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The complete classification of the finite simple groups that are (2, 3)-generated is a problem which is still open only for orthogonal groups. Here, we construct (2, 3)-generators for the finite odd-dimensional orthogonal groups Ω2k+1(q), k ≥ 4. As a byproduct, we also obtain (2, 3)-generators for Ω+4k(q) with k ≥ 3 and q odd, and for Ω± 4k+2(q) with k ≥ 4 and q ≡ ±1 (mod 4). [ABSTRACT FROM AUTHOR]

An automorphism u of a vector space is called unipotent of index 2 whenever (u − id) 2 = 0. Let b be a non-degenerate symmetric or skewsymmetric bilinear form on a vector space V over a field F of characteristic different from 2. Here, we characterize the elements of the isometry group of b that are the product of two unipotent isometries of index 2. In particular, if b is skewsymmetric and nondegenerate we prove that an element of the symplectic group of b is the product of two unipotent isometries of index 2 if and only if it has no Jordan cell of odd size for the eigenvalue −1. As an application, we prove that every element of a symplectic group is the product of three unipotent elements of index 2 (and no fewer in general). For orthogonal groups, the classification closely matches the classification of sums of two square-zero skewselfadjoint operators that was obtained in a recent article [7]. [ABSTRACT FROM AUTHOR]

We describe the foundations of a Hecke theory for the orthogonal group S O + (2 , n + 2). In particular we consider the Hermitian modular group of degree 2 as a special example of S O + (2 , 4). As an application we show that the attached Maaß space is invariant under Hecke operators. This implies that the Eisenstein series belongs to the Maaß space. If the underlying lattice is even and unimodular, our approach allows us to reprove the explicit formula of its Fourier coefficients. [ABSTRACT FROM AUTHOR]

We study commutative Schur rings over the symplectic groups Sp (n , 2) that contain the class C of symplectic transvections. We find the possible partitions of C determined by the Schur ring. We show how this restricts the possibilities for multiplicity one subgroups of Sp (n , 2). [ABSTRACT FROM AUTHOR]

We provide an algorithm to compute generators of the orthogonal group of the discriminant group associated to an integral quadratic lattice over the integers. We give a closed formula for its order. [ABSTRACT FROM AUTHOR]

We described a minimal separating set for the algebra of O 2 + (F q) -invariant polynomial functions of m -tuples of two-dimensional vectors over a finite field F q. [ABSTRACT FROM AUTHOR]

In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics. [ABSTRACT FROM AUTHOR]

Using Zhelobenko–Stern formulas for the action of the generators of orthogonal Lie algebra in corresponding Gelfand–Tsetlin basis, we derive Mellin–Barnes presentations for the wave functions of B n Toda lattice. They are in accordance with Iorgov–Shadura formulas. [ABSTRACT FROM AUTHOR]

LetXbe a compact connected Riemann surface of genusg≥2 equippedwith a holomorphic involutionσX, and letGbe a semisimple complex Liegroup which admits an outer involutionσ. A principal (G,σX,σ)-bundleoverXis a pair (E,ρ), whereEis a principalG-bundle overXandρ:E→σ∗X(σ(E)) is an isomorphism such that (σ∗Xρ)◦ρ:E→Eis anautomorphism of E which acts as the product by an element of the centerofG. In this paper, principal (G,σX,σ)-bundles overXare introduced andthe study is particularized to the case of G= SO(2n,C). It is shown thatthe stability and polystability conditions for a principal (SO(2n,C),σX,σ)-bundle coincide with those of the corresponding principal SO(2n,C)-bundle. Finally, the explicit form that a principal (SO(2n,C),σX,σ)-bundle takes isprovided, and the stability of these principal (SO(2n,C),σX,σ)-bundles isdiscussed. [ABSTRACT FROM AUTHOR]

In this article, we propose a novel alternating minimization scheme for finding completely positive factorizations. In each iteration, our method splits the original factorization problem into two optimization subproblems, the first one being a orthogonal procrustes problem, which is taken over the orthogoal group, and the second one over the set of entrywise positive matrices. We present both a convergence analysis of the method and favorable numerical results. [ABSTRACT FROM AUTHOR]

In this article, we compare the Dickson-Siegel-Eichler-Roy (DSER) elementary orthogonal group introduced by A. Roy in 1968 and the odd elementary hyperbolic unitary group introduced by V.A. Petrov in 2005. We prove that these two groups over a commutative ring with a pseudo-involution are conjugate subgroups of the general linear group. [ABSTRACT FROM AUTHOR]

The Grassmann manifold of linear subspaces is important for the mathematical modelling of a multitude of applications, ranging from problems in machine learning, computer vision and image processing to low-rank matrix optimization problems, dynamic low-rank decompositions and model reduction. With this mostly expository work, we aim to provide a collection of the essential facts and formulae on the geometry of the Grassmann manifold in a fashion that is fit for tackling the aforementioned problems with matrix-based algorithms. Moreover, we expose the Grassmann geometry both from the approach of representing subspaces with orthogonal projectors and when viewed as a quotient space of the orthogonal group, where subspaces are identified as equivalence classes of (orthogonal) bases. This bridges the associated research tracks and allows for an easy transition between these two approaches. Original contributions include a modified algorithm for computing the Riemannian logarithm map on the Grassmannian that is advantageous numerically but also allows for a more elementary, yet more complete description of the cut locus and the conjugate points. We also derive a formula for parallel transport along geodesics in the orthogonal projector perspective, formulae for the derivative of the exponential map, as well as a formula for Jacobi fields vanishing at one point. [ABSTRACT FROM AUTHOR]

We obtain new invariant Einstein metrics on the compact Lie groups SO (n) which are not naturally reductive. This is achieved by using the real flag manifolds SO (k 1 + ⋯ + k p) / SO (k 1) × ⋯ × SO (k p) and by imposing certain symmetry assumptions in the set of all left-invariant metrics on SO (n) . [ABSTRACT FROM AUTHOR]

An element of a group is called bireflectional when it is the product of two involutions of the group (i.e. elements of order 1 or 2). If an element is bireflectional then it is conjugated to its inverse. It is known that all elements of orthogonal groups of quadratic forms are bireflectional ([18] for fields of characteristic not 2, [7] for fields of characteristic 2). F. Bünger [3] has characterized the elements of unitary groups (over fields of characteristic not 2) that are bireflectional. Yet in symplectic groups over fields with characteristic different from 2, in general there are elements that are conjugated to their inverse but are not bireflectional (however, over fields of characteristic 2, every element of a symplectic group is bireflectional, see [7]). In this article, we characterize the bireflectional elements of symplectic groups in terms of Wall invariants, over fields of characteristic not 2: the result is cited without proof in Bünger’s PhD thesis, and attributed to Klaus Nielsen. We also take advantage of our approach to give a simplified proof of Wonenburger’s corresponding result for orthogonal groups and general linear groups. [ABSTRACT FROM AUTHOR]

We prove that over any perfect field, the plane Cremona group is generated by involutions. [ABSTRACT FROM AUTHOR]

Recently, Man and Du, in the context of classical texture analysis where the orientation distribution function (ODF) is defined on the rotation group SO(3), presented a systematic procedure by which the classical expansion of an ODF truncated at the order l=L can be directly rewritten as a tensorial Fourier expansion truncated at the same order. In classical texture analysis, the groups of crystallite and texture symmetries are assumed to be subgroups of SO(3), which is unreasonable, say, for aggregates of crystallites in a crystal class defined by an improper point group. In this paper, we consider ODFs defined on the orthogonal group O(3) and extend the aforementioned procedure to write down tensorial Fourier expansions for polycrystals with crystallite symmetry defined by any of the 21 improper point groups. Examples that illustrate the general procedure are given. This article is part of the theme issue 'Foundational issues, analysis and geometry in continuum mechanics'. [ABSTRACT FROM AUTHOR]

Geometric, robust-to-noise features of curves in Euclidean space are of great interest for various applications such as machine learning and image analysis. We apply Fels–Olver's moving-frame method (for geometric features) paired with the log-signature transform (for robust features) to construct a set of integral invariants under rigid motions for curves in R d from the iterated-integrals signature. In particular, we show that one can algorithmically construct a set of invariants that characterize the equivalence class of the truncated iterated-integrals signature under orthogonal transformations, which yields a characterization of a curve in R d under rigid motions (and tree-like extensions) and an explicit method to compare curves up to these transformations. [ABSTRACT FROM AUTHOR]

We show that the additive Borcherds lifts of vector-valued non-holomorphic Eisenstein series are orthogonal non-holomorphic Eisenstein series for O (2 , l). Using this we give another proof that they have a meromorphic continuation, calculate their Fourier expansion and show that they have a functional equation analogous to the classical case. Moreover, we will investigate the image of Borcherds lift and give a sufficient condition for surjectivity. [ABSTRACT FROM AUTHOR]

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 SO(n,c) and SP(2n,c) 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. [ABSTRACT FROM AUTHOR]

This article contains the proof of a theorem on orthogonal–Pin duality that was cited without proof in a previous article in this journal (Bulg. J. Phys. 48 (2021) 390-399). [ABSTRACT FROM AUTHOR]

We show that the hypergeometric groups associated to the pairs of the parameters (0 , 1 12 , 5 12 , 7 12 , 11 12) , (1 2 , 1 3 , 1 3 , 2 3 , 2 3) ; and (0 , 0 , 1 3 , 2 3) , (1 2 , 1 2 , 1 4 , 3 4) are arithmetic. [ABSTRACT FROM AUTHOR]

In this paper, we study the structures of the invariant subspaces under the action of orthogonal group O2ν(Fq,S){O}_{2\nu }\left({F}_{q},S). In particular, we give a detailed description of 2-codimensional invariant subspaces. Moreover, we show that the height of transfer ideal Im(TrO2ν(Fq,S)){\rm{Im}}({{\rm{Tr}}}^{{O}_{2\nu }\left({F}_{q},S)}) is 2 and give a primary decomposition for the radical ideal of this transfer ideal.

Humans can use acquired experience to learn new skills quickly and without forgetting the knowledge they already have. However, the neural network cannot do continual learning like humans, because it is easy to fall into the stability-plasticity dilemma and lead to catastrophic forgetting. Since meta-learning with the already acquired knowledge as a priori can directly optimize the final goal, this paper proposes LGCMLA (Lie Group Continual Meta Learning Algorithm) based on meta-learning, this algorithm is an improvement of CMLA (Continual Meta Learning Algorithm) proposed by Jiang et al. On the one hand, LGCMLA enhances the continuity between tasks by changing the inner-loop update rule (from using random initialization parameters for each task to using the updated parameters of the previous task for the subsequent task). On the other hand, it uses orthogonal groups to limit the parameter space and adopts the natural Riemannian gradient descent to accelerate the convergence speed. It not only corrects the shortcomings of poor convergence and stability of CMLA, but also further improves the generalization performance of the model and solves the stability-plasticity dilemma more effectively. Experiments on miniImageNet, tieredImageNet and Fewshot-CIFAR100 (Canadian Institute For Advanced Research) datasets prove the effectiveness of LGCMLA. Especially compared to MAML (Model-Agnostic Meta-Learning) with standard four-layer convolution, the accuracy of 1 shot and 5 shot is improved by 16.4% and 17.99% respectively under the setting of 5-way on miniImageNet. [ABSTRACT FROM AUTHOR]

This paper describes a novel multi-dimensional OCDMA coding technique with the objective of enhancing capacity of the optical communication network. In order to overcome the drawbacks associated with the conventional coding methods, multiple modes, time slots and multicarrier are utilized to construct the code set using prime sequences. The general procedure for code construction using the proposed method is described in this paper and the performance of the proposed coding technique has been analyzed for 36 users using 3 frequency channels and 5 fiber modes. By properly assigning 12 time slots to orthogonal group of users, the encoded data transmitted simultaneously by all 36 users have been correctly decoded without interference. The obtained results confirm the ability of the proposed method to improve the network capacity by supporting maximum number of concurrent users. Due to mode and frequency spreading, the data security against interceptor is also ensured in the proposed method than the conventional mode division multiplexing. [ABSTRACT FROM AUTHOR]

We derive an explicit isomorphism between the Hilbert modular group and certain congruence subgroups on the one hand and particular subgroups of the special orthogonal group SO(2, 2) on the other hand. The proof is based on an application of linear algebra adapted to number theoretical needs. [ABSTRACT FROM AUTHOR]

Let us say that a three-qubit state u 000|000âź© + u 001|001âź© + â‹Ż + u 111|111âź© is real if all its amplitudes u rst are real numbers. We will prove that for every real three-qubit state | Ď• âź©, there exist three angles θ 0, θ 1 and θ 2 such that R y (θ 2) ⊗ R y (θ 1) ⊗ R y (θ 0)| Ď• âź© is a three-qubit of the form λ 1|000âź© + λ 2|011âź© + λ 3|101âź© + λ 4|110âź© + λ 5|111âź© with the λ i real numbers. In contrast with the general case, the case of three-qubits with complex amplitudes, we proved that for three qubit states, the dimension of the real entanglement space (the space obtained by identifying real qubit states with local orthogonal gates, instead of local unitary gates) is 4 and in this paper we find four linearly independent polynomial invariants of degree 4 which are not possible to find for the different Schmidt representations of three qubit states. See (AcĂ-n et al 2000 Phys. Rev. Lett. 85 1560; AcĂ-n et al 2001 J. Phys. A: Math. Gen. 34 6725; Carteret et al 2000 J. Math. Phys. 41 7932; Sudbery 2001 J. Phys. A: Math. Gen. 34 643). [ABSTRACT FROM AUTHOR]

Sparse coding is a class of unsupervised methods for learning a sparse representation of the input data in the form of a linear combination of a dictionary and a sparse code. This learning framework has led to state-of-the-art results in various signal processing tasks. However, classical methods learn the dictionary and the sparse code based on alternating optimizations, usually without theoretical guarantees for either optimality or convergence due to the non-convexity of the problem. Recent works on sparse coding with a complete dictionary provide strong theoretical guarantees thanks to the development of non-convex optimization. However, initial non-convex approaches learned the dictionary in the sparse coding problem sequentially in an atom-by-atom manner, which led to a long execution time. More recent works  have sought to directly learn the entire dictionary at once, which substantially reduces the execution time. However, the associated recovery performance is degraded with a finite number of data samples. In this paper, we propose an efficient sparse coding scheme with a two-stage optimization. The proposed scheme leverages the global and local Riemannian geometry of the two-stage optimization problem and facilitates fast implementation for superb dictionary recovery performance by a finite number of samples. We further prove that, with high probability, the proposed scheme can exactly recover any atom in the target dictionary with a finite number of samples. Experiments on both synthetic and real-world data verify the efficiency and robustness of the proposed scheme.1 [ABSTRACT FROM AUTHOR]

The construction of integrity bases for invariant and covariant polynomials built from a set of three dimensional vectors under the SO (3) and O (3) symmetries is presented. This paper is a follow-up to our previous work that dealt with a set of two dimensional vectors under the action of the SO (2) and O (2) groups (Dhont and Zhilinskií in J Phys A Math Theor 46:455202, 2013). The expressions of the Molien generating functions as one rational function are a useful guide to build integrity bases for the rings of invariants and the free modules of covariants. The structure of the non-free modules of covariants is more complex. In this case, we write the Molien generating function as a sum of rational functions and show that its symbolic interpretation leads to the concept of generalized integrity basis. The integrity bases and generalized integrity bases for O (3) are deduced from the SO (3) ones. The results are useful in quantum chemistry to describe the potential energy or multipole moment hypersurfaces of molecules. In particular, the generalized integrity bases that are required for the description of the electric and magnetic quadrupole moment hypersurfaces of tetratomic molecules are given for the first time. [ABSTRACT FROM AUTHOR]

Since their introduction by Beilinson–Drinfeld (Opers, 1993. arXiv math/0501398; Quantization of Hitchin's integrable system and Hecke eigensheaves, 1991), opers have seen several generalizations. In Biswas et al. (SIGMA Symmetry Integr Geom Methods Appl 16:041, 2020), a higher rank analog was studied, named generalized B-opers, where the successive quotients of the oper filtration are allowed to have higher rank and the underlying holomorphic vector bundle is endowed with a bilinear form which is compatible with both the filtration and the oper connection. Since the definition did not encompass the even orthogonal groups, we dedicate this paper to study generalized B-opers whose structure group is SO (2 n , C) and show their close relationship with geometric structures on a Riemann surface. [ABSTRACT FROM AUTHOR]

We derive explicit isomorphisms between certain congruence subgroups of the Siegel modular group, the Hermitian modular group over an arbitrary imaginary-quadratic number field and the modular group over the Hurwitz quaternions of degree 2 and the discriminant kernels of special orthogonal groups S O 0 (2 , n) , n = 3 , 4 , 6. The proof is based on an application of linear algebra adapted to the number theoretical needs. [ABSTRACT FROM AUTHOR]

Using the embedded gradient vector field method (see P. Birtea, D. Comănescu (2015) [7]), we present a general formula for the Laplace-Beltrami operator defined on a constraint manifold, written in the ambient coordinates. Regarding the orthogonal group as a constraint submanifold of the Euclidean space of n × n matrices, we give an explicit formula for the Laplace-Beltrami operator on the orthogonal group using the ambient Euclidean coordinates. We apply this new formula for some relevant functions. [ABSTRACT FROM AUTHOR]

Suppose that is an odd integer, . We prove that a periodic group , saturated with finite simple orthogonal groups of odd dimension over fields of odd characteristic, is isomorphic to for some locally finite field of odd characteristic. In particular, is locally finite and countable. [ABSTRACT FROM AUTHOR]

Let V be a rational quadratic space of signature (m; 2). A conjecture of Kudla relates the arithmetic degrees of top degree special cycles on an integral model of a Shimura variety associated with SO(V) to the coefficients of the central derivative of an incoherent Siegel Eisenstein series of genus m + 1. We prove this conjecture for the coefficients of non-singular index T when T is not positive definite. We also prove it when T is positive definite and the corresponding special cycle has dimension 0. To obtain these results, we establish new local arithmetic Siegel-Weil formulas at the archimedian and non-archimedian places. [ABSTRACT FROM AUTHOR]

We consider the subgroups in a symplectic or orthogonal group over a finite field of odd characteristic such that for some odd prime . We obtain a refinement of the well-known Aschbacher Theorem on subgroups of classical groups for this case. [ABSTRACT FROM AUTHOR]

Rotations are among the simplest and most important of linear operations, and the group of rotations is a fundamental structure both for physics and geometry. In the article, we present an introductory discussion about random rotations, which can be seen as an example from the more general theory of random operators or random matrices, area which finds countless applications. We discuss the structure of the orthogonal group. an algorithm for generating random rotations and results for the probability distributions of matrix elements and eigenvalues. [ABSTRACT FROM AUTHOR]

Let R be a unital subring of a commutative ring S, which is a free R-module of rank m. In 1994 and then in 2017, V. A. Koibaev and we described normalizers of subgroups GL(n, S) and E(n, S) in G = GL(mn,R), and showed that they are equal and coincide with the set {g ∈ G:E(n, S)g ≤ GL(n, S)} = Aut(S=R) x GL(n, S). Moreover, for any proper ideal A of R, NG (E(n, S) E(mn,R,A)), = ... (NGL(mn,R/A)) (E(n,S/SA))). In the present paper, we prove similar results about normalizers of classical subgroups, namely, the normalizers of subgroups EO(n, S), SO(n, S),O(n, S) and GO(n, S) in G are equal and coincide with the set {g ∈ G: EO(n, S)g ≤ GO(n, S)} = Aut(S/R) x GO(n, S). Similarly, the ones of subgroups Ep(n, S), Sp(n, S), and GSp(n, S) are equal and coincide with the set {g ∈ G:Ep(n, S)g ≤ GSp(n, S)} = Aut(S/R) x GSp(n, S). Moreover, for any proper ideal A of R, NG(EO(n, S) E(mn,R,A)) = ... (NGL(mn,R/A)) (EO(n,S/SA))) and NG(Ep(n, S) E(mn,R,A)) = ... (NGL(mn,R/A)) (Ep(n,S/SA))). When R = S, we obtain the known results of N. A. Vavilov and V. A. Petrov. [ABSTRACT FROM AUTHOR]

This paper considers the fundamental problem of learning a complete (orthogonal) dictionary from samples of sparsely generated signals. Most existing methods solve the dictionary (and sparse representations) based on heuristic algorithms, usually without theoretical guarantees for either optimality or complexity. The recent ℓ¹-minimization based methods do provide such guarantees but the associated algorithms recover the dictionary one column at a time. In this work, we propose a new formulation that maximizes the ℓ4-norm over the orthogonal group, to learn the entire dictionary. We prove that under a random data model, with nearly minimum sample complexity, the global optima of the ℓ4-norm are very close to signed permutations of the ground truth. Inspired by this observation, we give a conceptually simple and yet effective algorithm based on "matching, stretching, and projection" (MSP). The algorithm provably converges locally and cost per iteration is merely an SVD. In addition to strong theoretical guarantees, experiments show that the new algorithm is significantly more efficient and effective than existing methods, including KSVD and ℓ¹-based methods. Preliminary experimental results on mixed real imagery data clearly demonstrate advantages of so learned dictionary over classic PCA bases. [ABSTRACT FROM AUTHOR]

We construct distance-regular graphs, including strongly regular graphs, admitting a transitive action of the Chevalley groups G 2 (4) and G 2 (5) , the orthogonal group O(7, 3) and the Tits group T = 2 F 4 (2) ′ . Most of the constructed graphs have more than 1000 vertices, and the number of vertices goes up to 28431. Some of the obtained graphs are new. [ABSTRACT FROM AUTHOR]

Let M be a set of finite groups. Given a group G, denote by M (G) the set of all subgroups of G isomorphic to the elements of M . A group G is said to be saturated with groups from M (saturated with M , for brevity) if each finite subgroup of G lies in an element of M (G) . We prove that a periodic group G saturated with M = { O 7 (q) | q = ± 3 (mod 8) } is isomorphic to O7(F) for some locally finite field F of odd characteristic. [ABSTRACT FROM AUTHOR]

We show that for all integers m ⩾ 2 , and all integers k ⩾ 2 , the orthogonal groups O ± (2 m , F 2 k) act on abstract regular polytopes of rank 2m, and the symplectic groups Sp (2 m , F 2 k) act on abstract regular polytopes of rank 2 m + 1 . [ABSTRACT FROM AUTHOR]

Using the embedded gradient vector field method, we explicitly compute the set of critical points of the shear-stretch strain energy of a Cosserat body model. We also formulate necessary and sufficient conditions for critical points in the abstract case of the special orthogonal group SO (n) . Each critical point is then characterized using an explicit formula for the Hessian operator of a cost function defined on the orthogonal group. We also give a positive answer to an open question posed by Borisov et al. (Z Angew Math Mech 99:e201800120, 2019), namely if all local minima of the optimization problem are global minima. [ABSTRACT FROM AUTHOR]