The paper classifies reversible elements in Seifert-fibered spaces and braid groups.
problem Classifying reversible elements in Seifert-fibered spaces and braid groups.
method Classification of reversible elements in Fuchsian groups, application to Seifert-fibered groups, and analysis of 3-torsion elements.
result Classification and analysis of reversible and 3-torsion elements in Seifert-fibered spaces and braid groups.
Let Mod_{g,b} denote the mapping class group of a surface of genus g with b punctures. Feng Luo asked in a recent preprint if there is a universal upper bound, independent of genus, for the number of torsion elements needed to generate Mod_{g,b}. We answer Luo's question by proving that 3 torsion elements suffice to ge…
In the Khovanov homology of links, presence of Z2-torsion is a very common phenomenon. Finite number of examples of knots with Zn-torsion for n>2 were also known, none for n>8. In this paper, we prove that there are infinite families of links whose Khovanov homology contains Zn-t…
We study SU(3)-structures induced on orientable hypersurfaces of seven-dimensional manifolds with G_2-structure. Taking Gray-Hervella types for both structures into account, we relate the type of SU(3)-structure and the type of G_2-structure with the shape tensor of the hypersurface. Additionaly, we show how to compute…
We develop techniques for studying fundamental groups and integral singular homology of symmetric Delta-complexes, and apply these techniques to study moduli spaces of stable tropical curves of unit volume, with and without marked points. As one application, we show that Delta_g and Delta_{g,n} are simply connected, fo…
We construct an invariant for non-spin 4-manifolds by using 2-torsion cohomology classes of moduli spaces of instantons on SO(3)-bundles. The invariant is an SO(3)-version of Fintushel-Stern's 2-torsion instanton invariant. We show that this SO(3)-torsion invariant is non-trivial for $2CP^2 # -CP^2$, while it is known …
Researchers prove a conjecture linking 1-loop invariants to torsion for fibered 3-manifolds.
problem Proving a conjecture about polynomial invariants and torsion for fibered 3-manifolds.
method Using combinatorial data of ideal triangulations and layered triangulations of fibered 3-manifolds with toroidal boundary, proving the conjecture for specific cases and confirming it for a large number of nonfibered manifolds.
result The conjecture linking 1-loop invariants to torsion for fibered 3-manifolds with toroidal boundary is proven.
The study proves representations for certain 3-manifolds using gauge theory.
problem Proving irreducible SL(2,C) representations for specific 3-manifolds.
method Instanton gauge theory applied to 3-manifolds with specific torsion properties.
result Irreducible SL(2,C) representations exist for certain 3-manifolds.
Classifies reciprocal elements in Hecke groups, generalizing Sarnak's work.
problem Classifying reciprocal elements in Hecke groups.
method Classifying and parametrizing reciprocal classes in Hecke groups Γp for p≥3. result Generalizes Sarnak's result on reciprocal elements in the modular group.
We know that any element A of the group SO(3) can be represented as A = A1 A2 A1', where A1, A1' are elements of SO1(2)={A is an element of SO(3) | Ae1=e1}, and SO2(2)={A is an element of SO(3) | Ae2=e2} . This fact is known as Euler's angle. When this situation, a matrix A is called the generator. In the present paper…
We define a new notion of contracting element of a group and we show that contracting elements coincide with hyperbolic elements in relatively hyperbolic groups, pseudo-Anosovs in mapping class groups, rank one isometries in groups acting properly on proper CAT(0) spaces, elements acting hyperbolically on the Bass-Serr…
In this paper, we establish that, for statistically convex-cocompact actions, contracting elements are exponentially generic in counting measure. Among others, the following exponential genericity results are obtained as corollaries for the set of hyperbolic elements in relatively hyperbolic groups, the set of rank-1 e…
The paper classifies reversible and strongly reversible elements in Hermitian isometry groups.
problem Classifying reversible and strongly reversible elements in Hermitian isometry groups.
method Classification through group theory and algebraic manipulation.
result New classification of strongly reversible elements in Sp(n).
Characterizes periodic elements in Artin-Tits groups via stability conditions.
problem Understanding periodic elements in Artin-Tits groups.
method Dynamical characterization via 2-Calabi-Yau category and stability conditions.
result An element is periodic if and only if it has a fixed point in the stability manifold.
A nontrivial element in a group is a generalized torsion element if some nonempty finite product of its conjugates is the identity. We prove that any generalized torsion element in a free product of torsion-free groups is conjugate to a generalized torsion element in some factor group. This implies that the fundamental…
New findings on generating mapping class groups using pseudo-Anosov elements.
problem Generating mapping class groups using specific types of elements.
method Proving the generation of mapping class groups by pseudo-Anosov elements and conjugate reducible but not periodic elements.
result The mapping class group can be generated by two conjugate pseudo-Anosov elements with arbitrarily large dilatations for surfaces of genus greater than or equal to nine.
This note shows that if two elements of equal trace (e.g., conjugate elements) generate an arithmetic two-bridge knot or link group, then the elements are parabolic. This includes the figure-eight knot and Whitehead link groups. Similarly, if two conjugate elements generate the trefoil knot group, then the elements are…
Classifies reversible and strongly reversible elements in quaternionic groups.
problem Classifying reversible and strongly reversible elements in quaternionic groups.
method Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
result Proves elements are reversible if and only if they are products of skew-involutions (resp. involutions).
Persistent elements are ubiquitous in knot groups, especially for hyperbolic knots.
problem Identifying persistent elements in knot groups under Dehn fillings.
method Combining techniques from knot theory and hyperbolic geometry, including Dehn fillings and automorphisms.
result Persistent elements are structurally pervasive in knot groups, not just rare exceptions.
New finite element method for complex forms in any dimension.
problem Discretization of complex forms in arbitrary dimensions.
method Finite element discretization of ℓ-form-valued k-forms on triangulations. result Generalizes existing finite element methods for various tensor fields.
The paper improves convergence rates of curvature approximations using Regge elements.
problem Improving convergence rates of curvature approximations using Regge elements.
method Investigates the interplay between polynomial degree of curvature lifting and metric tensor degree in Regge finite element space.
result Higher convergence rates are achieved by reducing the polynomial degree of curvature lifting and using linear Regge elements.
New group with non-loxodromic Morse element found.
problem Finding non-loxodromic Morse elements in groups.
method Small-cancellation techniques to construct a Morse local-to-global group.
result Found an infinite-order Morse element that is not loxodromic.
The paper solves a complex option pricing model using finite elements.
problem Risk-Adjusted Pricing Methodology (RAPM) Black-Scholes model with transaction costs.
method Spatial finite element models based on P1 and/or P2 elements, combined with a Crank-Nicolson-type temporal scheme.
result Results compare favorably with finite difference methods in the literature.
Paper introduces Deep Sets for Symmetric Elements (DSS) layers for learning sets of symmetric elements.
problem Learning sets of symmetric elements is underexplored.
method Characterized equivariant layers, showed DSS layers are universal approximators, and demonstrated their effectiveness.
result DSS layers improve set-learning architectures across various data types.
The study calculates the growth rate of reciprocal hyperbolic elements in Hecke groups.
problem Counting reciprocal hyperbolic elements in Hecke groups.
method Analyzes conjugacy classes of hyperbolic elements associated with reciprocal geodesics.
result Determines the asymptotic growth rate and limiting constant of primitive conjugacy classes of reciprocal hyperbolic elements.
The paper classifies 3-manifold groups with specific torsion elements.
problem Classifying 3-manifold groups with generalized torsion elements of order two.
method Analyzing the fundamental groups of 3-manifolds and their conjugates.
result 3-manifold groups with generalized torsion elements of order two have been classified.
New examples of hyperbolic links with generalized torsion elements found.
problem Finding generalized torsion elements in the fundamental groups of hyperbolic links.
method Analyzing the Weeks manifold, figure-eight sister manifold, and Whitehead sister link to identify generalized torsion elements.
result First examples of hyperbolic links with link groups admitting generalized torsion elements.
Proves mapping class group generated by two torsion elements for certain surfaces.
problem Generating mapping class group with two torsion elements.
method Analyzes surfaces of different genera and orders, proving generation by two elements of specific orders.
result Mapping class group generated by two torsion elements for g≥6 and other genera. Study shows Morse elements are common in acylindrically hyperbolic groups.
problem Understanding generic elements in acylindrically hyperbolic groups.
method Analyzing Morse elements and outer automorphisms.
result Morse elements are common in acylindrically hyperbolic groups.
Paper solves convertible bond valuation using finite elements with penalty method.
problem Valuation of convertible bonds under penalty TF model.
method Solves TF system of equations using P1 and P2 finite elements with penalty method.
result Numerical solutions compare favorably with finite difference method.
Constructs Poisson structures on gauge orbits of Maurer-Cartan elements.
problem Tackles constructing Poisson structures on gauge orbits of Maurer-Cartan elements.
method Constructs Poisson structures on gauge orbits of Maurer-Cartan elements of dgla L, associating a compatible Batalin-Vilkovisky algebra to each MC element.
result MCP structures yield a notion of hamiltonian flow of MC elements and define Lie algebroids on gauge orbits.
Loxodromic elements are pseudo-Anosov on specific graphs.
problem Characterizing loxodromic elements in specific groups.
method Analyzing subgroups acting on multiarc and curve graphs, and the handlebody group on disk graphs.
result Loxodromic elements are pseudo-Anosov on witness graphs.
The paper approximates Levi-Civita connection and curvature on 2D manifolds using finite elements.
problem Approximating Levi-Civita connection and curvature on 2D manifolds with finite elements.
method Using Regge finite elements, piecewise polynomial symmetric (0,2)-tensor fields, and distributional sense for non-regular tensors.
result Distributional quantities converge to their smooth counterparts under refinement of triangulation.
We show that the mapping class group of a closed oriented surface of genus at least three is generated by 3 elements of order 3 and by 4 elements of order 4. Note that the mapping class group cannot be generated by finitely many torsion elements of same order if genus is equal to one or two.
An algorithm is proposed that solves two decision problems for pseudo-Anosov elements in the mapping class group of a surface with at least one marked fixed point. The first problem is the root problem: decide if the element is a power and in this case compute the roots. The second problem is the symmetry problem: deci…
Characterizes stably elliptic elements in Lie groups and their properties.
problem Understanding stably elliptic elements in Lie groups and their geometric and algebraic properties.
method Characterization through fixed point algebra and Weyl group action; relates to maximal invariant cones and compactness of order intervals.
result Connected components of stably elliptic elements can be described using Weyl group action on a compactly embedded Cartan subalgebra.
Positive Thompson links are proven for oriented subgroup elements.
problem Proving properties of Thompson links with positive elements.
method Analyzing elements of the oriented subgroup of the Thompson group.
result Positive oriented Thompson links are established.
The paper classifies and decomposes quaternionic projective transformations.
problem Classifying and decomposing elements of the projective linear group PSL(3,H). method Algebraic characterization of dynamical types using reversibility, decomposition of elements into simple elements.
result Offered a complete classification for elements of SL(3,R). Let Sg be the closed oriented surface of genus g and let Mod(Sg) be the mapping class group. When the genus is at least 3, Mod(Sg) can be generated by torsion elements. We prove the follow results. For g≥4, Mod(Sg) can be generated by 4 torsion elements. Three generators are invo…
New loxodromic elements found in infinite-type surfaces.
problem Finding loxodromic elements in infinite-type surfaces.
method Adapting Thurston-Veech construction for infinite-type surfaces.
result Infinitely many loxodromic elements produced without leaving finite-type subsurfaces invariant.
We have been studying the index theory for some special infinite-dimensional manifolds with a "proper cocompact" actions of the loop group LT of the circle T, from the viewpoint of the noncommutative geometry. In this paper, we will introduce the LT-equivariant KK-theory and we will construct three KK-elements: the ind…
The paper constructs hyperbolic elements in multiple spaces.
problem Constructing hyperbolic elements in multiple Gromov-hyperbolic spaces.
method Explicit construction under minimal conditions.
result Set of simultaneously hyperbolic elements has strictly positive density.
Two elements generate extended mapping class groups of certain surfaces.
problem Generating extended mapping class groups with specific elements.
method Analyzing finite order elements and isotopy classes of homeomorphisms.
result Extended mapping class groups of certain surfaces are generated by two elements of finite order.
Paper maps Hamiltonians and line elements in manifolds.
problem Mapping among generalized Hamiltonians and line elements.
method Constructing Calabi's Riemannian Line Elements and solving matrix Riccati equations.
result Analytical and exact solutions of mapping between manifolds.
Finite element method applied to Leland's model for option pricing with transaction costs.
problem Option pricing with transaction costs using Leland's model.
method Spatial finite element models based on P1 and/or P2 elements combined with a Crank-Nicolson-type temporal scheme.
result Results compare favorably with finite difference methods in the literature.
We define the notion of a Kirby element of a ribbon category C (not necessarily semisimple). Kirby elements lead to 3-manifolds invariants. We characterize (in terms of the structure maps of some categorical Hopf algebra) a set of Kirby elements of C which is sufficiently large to recover the known quantum invariants c…
Constructs finite element spaces for (p,q)-forms, excluding one subspace.
problem Constructing finite element spaces for (p,q)-forms. method Piecewise polynomial finite element spaces for all natural subspaces of (p,q)-forms, excluding one subspace. result Recovers known finite element spaces and introduces new ones.
Two elements generate all mappings of a nonorientable surface.
problem Generating the mapping class group of a nonorientable surface.
method Proving two elements generate the mapping class group for g≥13. result The mapping class group of a nonorientable surface of genus g≥13 can be generated by exactly two elements.