Study Euler characteristic of manifolds with almost nonnegative curvature operator, showing nonnegativity under certain conditions.
problem Addressing the sign of Euler characteristic for manifolds with almost nonnegative curvature operator.
method Analyzing closed manifolds with uniform upper bounds on curvature operator and applying ANCO-type conditions.
result Nonnegative Euler characteristic for closed 2n-dimensional manifolds with almost nonnegative curvature operator and uniform upper bounds on curvature. 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.
Formula proves Euler characteristic of singularized surfaces.
problem Calculating the Euler characteristic of singularized surfaces.
method Three operations: collapsing, zipping, and double loop identification.
result Formula for Euler characteristic of singularized surfaces.
New operator reveals unique features of sl(N) link homology.
problem Understanding unique features of sl(N) link homology.
method Constructing an operator on sl(N) link homology with mod N coefficients.
result Structural features of sl(P) link homology are revealed.
New classifiers ensure fairness by adjusting a base classifier's operating characteristics.
problem Ensuring fairness in binary classification with multiple group constraints.
method Intervening directly on a base classifier's operating characteristics using group-wise ROC convex hulls and post-processing.
result Methods satisfy multiple fairness constraints (DP, EO, PP) with minimal interventions and near-oracle accuracy.
Assume that the compact Riemannian spin manifold (Mn,g) admits a G-structure with characteristic connection ∇ and parallel characteristic torsion (∇T=0), and consider the Dirac operator D1/3 corresponding to the torsion T/3. This operator plays an eminent role in the investigation of such man…
The study describes Nijenhuis operators with specific properties.
problem Characterizing Nijenhuis operators with functional independence and determinant constraints.
method Proving the general form and describing the specific case of Nijenhuis operators.
result Complete description of Nijenhuis operators with nondegenerate determinant.
New method assesses prediction intervals across different operating points.
problem Difficulty in comparing prediction intervals across studies.
method Operating characteristics curves and gain over a simple reference.
result A novel operating point agnostic assessment methodology for prediction intervals.
Classifies 3D non-degenerate left-symmetric algebras.
problem Classifying left-symmetric algebras in 3D.
method Using Nijenhuis geometry and algebraic independence of coefficients in characteristic polynomial.
result Classification of differentially non-degenerate LSA in dimension 3.
New connections found between curvature and Euler characteristic using Schrödinger operators.
problem Establishing relationships between curvature and topological invariants of Riemannian manifolds.
method Using twisted Dirac operators and scaling of potentials to analyze the kernel of these operators.
result Found conditions under which the Euler characteristic of a manifold can be zero or non-zero.
DOODL learns shared spectral dynamics across related dynamical systems.
problem Learning independent dynamical operators for each system limits discovery of shared structure.
method DOODL learns a dictionary of characteristic spectral dynamics on a manifold of related systems.
result DOODL achieves errors one to two orders of magnitude lower than independent operator estimation methods.
We define and study invariants which can be uniformly constructed for any gauge system. By a gauge system we understand an (anti-)Poisson supermanifold provided with an odd Hamiltonian self-commuting vector field called a homological vector field. This definition encompasses all the cases usually included into the noti…
We define the notion of characteristic classes for supermanifolds endowed with a homological vector field Q. These take values in the cohomology of the Lie derivative operator LQ acting on arbitrary tensor fields. We formulate a classification theorem for intrinsic characteristic classes and give their explicit de…
The paper generalizes CR invariants using renormalized characteristic forms.
problem Defining new CR invariants via renormalized characteristic forms.
method Introducing new curvatures for each renormalized characteristic form.
result The new curvatures' integrals match CR invariants constructed by Marugame.
Study stochastic processes on surfaces in contact sub-Riemannian manifolds using Riemannian approximations.
problem Analyzing stochastic processes on surfaces in contact sub-Riemannian manifolds.
method Employing Riemannian approximations, a second order partial differential operator is derived on the surface. The stochastic process moves along the characteristic foliation induced by the contact distribution.
result Elliptic characteristic points are inaccessible, while hyperbolic characteristic points are accessible from separatrices.
An odd vector field Q on a supermanifold M is called homological, if Q2=0. The operator of Lie derivative LQ makes the algebra of smooth tensor fields on M into a differential tensor algebra. In this paper, we give a complete classification of certain invariants of homological vector fields called character…
Study curvature operators in 4n-dimensional manifolds, finding new conformal invariants.
problem Analyzing curvature operators in oriented Riemannian 4n-manifolds.
method Examining finite systems of hafnian identities in eigenvalues, focusing on locally conformally flat cases.
result Discovering a new conformal invariant in dimensions 4n, related to nonnegativity of Euler characteristic.
In this note we study the relative Kervaire semi-characteristic and prove its invariance under cut-and-past operation. Our approach is analytic and follow very closely the method introduced by W. Zhang
New proof confirms operations on constructible functions match theory.
problem Matching operations on constructible functions with generalized valuations theory.
method Comparison with characteristic cycles approach.
result Operations on constructible functions match generalized valuations theory under mild assumptions.
For large-scale industrial processes under closed-loop control, process dynamics directly resulting from control action are typical characteristics and may show different behaviors between real faults and normal changes of operating conditions. However, conventional distributed monitoring approaches do not consider the…
We expound some results about the relationships between the Jacobi operators with respect to null vectors on a Lorentzian S-manifold M and the Jacobi operators with respect to particular spacelike unit vectors on M. We study the number of the eigenvalues of such operators in a φ-null Osserman Lorentzi…
We study a Laplacian operator related to the characteristic cohomology of a smooth manifold endowed with a distribution. We prove that this Laplacian does not behave very well: it is not hypoelliptic in general and does not respect the bigrading on forms in a complex setting. We also discuss the consequences of these n…
Formula calculates index for CR operators on surfaces with boundary punctures.
problem Computing the index for Cauchy-Riemann operators on surfaces with boundary punctures.
method Large antilinear deformations method, generalized to punctured surfaces.
result Involves a non-standard weighted count of boundary zeros in the Euler characteristic term.
We introduce the Γ-Euler-Satake characteristics of a general orbifold Q presented by an orbifold groupoid G, generalizing to orbifolds that are not necessarily global quotients the generalized orbifold Euler characteristics of Bryan-Fulman and Tamanoi. Each of these Euler characteristics is defined as t…
NOGaP uses neural operators and GPs to solve PDEs with uncertainty quantification.
problem Lack of uncertainty measures in neural operator solutions for PDEs.
method NOGaP combines neural operators with Gaussian Processes to provide probabilistic solutions.
result NOGaP offers improved prediction accuracy and uncertainty quantification.
Operators on the ring of algebraically constructible functions are used to compute local obstructions for a four-dimensional semialgebraic set to be homeomorphic to a real algebraic set. The link operator and arithmetic operators yield 243−43 independent characteristic numbers mod 2, which generalize the Akbulut-K…
A formula is given in terms of secondary characteristic classes for the leading order contribution to the spectral flow for a path of twisted Dirac operators on an odd dimensional, Riemannian manifold when the twisting is done by a path of unitary connections with large curvature.
New Hilbert bundles with ends defined from indexed bases.
problem Defining new structures in Hilbert bundles.
method Indexed bases and unitary operators of finite propagation.
result Characteristic classes of Hilbert bundles with ends.
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 predicts antimicrobial resistance in ICU patients quickly.
problem Delayed AMR testing in ICU leads to suboptimal treatment.
method Developed predictive models using clinical and microbiological data.
result Machine learning models predict AMR with higher accuracy than naive model.
An integer valued topological index of a Dirac operator is introduced for a pair of a 4n+2 dimensional open Spin^c manifold and a section of the determinant line bundle satisfying some property. We show a relation between the index and an index of a Dirac operator of its characteristic submanifold, by a localization of…
Develops an L^p theory for Dolbeault-Dirac operators on compact Kähler manifolds.
problem Analyzing Dolbeault-Dirac operators on compact Kähler manifolds with Banach space coefficients.
method Establishes an L^p theory for Dolbeault-Dirac operators, proving bisectoriality, H^\infty functional calculus, and Gaffney-type estimates.
result Identifies the index of the associated Fredholm operator with the holomorphic Euler characteristic, independent of p.
This study links blockchain design to cryptos' distributional characteristics.
problem Understanding the relationship between blockchain design and cryptos' distributional characteristics.
method Used spectral clustering to cluster cryptos based on their blockchain mechanisms and operational features.
result Clusters of cryptos share similar blockchain mechanisms, supporting the hypothesis.
In this paper we present a technique for using the bootstrap to estimate the operating characteristics and their variability for certain types of ensemble methods. Bootstrapping a model can require a huge amount of work if the training data set is large. Fortunately in many cases the technique lets us determine the eff…
We construct certain operations on stable moduli spaces and use them to compare cohomology of moduli spaces of closed manifolds with tangential structure. We obtain isomorphisms in a stable range provided the p-adic valuation of the Euler characteristics agree, for all primes p not invertible in the coefficients fo…
A new method prunes neural network channels based on operation characteristics.
problem Compressing deep neural networks efficiently and maintaining accuracy.
method Differentiable masks for channel pruning considering BN and ReLU.
result Outstanding performance in accuracy with less resources compared to state-of-the-art methods.
This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature.
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. Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
We propose a novel classifier accuracy metric: the Bayesian Area Under the Receiver Operating Characteristic Curve (CBAUC). The method estimates the area under the ROC curve and is related to the recently proposed Bayesian Error Estimator. The metric can assess the quality of a classifier using only the training datase…
Efficient algorithm for evaluating hierarchical classification methods at multiple operating points.
problem Evaluating hierarchical classification methods at multiple operating points.
method Efficient algorithm to produce operating characteristic curves for any method that assigns scores to every class in the hierarchy.
result Top-down classifiers are dominated by a naive flat softmax classifier across the entire operating range.
The study examines differential singularities in 3D Nijenhuis operators.
problem Characterizing differential singularities in three-dimensional Nijenhuis operators.
method Examined cases of proportional differentials of invariants and fold-type singularities.
result New examples of Nijenhuis operators with specified singularities constructed.
We provide evidence for the conjecture that the Wodzicki-Chern classes vanish for all bundles with the group Z of invertible zeroth order pseudodifferential operators as structure group. In particular, we prove this vanishing if the structure group reduces to pseudodifferential operators with leading order symbol the i…
Based on a pair of cohomology operations on so called δ−2-formal spaces, we construct the integral cohomology rings of the classifying spaces of the Lie groups Spin(n) and Spinc(n). As applications, we introduce characteristic classes for the reduced topological KSpin theory, determine the ring of integra…
On any manifold M^n, the de Rham operator D=d+d^* (with respect to a complete Riemannian metric), with the grading of forms by parity of degree, gives rise by Kasparov theory to a class [D] in KO_0(M), which when M is closed maps to the Euler characteristic chi(M) in KO_0(point) = Z. The purpose of this note is to give…
We introduce new symplectic cut-and-paste operations that generalize the rational blowdown. In particular, we will define k-replaceable plumbings to be those that, heuristically, can be symplectically replaced by Euler characteristic k 4-manifolds. We will then classify 2-replaceable linear plumbings, construct 2-r…
The paper proves positivity of characteristic forms for certain vector bundles.
problem Characterizing positivity conditions for vector bundles.
method Operator theory, pushforward identities, and differential forms.
result Schur polynomials in Chern forms of Nakano and Griffiths positive vector bundles are positive as differential forms.
We revisit the cohomological index theorem for elliptic elements in the universal enveloping algebra of a Lie groupoid previously proved by the authors. We prove a Thom isomorphism for Lie algebroids which enables us to rewrite the "topological side" of the index theorem. This results in index formulae for Lie groupoid…