This paper provides a structural decomposition for extended function groups.
problem No existing literature states a structural decomposition for extended function groups.
method Uses the Klein-Maskit combination theorems.
result States and proves a structural decomposition for extended function groups.
The G-function associated to the semi-simple Frobenius manifold C^n/W (where W is a Coxeter group or an extended affine Weyl group) is studied. The general form of the G function is given in terms of a logarithmic singularity over caustics in the manifold. The main result in this paper is a universal formula for the G-…
Extends functions on symmetric spaces to analytic functions.
problem Extending functions on symmetric spaces to analytic functions.
method Harmonic analysis on symmetric spaces and representation theory of groups.
result Proves Whitney type extension theorems for symmetric spaces.
We study the Dehn function of connected Lie groups. We show that this function is always exponential or polynomially bounded, according to the geometry of weights and of the 2-cohomology of their Lie algebras. Our work, which also addresses algebraic groups over local fields, uses and extends Abels' theory of multiamal…
The existence of nonconstant harmonic Dirichlet functions on a Cayley graph of a discrete group is equivalent to the nonvanishing of the first L2-cohomology of the given group. It was first proven by Cheeger and Gromov that such functions do not exists on the Cayley-graph of an amenable group. The result was extended u…
New distance function proves rigidity in geodesic lamination space.
problem Proving rigidity in geodesic lamination space.
method Introduced left Hausdorff distance function and proved rigidity result.
result Extended mapping class group is isomorphic to bijections preserving left Hausdorff convergence.
Equations of motion for linear Hamiltonians in the real Jacobi group
problem Equations of motion for linear Hamiltonians in the real Jacobi group
method Using the energy function on the extended Siegel-Jacobi upper half space
result Equations of motion attached to linear Hamiltonians in the generators of the real Jacobi group
We define for arbitrary modules over a finite von Neumann algebra $\cala$ a dimension taking values in [0,∞] which extends the classical notion of von Neumann dimension for finitely generated projective $\cala$-modules and inherits all its useful properties such as additivity, cofinality and continuity. This all…
Research extends geodesic length function study to three holed sphere.
problem Geodesic length function on orbifolds.
method Extending previous work on punctured torus to three holed sphere and related orbifolds.
result Extension to three holed sphere and related orbifolds.
We define an extended Bloch group and show it is isomorphic to H3(PSL(2,C)δ;Z). Using the Rogers dilogarithm function this leads to an exact simplicial formula for the universal Cheeger-Simons class on this homology group. It also leads to an independent proof of the analytic relationship between volume and Chern-S…
Chevalley theorems extended to isotropic functions on matrix spaces.
problem Extending Chevalley theorems to isotropic functions on matrix spaces.
method Proving ultradifferentiable Chevalley restriction theorems for various ultradifferentiable classes.
result Isotropic functions on symmetric matrices have ultradifferentiable regularity if and only if their diagonal restrictions do.
Extends Calabi's correspondence to φ-minimal graphs in R3.
problem Extending Calabi's correspondence to φ-minimal graphs. method Using a two-parametric group of translations for φ-invariant functions. result New applications in studying φ-minimal graphs. We define an extended Bloch group and show it is naturally isomorphic to H_3(PSL(2,C)^δ;Z). Using the Rogers dilogarithm function this leads to an exact simplicial formula for the universal Cheeger-Chern-Simons class on this homology group. It also leads to an independent proof of the analytic relationship between volu…
The paper extends spectral results to non-abelian groups acting on compact Riemannian manifolds.
problem Determining potential functions from spectral data for non-abelian group actions.
method Generalized Legendrian relations and spectral invariants.
result Potential functions are determined by the equivariant spectrum for certain Schrödinger operators.
Extends Fried's result to arbitrary representations of compact hyperbolic manifolds.
problem Behavior of dynamical zeta functions at the origin for compact hyperbolic manifolds.
method Uses complex-valued torsion instead of Ray-Singer analytic torsion.
result Holomorphicity and value at s=0 for twisted Ruelle zeta function for arbitrary representations.
Abstract: Studies differential systems on compact Lie groups, extending Greenfield and Wallach's methods.
problem Global properties of left-invariant differential systems on compact Lie groups.
method Abstract: Extends Greenfield and Wallach's methods to systems, obtaining characterizations for regularity, range closeness, and cohomology spaces.
result Abstract: Derives generalizations of results and global versions of Caetano and Cordaro's result.
A novel method for estimating group-representative functional networks from multi-subject fMRI data.
problem Estimating common neuronal characteristics in a population from multi-subject fMRI data.
method Two-phase approach: clustering-based ICA for component maps, MAP-MRF labeling for group-representative map estimation.
result Demonstrated the viability of the proposed method in extracting group-representative functional networks from simulated fMRI data.
Extends curve functions to geodesic currents with a simple criterion.
problem Continuous extension of curve functions to geodesic currents.
method Simple criterion based on smoothing property.
result Extends known curve functions and introduces new examples.
We extend a work of Bartsch, Clapp and Puppe on the Mountain pass theorems. We consider functionals invariant with respect to infinite discrete groups satisfying a maximality condition on the finite subgroups.
The Hodge series of a finite matrix group is the generating function for invariant exterior forms of specified order and degree. Lauret, Miatello, and Rossetti gave examples of pairs of non-conjugate cyclic groups having the same Hodge series; the corresponding space forms are isospectral for the Laplacian on p-forms f…
In his 1992 article on generating functions Viterbo constructed a bi-invariant metric on the group of compactly supported Hamiltonian symplectomorphisms of R^2n. Using the set-up of arXiv:0901.3112 we extend the Viterbo metric to the group of compactly supported contactomorphisms of R^2n x S^1 isotopic to the identity.…
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.
This paper describes dihedral extended Schottky groups and their symmetries.
problem Understanding symmetries of handlebodies with Schottky structures.
method Geometrical structural description of dihedral extended Schottky groups using Klein-Maskit combination theorems.
result Sharp upper bounds for the fixed points of symmetries in handlebodies.
Improved bounds on acylindricity for right-angled Artin groups.
problem Bounding the acylindrical action of right-angled Artin groups on their extension graphs.
method Exploring lattice properties, studying prefixes of powers, and extending quasi-root uniqueness.
result Cardinality of r-quasi-stabilizer is bounded by a linear function of r. Extends coarse index theory to locally compact groups for Callias operators.
problem Developing an equivariant coarse index theory for non-cocompact actions.
method Using admissible modules and localised K-theory of group C∗-algebras. result Equivariant index for Callias operators is a special case of the localised index.
The theory of geometric zeta functions for locally symmetric spaces as initialized by Selberg and continued by numerous mathematicians is generalized to the case of higher rank spaces. We show analytic continuation, describe the divisor in terms of tangential cohomology and in terms of group cohomology which generalize…
Quantum groups applied to finance models, extending classical economics.
problem Establishing the relationship between expectation and price in finance.
method Developing quantum group operations and axioms in stochastic and functional calculus.
result Two distinct economic models emerge from the same valuations, extending classical economics.
The study extends group actions from surfaces to 3-manifolds using G-cobordisms.
problem When a finite group action on a surface extends to a 3-manifold.
method Using Schur multiplier and G-cobordisms, the study examines principal actions. result Affirmative extension for abelian, dihedral, symmetric, and alternating groups.
Generalizes Fenchel conjugation to nonlinear functions on arbitrary sets.
problem Extending Fenchel conjugation to functions on arbitrary sets without structure.
method Replacing linear test functions with nonlinear ones, investigating properties including biconjugation.
result Derived further results on smooth manifolds and Lie groups, relating to convexity.
The paper extends Johnson homomorphisms to groups with extended N-series.
problem Johnson homomorphisms for mapping class groups.
method Developed a theory of Johnson homomorphisms for groups with extended N-series.
result Many known and new variants of Johnson homomorphisms.
Let Σg(g>1) be a closed surface embedded in S3. If a group G can acts on the pair (S3,Σg), then we call such a group action on Σg extendable over S3. In this paper we show that the maximum order of extendable cyclic group actions is 4g+4 when g is even and 4g−4 when g is odd; the maximum ord…
The paper describes a structural decomposition of a specific type of Schottky groups.
problem Understanding the structure of extended Z2n-Schottky groups. method Using Klein-Maskit's combination theorems.
result A structural decomposition theorem for extended Z2n-Schottky groups. Extended mapping class groups can be generated by two elements for low genus cases.
problem Generating finite order elements to represent extended mapping class groups.
method Proving finite order generating sets for specific genus cases.
result Extended mapping class groups can be generated by two elements for genus 3 and 4, but not for genus 1.
Sharp bounds found for fixed points of symmetries in handlebodies.
problem Finding upper bounds for fixed points of symmetries in handlebodies.
method Structural description of dihedral extended Schottky groups and their fixed points.
result Sharp upper bounds for the number of fixed points components of two and three symmetries of handlebodies.
We start by studying the distribution of (cyclically reduced) elements of the free groups Fn with respect to their abelianization (or equivalently, their integer homology class. We derive an explicit generating function, and a limiting distribution, by means of certain results (of independent interest) on Chebyshev pol…
For a compact contact manifold it is shown that the anisotropic Folland-Stein function spaces form an algebra. The notion of anisotropic regularity is extended to define the space of Folland-Stein contact diffeomorphisms, which is shown to be a topological group under composition and a smooth Hilbert manifold. These re…
Novel method solves group synchronization with robust corruption tolerance.
problem Group synchronization with high corruption tolerance.
method Quadratic programming formulation exploiting cycle consistency.
result Global minimum recovers corruption levels under mild conditions.
Extended logarithm for solvable elements in mapping class groups.
problem Logarithm of Johnson map extension to solvable elements.
method Extension to exponential solvable elements in mapping class groups using solvable Lie groups.
result Solvability of extended logarithm.
Holonomy-preserving transformations help recover Alexander polynomials from graph zeta functions.
problem Recovering Alexander polynomials from graph zeta functions.
method Introducing holonomy to preserve zeta functions of matrix-weighted graphs and extending to group elements and quandles.
result Holonomy-preserving transformations correspond to transformations of group presentations and preserve the twisted Alexander polynomial.
Scalable multi-task regression via sparse Gaussian process priors.
problem Efficiently modeling and predicting multiple related tasks.
method Direct Cholesky factorization for sparse parameterization of Gaussian process priors.
result Sparse parameterization improves scalability and accuracy in multi-task regression.
Many normal subgroups of mapping class groups are geometric.
problem Characterizing normal subgroups of mapping class groups of surfaces with punctures.
method Proving automorphism and commensurator groups of certain subgroups are isomorphic to the mapping class group, using simplicial complexes.
result Many normal subgroups of mapping class groups are geometric.
In this note, we show that some F-harmonic maps into spheres are global maxima of the variations of their energy functional on the conformal group of the sphere. Our result extends partially those obtained in [15] and [17] for harmonic and p-harmonic maps.
GTBO uses group testing to optimize high-dimensional functions efficiently.
problem Challenges in optimizing high-dimensional, expensive functions due to the curse of dimensionality.
method GTBO combines testing and optimization phases to identify active variables and guide efficient optimization.
result GTBO outperforms state-of-the-art methods on high-dimensional optimization tasks.
Extended mapping class group can be generated by three involutions for certain surfaces.
problem Generating the extended mapping class group using involutions.
method Proving generation by three involutions for specified surface conditions.
result Extended mapping class group can be generated by three involutions for specified genus and puncture conditions.
The study extends calibrated geometry to smooth maps and finds energy bounds.
problem Finding energy bounds for smooth maps between Riemannian manifolds.
method Generalizing calibrated submanifolds to smooth maps and applying to energy functional.
result Lower bounds to the energy of smooth maps in homotopy classes.
Finite groups act freely on surfaces but not on 3-manifolds.
problem Understanding finite group actions on surfaces and 3-manifolds.
method Analyzing homeomorphisms of surfaces and 3-manifolds.
result Finite groups can act freely on surfaces but not on handlebodies.
Researchers prove any graph can be realized as Reeb graph, linking it to manifold properties.
problem Realizing graphs as Reeb graphs on manifolds.
method Proving graphs can be realized as Reeb graphs under natural conditions, linking Reeb number to fundamental group corank.
result Reeb number equals corank of fundamental group, extending previous results.
The 3D Index is extended to meromorphic functions on triangulated 3-manifolds.
problem Extending the 3D Index to a broader class of triangulated 3-manifolds.
method Assigning a meromorphic function to each ideal triangulation, invariant under Pachner moves, and expanding it into a Laurent series.
result The meromorphic function can be computed from gluing equations and coincides with the 3D Index for ideal triangulations with strict angle structures.