Derives integral formulae on weighted manifolds.
problem No specific problem stated; focuses on mathematical derivations.
method Introduces weighted mean sigma-r curvature and uses weighted Newton transformations.
result Derives integral formulae generalizing previous work.
Study geometric and topological properties of Finsler manifolds with weighted Ricci curvature bounds.
problem Geometric and topological properties of Finsler metric measure manifolds with integral weighted Ricci curvature bounds.
method Establish Laplacian comparison theorem, volume comparison theorems, volume growth estimate, Gromov pre-compactness, local Dirichlet isoperimetric constant estimate.
result First Dirichlet eigenvalue estimate and gradient estimate for harmonic functions.
Sharp upper bounds derived for Alexandrov-Fenchel deficit using weighted Minkowski integral formulas.
problem Deriving upper bounds for the Alexandrov-Fenchel deficit.
method Using weighted Minkowski integral formulas and an integral formula for the deficit in Jensen's inequality.
result Quantitative estimates under weaker convexity assumptions, including a distance term.
Study extends compactness theorems to weighted manifolds with integral curvature bounds.
problem Estimating diameter of weighted manifolds under curvature constraints.
method Extended Sprouse's compactness theorems to weighted manifolds with integral curvature bounds. Used ε-range to handle specific cases. Extended segment inequality to weighted manifolds.
result Proved theorems for weighted manifolds with effective dimension ≤ 1 and ≥ dimension.
Enhanced visual feature attribution via adaptive baseline weighting.
problem IG's sensitivity to baseline images leads to noisy or unstable explanations.
method Weighted Integrated Gradients (WG) evaluates and weights baselines for improved reliability.
result WG improves over Expected Gradients (EG) by up to 36% across various models.
We prove that square integrable holomorphic functions (with respect to a plurisubharmonic weight) can be extended in a square integrable manner from certain singular hypersurfaces (which include uniformly flat, normal crossing divisors) to entire functions in affine space. This provides evidence for a conjecture regard…
The paper explores inequalities on weighted Riemannian manifolds with boundary.
problem Developing inequalities on weighted Riemannian manifolds with boundary.
method Using a Reilly type integral formula associated with the φ-Laplacian.
result Provided inequalities of Brascamp-Lieb type and Colesanti type.
Researchers extend asymptotic analysis to Bergman projections with Gevrey weights.
problem Analyzing Bergman projections with Gevrey weights.
method Extending direct approach to semiclassical asymptotics to Gevrey weights using Fourier integral operators.
result Gevrey symbol amplitude of asymptotic Bergman projection with Gevrey weights and Gevrey-type growth rate.
Develops theory of weightings for Lie groupoids and algebroids.
problem Understanding differential geometry of weightings for Lie groupoids and algebroids.
method Extending work on weighted manifolds, defining weighted submanifolds, and developing theories of linear weightings and multiplicative weightings.
result Characterizes infinitesimally multiplicative weightings for Lie algebroids and classifies multiplicative weightings of Lie groupoids.
A new type of quadrature is developed. The Gaussian quadrature, for a given measure, finds optimal values of a function's argument (nodes) and the corresponding weights. In contrast, the Lebesgue quadrature developed in this paper, finds optimal values of function (value-nodes) and the corresponding weights. The Gaussi…
Derives integral formula for differential forms on compact spaces with applications.
problem Integral formula for differential forms on compact spaces with boundary.
method Derives a weighted Reilly type integral formula.
result Lower bounds for spectrum and eigenvalues of differential forms.
We show that integration over a G-manifold M can be reduced to integration over a minimal section Σ with respect to an induced weighted measure and integration over a homogeneous space G/N. We relate our formula to integration formulae for polar actions and calculate some weight functions. In case of a compact …
Counting HCMU sphere components using weighted trees.
problem Counting components of moduli space of HCMU spheres.
method Using weighted plane trees to characterize HCMU spheres with a single integral conical angle, and an explicit counting formula is derived.
result An explicit counting formula for the components of the moduli space of HCMU spheres.
A new method integrates forms on Riemann surfaces, leading to modular forms.
problem Integrating differential forms with poles on Riemann surfaces.
method Simple procedure to integrate differential forms with arbitrary holomorphic poles, establishing an analytic theory for integrals over configuration spaces.
result Regularized graph integrals on elliptic curves are almost-holomorphic modular forms.
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
This research integrates attention into XAI frameworks for better model explanations.
problem Improving the interpretability of transformer models.
method Developed two novel explanation methods: Shapley value decomposition and token-level directional derivatives.
result Attention weights can be meaningfully incorporated into XAI frameworks, enhancing transformer explainability.
The paper proposes a new method for covariate balancing using IPM to improve causal inference.
problem Covariate imbalance in causal inference weighting methods, especially when models are not correctly specified.
method The integral probability metric (IPM) is used to determine optimal weights for treated and control groups.
result The proposed method can be consistent without specifying either the propensity score or outcome regression model.
We study the weighted light ray transform L of integrating functions on a Lorentzian manifold over lightlike geodesics. We analyze L as a Fourier Integral Operator and show that if there are no conjugate points, one can recover the spacelike singularities of a function f from its the weighted light ray transform …
In his seminal paper, A. N. Varchenko precisely investigates the leading term of the asymptotic expansion of an oscillatory integral with real analytic phase. He expresses the order of this term by means of the geometry of the Newton polyhedron of the phase. The purpose of this paper is to generalize and improve his re…
Econometric framework integrates heavy-tailed distributions with behavioral probability weighting for better asset pricing.
problem Underestimation of Value-at-Risk by traditional models in asset pricing.
method Developed an econometric framework combining heavy-tailed Student's t distributions with behavioral probability weighting. result Student's t specifications outperform Gaussian models in 88.4% of cases, reducing underestimation of Value-at-Risk by 16.5 percentage points. The paper proves lower bounds for Gaussian-weighted curvature integrals of self-shrinkers.
problem Proving lower bounds for Gaussian-weighted \(L^2\)-curvature integrals of self-shrinkers.
method Combining normal coordinate functions with weighted Poincaré inequalities and first-eigenvalue estimates.
result Explicit lower bounds in terms of entropy for closed self-shrinkers, leading to curvature gaps.
New weighted surface area measures for convex bodies with applications.
problem Generalizing surface area measures to weighted Borel measures.
method Formulating and analyzing weighted surface area measures, proving integral formula and Bézout-type inequality.
result New integral formula for mixed measure of three bodies, generalizing Bézout-type inequality.
Study connects weighted isoperimetric problems to nonlocal elliptic operator extensions.
problem Sharp inequalities for weighted Poisson integrals and their extremizers.
method Formulates variational problem on conformal metric measure space.
result Sharp inequalities are linked to variational problem on CCE manifolds.
Algorithm selects variables and bandwidths for geographically weighted regression.
problem Estimating variable subsets and bandwidths for geographically weighted regression.
method Mathematical programming-based approach integrating variable selection and bandwidth estimation.
result Proposed algorithm provides stable spatially varying patterns with competitive explanatory power.
We study the weighted ray transform of integrating functions on a Lorentzian manifold over lightlike geodesics. We prove support theorems if the manifold and the weight are analytic.
Integral simplicial volume is a homotopy invariant of oriented closed connected manifolds, defined as the minimal weighted number of singular simplices needed to represent the fundamental class with integral coefficients. We show that odd-dimensional spheres are the only manifolds with integral simplicial volume equal …
Sharp inequalities for weighted log canonical thresholds derived.
problem Understanding weighted log canonical thresholds in complex analysis.
method Combining integrability estimates, complex line restrictions, and pluripotential theory.
result Uniform control of difference quotients and explicit lower bounds derived.
The paper proves gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
problem Gradient estimates for a weighted p-Laplacian equation on Riemannian manifolds.
method Assumes a Sobolev inequality and integral Ricci bounds, proving local gradient estimates and Liouville type results.
result Proves local gradient estimates and Liouville type results on manifolds with lower bounds of Ricci curvature.
The paper studies variations of weighted curvature on submanifolds.
problem Variational properties of weighted curvature on submanifolds.
method Analysis of a functional with integrant r-th weighted curvature.
result Applications to hypersurfaces in Euclidean space and the unit sphere.
Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature
problem Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature
method Use Bézout estimates and a Lipschitz weight with finite Monge-Ampère mass
result Prove top wedge power of Ricci form has finite integral for Kähler manifolds with positive sectional curvature
The paper integrates behavioral distortions into portfolio optimization using implied probability weighting functions.
problem Behavioral distortions in probability weighting affect portfolio optimization under different return distributions.
method Developed a unified framework to extract probability weighting functions from optimal portfolios modeled under Gaussian and NIG distributions.
result Increasing tail fatness amplifies behavioral distortions, and shifts in risk-free rates alter the curvature of these distortions.
Derives formulas for differential forms on weighted manifolds.
problem Developing formulas for differential forms on weighted manifolds.
method Derives a Reilly formula and explores its applications.
result Proves a Poincaré-type inequality and obtains new eigenvalue estimates.
Optimizes sliding window approach for tracking Gaussian densities.
problem Improving tracking performance of Gaussian density estimation.
method Theoretical analysis of sliding window Gaussian Kernel Density Estimators.
result Empirical evidence shows improved tracking performance with optimal weight sequence.
Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.
problem Cohomology of relative Rota-Baxter operators on Lie algebras and groups.
method Cohomology construction, infinitesimal deformations study, differentiation and integration of Rota-Baxter operators.
result Integration of Rota-Baxter operators on Lie groups and Lie algebras.
A well known question in differential geometry is to control the constant in isoperimetric inequality by intrinsic curvature conditions. In dimension 2, the constant can be controlled by the integral of the positive part of the Gaussian curvature. In this paper, we showed that on simply connected conformal flat manifol…
PS-IG improves feature attribution by reducing noise and variance.
problem Improving feature attribution in machine learning models.
method Path-sampled integrated gradients (PS-IG) computes expected value over sampled baselines.
result PS-IG reduces attribution variance by a factor of 1/3 under uniform sampling.
Enhances linear regression with Kalman filter for loss minimization.
problem Minimizing loss in linear regression models.
method Integrates Kalman filter and SGD for optimal weight updates.
result Develops optimal linear regression equation with minimum area under curve.
We define invariants for colored oriented spatial graphs by generalizing CM invariants, which were defined via non-integral highest weight representations of Uq(sl2). We apply the same method to define Yokota's invariants, and we call these invariants Yokota type invariants. Then we propose a volume conjecture of t…
We compute many dimensions of spaces of finite type invariants of virtual knots (of several kinds) and the dimensions of the corresponding spaces of "weight systems", finding everything to be in agreement with the conjecture that "every weight system integrates".
DW-KNN improves KNN by integrating distance and neighbor reliability for better prediction accuracy.
problem Standard KNN assumes all neighbors are equally reliable, leading to unreliable predictions in heterogeneous feature spaces.
method DW-KNN integrates exponential distance with neighbor validity, providing instance-level interpretability and reducing hyperparameter sensitivity.
result DW-KNN achieves 0.8988 average accuracy, ranks 2nd among six methods, and has the lowest cross-validation variance.
Using the monotonicity formulas of Colding and Minicozzi, we prove that on any complete, non-parabolic Riemannian manifold (M3,g) with non-negative Ricci curvature, the asymptotic weighted scaling invariant integral of scalar curvature has an explicit bound in form of asymptotic volume ratio.
The paper proves geometric inequalities in sphere using locally constrained flows.
problem Deriving geometric inequalities in sphere.
method Established the longtime existence and convergence of a locally constrained flow.
result Proved new families of three-term geometric inequalities in sphere.
New insights into choosing between two data integration methods based on SVD.
problem Choosing between two data integration methods (Stack-SVD and SVD-Stack) for shared latent structure across multiple datasets.
method Derive exact expressions for the asymptotic performance and phase transitions of Stack-SVD and SVD-Stack, and develop optimal weighting schemes.
result Optimally weighted Stack-SVD outperforms optimally weighted SVD-Stack in the asymptotic regime.
We consider a conjecture made by Ge, Wang and Wu regarding weighted Alexandrov-Fenchel inequalities for horospherically convex hypersurfaces in hyperbolic space (a bound, for some physically motivated weight function, of the weighted integral of the kth mean curvature in terms of the area of the hypersurf…
Global approximation for piecewise linear paths via signatures.
problem Global approximation theorems for piecewise linear paths.
method Using signatures of piecewise linear paths and their density in Lp-norms. result Linear functionals of signatures are dense in Lp-norms under an integrability condition. We propose a methodology for computing single and multi-asset European option prices, and more generally expectations of scalar functions of (multivariate) random variables. This new approach combines the ability of Monte Carlo simulation to handle high-dimensional problems with the efficiency of function approximation…
Mathematical study of learning long-term integration in linear RNNs.
problem How do linear recurrent neural networks learn to integrate over long timescales?
method Analytical study of linear RNNs trained to integrate white noise and damped oscillatory filters.
result Learning dynamics are described by low-dimensional effective equations for outlier eigenvalues.
The paper integrates multiple Gaussian process predictions using Monte Carlo sampling.
problem Accurate prediction of variables using multiple models.
method Log-linear pooling of Gaussian process predictions, combined with Monte Carlo sampling.
result The log-linear pooling method improves prediction accuracy compared to linear pooling.