Virtual twin groups map to symmetric groups, revealing automorphism structure.
problem Understanding homomorphisms between virtual twin groups and symmetric groups.
method Using irreducible right-angled Coxeter groups and right-angled Artin groups.
result A complete description of homomorphisms between virtual twin groups and symmetric groups, including the structure of the automorphism group of VTn. Paper studies pure virtual twin groups and their automorphisms.
problem Understanding the structure and automorphisms of pure virtual twin groups.
method Analyzes stable isotopy classes of immersed circles on surfaces, extending classical knot theory.
result Proves PVTn is an irreducible right-angled Artin group with trivial center and gives its precise presentation. The paper characterizes crystallographic groups derived from virtual braid and twin groups.
problem Characterizing crystallographic groups from virtual braid and twin groups.
method Analyzing quotients of virtual braid and twin groups by their commutator subgroups.
result The quotients of virtual braid and twin groups by their commutator subgroups are crystallographic groups.
Proves Alexander and Markov theorems for higher genus virtual doodles.
problem Classifying isotopy classes of immersed circles on surfaces.
method Introduces virtual twin groups to extend twin groups and prove theorems for higher genus.
result Proves Alexander and Markov theorems for higher genus virtual doodles.
Paper defines doodles on closed surfaces, unifying classical and virtual theories.
problem Classifying doodles on closed surfaces, especially non-orientable ones.
method Introducing twisted virtual doodles, defining twin groups, and proving Alexander- and Markov-type theorems.
result Unified theory of doodles, showing trivial center and residually finite properties.
New groups and subgroups classified with representations and properties.
problem Classifying representations of new groups and subgroups.
method Introduced new groups and subgroups, classified representations into GL_n(C), investigated properties.
result Classified homogeneous 2-local representations of M_kVT_n into eight distinct types.
Study of commutator subgroups and crystallographic quotients of virtual groups.
problem Investigate commutator subgroups and crystallographic quotients of virtual groups.
method Derived explicit finite presentations and proved crystallographic properties.
result Explicit finite presentations of commutator subgroups and crystallographic quotients.
Unified algebraic framework for virtual braid structures with strong structural consequences.
problem Unified algebraic framework for virtual braid structures with various types of crossings.
method Introducing the universal virtual braid group UVn(c) and proving its properties. result Strong structural consequences including residual finiteness, linearity, and solvability of conjugacy problems.
Generative model calibrates 3D battery cathode morphologies from 2D images.
problem Calibrate 3D morphologies of all-solid-state battery cathodes from 2D microscopy images.
method Combining GANs with excursion sets of Gaussian random fields.
result Calibrated digital twins enable systematic exploration of morphological scenarios.
Survey on fibring in manifolds and groups, focusing on recent developments and conjectures.
problem Understanding fibring phenomena in manifolds and groups.
method Discussion of state of the art and explanation of recent developments.
result Many conjectures about fibring in higher dimensions, some plausible, some dubious.
The paper studies congruence subgroups and crystallographic quotients of small Coxeter groups.
problem The congruence subgroup property for small Coxeter groups.
method Analyzes the properties of small Coxeter groups, proving the failure of the congruence subgroup property for certain groups.
result Proves the failure of the congruence subgroup property for infinite small Coxeter groups which are not virtually abelian.
This paper explores Brunnian twin groups and their properties.
problem Alexander-Markov correspondence for isotopy classes of immersed circles.
method Diagrammatic representations and combinatorial structures.
result Brunnian twin groups are free groups when more than two strands are involved.
Study uses GPLFM to create Digital Twin for ferry quay health monitoring.
problem Deterioration of ferry quays due to harsh maritime environments and impacts.
method Gaussian Process Latent Force Model (GPLFM) integrating physics-based model and machine learning.
result GPLFM provides accurate acceleration response estimates, even under simplifying assumptions.
The twin group Tn is a right angled Coxeter group generated by n−1 involutions and having only far commutativity relations. These groups can be thought of as planar analogues of Artin braid groups. In this note, we study some properties of twin groups whose analogues are well-known for Artin braid groups. We give…
Study of permutational wreath pullbacks and their properties.
problem Structural study of permutational wreath pullbacks and their properties.
method Systematic structural study of permutational wreath pullbacks, focusing on center, abelianization, and functorial behavior.
result Established a criterion for the abelian kernel to be characteristic and for the wreath product to inherit the R-infinity property.
The relationship between minimal algebraic Kac-Moody groups and twin buildings is well known as is the relationship between formal completions in one direction and affine buildings. Nevertheless, as the completion of a Kac-Moody group in one direction destroys the opposite BN-pair, there exists no longer a twin buildin…
For spherical Tits buildings of the classical types there are well-known explicit descriptions as flag complexes. Similarly for affine buildings of the classical types there are explicit constructions in terms of lattices. In this article we generalize the flag complex description to twin cities, a generalization of tw…
Generators of the 4-sphere's smooth mapping class group via diffeomorphisms of Montesinos twins.
problem Understanding the smooth mapping class group of the 4-sphere.
method Homomorphism from loops of 2-spheres to smooth mapping class group, generators as twists along Montesinos twins.
result Generators of the image of the homomorphism as diffeomorphisms of Montesinos twins.
The twin group Tn is a right angled Coxeter group generated by n−1 involutions and the pure twin group PTn is the kernel of the natural surjection from Tn onto the symmetric group on n symbols. In this paper, we investigate some structural aspects of these groups. We derive a formula for the number of conj…
Alexander invariant created for doodles, vanishes on unlinked doodles.
problem Creating an Alexander type invariant for doodles.
method Deformation of Tits representation and Chebyshev polynomials of second kind.
result Invariant vanishes on unlinked doodles with more than one component.
New curves share invariant up to any fixed order.
problem Finite-order rondle invariants and lower central series.
method Formulation and construction of infinite sequences of curves.
result Infinitely many curves share invariants up to any fixed order.
Let TWn be the twin group on n arcs, n≥2. The group TWm+2 is isomorphic to Grothendieck's m-dimensional cartographical group Cm, m≥1. In this paper we give a finite presentation for the commutator subgroup TWm+2′, and prove that TWm+2′ has rank 2m−1. We derive that $TW_…
New findings about twists in 4-sphere diffeomorphisms.
problem Understanding isotopy classes of diffeomorphisms in 4-sphere.
method Using Cerf theory and twists along Montesinos twins.
result The subgroup of twists along Montesinos twins is trivial or cyclic of order two.
New virtual version of Thompson's group created to handle virtual knots.
problem Creating a mathematical framework for virtual knots.
method Defining a virtual version of Thompson's group and proving its properties.
result Any virtual link can be constructed from an element of the new virtual Thompson's group.
Study virtual braid groups, proving a key subgroup result.
problem Understanding congruence subgroups in virtual braid groups.
method Using an extension of the integral Burau representation.
result Proved level 2 congruence subgroup of virtual braid group is pure virtual braid group.
We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…
Classic braids embed in virtual braids.
problem Embedding classic braids in virtual braids.
method Elementary proof using generalizations of braid groups.
result Classical braid group injects into virtual braid group.
Twinning splits data into fast, statistically similar sets.
problem Creating statistically similar data splits for Big Data.
method Twinning is a method based on SPlit for fast, model-independent dataset splitting.
result Twinning is orders of magnitude faster than SPlit.
CMCO provides robust uncertainty estimates for neural operators without retraining.
problem Uncertainty quantification in deep learning for real-time virtual sensing.
method Unified Monte Carlo dropout and split conformal prediction in DeepONet.
result Near-nominal empirical coverage in diverse applications.
The paper defines new representations and groups related to virtual links.
problem Defining and studying new representations of virtual braid groups and link groups.
method Introducing virtually symmetric representations, virtual link groups, and marked Gauss diagrams.
result Established equivalence of many known representations to virtually symmetric representations.
The paper introduces twins in Kähler and Sasaki geometry, generalizing known concepts.
problem Understanding twins in Kähler and Sasaki geometry.
method Introducing weighted extremal Kähler twins and extremal Sasaki twins, studying their properties.
result Many twins appear in the Kähler setting and more than one extremal ray in the Sasaki cone.
We report some minimal surfaces that can be seen as copies of a triply periodic minimal surface (TPMS) related by reflections in parallel mirrors. We call them minimal twin surfaces for the resemblance with twin crystal. Brakke's Surface Evolver is employed to construct twinnings of various classical TPMS, including Sc…
We study groups of some virtual knots with small number of crossings and prove that there is a virtual knot with long lower central series which, in particular, implies that there is a virtual knot with residually nilpotent group. This gives a possibility to construct invariants of virtual knots using quotients by term…
The paper examines subgroup separability for surface and virtual braid groups.
problem Subgroup separability of surface and virtual braid groups.
method Study of subgroup separability (LERF) properties.
result Properties of subgroup separability for surface and virtual braid groups are explored.
This paper gives a new interpretation of the virtual braid group in terms of a strict monoidal category SC that is freely generated by one object and three morphisms, two of the morphisms corresponding to basic pure virtual braids and one morphism corresponding to a transposition in the symmetric group. The key to this…
The paper constructs representations of flat virtual braids by free group automorphisms.
problem Representing flat virtual braids by automorphisms of free groups.
method Construction of representations of flat virtual braid groups FVBn by automorphisms of free groups of rank 2n. result Established conditions of faithfulness and properties of the kernel for n≥3. Improved aircraft structure prediction using derivative-enhanced sparse Cholesky GP method.
problem Accurate real-time prediction of aircraft structure performance.
method Combining derivative data with a modified dynamic sparse Cholesky linear system solver.
result Improved prediction accuracy of aircraft structure performance.
The study enumerates virtual quandles up to isomorphism.
problem Classifying virtual quandles up to isomorphism.
method Computer search and classification based on conjugacy class structures of rack automorphism groups.
result Classifications of virtual racks and quandles up to order 8.
We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.
Wise's Quasiconvex Hierarchy Theorem classifying hyperbolic virtually compact special groups in terms of quasiconvex hierarchies played an essential role in Agol's proof of the Virtual Haken Conjecture. Answering a question of Wise, we construct a new virtual quasiconvex hierarchy for relatively hyperbolic virtually co…
The study shows exponential distortion in virtually special groups containing free subgroups.
problem Understanding distortion in virtually special groups containing free subgroups.
method Constructing examples of virtually special groups with finite rank free subgroups.
result Distortion functions grow like exp^k(x^m) and can be superexponential.
We describe some buildings related to complex Kac-Moody groups. First we describe the spherical building of SLn(C) (i.e. the projective geometry PG(Cn)) and its Veronese representation. Next we recall the construction of the affine building associated to a discrete valuation on the rational function field C(z). Then …
Study on virtual singular braid groups with algebraic properties and homomorphisms.
problem Algebraic properties and homomorphisms of virtual singular braid groups.
method Numerical invariants, homomorphisms, semi-direct product decompositions, presentations, and quotients.
result Determined all group homomorphisms from VSGn to Sn and obtained corresponding semi-direct product decompositions. Generative models create personalized patient health simulations.
problem Creating accurate digital twins for personalized medicine.
method Neural network architecture for conditional generative models of clinical trajectories.
result Same architecture generates accurate twins across 13 indications.
Study classifies 2-manifolds with special homeomorphism groups.
problem Classifying 2-manifolds with specific homeomorphism groups.
method Introduced virtual Rokhlin property; classified homeomorphism groups of 2-manifolds.
result 2-manifolds with the virtual Rokhlin property have specific homeomorphism groups.
Virtual singular braids embed in a group with normal form.
problem Embedding virtual singular braids into algebraic structures.
method Presented a semi-direct product structure and provided a normal form.
result Virtual singular braid group VSGn is a semi-direct product of VSPGn and Sn. Virtual knots, defined by Kauffman, provide a natural generalization of classical knots. Most invariants of knots extend in a natural way to give invariants of virtual knots. In this paper we study the fundamental groups of virtual knots and observe several new and unexpected phenomena. In the classical setting, if the…
This paper stems from the observation (arising from work of T. Delzant) that "most" Kähler groups virtually algebraically fiber, i.e. admit a finite index subgroup that maps onto Z with finitely generated kernel. For the remaining ones, the Albanese dimension of all finite index subgroups is at most one, i.e. t…