In this paper we extend DDVV-type inequalities involving the Frobenius norm of commutators from real symmetric and skew-symmetric matrices to Hermitian and skew-Hermitian matrices.
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A sufficient knowledge of the demographics of a commuting public is essential in formulating and implementing more targeted transportation policies, as commuters exhibit different ways of traveling. With the advent of the Automated Fare Collection system (AFC), probing the travel patterns of commuters has become less i…
New neural networks for non-commutative data.
problem No existing neural networks suitable for non-commutative data.
method Developed compact matrix quantum group equivariant neural networks.
result Characterized weight matrices for easy compact matrix quantum groups.
This paper explores how Transformers predict next tokens in autoregressive tasks.
problem Understanding the success of Transformers in autoregressive learning.
method Trained a Transformer on a next-token prediction task, focusing on commuting orthogonal matrices.
result Trained Transformers can be seen as implementing gradient descent for a specific objective function.
We represent algebraic curves via commuting matrix polynomials. This allows us to show that the Hilbert scheme of cohomologically stable twisted rational curves of degree in is isomorphic to a complexified hyperkähler quotient of an open subset of a vector space by a non-reductive …
New invariants derived from random matrices for words in free groups.
problem Defining and understanding new topological invariants for words in free groups.
method Defining and analyzing invariants from -random matrices and permutations.
result Presented new topological, combinatorial, and algebraic invariants of words.
We develop the theory of linear algebra over a (Z_2)^n-commutative algebra (n in N), which includes the well-known super linear algebra as a special case (n=1). Examples of such graded-commutative algebras are the Clifford algebras, in particular the quaternion algebra H. Following a cohomological approach, we introduc…
The paper extends ternary algebra concepts using cube roots of unity.
problem Extending algebraic structures from binary to ternary multiplication.
method Introducing ternary associator, commutator, and Lie algebra at cube roots of unity.
result Derived an identity for ternary commutator based on .
We define the notions of trace, determinant and, more generally, Berezinian of matrices over a (Z_2)^n graded commutative associative algebra. The applications include a new approach to the classical theory of matrices with coefficients in a Clifford algebra, in particular of quaternionic matrices. In a special case, w…
A Non-commutative Extension of Lee-Seung's Algorithm for Positive Semidefinite Factorizationsmath.OC
An algorithm for computing positive semidefinite factorizations of matrices.
problem Computing positive semidefinite factorizations of matrices.
method Non-commutative extension of Lee-Seung's algorithm (Matrix Multiplicative Update, MMU).
result The MMU algorithm ensures PSD updates and achieves critical points.
In this paper we show how generalized quaternions, including 2X2 matrices, can be used to find solutions of a non-commuting equation intimately connected with braid groups. These solutions can then be used to find polynomial invariants of virtual knots and links.
Classifies cobounded hyperbolic actions of metabelian groups.
problem Classify cobounded hyperbolic actions of metabelian groups.
method Builds connections between hyperbolic geometry and commutative algebra to classify actions.
result Classifies cobounded hyperbolic actions of many abelian-by-cyclic groups.
The space of tensors of metric curvature type on a Euclidean vector space carries a two-parameter family of orthogonally invariant commutative nonassociative multiplications invariant with respect to the symmetric bilinear form determined by the metric. For a particular choice of parameters these algebras recover the p…
The paper presents a new algebraic structure for planar surfaces.
problem Understanding the algebraic structure of planar surfaces.
method Explicitly defined generators and relations for Kauffman bracket skein algebras of planar surfaces.
result A new independent presentation of Kauffman bracket skein algebras for planar surfaces.
Study cohomology of GL₂n(Z) and graph complexes using Pfaffian forms.
problem Cohomology of GL₂n(Z) and related graph complexes.
method Use Pfaffian forms on symmetric spaces and dual Laplacians on graphs.
result First cocycle gives a non-trivial class in H⁻⁶(GC₃).
In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 22 matrices with entries in a possibly non-commutative ring, for example the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from …
Formula for sectional curvatures on matrix groups.
problem Calculating curvatures on matrix groups.
method Simple formula derivation for sectional curvatures.
result Valid formula for general linear and reductive Lie groups.
Mirror descent algorithm recovers low-rank matrices in matrix sensing.
problem Matrix sensing with low-rank matrices under certain conditions.
method Discrete-time mirror descent applied to empirical risk with Bregman divergence analysis.
result Mirror descent converges to a matrix minimizing a specific nuclear norm-related quantity.
Investigates O(n)-invariant metrics on SPD matrices, extending kernel metrics.
problem Limited coverage of O(n)-invariant metrics by kernel metrics.
method Characterization of O(n)-invariant metrics, intermediate classes construction.
result Introduction of cometric-stability as a key property for geodesics.
We provide bounds for kernel matrices and new approximations for high-dimensional data.
problem Approximating high-dimensional empirical kernel matrices.
method Decoupling results for U-statistics and non-commutative Khintchine inequality.
result New tighter approximations for inner-product kernel matrices.
This note classifies splittable lattices in a specific Lie group.
problem Classifying splittable lattices in a metabelian solvable Lie group.
method Description and classification of splittable lattices in .
result Classification of splittable lattices in the specified Lie group.
Starting from considering deeper relationship between conjugacy classes and irreducible representations of a finite group , we find some quite simple matrice defined by using finite groups. This construction produces many sets (or topological spaces) admitting braid group actions. We introduce conceptions "exten…
There has been much recent progress in forecasting the next observation of a linear dynamical system (LDS), which is known as the improper learning, as well as in the estimation of its system matrices, which is known as the proper learning of LDS. We present an approach to proper learning of LDS, which in spite of the …
There is an equivalence relation on the set of smooth maps of a manifold into the stable unitary group, defined using a Chern-Simons type form, whose equivalence classes form an abelian group under ordinary block sum of matrices. This construction is functorial, and defines a differential extension of odd K-theory, fit…
Study integrability of quantized six-vertex model on torus.
problem Integrability of a specific lattice model on a torus.
method Defined layer transfer matrices and tetrahedron equations for admissible graphs.
result Established commutativity of transfer matrices and derived quantum Hamiltonians.
Convexity proven for sums of angles of unitary paths.
problem Proving convexity of sums of eigenvalues of unitary matrices.
method Analyzing paths of unitary matrices and their angles, using operator norms.
result Sum of first m angles of unitary path is convex.
The paper studies matrix normalization and graph balancing using a new functional and gradient descent.
problem Matrix normalization and graph balancing.
method A new functional called the non-normal energy, and gradient descent.
result Gradient descent of the non-normal energy converges to balanced graphs and preserves spectra and realness of weights.
Innovates rotation index for matrix pairs, solving group action problems.
problem Solving group actions problems, especially Nielsen realization and higher-rank Anosov actions.
method Rotation index and Milnor--Munkres--Novikov pairing applied to group actions.
result Solved specific group action problems using new matrix pair invariant.
The paper establishes correspondences between quaternionic spinors, Minkowski flags, and hyperbolic horospheres.
problem Understanding geometric correspondences in 4D hyperbolic geometry.
method Explicit bijective correspondences using Clifford matrices and bilinear forms.
result Lambda lengths generalize to quaternionic values in 4D hyperbolic space and satisfy a non-commutative Ptolemy equation.
Let be invertible, non-commuting elements of a ring . Suppose that is also invertible and that the equation called the fundamental equation is satisfied. Then an invariant -module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case hav…
New algebraic structures on manifolds generalize supergeometry concepts.
problem Developing algebraic structures for non-commutative manifolds.
method Introducing -commutative manifolds, Q-manifolds, and modular classes.
result Generalized modular classes for non-commutative spaces.
For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers and a theorem of S. A.…
Each element of the commutator subgroup of a group can be represented as a product of commutators. The minimal number of factors in such a product is called the commutator length of the element. The commutator length of a group is defined as the supremum of commutator lengths of elements of its commutator subgroup. We …
Since the 1970's, physicists and mathematicians who study random matrices in the GUE or GOE models are aware of intriguing connections between integrals of such random matrices and enumeration of graphs on surfaces. We establish a new aspect of this theory: for random matrices sampled from the group $\mathcal{U}\left(n…
Doodles link to commutator identities in a 2-sphere.
problem Understanding commutator identities in free groups via doodles.
method Analyzing doodles with proper noose systems and establishing bijections.
result A bijection between doodles and commutator identities.
Study on deformation cohomology for braided commutative structures.
problem Classifying and understanding deformations of braided commutative algebras.
method Extending Yang-Baxter Hochschild cohomology to braided commutative deformations.
result Classifies infinitesimal deformations of braided algebras that are braided commutative.
Examining singularities of commuting vector fields on submanifolds.
problem Understanding singularities of commuting vector fields on submanifolds.
method Analyzing real submanifolds within Kähler manifolds.
result Characterized singularities of commuting vector fields.
Study on infinite-type surfaces shows stable commutator length is continuous and defines open subgroups.
problem Understanding stable commutator length on infinite-type surfaces.
method Analyzing mapping class groups of infinite-type surfaces, showing continuity and openness of commutator subgroups.
result Stable commutator length defines a continuous function on commutator subgroups of infinite-type mapping class groups.
The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.
problem Defining and studying the geometric mean for tensors.
method Generalized geometric mean for tensors using T-product, verified properties, and investigated Riemannian manifold.
result Geometric mean of T-positive definite tensors is a unique solution of algebraic Riccati tensor equations and a midpoint of geodesics.
New proof and description of commutator subgroups for free and surface groups.
problem Understanding commutator subgroups of free and surface groups.
method Geometric proof and representation-theoretic description.
result New free generating sets and structure descriptions for commutator subgroups.
Formulae for non-symmetric connections derived from covariant derivatives.
problem Deriving commutation formulae for non-symmetric affine connections.
method Covariant derivatives of tensors with respect to symmetric and non-symmetric affine connections.
result Formulae for non-symmetric connections derived from covariant derivatives.
We study the Newton polytopes of determinants of square matrices defined over rings of twisted Laurent polynomials. We prove that such Newton polytopes are single polytopes (rather than formal differences of two polytopes); this result can be seen as analogous to the fact that determinants of matrices over commutative …
The paper studies Riemannian metrics on Lie groups with specific commutator subgroups.
problem Investigating Riemannian metrics on Lie groups with commutator subgroups of dimensions 1 and 2.
method Explicitly provided Levi-Civita connection, sectional curvature, and Ricci curvature; computed necessary and sufficient conditions for Ricci solitons; characterized Ricci solitons on Lie groups with one-dimensional commutator subgroups; examined indecomposable Lie groups with two-dimensional commutator subgroups.
result Characterization of all Ricci solitons on Lie groups with one-dimensional commutator subgroups and examination of indecomposable Lie groups with two-dimensional commutator subgroups.
Homotopy commutativity in quasitoric manifolds depends on polytope structure and characteristic matrix type.
problem Conditions for homotopy commutativity in quasitoric manifolds.
method Analyzing characteristic matrices and polytope structures.
result Homotopy commutativity is determined by specific polytope and matrix conditions.
Covariance as a commutatormath.ST
Covariance is shown as a commutator in random variable calculus.
problem Expressing covariance as a commutator of operators.
method Demonstrated through commutator identities involving expectations and products of functions.
result Revealed the underlying Lie algebraic structure in efficient influence curve calculus.
Every homeomorphism of Euclidean space is a commutator of two homeomorphisms.
problem Understanding the structure of homeomorphisms in Euclidean space.
method Proving every orientation-preserving homeomorphism can be written as a commutator of two such homeomorphisms.
result Every orientation-preserving homeomorphism of Euclidean space is a commutator of two homeomorphisms.
We show that solutions of Thurston equation on triangulated 3-manifolds in a commutative ring carry topological information. We also introduce a homogeneous Thurston equation and a commutative ring associated to triangulated 3-manifolds.
We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations. For ma…