai pill (ZYP), a well-known traditional Chinese patent medicine, has been widely used to prevent recurrent miscarriage and treat threatened abortion. However, the chemical components of ZYP are unknown, and there is no convincing quality control method applied on ZYP. Although ZYP has been found to promote endometrial receptivity and treat impending abortion, the substantial basis of the therapeutic effects is unclear. The aim of this study was to clarify the quality markers correlated with the potential medicinal activities and provide a theoretical foundation for scientific quality control and product quality improvement of ZYP. Methods: The chemical constituents of ZYP were comprehensively analyzed by offline two-dimensional liquid chromatography-mass spectrometry (2DLC-LTQ-Orbitrap-MS). The efficacy of the 27 ZYP orthogonal groups was investigated using the HTR-8/SVneo oxidative damage model and migration model in vitro, as well as the endometrial receptivity disorder mouse model and premature ovarian failure mouse model in vivo. Based on the efficacy and mass spectral results, spectrum-effect relationship analysis was used to identify the chemical components with corresponding pharmacological activities. Results: A total of 589 chemical components were found in ZYP, of which 139 were not identified in the literature. The potential quality markers for ZYP were successfully identified through orthogonal design and spectrum-effect relationship analysis. By combining mass spectrum data and pharmacological results of 27 orthogonal groups, 39 substances were identified as potential quality markers. Conclusion: The approaches used in this study will provide a feasible strategy for the discovery of quality markers with bioactivity and further investigation into the quality evaluation of TCM.

Customizable Supraparticles Constructed from Catechol-Terminated Molecular Building Blocks with Controllable Intermolecular Interactions.

Colloidal supraparticles integrated with multicomponent primary particles come with emerging or synergetic functionalities. However, achieving the functional customization of supraparticles remains a great challenge because of the limited options of building blocks with tailorability and functional extensibility. Herein, we developed a universal approach to construct customizable supraparticles with desired properties from molecular building blocks obtained by the covalent conjugation of catechol groups with a series of orthogonal functional groups. These catechol-terminated molecular building blocks can assemble into primary particles driven by various intermolecular interactions (i.e. metal-organic coordination, host-guest, and hydrophobic interactions), and then further assemble into supraparticles governed by catechol-mediated interfacial interactions. Our strategy enables the formation of supraparticles with diverse functionalities, such as dual-pH responsiveness, light-controllable permeability, and non-invasive fluorescence labeling of living cells. The ease with which these supraparticles can be fabricated, and the ability to tailor their chemical and physical properties through the choice of metals and orthogonal functional groups used, should enable a variety of applications.

Customizable Supraparticles Constructed from Catechol‐Terminated Molecular Building Blocks with Controllable Intermolecular Interactions**

AbstractColloidal supraparticles integrated with multicomponent primary particles come with emerging or synergetic functionalities. However, achieving the functional customization of supraparticles remains a great challenge because of the limited options of building blocks with tailorability and functional extensibility. Herein, we developed a universal approach to construct customizable supraparticles with desired properties from molecular building blocks obtained by the covalent conjugation of catechol groups with a series of orthogonal functional groups. These catechol‐terminated molecular building blocks can assemble into primary particles driven by various intermolecular interactions (i.e. metal‐organic coordination, host–guest, and hydrophobic interactions), and then further assemble into supraparticles governed by catechol‐mediated interfacial interactions. Our strategy enables the formation of supraparticles with diverse functionalities, such as dual‐pH responsiveness, light‐controllable permeability, and non‐invasive fluorescence labeling of living cells. The ease with which these supraparticles can be fabricated, and the ability to tailor their chemical and physical properties through the choice of metals and orthogonal functional groups used, should enable a variety of applications.

The classification of 2-reflective modular forms

The classification of reflective modular forms is an important problem in the theory of automorphic forms on orthogonal groups. In this paper, we develop an approach based on the theory of Jacobi forms to give a full classification of 2-reflective modular forms. We prove that there are only 3 lattices of signature (2, n) having 2-reflective modular forms when n ≥ 14. We show that there are exactly 51 lattices of type 2U ⊕ L(−1) which admit 2-reflective modular forms and satisfy that L has 2-roots. We further determine all 2-reflective modular forms giving arithmetic hyperbolic 2-reflection groups. This is the first attempt to classify reflective modular forms on lattices of arbitrary level.