Simplicial spheres have a weak Lefschetz property in characteristic 2.
problem Proving the weak Lefschetz property for simplicial spheres.
method Using bistellar moves to show the property is preserved.
result The weak Lefschetz property is preserved by bistellar moves for PL-spheres.
The paper defines and calculates Euler characteristics for quandles.
problem Defining and calculating Euler characteristics for quandles.
method Definition and calculation of Euler characteristics for quandles.
result The quandle Euler characteristic of a compact connected Riemannian symmetric space coincides with the topological Euler characteristic.
The paper extends a theorem to number fields without infinite places.
problem Finiteness properties of arithmetic approximate lattices.
method Geometric and homological finiteness properties for countable approximate groups.
result The finiteness length is finite and can be computed explicitly.
Characterizes discrete nets with characteristic properties on larger parameter rectangles.
problem Classify discrete and smooth surfaces with specific properties.
method Imposes characteristic properties on larger parameter rectangles for nets.
result Characterizes discrete multi-nets leading to classical smooth surfaces.
Study on symplectic semi-characteristic using cohomology and vector fields.
problem Defining and calculating the symplectic semi-characteristic of symplectic manifolds.
method Defined using even-degree primitive cohomology and proved a counting formula using vector fields.
result Established a counting formula for symplectic semi-characteristic and derived vanishing properties.
The paper studies spectral properties of Jacobi operator for surfaces with nonpositive Euler characteristic.
problem Investigating spectral properties of the Jacobi operator for surfaces with nonpositive Euler characteristic.
method Proving a sharp upper bound for the second eigenvalue of the Jacobi operator and classifying surfaces attaining this bound.
result Totally geodesic tori maximize the second eigenvalue among compact orientable surfaces with positive genus.
The paper explores tensor product kernels and their characteristic properties.
problem Understanding when HSIC characterizes independence and MMD with tensor product kernel discriminates probability distributions.
method Study of various notions of characteristic property of tensor product kernels.
result The paper answers questions about the characteristic and universal properties of tensor product kernels.
Linear groups without infinite order unipotents have good properties.
problem Properties of linear groups without unipotent elements of infinite order.
method Analyzing the structure and properties of linear groups without unipotent elements of infinite order.
result Linear groups without unipotent elements of infinite order have good properties, including centralisers virtually splitting and finitely generated abelian subgroups being undistorted.
It is well-known that 7-dimensional 3-Sasakian manifolds carry a one-parametric family of compatible G_2 structures and that they do not admit a characteristic connection. In this note, we show that there is nevertheless a distinguished cocalibrated G_2 structure in this family whose characteristic connection along wit…
Study G-H limits of surfaces with boundary, focusing on same Euler characteristic.
problem Investigate Gromov-Hausdorff limits of compact surfaces with boundary.
method Focus on surfaces with same Euler characteristic, build on previous work on closed surfaces.
result Complete description and topological properties of limit spaces.
The abstract discusses braided surfaces and their characteristic maps, linking them to algebraic and geometric properties.
problem Characterizing braided surfaces and their characteristic maps.
method Using branched coverings and factorization through free groups, the abstract explores the algebraic and geometric properties of braided surfaces.
result The abstract establishes a connection between braided surfaces and characteristic maps, providing new insights into their structure.
In Karatzas and Kardaras's paper on semimartingale financial models, it is proved that the NUPBR condition is a property of the local characteristic of the asset process alone. In Takaoka's paper on NUPBR, it is proved that the NUPBR condition is equivalent to the existence of a simga-martingale deflator. However, Taka…
The paper defines and proves a new property for symplectic manifolds.
problem The study introduces a new property for symplectic manifolds.
method Defines and proves the L2-hard Lefschetz property for complete symplectic manifolds. result Proves that a complete symplectic manifold satisfies the L2-hard Lefschetz property if and only if every class of L2-harmonic forms contains a L2 symplectic harmonic form. Study geometric properties of SGL submanifolds in a specific manifold.
problem Analyzing geometric characteristics of SGL submanifolds.
method Examines integrability conditions and parallelism properties of distributions.
result Provides insights into geometric behavior of SGL submanifolds.
If a real value invariant of compact combinatorial manifolds (with or without boundary) depends only on the number of simplices in each dimension on the manifold, then the invariant is completely determined by Euler characteristics of the manifold and its boundary. So essentially, Euler characteristic is the unique inv…
Study examines how statistical properties of deep learning representations can be adjusted.
problem Improving performance in deep learning models.
method Investigated eight representation regularization methods, including two new rank regularizers.
result Manipulating statistical properties of representations can indirectly improve model performance.
In this article we construct a minimal symplectic 4-manifold R that has small Euler characteristic (e(R)=8) and two essential Lagrangian tori with nice properties. These properties make R particularly suitable for constructing interesting examples of symplectic manifolds with small Euler characteristic. In particular, …
We study geometric properties of characteristic classes of surfaces bundles. In particular, we show that oriented surface bundles over bases with amenable fundamental groups and dimension at least 2 have trivial simplicial volume. We show furthermore that all MMM-classes are hyperbolic in the sense of Gromov, verifying…
We show here that the Nielsen core of the bumping set of the domain of discontinuity of a Kleinian group Γ is the boundary of the characteristic submanifold of the associated 3-manifold with boundary. Some examples of interesting characteristic submanifolds are given. We also give a construction of the characteristic…
Classifies certain types of geometric shapes with specific algebraic properties.
problem Classifying geometric shapes with specific algebraic properties.
method Classifies simply connected, closed cohomogeneity one manifolds with specific rational cohomology properties.
result Classifies manifolds with singly generated or 4-periodic rational cohomology.
The paper extends macroscopic market making to stochastic games, revealing properties and solving equations.
problem Price competition among market makers in a stochastic game setting.
method Extension of macroscopic market making framework to stochastic games, introducing multidimensional characteristic equations.
result New well-posedness results for forward-backward stochastic differential equations.
We examine groups whose resonance varieties, characteristic varieties and Sigma-invariants have a natural arithmetic group symmetry, and we explore implications on various finiteness properties of subgroups. We compute resonance varieties, characteristic varieties and Alexander polynomials of Torelli groups, and we sho…
Study shows how to compute manifold's characteristic numbers from flow properties.
problem Computing characteristic numbers of manifolds.
method Analyzes singular Riemannian flows and their foliations.
result Characteristic numbers computed as residues of infinitesimal foliations.
Extends transversality to supergeometry, proving stability and genericity.
problem Transversality in supergeometry.
method Extending Sard's theorem to supergeometry.
result Proves stability and genericity of supertransversality.
Study local expansions of continuous-time processes using Ito signature properties.
problem Analyzing local expansions of continuous-time processes and their moments.
method Using the Ito signature, a basis of iterated integrals, to conduct expansions of the process' characteristic function.
result Explicit coefficients and stochastic representations for asymptotics as time shrinks or diverges.
By studying modular invariance properties of some characteristic forms, we obtain twisted anomaly cancellation formulas. We apply these twisted cancellation formulas to study divisibilities on spin manifolds and congruences on spinc manifolds. Especially, we get twisted Rokhlin congruences for 8k+4 dimensional spi…
We compute the transgressed forms of some modularly invariant characteristic forms,which are related to the twisted elliptic genera. We study the modularity properties of these secondary characteristic forms and relations among them. We also get some twisted anomaly cancellation formulas on some odd dimensional manifol…
We investigate some characteristic properties of specific Weingarten surfaces in the three-dimensional Euclidean space using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.
The paper shows how to deform foliations to prove geometric properties of manifolds.
problem Proving geometric properties of manifolds with Killing foliations.
method Deforming foliations to maintain transverse geometric properties and applying Riemannian geometry of orbifolds.
result Positive basic Euler characteristic for positively curved Killing foliations.
New Euler characteristics for groupoids generalize orbifold Euler characteristics.
problem Generalizing orbifold Euler characteristics to non-orbifold groupoids.
method Introducing two Euler characteristics for groupoids, using o-minimal structures, and relating them to orbifold Euler characteristics.
result The two new Euler characteristics coincide and generalize orbifold Euler characteristics.
We compute the Chern-Simons transgressed forms of some modularly invariant characteristic forms, which are related to the elliptic genera. We study the modularity properties of these secondary characteristic forms and the relations among them. We also compute the Chern-Simons forms of some vector bundles over free loop…
Universal Euler characteristic defined for V-manifolds.
problem Defining a universal Euler characteristic for V-manifolds.
method Generalizing the Euler characteristic to V-manifolds and equivariant CW-complexes.
result The universal Euler characteristic is a universal additive invariant for V-manifolds and equivariant CW-complexes.
We define parametrized cobordism categories and study their formal properties as bivariant theories. Bivariant transformations to a strongly excisive bivariant theory give rise to characteristic classes of smooth bundles with strong additivity properties. In the case of cobordisms between manifolds with boundary, we pr…
We use some basic properties of binomial and Stirling numbers to prove that the Euler characteristic is, essentially, the unique numerical topological invariant for compact polyhedra which can be expressed as a linear combination of the numbers of faces of triangulations. We obtain this result converting it into an eig…
We assign to a finite CW-complex and an element in its first cohomology group a twisted version of the L2-Euler characteristic and study its main properties. In the case of an irreducible orientable 3-manifold with empty or toroidal boundary and infinite fundamental group we identify it with the Thurston norm. W…
By studying modular invariance properties of some characteristic forms, we get some new anomaly cancellation formulas on (4r−1) dimensional manifolds. As an application, we derive some results on divisibilities of the index of Toeplitz operators on (4r−1) dimensional spin manifolds and some congruent formulas on ch…
Study curvature properties of G2 connections with skew-symmetric torsion.
problem Investigate curvature identities and solitons on G2 manifolds. method Analyzes curvature identities and properties of G2 connections with skew-symmetric torsion. result Characterizes conditions for curvature to be symmetric and Ricci flat.
Simplified derivation and simulation of Feller Diffusion.
problem Deriving the probability density function of Feller Diffusion.
method Fourier Transform and Method of Characteristics for derivation; simulation algorithms for validation.
result Confirmation of hitting time probabilities via simulation.
The paper reveals a property of chromatic homology for complete graphs.
problem Understanding the chromatic homology of complete graphs.
method Introduced a combinatorial description of enhanced states and used it to analyze the homology.
result Showed a splitting property of the chromatic homology for certain graphs.
In this paper we give explicit formulas of differential characteristic classes of principal G-bundles with connections and prove their expected properties. In particular, we obtain explicit formulas for differential Chern classes, differential Pontryagin classes and differential Euler class. Furthermore, we show that…
Study on characteristic classes for foliation deformations.
problem Characterizing and understanding characteristic classes for foliation deformations.
method Introduced a differential graded algebra (DGA) to recover Bott vanishing and formulae, and discussed properties of its cohomology.
result Discovered new classes that cannot be described by existing classes like Godbillon--Vey and Fuks--Lodder--Kotschick.
Uniformizes surfaces with boundaries, focusing on triple junctions.
problem Uniformization of surfaces with boundaries, especially triple junctions.
method Extends conformal structure results to triple junction surfaces.
result Weak uniformization results for triple junction surfaces.
We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,∇) where B ⊂ A are regular Lie algebroids, both over the same regular foliated manifold (M,…
Computes monopole Floer homology for three-manifolds.
problem Computing monopole Floer homology for three-manifolds.
method Develops a new framework to study homotopical properties of dga twisted with a specific kind of Maurer-Cartan element, and computes higher operations for the torus.
result Explicit computation of HM∗ for the torus. The paper explores the geometry and topology of DNN decision boundaries.
problem Understanding the geometric and topological properties of DNN decision boundaries.
method Differential geometry and the Gauss-Bonnet-Chern theorem.
result Computed the Euler characteristics of compact decision boundaries.
This is an expository paper, which provides a first introduction to geometric structures on TM⊕T∗M. The paper contains definitions and characteristic properties of generalized complex, generalized Kaehler, generalized (normal, almost) contact and generalized Sasakian structures. A few of these properties are n…
The paper extends a conjecture about Euler characteristics of perverse sheaves on certain Kähler manifolds.
problem The conjecture about Euler characteristics of perverse sheaves on compact aspherical Kähler manifolds.
method The method involves expressing the Euler characteristic as an intersection number involving the characteristic cycle, and using curvature conditions to deduce non-negativity. For the second result, the local system is shown to underlie a complex variation of Hodge structures, leading to the desired inequality from curvature properties of the period map.
result The conjecture holds for compact aspherical Kähler manifolds with non-positive holomorphic bisectional curvature and for projective manifolds with a faithful semi-simple rigid local system.
Study compatible and associated metrics for contact-symplectic structures, showing geodesic integral curves and minimal leaf properties.
problem Characteristics foliations of metric contact-symplectic structures.
method Analysis of compatible and associated metrics, study of geodesic integral curves, and minimal leaf properties.
result Integral curves of the Reeb vector field are geodesics for any compatible metric, and associated metrics share a common volume element.