Study limits of convex domains in projective plane, proving specific results.
problem Understanding limits of convex domains in projective plane.
method Analyzing sequences of properly convex domains with bounded multiplicity.
result Determined all Hausdorff limit domains after normalization.
Minimal covolume group found in hyperbolic 3-space.
problem Finding groups with minimal covolume in hyperbolic 3-space.
method Proved existence of a specific group with minimal covolume.
result Minimal covolume group found with covolume equal to Catalan's constant.
In this paper, we classify all of the five-sided three-dimensional hyperbolic polyhedra with one ideal vertex, which have the shape of a triangular prism. We show how to find each such polyhedron in the upper half-space model by considering lines and circles in the plane. Finally, we give matrix generators in $\mathrm{…
We describe a procedure for constructing a generalized Thompson group out of a family of groups that is equipped with what we call a cloning system. The previously known Thompson groups F, V, Vbr and Fbr arise from this procedure using, respectively, the systems of trivial groups, symmetric groups, braid groups and pur…
Hyperbolic knots decompose into prism orbifolds.
problem Understanding hyperbolic knot complements and their geometric properties.
method Analyzing knot complements as quotients of H3 by discrete groups of reflections in polyhedra with triangular prism combinatorial type. result Knot complements decompose into hidden symmetries and contain closed, embedded, totally geodesic surfaces.
An edge tessellation is a tiling of the plane generated by reflecting a polygon in its edges. We prove that a polygon generating an edge tessellation is one the following eight types: a rectangle; an equilateral, 60-right, isosceles right, or 120-isosceles triangle; a 120-rhombus; a 60-90-120 kite; or a regular hexagon…
Multifractal detrended cross-correlation methodology is described and applied to Foreign exchange (Forex) market time series. Fluctuations of high frequency exchange rates of eight major world currencies over 2010-2018 period are used to study cross-correlations. The study is motivated by fundamental questions in compl…
The study investigates linearizability of Poisson structures on groupoids.
problem Linearizing Poisson structures on groupoids around the unit section.
method Extending the Lagrangian neighbourhood theorem to cosymplectic Lie algebroids, integrating triangular Lie bialgebras to symplectic LA-groupoids.
result Poisson structures on groupoids are linearizable under certain conditions.
New basis confirms Thurston's conjecture and reveals knot configurations.
problem Understanding cluster algebras and their bases from surfaces.
method Topological construction of band basis and comparison with Kazhdan-Lusztig type basis.
result Common triangular basis matches band basis in quantum cluster algebras.
Let X be a simply connected compact Riemannian symmetric space, let U be the universal covering group of the identity component of the isometry group of X, and let \g denote the complexification of the Lie algebra of U, \g=\u^\C. Each \u-compatible triangular decomposition \g=\n_- + \h + \n_+ determines a Poisson Lie g…
It is known, since works of Burde and de Rham, that one can detect the roots of the Alexander polynomial of a knot by the study of the representations of the knot group into the group of the invertible upper triangular 2x2 matrices. In this work, we propose to generalize this result by considering the representations…
Researchers compute the cohomology ring of a foliation defined by a group action.
problem Computing the cohomology ring of a specific foliation defined by a group action.
method Non-abelian harmonic analysis on G to compute the leafwise cohomology ring. result Computed the leafwise cohomology ring H∗(FP). As a type of pseudoinverse learning, extreme learning machine (ELM) is able to achieve high performances in a rapid pace on benchmark datasets. However, when it is applied to real life large data, decline related to low-convergence of singular value decomposition (SVD) method occurs. Our study aims to resolve this issu…
Smooth resolutions found for quotient of R^2 by infinite discrete groups.
problem Symplectic resolutions of quotient spaces by infinite discrete subgroups.
method Constructing smooth symplectic resolutions for R^2 under infinite discrete subgroups of GL_2(R).
result Minimal resolutions of Du Val singular varieties are symplectic resolutions of R^2/G.
Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.
Paper presents a new triangular form for flat systems.
problem Designing flat systems with two inputs.
method Geometric characterization and static feedback equivalence.
result Sufficient condition for affine input systems to be flat.
The fundamental group of every surface that is not the projective plane or Klein bottle has a representation to a torsion-free group of upper-triangular matrices in SL(2,R) with no simple loop (i.e. a nontrivial element representing a simple closed curve) in the kernel.
Paper presents a new flat triangular form for systems.
problem Creating a structurally flat triangular form for systems.
method Developed a new triangular form based on the extended chained form with conditions for static feedback equivalence.
result Provided conditions for affine input systems to be static feedback equivalent to the new triangular form.
We show that for any coboundary Poisson Lie group G, the Poisson structure on G^* is linearizable at the group unit. This strengthens a result of Enriquez-Etingof-Marshall, who had established formal linearizability of G^* for quasi-triangular Poisson Lie groups G. We also prove linearizability properties for the group…
New reflection groups derived from torus knots with finite meridians.
problem Understanding reflection groups derived from torus knot groups with finite meridians.
method Using the theory of J-groups and Coxeter groups, study quotients of torus knot groups.
result Classification of toric reflection groups and their properties.
The paper defines projective structures for Lie bialgebras and Poisson-Lie groups.
problem Defining projective analogues of Lie bialgebras and Poisson-Lie groups.
method Introducing projective tensor products and adapting classical notions to these structures.
result Every quasi-triangular projective r-matrix gives rise to a projective Banach Lie bialgebra.
We consider the problem of approximate joint triangularization of a set of noisy jointly diagonalizable real matrices. Approximate joint triangularizers are commonly used in the estimation of the joint eigenstructure of a set of matrices, with applications in signal processing, linear algebra, and tensor decomposition.…
We study the de Rham 1-cohomology H^1_{DR}(M,G) of a smooth manifold M with values in a Lie group G. By definition, this is the quotient of the set of flat connections in the trivial principle bundle M×G by the so-called gauge equivalence. We consider the case when M is a compact Kähler manifold and G is a solv…
The paper presents a classification theorem for the class of flat connections with triangular (0,1)-components on a topologically trivial complex vector bundle over a compact Kahler manifold. As a consequence we obtain several results on the structure of Kähler groups, i.e., the fundamental groups of compact Kahler man…
New combinatorial type helps distinguish plane curve topologies.
problem Distinguishing the topology of plane curves.
method Introducing G-combinatorial type using modified plumbing graphs.
result Invariant of G-combinatorial type under certain homeomorphisms.
Study thin hyperbolic reflection groups and their properties.
problem Characterize and enumerate thin hyperbolic reflection groups.
method Analyze Zariski dense subgroups of hyperbolic isometries, apply Vinberg algorithm.
result All thin hyperbolic reflection groups are enumerable.
A discrete subgroup of the group of isometries of the hyperbolic space is called reflective if up to a finite index it is generated by reflections in hyperplanes. The main result of this paper is a complete classification of the reflective (and quasi-reflective) subgroups among the Bianchi groups and their extensions.
We show that there are infinitely many commensurability classes of pseudomodular groups, thus answering a question raised by Long and Reid. These are Fuchsian groups whose cusp set is all of the rationals but which are not commensurable to the modular group. We do this by introducing a general construction for the fund…
New link groups are derived from torus necklaces, connecting braid groups to reflection groups.
problem Understanding the relationship between braid groups and reflection groups.
method Constructing torus necklaces and linking them to braid groups of J-reflection groups. result Link groups of torus necklaces are precisely braid groups of J-reflection groups, with meridians as braid reflections. Introduces triangular transport for uncertain data.
problem Uncertainty in complex systems without known probabilistic representations.
method Characterizes and manipulates unknown probability distributions using triangular transport maps.
result Triangular transport guarantees desirable mathematical and computational properties.
Survey explores interactions between four conformal dynamics branches.
problem Understanding complex dynamics through different mathematical concepts.
method Examples and general results with technical tools.
result Dynamical relations between Schwarz reflection parameter spaces and anti-rational maps/ reflection groups.
Solves problem of describing transformations for upper triangular Toeplitz operators.
problem Describing coordinate transformations preserving upper triangular Toeplitz form of operator fields.
method Implicit formulas involving matrix-valued functions for describing transformations and Nijenhuis operators.
result Formulas for coordinate transformations and Nijenhuis operators in upper triangular Toeplitz form.
We study the large scale geometry of the upper triangular subgroup of PSL(2,Z[1/n]), which arises naturally in a geometric context. We prove a quasi-isometry classification theorem and show that these groups are quasi-isometrically rigid with infinite dimensional quasi-isometry group. We generalize our results to a lar…
Triangular flows ensure statistical consistency and fast rates in generative modeling.
problem Ensuring statistical consistency and fast rates in generative models.
method Statistical guarantees and sample complexity bounds for triangular flow models using empirical process theory.
result Established statistical consistency and finite sample convergence rates for Kullback-Leibler estimator of Knöthe-Rosenblatt measure coupling.
A hyperbolic reflection group is a discrete group generated by reflections in the faces of an n-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis on the results that were obtained in the last ten years and on the open problems.
Study of generalized J-groups and their presentations.
problem Understanding the structure of generalized J-groups and their presentations.
method Determine finitely generated groups, classify up to reflection isomorphism, and derive explicit presentations.
result Generalized J-groups coincide with rank 2 complex reflection groups and their torsion quotients.
Picard modular groups are shown to be generated by complex reflections.
problem Understanding the structure of Picard modular groups using reflections.
method Using presentations from previous works to show generation by reflections.
result Picard modular groups mPU(2,1,Od) are generated by complex reflections. Study on metrics on specific nilmanifolds, finding new examples and properties.
problem Characteristically solvable nilmanifolds and their metrics.
method Explicit determination of left-invariant metrics and their properties.
result First known examples of Lie groups without positive index of symmetry.
Paper uses GNNs to efficiently detect profitable triangular arbitrage opportunities.
problem Detecting profitable triangular arbitrage opportunities in dynamic markets.
method Formulate the problem as a graph-based optimization task and use a GNN architecture to capture complex relationships.
result GNN-based method achieves higher average yield with reduced computational time compared to traditional methods.
Defines fundamental racks for braid spaces of complex reflection groups.
problem Understanding fundamental racks for braid spaces of complex reflection groups.
method Defines an augmented rack associated to the orbifold fundamental group.
result Yields representations of the orbifold fundamental group on the cohomology of the rack space.
We create a smooth manifold of triangular meshes with a geodesically complete metric.
problem Representing and manipulating 2D shapes as triangular meshes.
method Developed a geodesically complete Riemannian metric for triangular meshes.
result The metric preserves mesh connectivity and avoids mesh degradation.
Paper studies viscosity solutions in unique Martinet spaces.
problem Properties of viscosity solutions in Martinet spaces.
method Established properties and proved uniqueness of solutions.
result Uniqueness of viscosity solutions in Martinet spaces.
Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.
problem Computing isotropy subgroups of orthogonal similarity on symmetric matrices.
method Algorithmic procedure solving a Toeplitz matrix equation.
result Structure of isotropy subgroups described.
Uniform diameter bound for reflection group disk patterns.
problem Uniform bounded diameter conjecture for reflection groups.
method Skinning map analysis and discrete extremal width on Coxeter graph.
result Diameter of skinning image is bounded by a constant.
The paper connects reflection groups to maps with specific dynamical properties.
problem Understanding the relationship between reflection groups and anti-rational maps.
method Established a correspondence using planar graphs.
result Complete answers to geometric mating problems for anti-rational maps.
Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
problem Computing isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
method Algorithm for solving a matrix equation to compute isotropy subgroups.
result Computed isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
This paper studies parabolic quasi-Coxeter elements in complex reflection groups and their combinatorial properties.
problem Characterizing and studying parabolic quasi-Coxeter elements in complex reflection groups.
method Defining and characterizing parabolic quasi-Coxeter elements, studying collections of reduced reflection factorizations and relative generating sets.
result Computing cardinalities of collections of reduced reflection factorizations and relative generating sets for large families of parabolic quasi-Coxeter elements.
Discretizes Helfrich-type energies on surfaces using triangular complexes.
problem Discretizing curvature energies on surfaces of specific type.
method Asymptotic lower bound combined with recovery sequence of triangulations and edge director fields.
result Valid discrete versions of integral curvature energies on surfaces.