Daily Papers

We provide a full characterisation of all of the possible group equivariant neural networks whose layers are some tensor power of R^{n} for three symmetry groups that are missing from the machine learning literature: O(n), the orthogonal group; SO(n), the special orthogonal group; and Sp(n), the symplectic group. In particular, we find a spanning set of matrices for the learnable, linear, equivariant layer functions between such tensor power spaces in the standard basis of R^{n} when the group is O(n) or SO(n), and in the symplectic basis of R^{n} when the group is Sp(n).

The learnable, linear neural network layers between tensor power spaces of R^{n} that are equivariant to the orthogonal group, O(n), the special orthogonal group, SO(n), and the symplectic group, Sp(n), were characterised in arXiv:2212.08630. We present an algorithm for multiplying a vector by any weight matrix for each of these groups, using category theoretic constructions to implement the procedure. We achieve a significant reduction in computational cost compared with a naive implementation by making use of Kronecker product matrices to perform the multiplication. We show that our approach extends to the symmetric group, S_n, recovering the algorithm of arXiv:2303.06208 in the process.

We present, for the first time, a novel theoretical approach to address the problem of correspondence free multivector cloud registration in conformal geometric algebra. Such formalism achieves several favorable properties. Primarily, it forms an orthogonal automorphism that extends beyond the typical vector space to the entire conformal geometric algebra while respecting the multivector grading. Concretely, the registration can be viewed as an orthogonal transformation (\it i.e., scale, translation, rotation) belonging to SO(4,1) - group of special orthogonal transformations in conformal geometric algebra. We will show that such formalism is able to: (i) perform the registration without directly accessing the input multivectors. Instead, we use primitives or geometric objects provided by the conformal model - the multivectors, (ii) the geometric objects are obtained by solving a multilinear eigenvalue problem to find sets of eigenmultivectors. In this way, we can explicitly avoid solving the correspondences in the registration process. Most importantly, this offers rotation and translation equivariant properties between the input multivectors and the eigenmultivectors. Experimental evaluation is conducted in datasets commonly used in point cloud registration, to testify the usefulness of the approach with emphasis to ambiguities arising from high levels of noise. The code is available at https://github.com/Numerical-Geometric-Algebra/RegistrationGA . This work was submitted to the International Journal of Computer Vision and is currently under review.

There exist a number of well known multiplicative generating functions for series of Schur functions. Amongst these are some related to the dual Cauchy identity whose expansion coefficients are rather simple, and in some cases periodic in parameters specifying the Schur functions. More recently similar identities have been found involving expansions in terms of characters of the symplectic group. Here these results are extended and generalised to all classical Lie groups. This is done through the derivation of explicit recurrence relations for the expansion coefficients based on the action of the Weyl groups of both the symplectic and orthogonal groups. Copious results are tabulated in the form of explicit values of the expansion coefficients as functions of highest weight parameters. An alternative approach is then based on dual pairs of symplectic and/or orthogonal groups. A byproduct of this approach is that expansions in terms of spin orthogonal group characters can always be recovered from non-spin cases.

Building on the iHopf algebra realization of quasi-split universal iquantum groups developed in a prequel, we construct the dual canonical basis for a universal iquantum group of arbitrary finite type, which are further shown to be preserved by the ibraid group action; this recovers the results of Lu-Pan in ADE type obtained earlier in a geometric approach. Moreover, we identify the dual canonical basis for the Drinfeld double quantum group of arbitrary finite type, which is realized via iHopf algebra on the double Borel, with Berenstein-Greenstein's double canonical basis, settling several of their conjectures.

This note discusses our formalisation in Lean 4 of the classification of groups of order p^3 for a prime number p, using mathlib4. We present the five isomorphism classes and give a detailed account of the formalisation, with particular emphasis on the non-abelian case, which requiring the most substantial formal development. For odd~p, the non-abelian groups are the Heisenberg group Heis(Z/pZ) and the semidirect product Z/p^2ZrtimesZ/pZ; for p=2, they are D_4 and Q_8. We describe the construction of these concrete groups, the structural lemmas about centers, commutators, and exponents, and the explicit isomorphism constructions that classify an arbitrary non-abelian p^3-group.

We use the extrapolate dictionary to revisit the spectrum of operators in Celestial CFT. Under the Celestial CFT map, each state in the 4D Hilbert space should map to one in the 2D Hilbert space. This implies that, beyond the familiar single particle states/operators, there should be multiparticle operators appearing in the celestial OPE. We extend the existing flat-space dictionary by constructing composite primaries from both Carrollian and Celestial perspectives. In the process, we demonstrate some subtleties in deriving the Poincaré primary condition from the Carrollian limit, clarify the compatibility of principal series representations with highest weight representations and unitarity in Celestial CFT, and discuss how the celestial OPE block expansion emerges from a 2D CFT standpoint.

If h is a ring-valued function on a simplicial complex G we can define two matrices L and g, where the matrix entries are the h energy of homoclinic intersections. We know that the sum over all h values on G is equal to the sum of the Green matrix entries g(x,y). We also have already seen that that the determinants of L or g are both the product of the h(x). In the case where h(x) is the parity of dimension, the sum of the energy values was the standard Euler characteristic and the determinant was a unit. If h(x) was the unit in the ring then L,g are integral quadratic forms which are isospectral and inverse matrices of each other. We prove here that the quadratic energy expression summing over all pairs h(x)^* h(y) of intersecting sets is a signed sum of squares of Green function entries. The quadratic energy expression is Wu characteristic in the case when h is dimension parity. For general h, the quadratic energy expression resembles an Ising Heisenberg type interaction. The conjugate of g is the inverse of L if h takes unit values in a normed ring or in the group of unitary operators in an operator algebra.

For any integer d geq 2 and prime power q, we construct unexpectedly large induced matchings in the point-line incidence graph of F_{q}^{d} by leveraging a new connection with the Furstenberg-Sárközy problem from arithmetic combinatorics. In particular, we significantly improve the previously well-known baselines when q is prime, showing that F_{q}^{2} contains matchings of size q^{1.233} and F_{q}^{d} contains matchings of size q^{d-o_{d}(1)}. These results and their proofs have several applications. First, we also obtain new constructions for finite field Nikodym sets in dimension d geq 2, improving recent results of Tao by polynomial factors. For example, when q is prime, we show the existence of Nikodym sets in F_q^d of size q^d - q^{d - o_d(1)}. Second, we construct a new minimal blocking set in PG(2,q), solving a longstanding problem in finite geometry. Third, we obtain new constructions for the minimal distance problem (in R^{2} and also in higher dimensions), improving a recent result of Logunov-Zakharov. We also obtain analogous results for general finite fields with large characteristics. In particular, in one of our constructions we introduce a new special set of points inside the norm hypersurface in F_{q}^{d}, which directly generalizes the classical Hermitian unital and which may be of independent interest for applications.

For an algebraic number field K, the Pólya group of K, denoted by Po(K), is the subgroup of the ideal class group Cl_{K} generated by the ideal classes of the products of prime ideals of same norm. The number field K is said to be Pólya if Po(K) is trivial. Motivated by several recent studies on the group Po(K) when K is a totally real bi-quadratic field, we investigate the same with five distinct odd primes ramifying in K/Q. This extends the previous results on this problem, where the number of distinct ramified primes was at most four.

We give a complete characterization of the holonomies of strictly convex cusps and of round cusps in convex projective geometry. We build families of generalized cusps of non-maximal rank associated to each strictly convex or round cusp. We also extend Ballas-Cooper-Leitner's definition of generalized cusp to allow for virtually solvable fundamental group, and we produce the first such example with non-virtually nilpotent fundamental group. Along with a companion paper, this allows to build strictly convex cusps and generalized cusps whose fundamental group is any finitely generated virtually nilpotent group. This also has interesting consequences for the theory of relatively Anosov representations.

The problem of computing optimal orthogonal approximation to a given matrix has attracted growing interest in machine learning. Notable applications include the recent Muon optimizer or Riemannian optimization on the Stiefel manifold. Among existing approaches, the Newton-Schulz iteration has emerged as a particularly effective solution, as it relies solely on matrix multiplications and thus achieves high computational efficiency on GPU hardware. Despite its efficiency, the method has inherent limitations - its coefficients are fixed and thus not optimized for a given matrix. In this paper we address this issue by proposing a Chebyshev-optimized version of Newton-Schulz (CANS). Based on the Chebyshev's alternance theorem, we theoretically derive optimal coefficients for the 3-rd order Newton-Schulz iteration and apply a Remez algorithm to compute optimal higher-degree polynomials. We leverage these polynomials to construct controlled approximate orthogonalization schemes, which is of interest in deep learning applications. Practically, we demonstrate the method's effectiveness in two key applications: orthogonalization in the Muon optimizer, and providing an efficient retraction alternative for Riemannian optimization on the Stiefel manifold.

We study the matrix models pi:C(S_N^+)to M_N(C(X)) which are flat, in the sense that the standard generators of C(S_N^+) are mapped to rank 1 projections. Our first result is a generalization of the Pauli matrix construction at N=4, using finite groups and 2-cocycles. Our second result is the construction of a universal representation of C(S_N^+), inspired from the Sinkhorn algorithm, that we conjecture to be inner faithful.

For any {rm E}-rigid presentation e, we construct an orthogonal projection functor to {rm rep}(e^perp) left adjoint to the natural embedding. We establish a bijection between presentations in {rm rep}(e^perp) and presentations compatible with e. For quivers with potentials, we show that {rm rep}(e^perp) forms a module category of another quiver with potential. We derive mutation formulas for the delta-vectors of positive and negative complements and the dimension vectors of simple modules in {rm rep}(e^perp), enabling an algorithm to find the projected quiver with potential. Additionally, we introduce a modified projection for quivers with potentials that preserves general presentations. For applications to cluster algebras, we establish a connection to the stabilization functors.

We propose Equivariance by Contrast (EbC) to learn equivariant embeddings from observation pairs (y, g cdot y), where g is drawn from a finite group acting on the data. Our method jointly learns a latent space and a group representation in which group actions correspond to invertible linear maps -- without relying on group-specific inductive biases. We validate our approach on the infinite dSprites dataset with structured transformations defined by the finite group G:= (R_m times Z_n times Z_n), combining discrete rotations and periodic translations. The resulting embeddings exhibit high-fidelity equivariance, with group operations faithfully reproduced in latent space. On synthetic data, we further validate the approach on the non-abelian orthogonal group O(n) and the general linear group GL(n). We also provide a theoretical proof for identifiability. While broad evaluation across diverse group types on real-world data remains future work, our results constitute the first successful demonstration of general-purpose encoder-only equivariant learning from group action observations alone, including non-trivial non-abelian groups and a product group motivated by modeling affine equivariances in computer vision.

Invariance and equivariance to geometrical transformations have proven to be very useful inductive biases when training (convolutional) neural network models, especially in the low-data regime. Much work has focused on the case where the symmetry group employed is compact or abelian, or both. Recent work has explored enlarging the class of transformations used to the case of Lie groups, principally through the use of their Lie algebra, as well as the group exponential and logarithm maps. The applicability of such methods to larger transformation groups is limited by the fact that depending on the group of interest G, the exponential map may not be surjective. Further limitations are encountered when G is neither compact nor abelian. Using the structure and geometry of Lie groups and their homogeneous spaces, we present a framework by which it is possible to work with such groups primarily focusing on the Lie groups G = GL^{+}(n, R) and G = SL(n, R), as well as their representation as affine transformations R^{n} rtimes G. Invariant integration as well as a global parametrization is realized by decomposing the `larger` groups into subgroups and submanifolds which can be handled individually. Under this framework, we show how convolution kernels can be parametrized to build models equivariant with respect to affine transformations. We evaluate the robustness and out-of-distribution generalisation capability of our model on the standard affine-invariant benchmark classification task, where we outperform all previous equivariant models as well as all Capsule Network proposals.

We unify the discrete Fourier transform (DFT), discrete cosine transform (DCT), Walsh-Hadamard, Haar wavelet, Karhunen-Loève transform, and several others along with their continuous counterparts (Fourier transform, Fourier series, spherical harmonics, fractional Fourier transform) under one representation-theoretic principle: each is the eigenbasis of every covariance invariant under a specific finite or compact group, with columns constructed from the irreducible matrix elements of the group via the Peter-Weyl theorem. The unification rests on the Algebraic Diversity (AD) framework, which identifies the matched group of a covariance as the foundational object of second-order signal processing. The data-dependent KLT emerges as the trivial-matched-group limit; classical transforms emerge as the cyclic, dihedral, elementary abelian, iterated wreath, and hybrid wreath cases. Composition rules cover direct, wreath, and semidirect products. The Reed-Muller and arithmetic transforms appear as related change-of-basis transforms on the matched group of Walsh-Hadamard. A polynomial-time algorithm for matched-group discovery, the DAD-CAD relaxation cast as a generalized eigenvalue problem in double-commutator form, closes the operational loop: the matched group of any empirical covariance is discovered without expert judgment, with noise-aware variants via the commutativity residual δ and algebraic coloring index α for finite-SNR settings. The fractional Fourier transform is treated as the metaplectic SO(2) case with Hermite-Gauss matched basis, and a structural principle relates matched group size inversely to transform resolution. Modern applications (massive-MIMO, graph neural networks, transformer attention, point cloud and 3D vision, brain connectivity, single-cell genomics, quantum informatics) are sketched with their matched groups.

The algebraic diversity framework generalizes temporal averaging over multiple observations to algebraic group action on a single observation for second-order statistical estimation. The central open problem in this framework is group selection: given an M-dimensional observation with unknown covariance structure, find the finite group whose spectral decomposition best matches the covariance. Naive enumeration of all subgroups of the symmetric group S_M requires exponential time in M. We prove that this combinatorial problem reduces to a generalized eigenvalue problem derived from the double commutator of the covariance matrix, yielding a polynomial-time algorithm with complexity O(d^2M^2 + d^3), where d is the dimension of a generator basis. The minimum eigenvector of the double-commutator matrix directly constructs the optimal group generator in closed form, with no iterative optimization. The reduction is exact: the double-commutator minimum eigenvalue is zero if and only if the optimal generator lies in the span of the basis, and its magnitude provides a certifiable optimality gap when it does not. This problem does not appear in the standard catalogs of computational complexity (Garey and Johnson, 1979) and represents a new class linking group theory, matrix analysis, and statistical estimation. We establish connections to independent component analysis (JADE), structured matrix nearness problems, and simultaneous matrix diagonalization, and we show that the double-commutator formulation is the unique approach that is simultaneously polynomial-time, closed-form, and certifiable. We extend the framework to non-Abelian symmetry recovery via a Sequential GEVP with deflation, and add two identifiability theorems characterizing the commutant-lattice ambiguity and the dichotomy on whether Aut(R) recovers a generative subgroup or only a supergroup.

The question of whether there exists a finite group of order at least three in which every element except one is a commutator has remained unresolved in group theory. In this article, we address this open problem by developing an algorithmic approach that leverages several group theoretic properties of such groups. Specifically, we utilize a result of Frobenius and various necessary properties of such groups, combined with Plesken and Holt's extensive enumeration of finite perfect groups, to systematically examine all finite groups up to a certain order for the desired property. The computational core of our work is implemented using the computer system GAP (Groups, Algorithms, and Programming). We discover two nonisomorphic groups of order 368,640 that exhibit the desired property. Our investigation also establishes that this order is the minimum order for such a group to exist. As a result, this study provides a positive answer to Problem 17.76 in the Kourovka Notebook. In addition to the algorithmic framework, this paper provides a structural description of one of the two groups found.

For w in the symmetric group, we provide an exact formula for the smallest positive power q^{h(w)} appearing in the Kazhdan-Lusztig polynomial P_{e,w}(q). We also provide a tight upper bound on h(w) in simply-laced types, resolving a conjecture of Billey-Postnikov from 2002.

Developing equivariant neural networks for the E(3) group plays an important role in modeling 3D data across real-world applications. Enforcing this equivariance primarily involves the tensor products of irreducible representations (irreps). However, the computational complexity of such operations increases significantly as higher-order tensors are used. In this work, we propose a systematic approach to substantially accelerate the computation of the tensor products of irreps. We mathematically connect the commonly used Clebsch-Gordan coefficients to the Gaunt coefficients, which are integrals of products of three spherical harmonics. Through Gaunt coefficients, the tensor product of irreps becomes equivalent to the multiplication between spherical functions represented by spherical harmonics. This perspective further allows us to change the basis for the equivariant operations from spherical harmonics to a 2D Fourier basis. Consequently, the multiplication between spherical functions represented by a 2D Fourier basis can be efficiently computed via the convolution theorem and Fast Fourier Transforms. This transformation reduces the complexity of full tensor products of irreps from O(L^6) to O(L^3), where L is the max degree of irreps. Leveraging this approach, we introduce the Gaunt Tensor Product, which serves as a new method to construct efficient equivariant operations across different model architectures. Our experiments on the Open Catalyst Project and 3BPA datasets demonstrate both the increased efficiency and improved performance of our approach.

In this paper, we sharpen results obtained by the author in 2023. The new results reduce the Mathieu Conjecture on SU(N) (formulated for all compact connected Lie groups by O. Mathieu in 1997) to a conjecture involving only functions on R^ntimes (S^1)^m with n,m non-negative integers instead of involving functions on R^ntimes (S^1setminus{1})^m. The proofs rely on a more recent work of the author (2024) and a specific KAK decomposition. Finally, with these results we can also improve the results on the groups Sp(N) and G_2 in the latter paper, since they relied on the construction introduced in the 2023 paper.

For a finite group, it is interesting to determine when two ordinary irreducible representations have the same p-modular reduction; that is, when two rows of the decomposition matrix in characteristic p are equal, or equivalently when the corresponding p-modular Brauer characters are the same. We complete this task for the double covers of the symmetric group when p=2, by determining when the 2-modular reduction of an irreducible spin representation coincides with a 2-modular Specht module. In fact, we obtain a more general result: we determine when an irreducible spin representation has 2-modular Brauer character proportional to that of a Specht module. In the course of the proof, we use induction and restriction functors to construct a function on generalised characters which has the effect of swapping runners in abacus displays for the labelling partitions.

In this article, we continue the development of the Riemann-Hilbert formalism for studying the asymptotics of Toeplitz+Hankel determinants with non-identical symbols, which we initiated in GI. In GI, we showed that the Riemann-Hilbert problem we formulated admits the Deift-Zhou nonlinear steepest descent analysis, but with a special restriction on the winding numbers of the associated symbols. In particular, the most natural case, namely zero winding numbers, is not allowed. A principal goal of this paper is to develop a framework that extends the asymptotic analysis of Toeplitz+Hankel determinants to a broader range of winding-number configurations. As an application, we consider the case in which the winding numbers of the Szego-type Toeplitz and Hankel symbols are zero and one, respectively, and compute the asymptotics of the norms of the corresponding system of orthogonal polynomials.

Given two homogeneous spaces of the form G_1/K and G_2/K, where G_1 and G_2 are compact simple Lie groups, we study the existence problem for G_1xG_2-invariant Einstein metrics on the homogeneous space M=G_1xG_2/K. For the large subclass C of spaces having three pairwise inequivalent isotropy irreducible summands (12 infinite families and 70 sporadic examples), we obtain that existence is equivalent to the existence of a real root for certain quartic polynomial depending on the dimensions and two Killing constants, which allows a full classification and the possibility to weigh the existence and non-existence pieces of C.

We present an algorithm to compute the torsion component Pic^τX of the Picard scheme of a smooth projective variety X over a field k. Specifically, we describe Pic^τX as a closed subscheme of a projective space defined by explicit homogeneous polynomials. Furthermore, we compute the group scheme structure on Pic^τX. As applications, we provide algorithms to compute various homological invariants. Among these, we compute the abelianization of the geometric étale fundamental group π^{{et}}_1(X_{k}, x)^{ab}. Moreover, we determine the Galois module structure of the first étale cohomology groups H^1_{{et}}(X_{k}, Z/nZ) without requiring n to be prime to the characteristic of k.

Symmetries play a central role in quantum many-body physics, yet uncovering them systematically remains challenging. We introduce a bootstrap framework designed to reconstruct the representation theory of hidden finite group symmetries of quantum many-body lattice Hamiltonians, using only a known symmetry subgroup N and spectral correlations between its symmetry sectors. We introduce a novel variant of the spectral form factor, the cross spectral form factor (xSFF), which we compute via exact diagonalization to seed the bootstrap algorithm. By applying the constraints derived from these data alongside the algebraic conditions of the fusion rules, our bootstrap procedure sharply restricts the set of candidate groups G. Remarkably, without any prior assumptions regarding the full symmetry group G, our method can systematically recover its representation-theoretic data, including the number and dimensions of the irreducible representations, their branching rules with respect to N, the fusion algebra, and the full character table. This framework applies equally well to chaotic and integrable many-body systems and accommodates both unitary and anti-unitary symmetries. Through various examples, we demonstrate that the underlying group G can be uniquely identified. In particular, our bootstrap independently recovers the Z_4 symmetry at the self-dual point of the three-state quantum torus chain, detects signatures of projective representations in the effective Hamiltonian of the driven Bose-Hubbard model, and rediscovers the η-pairing SO(4) symmetry of the one-dimensional Fermi-Hubbard model. Our framework thus establishes a practical route to identify symmetries directly from dynamical spectral observables.

For a finite family of representations of a reductive group, we define a Laurent polynomial on the group. The exponential sum associated this Laurent polynomial is called a hypergeometric exponential sum. We introduce an arithmetic hypergeometric mathcal D-module to study the hypergeometric exponential sum. It is an overholonomic arithmetic mathcal D-module with a Frobenius structure so that the trace of the Frobenius at a rational point is the exponential sum. Over the locus where the Laurent polynomial is nondegenerate, the arithmetic hypergeometric D-module defines an F-isocrystal overconvergent along the degenerate locus. As an application, we get an estimation of the hypergeometric exponential sum.

We investigate the exceptional points (EPs) and their pseudospectra in black hole perturbation theory. By considering a Gaussian bump modification to the Regge-Wheeler potential with variable amplitude, position, and width parameters, (varepsilon,d,σ_0), a continuous line of EPs (exceptional line, EL) in this three-dimensional parameter space is revealed. We find that the vorticity ν=pm1/2 and the Berry phase γ=π for loops encircling the EL, while ν=0 and γ=0 for those do not encircle the EL. Through matrix perturbation theory, we prove that the ε-pseudospectrum contour size scales as ε^{1/q} at an EP, where q is the order of the largest Jordan block of the Hamiltonian-like operator, contrasting with the linear ε scaling at non-EPs. Numerical implements confirm this observation, demonstrating enhanced spectral instability at EPs for non-Hermitian systems including black holes.

For a locally linear graph G, which is a graph built out of triangles, it is possible to construct another graph G^* that would consist of triangles of G as vertices, while sharing (or not sharing) a common vertex between a pair of triangles would define a binary relation for edges of G^*. In this paper we show that the spectrum of G^* is uniquely defined by G. We will also show some structural similarities of these graphs; in particular, that the number of quadrilaterals and pentagons in both graphs are the same; that G^* does not contain K_4-e and K_{1,4}; and that G can be reconstructed from G^*.

Hyper-Connections (HC) replace the single residual stream of a Transformer with n parallel ones, mixing them at every layer with a learned n times n residual matrix. Leaving that matrix unconstrained places no limit on the factor by which the mixing step rescales the residual streams, and that factor compounds across layers, which destabilizes training. Manifold-constrained Hyper-Connections (mHC) address this by restricting the matrix to the doubly stochastic matrices. That caps the factor at one, so the mixing can no longer amplify any direction, but nothing bounds it from below. We prove that inside this set the mixing step can reduce the norm of the residual streams only by shrinking the differences between the streams, while their mean is left unchanged; and since the reduction accumulates over layers, the streams grow more alike and their diversity is spent with depth. We therefore propose Orthogonal Hyper-Connections (oHC), restricting the residual matrix to the rotation group SO(n), so that the mixing step can neither amplify nor attenuate the residual streams in any direction, which keeps training stable and no longer forces the differences between the streams to contract. Specifically, at the four streams used by recent HC models we parameterize the group in closed form by a pair of unit quaternions, which adds no parameters, replaces the iterative projection with a fixed pattern of signed additions, and can be constructed faster than mHC. We evaluate oHC across a comprehensive set of downstream tasks, where it outperforms the single-stream residual baseline, mHC and iHC, which fixes the residual matrix to the identity.

In this paper, we study the mathbb J-stratification of basic affine Deligne-Lusztig varieties for a minuscule cocharacter μ. This stratification was introduced by Chen-Viehmann and has been expected to serve as an interesting tool for studying basic loci in Shimura varieties. We parametrize the mathbb J-strata whose stabilizers in the Frobenius-twisted centralizer group are parahoric by constructing a natural bijection to combinatorial invariants called small cocharacters. We further prove that the cardinality of these sets is equal to that of a certain subset of the Weyl group orbit of μ. A relationship with the weakly fully Hodge-Newton decomposability of Chen-Tong is also discussed.

The tensor rank of a bilinear map is the least number of multiplications any bilinear algorithm needs to compute it; for the multiplication of an algebra it measures how cheaply the algebra can be multiplied at all. For the even-dimensional real normed division algebras it is 3 for the complex numbers and 8 for the quaternions, both classical, while for the octonions O only a range was known: at least 15 (Fiduccia and Zalcstein, 1977) and at most 30 (Cariow and Cariowa). We prove $18 le R_{R}(T_{O}) le 25. The lower bound peels the eight slices of T_{O} down to two and bounds the rank of the surviving pencil through the octonion norm. Nothing in it is special to dimension 8: the same steps give R_{R}(T_A) \ge 5{2}n - 2 for every real normed division algebra A of even dimension n, sharp for C and H and the best bound we know for O. The upper bound is a separate construction, an explicit rank-25 decomposition certified by a Krawczyk argument, in exact rational arithmetic, to sit within 10^{-6} of an exact one. The same two arguments pin down the rank of a smaller three-slice quaternion tensor τ, giving R_{R}(τ) = 7$. The Lean 4 kernel checks the lower bounds and the Krawczyk existence principle; the accompanying scripts check the certificate's finitely many exact-rational inequalities.

We identify the quadratic symmetry algebra of the two-dimensional Smorodinsky--Winternitz II system with a Laguerre-type confluent Heun algebra. The system is separable in Cartesian and parabolic coordinates. The complementary Cartesian separation operator \[ Y=\partial_y^2-ω^2y^2+1/4-c^2{y^2} \] is of Laguerre type, while the parabolic integral \(W=L_2\) is its algebraic Heun partner. With \(Z=[Y,W]\), the defining relations are \[ [Y,Z]=16ω^2W-2bY,\qquad [W,Z]=6Y^2-4HY+2bW+8ω^2(1-c^2), \] where \(H\) is central. This gives a direct superintegrable realization of the Laguerre--Heun algebra.

As neural networks become deeper, the redundancy within their parameters increases. This phenomenon has led to several methods that attempt to reduce the correlation between convolutional filters. We propose a computationally efficient regularization technique that encourages orthonormality between groups of filters within the same layer. Our experiments show that when incorporated into recent adaptation methods for diffusion models and vision transformers (ViTs), this regularization improves performance on downstream tasks. We further show improved robustness when group orthogonality is enforced during adversarial training. Our code is available at https://github.com/YoavKurtz/GOR.