Formula for manifold Euler characteristic using even faces.
problem Calculating Euler characteristic of triangulated manifolds.
method Formula based on even-dimensional faces.
result Universal coefficients for Euler characteristic.
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48 results for “even dimension”
We show that a set of conformally invariant equations derived from the Fefferman-Graham tensor can be used to construct global solutions of the vacuum Einstein equations, in all even dimensions. This gives, in particular, a new, simple proof of Friedrich's result on the future hyperboloidal stability of Minkowski space…
The paper studies a flow equation on even-dimensional manifolds, proving convergence under critical conditions.
problem Proving convergence of the prescribed -curvature flow equation in critical cases.
method Analyzes the flow equation on arbitrary even-dimensional closed Riemannian manifolds, proving convergence under specific geometric hypotheses.
result Proves convergence of the flow equation when the integral of equals , extending previous results.
We study manifolds endowed with an (almost) even Clifford (hermitian) structure and admitting a large automorphism group. We classify them when they are simply connected and the dimension of the automorphism group is maximal, and also prove a gap theorem for the dimension of the automorphism group.
A new proof of Friedrich's theorem on the existence and stability of asymptotically de Sitter spaces in 3+1 dimensions is given, which extends to all even dimensions. In addition, we characterize the possible limits of spaces which are globally asymptotically de Sitter, to the past and future.
Study circle actions on manifolds with 3 fixed points, finding dimension constraints and unique structures.
problem Characterize circle actions on oriented manifolds with exactly 3 fixed points.
method Analyzes manifold dimensions, isotropy submanifolds, and uses quaternionic projective space as a reference.
result For a manifold with three fixed points, its dimension must be a multiple of 4, and specific weights are unique.
The paper improves estimates for asymptotically hyperbolic Einstein manifolds in even dimensions.
problem Estimating the boundary regularity of asymptotically hyperbolic Einstein manifolds.
method Analyzing the -th derivative of scalar curvature and using Hölder continuity.
result The AHE metric is conformally compact under certain conditions.
Eigenvalue Estimates For The Dirac Operator On Kaehler-Einstein Manifolds Of Even Complex Dimensionmath.DG
In Kähler-Einstein case of positive scalar curvature and even complex dimension, an improved lower bound for the first eigenvalue of the Dirac operator is given. It is shown by a general construction that there are manifolds for which this new lower bound itself is the first eigenvalue.
Any Kaehler metric on the ball which is strongly asymptotic to complex hyperbolic space and whose scalar curvature is no less than the one of the complex hyperbolic space must be isometrically biholomorphic to it. This result has been known for some time in odd complex dimension and we provide here a proof in even dime…
Study of conformally compact metrics and Lovelock tensors in even dimensions.
problem Understanding conformally compact metrics satisfying Lovelock equations.
method Polyhomogeneous expansions and formal solutions to singular Yamabe-(2q) problem.
result Identification of a boundary obstruction in even dimensions that generalizes the ambient obstruction tensor.
We show that C^2 conformally compact Riemannian Einstein metrics have conformal compactifications that are smooth up to the boundary in dimension 3 and all even dimensions, and polyhomogeneous in odd dimensions greater than 3.
New property: polygons have a fixed dimension regardless of ambient space dimensions.
problem Understanding the dimension of polygon moduli spaces.
method Generalizing the square bending example to polygons of arbitrary edge lengths.
result There are only finitely many moduli spaces of polygons with given edge lengths, even as ambient dimension increases.
This study shows that certain cohomology groups of symplectic manifolds are always even-dimensional.
problem Understanding the cohomology structure of symplectic manifolds.
method Constructing and deforming a skew-adjoint operator to prove the vanishing property.
result The even dimensionality of even-degree cohomology groups in (4n+2)-dimensional symplectic manifolds.
Computes spectral Einstein functional for Witten deformation on even-dimensional spin manifolds.
problem Calculating the spectral Einstein functional for a specific deformation.
method Computes the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
result Computed the spectral Einstein functional for the Witten deformation on even-dimensional spin manifolds.
Density mode clustering is a nonparametric clustering method. The clusters are the basins of attraction of the modes of a density estimator. We study the risk of mode-based clustering. We show that the clustering risk over the cluster cores --- the regions where the density is high --- is very small even in high dimens…
Proves harmonic coordinates for weak immersions in even dimensions.
problem Existence of harmonic coordinates for weak immersions in Sobolev spaces.
method Analyzes weak immersions in critical Sobolev spaces and uses smallness conditions on the second fundamental form.
result Global harmonic coordinates exist for weak immersions in even dimensions under certain conditions.
On the realization of symplectic algebras and rational homotopy types by closed symplectic manifoldsmath.AT
Study realizes symplectic algebras and homotopy types on manifolds.
problem Realizing symplectic algebras and homotopy types on manifolds.
method Addressing questions on realizability of symplectic algebras and rational homotopy types by closed symplectic manifolds.
result Realization of symplectic algebras and homotopy types in various dimensions.
This paper introduces even triangulations of n-dimensional pseudo-manifolds and links their combinatorics to the topology of the pseudo-manifolds. This is done via normal hypersurface theory and the study of certain symmetric representation. In dimension 3, necessary and sufficient conditions for the existence of even …
The classification of even-homogeneous complex supermanifolds of dimension 1|m, m\leq 3, on CP^1 up to isomorphism is given. An explicit description of such supermanifolds in terms of local charts and coordinates is obtained.
We generalize the methods in previous work to provide a program for proving Singer's Conjecture for Coxeter systems. Specifically, we consider even Coxeter systems with nerves that are flag triangulations of $\BS^{n-1}$, . We prove that Singer's Conjecture in dimensions and , along with the vanishing o…
Establishes scattering theory for de Sitter vacuum solutions in even dimensions.
problem Quantitative nonlinear scattering theory for asymptotically de Sitter vacuum solutions in even spatial dimensions.
method Geometric Littlewood-Paley decomposition of the solution, constructing the scattering map.
result Existence and uniqueness of scattering states, asymptotic completeness, and invertible scattering map with quantitative control.
Overfitting Behaviour of Gaussian Kernel Ridgeless Regression: Varying Bandwidth or Dimensionalitycs.LG
Study shows that ridgeless Gaussian kernel regression overfits even with varying bandwidth or dimensionality.
problem Analyzing overfitting in Gaussian kernel ridgeless regression with varying bandwidth or dimensionality.
method Examined the behavior of minimum norm interpolating solutions for fixed and increasing dimensions under varying bandwidth and sample size.
result Ridgeless solutions are never consistent and can be worse than null predictor with large enough noise, even with varying bandwidth or dimensionality.
An extension of the ambient metric construction of Fefferman-Graham to infinite order in even dimensions is described. The main ingredients are the introduction of "inhomogeneous ambient metrics" with asymptotic expansions involving the logarithm of a defining function homogeneous of degree 2, and an invariant procedur…
Conservation laws for even order elliptic systems in the critical dimension -- a new approachmath.AP
New conservation laws found for polyharmonic maps in critical dimension.
problem Existence of conservation laws for polyharmonic maps in critical dimension.
method Small perturbation of Uhlenbeck's gauge fixing matrix.
result Existence of conservation laws for elliptic systems of even order in critical dimension.
Study online multiclass classification under bandit feedback, extending previous results.
problem Online multiclass classification with bandit feedback, focusing on label space unboundedness.
method Extend Daniely and Helbertal's results, show necessity and sufficiency of Bandit Littlestone dimension for learnability.
result Sequential uniform convergence is necessary but not sufficient for bandit online learnability.
Study proves rigidity and gap theorems for specific metrics.
problem Existence and properties of self-dual and even Poincaré-Einstein metrics in 4D.
method Rigorous mathematical proofs, including gap theorems and rigidity results.
result Obtained new scalar conformal invariants and identified obstructions to metric existence.
Higher-dimensional spacetimes have well-behaved boundaries.
problem Understanding boundaries of higher-dimensional spacetimes.
method Analyzing -dimensional Myers-Perry metrics at spacelike infinity.
result Optimal conformal completion at spacelike infinity for differentiability class.
Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
problem Characterizing manifolds that are both rational homology spheres and double disk bundles.
method Analyzing the structure of manifolds as unions of disk bundles and using properties of rational homology and cohomology.
result Even-dimensional simply connected manifolds that are rational homology spheres and double disk bundles are homeomorphic to spheres.
We study boundary regularity for conformally compact Einstein metrics in even dimensions by generalizing the ideas of Michael Anderson. Our method of approach is to view the vanishing of the Ambient Obstruction tensor as an nth order system of equations for the components of a compactification of the given metric. This…
The paper studies the ergodicity of frame flow on even-dimensional manifolds.
problem Understanding the ergodicity of frame flow on even-dimensional manifolds.
method Analyzing pinching conditions to determine ergodicity.
result The frame flow is ergodic under specific pinching conditions for even-dimensional manifolds.
The paper provides a differential form interpretation of a theorem about the dimensions of rational homotopy groups of Diff(D^4).
problem Lower bounds of dimensions of rational homotopy groups of Diff(D^4) in terms of graph homology.
method Differential form interpretation and extension to arbitrary even dimensions.
result The proof can be extended to arbitrary even dimensions and made accessible to more readers.
New method estimates deep neural network's intrinsic dimension for better generalization.
problem Estimating intrinsic dimension of deep neural networks for generalization.
method Topological data analysis (TDA) and persistent homology (PHD).
result Efficient algorithm to estimate PHD in deep neural networks.
We construct a family of non-collapsed, non-Kähler, non-Einstein steady Ricci solitons in even dimensions greater or equal to four. These solitons exist on complex line bundles over Kähler-Einstein manifolds of positive scalar curvature. They include a four-dimensional -invariant, non-collapsed Riemannian steady …
We study the linearization of the Dirichlet-to-Neumann map for Poincaré-Einstein metrics in even dimensions on an arbitrary compact manifold with boundary. By fixing a suitable gauge, we make the linearized Einstein equation elliptic. In this gauge the linearization of the Dirichlet-to-Neumann map appears as the scatte…
We derive a priori interior Hessian estimates for special Lagrangian equation with critical and supercritical phases in general higher dimensions. Our unified approach leads to sharper estimates even for the previously known three dimensional and convex solution cases.
The paper finds manifolds with many Rarita-Schwinger fields.
problem Finding large spaces of Rarita-Schwinger fields.
method Construction of compact manifolds with specific properties.
result Existence of manifolds with dimension of Rarita-Schwinger field space tending to infinity.
We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the dimens…
Non-rigidity degree of a lattice , nrd, is dimension of the L-type domain to which belongs. We complete here the table of nrd's of all root lattices and their duals; namely, the hardest remaining case of , and the case of are decided. We describe explicitly the -type domain …
We study high-dimensional distribution learning in an agnostic setting where an adversary is allowed to arbitrarily corrupt an -fraction of the samples. Such questions have a rich history spanning statistics, machine learning and theoretical computer science. Even in the most basic settings, the only known…
New examples show some manifolds can't be decomposed.
problem Existence of open book decompositions for certain manifolds.
method Using signature non-multiplicativity in fibre bundles.
result Closed manifolds with no open book decomposition exist.
Study proves higher-order conformal forms don't exist in odd dimensions.
problem Proving non-existence of higher-order conformal forms in odd dimensions.
method Analyzing conformal hypersurface embeddings and differential order invariants.
result General non-existence of higher-order conformal forms in odd dimensions.
Simplicial volume vanishes for 4-manifolds with open book decompositions.
problem Characterizing open book decompositions in dimension 4.
method Adapted proof by Bucher and Neofytidis.
result Simplicial volume vanishes for 4-manifolds with open book decompositions.
A geometrical structure on even-dimensional manifolds is defined which generalizes the notion of a Calabi-Yau manifold and also a symplectic manifold. Such structures are of either odd or even type and can be transformed by the action of both diffeomorphisms and closed 2-forms. In the special case of six dimensions we …
The Hopf conjecture states that an even-dimensional, positively curved Riemannian manifold has positive Euler characteristic. We prove this conjecture under the additional assumption that a torus acts by isometries and has dimension bounded from below by a logarithmic function of the manifold dimension. The main new to…
We define and discuss an extension of the SpinC quantization concept to odd-dimensional manifolds. After that we describe its relation to (the usual) even-dimensional SpinC quantization and how its famous properties like "Quantization commutes with reduction" can be regained in odd dimensions. At the end, we analyze th…
Study of geodesics on SL(n) with Hilbert-Schmidt metric, revealing complex dynamics in higher dimensions.
problem Geodesics on SL(n) with Hilbert-Schmidt metric.
method Analysis of geodesics, use of Virial-identity-based criterion, study of explicit families of solutions, classification of geodesics.
result Complex dynamics in higher dimensions, existence of bounded geodesic motions in even dimensions, instability of swirling and shear flows in even dimensions.
Positive curvature manifolds from smaller ones using division algebras and geodesic flow.
problem Understanding positive curvature manifolds and their properties.
method Using Conner-Kobayashi reductions and division algebras to build up positive curvature manifolds.
result Existence of two linearly independent harmonic k-forms on positive curvature manifolds.
It is well-known that odd-dimensional manifolds have Euler characteristic zero. Furthemore orientable manifolds have an even Euler characteristic unless the dimension is a multiple of . We prove here a generalisation of these statements: a -orientable manifold (or more generally Poincaré complex) has even Euler c…