The study infers passenger types from commute data.
problem Classifying commuters based on travel patterns.
method Constructing eigentravel matrices and using gradient boosting for prediction.
result Gradient boosting method achieves 76% accuracy in passenger type prediction.
Paper extends matrix inequality to Hermitian matrices.
problem Extending inequalities to Hermitian matrices.
method Using Frobenius norm of commutators for real and skew matrices, extending to Hermitian and skew-Hermitian.
result DDVV-type inequalities now apply to Hermitian matrices.
New representation of curves helps prove complex geometry result.
problem Understanding cohomologically stable curves in projective space.
method Using commuting matrix polynomials to represent curves and show isomorphism to hyperkähler quotient.
result Hilbert scheme isomorphic to a hyperkähler quotient.
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.
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 w-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 GA(1,5). 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…
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.
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).
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.
In this paper we define and give examples of a family of polynomial invariants of virtual knots and links. They arise by considering certain 2×2 matrices with entries in a possibly non-commutative ring, for example the quaternions. These polynomials are sufficiently powerful to distinguish the Kishino knot from …
Study of curvature tensor algebra with nonstandard multiplications.
problem Understanding curvature tensor algebra structures.
method Investigates orthogonally invariant commutative nonassociative multiplications on curvature tensors.
result Characterizes curvature tensor algebras in low dimensions and proves their simplicity in higher dimensions.
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 G:=RntimesηRm. result Classification of splittable lattices in the specified Lie group.
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.
Abstract: New invariants for links and three-manifolds from finite groups.
problem Creating invariants for links and three-manifolds.
method Using finite groups to define R−matrices and extended R−matrices, constructing invariants through colored tangle categories. result Invariants for links and three-manifolds are group invariants.
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 Z2 group actions. result Solved specific group action problems using new matrix pair invariant.
New theory connects random matrices to surface graphs and mapping class groups.
problem Understanding moments of measures induced by free words on unitary matrices.
method Study measures induced by free words on U(n) and relate to surfaces and mapping class groups.
result Every moment of the measure on U(n) is determined by pairs (Σ, f) involving surfaces and maps.
New method for accurately predicting linear dynamical systems.
problem Forecasting and estimating system matrices of linear dynamical systems.
method Non-convex polynomial optimization approach with global convergence guarantee.
result Global convergence of numerical solutions to a least-squares estimator.
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.
Study shows L2-Betti numbers vanish for certain matrix groups over rings, proving non-acylindrical hyperbolicity.
problem Investigating L2-Betti numbers and acylindrical hyperbolicity for matrix groups over various rings. method Utilizing n-rigid rings, the study proves vanishing L2-Betti numbers and non-acylindrical hyperbolicity for specified groups. result Matrix groups over certain rings have vanishing L2-Betti numbers and are not acylindrically hyperbolic. Let A,B be invertible, non-commuting elements of a ring R. Suppose that A−1 is also invertible and that the equation [B,(A−1)(A,B)]=0 called the fundamental equation is satisfied. Then an invariant R-module is defined for any diagram of a (virtual) knot or link. Solutions in the classic quaternion case hav…
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.…
New algebraic structures on manifolds generalize supergeometry concepts.
problem Developing algebraic structures for non-commutative manifolds.
method Introducing ρ-commutative manifolds, Q-manifolds, and modular classes. result Generalized modular classes for non-commutative spaces.
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 …
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.
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.
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.
Two commutators generate mapping class groups of surfaces with genus ≥5.
problem Generating mapping class groups efficiently.
method Analyzing commutators in mapping class groups of surfaces.
result Mapping class groups of surfaces with genus ≥5 are generated by two commutators.
New mathematical invariants derived from polytopes of matrices over rings.
problem Understanding Bieri-Neumann-Strebel invariants via algebraic structures.
method Investigating Newton polytopes of determinants of matrices over rings of twisted Laurent polynomials.
result Established a connection between Bieri-Neumann-Strebel invariants and Newton polytopes.
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.
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.
Study on commutator subgroups of welded braid groups, proving their finiteness and perfection.
problem Investigating the structure of commutator subgroups in welded braid groups.
method Proved finiteness and Hopfian property, showed perfection for n≥5, computed finite presentations. result Commutator subgroups of welded braid groups are finitely generated, Hopfian, and perfect for n≥5. 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.
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.
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…
Analytic torsion form constructed for non-commutative spaces.
problem Constructing analytic torsion form on non-commutative spaces.
method Using fiber bundles, flat vector bundles, and crossed product algebras, we define a non-commutative deRham differential form.
result The constructed torsion form appears in a transgression formula and leads to an index formula.