In this paper, we observe the special orthogonal matrix Lie group containing of all 2x2 real matrices, denoted by SO(2), which can be geometrically visualized as the one-dimensional torus S1 which is nothing but the unit circle. A Brownian motion on SO(2) can be constructed and represented by a stochastic differential equation defined over a dynamic state space. The research aims to derive a short-term interest rate model on SO(2) through Brownian motion analysis which is a geometric approach. We employ a qualitative methodology, including a literature review of Brownian motion, stochastic differential equations, and dynamical state-space techniques on SO(2). Firstly, we prove the isomorphism SO(2) is isomorphic to S1, secondly, we determine Brownian motion on SO(2) and its equivalent, and thirdly, we formulate the corresponding stochastic differential equation, and the last, determine the short-term interest rate equation on SO(2). In this study, it is confirmed Lim and Privault’s work that the interest rate equation on SO(2) is given by rt = beta + 2 gamma cos(Wt) with beta, gamma is constant and Wt is standard Brownian motion. To clarify the obtained results, this study also gave a quantitative approach that is Python simulation of interest rate calculation using the matrix Lie group interest rate and other equations. The interest rate equation uses the matrix Lie group SO(n) with n greater than or equal to three still open to further research that can be applied to long-term interest rates.

Enzymatic catalysis promises the achievement of functional group reactivity selectivity with the conjunctional expansive acquirement of reactivity diversity through the conformal mutagenesis tuning of catalytic cavity. However, essentially all of the functional group reactivity diversification systems demonstrated thus far are centered on the intersubstrate-shift regime, with the enzymatic catalysis adapted to alternative substrate and associated target functional group in totality. Herein, we report an enzymatic stringency-relaxation strategy for effecting reactivity diversification into the intrasubstrate orthogonal functional group reactivity selectivity regime. The stringency-relaxation strategy operates in either the amino acid relaxation or functional group relaxation format by a working sequence of initial stringency amino acid catalytic access to the catalytically most demanding functional group and subsequent relaxation amino acid catalytic access to the catalytically less demanding functional group. In particular, herein, through this strategy, α,β-unsaturated carbonyls have been reduced with ketoreductase, in a selective manner, at either the carbonyl group site or the alkenyl group site. A broad substrate scope has been established for the alkenyl reduction of α-cyano-α,β-unsaturated esters, showcasing enzymatic stringency-relaxation as a prospective platform for programming reactivity diversification and reaction development.

In this paper, we present the spinor structure associated with the Hyperbolic Clifford algebra of a real n-dimensional vector space V, which is denoted by ClHV. Unlike the standard Clifford algebra, the Hyperbolic Clifford algebra Cl(HV) simultaneously accommodates both multiforms and multivectors in a single algebraic structure, making it the natural framework—known as the “mother algebra”—for the study of superfields in theoretical physics and for generalizing the Clifford bundle formalism to hyperbolic structures arising in gravitational theories. The orthogonal groups and orthogonal transformations associated to the hyperbolic space HV are presented. The Clifford–Lipschitz group and the Pin and Spin groups associated with ClHV are defined. Then, the frame bundle and spinor structure associated to Hyperbolic Clifford algebra is derived.

Leaf orientation (posture) influences photosynthesis and plant responses to environmental cues. However, existing methods for quantifying posture typically compress its inherently three-dimensional structure into scalar values or single vectors, leaving the 3D aspects of leaf movement poorly understood. For a more complete geometric description, we propose representing leaf-blade posture by an orthonormal basis (ONB), defined as three perpendicular unit vectors aligned with the developmental axes of the leaf. This ONB serves as a local coordinate system and corresponds to rotation matrices used to represent orientation in three-dimensional space, embedding leaf posture within the mathematical structure of the special orthogonal group SO(3). Using three-dimensional point-cloud data, we reconstructed an ONB aligned with the three axes of the leaf blade and quantified elevation and azimuth angles. When applied to diurnal posture changes, the resulting angular patterns were consistent with previous observations. We then visualized posture changes after gravitational perturbation as continuous rotational trajectories. These trajectories could be compared with mathematically defined geodesic shortest paths and used to simulate alternative reorientation routes that satisfy constraints. The rotational trajectories could also be separated into swing and twist components, which reflect distinct deformation modes. ONB can be obtained not only from three-dimensional point clouds but also from other measurement tools, making the approach broadly applicable. Conceptually, ONB representation places leaf posture within the geometric structure of SO(3), enabling the use of well-established mathematical tools such as rotational distance and geodesics for analyzing leaf reorientation.

Abstract We construct an explicit Cartan–Dieudonné decomposition for orthogonal group elements of binary quadratic forms over non-archimedean local fields of characteristic zero, expressing each group element as a product of reflections defined by vectors. A key feature of our construction is its stability: we establish quantitative control on how the reflection matrices vary under small perturbations of the underlying vectors. Using this decomposition, we establish an effective result for the equivalence of binary quadratic forms over number fields. Specifically, let K be a number field and S a finite set of non-archimedean places of K . Given two K -equivalent binary quadratic forms integrally equivalent at every prime in S , we provide an explicit search bound for finding a K -equivalence that are integral at all primes in S .

Background: Intra articular distal humerus fractures are complex injuries that can comsiderably limit elbow function if not treated properly. In the treatment of distal humerus intra articular fractures double plate osteosynthesis is the standard treat ment method. However, controversy still exists concerning the plate positions in terms of providing optimal stability of these kind of fractures. This study is aimed at finding any significant difference in outcome of these orthogonal and parallel plat ing methods of fracture fixation. Materials and methods: This quasi experimental study was undertaken to compare the differences between these two dual plating methods. It was conducted in Department of Orthopedics and Traumatology of Chittagong Medical College Hospital, Chattogram, for a period of fifteen months. 36 patients were enrolled conveniently according to inclusion and exclusion criteria. Patients were followed up for 6 months and overall outcomes were measured by MEPS. Depending on the overall functional outcome patients were grouped as having excellent to good outcome and fair to poor outcome. Results: At 6th month follow up, mean MEPS in the parallel plating group was 87.89±6.50 (Mean ± SD) and in the orthogonal group it was 83.72±7.45. There was no significant (p=0.785) difference between two groups in functional outcome but in parallel plating group MEPS was better. The mean union time was 6.17±0.383 weeks in parallel plating group and 6.22±0.548 weeks in orthogonal plating group. p value was 0.549. Regarding complications, elbow stiffness developed in 3(13.89%) in parallel and 4(19.44%) in orthogonal group, superficial infection developed 1 in parallel and 3 in orthogonal group. Conclusion: This study shows that, parallel plating group was superior to the orthogonal group, as of better functional outcome, less union time and less complications Chatt Maa Shi Hosp Med Coll J; Vol.24 (2); July 2025; Page 36-41

Bioluminescence tomography (BLT) is a promising molecular imaging modality with significant potential in preclinical research, enabling three-dimensional quantitative reconstruction of internal bioluminescent sources. However, the low absorption and severe photon scattering effects in biological tissues render the BLT inverse problem highly ill-conditioned, often leading to unstable and inaccurate reconstruction results. In this study, an adaptive orthogonal matching pursuit with group K-singular value decomposition (AOMP-GKSVD) algorithm is proposed within a dictionary learning framework. An adaptive sparsity estimation mechanism is incorporated into the sparse coding phase to infer the sparsity level from the measured data and the system matrix, thereby better capturing the intrinsic sparsity characteristics of bioluminescent sources. This overcomes the limitation of fixed sparsity settings in conventional OMP, enhancing reconstruction accuracy and robustness, while saving time and effort by eliminating the need for manual sparsity tuning under varying noise and source conditions. During the dictionary update phase, a grouping strategy based on a discretized tetrahedral mesh is employed, in which atoms are updated collectively at the group level, exploiting spatial adjacency to maintain coherence and efficiently reconstruct clustered source regions. The performance of AOMP-GKSVD was validated through a series of numerical simulations and a light source implantation experiment. The experimental results demonstrate that AOMP-GKSVD achieves superior performance in terms of localization accuracy, morphological recovery, and robustness, highlighting its potential to advance the practical application of BLT in preclinical optical molecular imaging.

Optical mode-sorting and generation are used for a wide range of quantum, sensing and communication applications. High-speed switching between mode sets would allow photonics systems to react to changes in propagation channel in real time. Photonic-Integrated-Circuits (PICs) with phased arrays can rapidly reconfigure their function at MHz depending on installed modulators. As a reconfigurable photonic system, PIC can reconfigure significantly fast than spatial light modulators that are limited to less than 1 kHz. However, phased-arrays are bound by two-dimensional Nyquist-sampling limit and spacing between array elements leads to grating lobe formation, where many additional array elements are needed to mitigate these effects. We leverage a passive Multiple-Plane-Light-Converters (MPLC) as an Optical Mode-Selective Interface (OMSI), coupling a basis set of free-space Hermite Gaussian (HG) modes directly into an active optical mesh implemented on a PIC. The active mesh can generate or sort any free-space mode-set that can be created by a linear superposition of 15 HG modes. Without changing the physical system, we demonstrate the generation and sorting of four orthogonal mode groups, each with 15 modes, achieving a mean intermodal crosstalk for sorting of -22 dB. This approach can allow for rapidly reconfigurable mode-sorters that could be used for quantum or classical communications or sensing applications.

Abstract We establish asymptotic formulae for general joint moments of characteristic polynomials and their higher‐order derivatives associated with matrices drawn randomly from the groups and in the limit as . This relates the leading‐order asymptotic contribution in each case to averages over the Laguerre ensemble of random matrices. We uncover an exact connection between these joint moments and a solution of the ‐Painlevé V equation, valid for finite matrix size, as well as a connection between the leading‐order asymptotic term and a solution of the ‐Painlevé equation in the limit as . These connections enable us to derive exact formulae for joint moments for finite matrix size and for the joint moments of certain random variables arising from the Bessel point process in a recursive way. As an application, we provide a positive answer to a question proposed by Altuğ et al. [Q. J. Math. 65 (2014), 1111–1125].

A bstract We discuss the holographic correspondence between 3d “Chern-Simons gravity” and an ensemble of 2d Narain code CFTs. Starting from 3d abelian Chern-Simons theory, we construct an ensemble of boundary CFTs defined by gauging all possible maximal subgroups of the bulk one-form symmetry. Each maximal non-anomalous subgroup is isomorphic to a classical even self-dual error-correcting code over ℤ p × ℤ p , providing a way to define a boundary “code CFT.” The average over the ensemble of such theories is holographically dual to Chern-Simons gravity, a bulk theory summed over 3d topologies sharing the same boundary. In the case of prime p , the sum reduces to that over handlebodies, i.e. becomes the Poincaré series akin to that in semiclassical gravity. As the main result of the paper, we show that the mathematical identity underlying this holographic duality can be understood and rigorously proven using the framework of Howe duality over finite fields. This framework is concerned with the representation theory of two commuting groups forming a dual pair: the symplectic group of modular transformations of the boundary, and an orthogonal group mapping codes to each other. Finally, we reformulate the holographic duality as an identity between different averages over quantum stabilizer states, providing an interpretation in terms of quantum information theory.

This paper proposes a hybrid control algorithm that combines geometric tracking control (GTC) with special orthogonal group SO(3) for position tracking and quaternion orientation-based attitude tracking control (QOATC) in order to achieve trajectory tracking with high attitude maneuverability for quadcopter systems suffering from aggressive orientation. Firstly, to solve the problem of the limitation of executing large-angle maneuvers in a quadrotor system caused by the lack of globality or uniqueness in Euler angles models, an improved geometric controller is constructed on SO(3). Secondly, an advanced attitude quaternion-based controller guarantees the tracking performances in finite time and addresses the persisting issue associated with Euler angle-based solutions. The proposed controller guarantees stability and precision by leveraging nonlinear control methods, with thrust and torque computed directly in the body frame. Furthermore, the theoretical research assures the asymptotic tracking and boundedness of all signals in the closed loop system by utilizing Lyapunov's stabilization theory on SO(3). Several numerical simulations are presented to demonstrate the superiority of the proposed approach, showing accurate trajectory tracking, reduced position and velocity errors, and robust performance under aggressive conditions. The hybrid controller effectively avoids singularities and ensures global stability, making it suitable for real-world UAV applications requiring high performance.

A converse theorem for quasi-split even special orthogonal groups over finite fields

ABSTRACT The handling of manifold‐valued data, for instance, plays a central role in color restoration tasks relying on circle‐ or sphere‐valued color models, in the study of rotational or directional information related to the special orthogonal group, and in Gaussian image processing, where the pixel statistics are interpreted as values on the hyperbolic sheet. Especially to denoise these kinds of data, several generalizations of total variation (TV) and Tikhonov‐type denoising models incorporating the underlying manifolds have been proposed. Recently, a novel, numerically efficient denoising approach has been introduced, where the data are embedded in a Euclidean ambient space, the non‐convex manifolds are encoded by a series of positive semi‐definite, fixed‐rank matrices, and the rank constraint is relaxed to obtain a convexification that can be solved using standard algorithms from convex analysis. The aim of the present paper is to extend this approach to new kinds of data like multi‐binary and Stiefel‐valued data. Multi‐binary data can, for instance, be used to model multi‐color QR codes, whereas Stiefel‐valued data occur in image and video‐based recognition. For both new data types, we propose TV‐ and Tikhonov‐based denoising models together with easy‐to‐solve convexification. All derived methods are evaluated on proof‐of‐concept, synthetic experiments.

Nucleophilic aromatic substitution (SNAr) reactions are critical methods for forming C─N bonds in synthetic campaigns, but limitations in electrophile electronics restrict access to a large portion of chemical space. Photochemical oxidation of fluoroarenes has emerged as an attractive strategy to activate fluoroarenes toward nucleophilic addition, but back-electron transfer to solution-phase reduced photocatalysts limit the scope and efficiency of these methods. Herein, we describe an electrochemical strategy to overcome this obstacle by spatially separating redox events at electrode surfaces, extending the lifetime of the activated electrophile and enabling the azolation of electron-rich alkoxyfluoroarenes. Through stabilization of the oxidized product with voltage control and HFIP solvent, the reaction proceeds with catalytic charge via a proposed uphill redox chain mechanism. A wide range of electron-rich fluoroarenes and azoles are tolerated-including those with orthogonal functional group handles. The redox catalytic nature of this e-SNAr reaction enables energy and mass efficient syntheses and facile scaling in a simple batch setup.

Multispectral pyrometers are still limited in instrument development to prism based single point measurement, while multi-point line temperature measurement requires complete replication of the optical and circuit systems of a single point, with complex equipment, high costs, and can only achieve measurement of a limited number of temperature points. Due to the different distances from each measured point to the optical axis in multi-point line temperature measurement, there are significant off-axis aberrations caused by far-field transmission and prism splitting, which affects the extraction of radiation energy from the specified area of the target filled with specified solid angles. In order to solve the problem of severe aberrations in the far-field transmission of multi-point radiation in line measurement, a multispectral line pyrometer based on orthogonal cylindrical lens group was proposed, which achieved the best matching between the radiation at each point on the line and the detector array. By utilizing a high-speed synchronous data acquisition system, high-quality radiation information was obtained. Combined with the proposed improved 2D graphical multispectral radiation temperature inversion algorithm based on improved CCPCANet, the goal of achieving accurate measurement of multi-point line temperature simultaneously with only one optical path system was achieved.

The cone of a classical group \(G\) is an affine \(G\times G\)-variety. The aim of this note is to initiate its combinatorial study in the cases when \(G\) is the complex orthogonal or symplectic group (the case of the general linear group being well documented in the literature). The coordinate ring of the cone of G is a finitely generated commutative graded algebra. First the \(G\times G\)-module structure of its homogeneous components is determined. This is used to compute the Hilbert series of this coordinate ring in the cases when G is the orthogonal group \(\mathrm{O}(3)\), \(\mathrm{O}(4)\), the special orthogonal group \(\mathrm{SO}(4)\), and when \(G\) is the symplectic group \(\mathrm{Sp}(4)\). It is concluded that the coordinate ring of the cone of \(\mathrm{O}(3)\) is not Koszul, hence the vanishing ideal of this cone has no quadratic Gr¨obner basis (although it is minimally generated by quadratic elements).

The concept of a supercharacter theory for a finite group was introduced in 2008 by Diaconis and Iasaacs in [6]. In their article the notion of irreducible characters and conjugacy classes is generalized to superchacters and superclasses while still maintaining important information about the group. This article continues an investigation of a specific supercharacter theory where the supercharacters are taken to be sums of irreducible characters of the same degree. We show this supercharacter theory construction can be done for all projective special linear groups PSL(2,q) and all special orthogonal groups SO(3,q) where q is any power of an (even or odd) prime.

Theta liftings of non-generic representations on double covers of orthogonal groups

The purpose of this paper is to contribute part of the classification of nontrivial 2-design D admitting a flag-transitive and point-primitive automorphism group G of almost simple type whose socle X is an orthogonal simple groups of odd dimension.

Abstract This paper presents the SpHelico , a novel aerial platform composed of a coaxial rotor with dual control vanes, enclosed within a freely actuated spherical gimbal. This mechanical design ensures protection against collisions while maintaining full maneuverability. A bounded attitude tracking controller is developed on the Special Orthogonal Group , accounting for actuator saturation and external disturbances. To achieve full six‐degree‐of‐freedom control, a hierarchical control scheme is proposed that incorporates a geometric position controller on , generating the desired orientations and thrust vectors required for translational tracking. Theoretical guarantees of exponential convergence are provided for both attitude and position tracking errors. Extensive simulations validate the proposed strategy under realistic conditions, including perturbations and actuator limitations, and benchmark its performance against a sliding mode controller. Additionally, the paper details the complete mechanical design and provides a video demonstration using a real prototype to illustrate the practical feasibility of the approach.