We show that a weighted homogeneous complex surface singularity is metrically conical (i.e., bi-Lipschitz equivalent to a metric cone) only if its two lowest weights are equal. We also give an example of a pair of weighted homogeneous complex surface singularities that are topologically equivalent but not bi-Lipschitz …
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48 results for “homogeneous weights”
In the present paper, we study deformations of polar weighted homogeneous polynomials which are also polar weighted homogeneous polynomials. We describe a round handle decomposition of the Milnor fibration of a deformation of a polar weighted homogeneous polynomial concretely and give the number of round handles by the…
Study weightings from singular Lie filtrations.
problem Generalize constructions for singular Lie filtrations.
method Study weightings arising from singular Lie filtrations.
result Generalizes constructions for (regular) Lie filtrations.
We classify the resolution graphs of weighted homogeneous surface singularities which admit rational homology disk smoothings. The nonexistence of rational homology disk smoothings is shown by symplectic geometric methods, while the existence is verified via smoothings of negative weights. In particular, it is shown th…
The paper analyzes neural network dynamics after weights escape the origin.
problem Understanding gradient flow dynamics of neural networks after the origin.
method Analyzes gradient flow of homogeneous neural networks with locally Lipschitz gradients.
result Characterizes the first saddle point encountered after escaping the origin.
Flat semigroups can represent normal weighted homogeneous surface singularities.
problem Representability of flat semigroups in normal weighted homogeneous surface singularities.
method Study of numerical semigroups associated with surface singularities and prove representability conditions.
result A numerical semigroup is representable if and only if it can be written as a quotient of a flat semigroup.
In the present paper, we deform isolated singularities of a certain class of polar weighted homogeneous mixed polynomials, and show that there exists a deformation which has only definite fold singularities and mixed Morse singularities.
No stable discrete maps into certain curved spaces exist.
problem Stability of discrete maps into curved spaces.
method Analysis of weighted length or energy functionals on graphs.
result Non-existence of stable discrete minimal immersions or harmonic maps into specific homogeneous spaces.
Study resolves polynomial germs, proving no mixed critical points and strict transform properties.
problem Resolving mixed critical points and properties of strict transforms of polynomial germs.
method Toric resolutions and modifications of weighted homogeneous polynomials.
result No mixed critical points and strict transform properties as germs.
Introduces VB-structures for geometric objects on manifolds.
problem Properties of higher tangent lifts of geometric structures.
method Introduces weighted structures for various geometric objects on a manifold with a homogeneity structure.
result Proves interesting properties of various weighted structures.
Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densitiesmath.DG
We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…
The paper explores the Rumin complex and spectral sequence on Carnot groups.
problem Understanding the relationship between Rumin complex and spectral sequence on Carnot groups.
method Investigates the Rumin complex and spectral sequence on Carnot groups, focusing on the filtration by homogeneous weights.
result Provides a detailed insight into the relationship between the Rumin complex and the spectral sequence on Carnot groups.
New regularizer improves neural network robustness and generalization.
problem Ineffective weight decay for networks with homogeneous activation functions.
method Proposes an invariant regularizer to penalize intrinsic weight norms.
result Improves generalization and adversarial robustness on various datasets.
Normal forms and moduli stacks for flat connections on complex manifolds.
problem Understanding singular flat connections on complex manifolds.
method Introducing homogeneous Lie groupoids and studying their representation theory to prove normal form theorems and moduli space structures.
result Moduli spaces of singular flat connections admit the structure of algebraic quotient stacks.
Introduces homogeneity supermanifolds for studying graded structures.
problem Graded structures on supermanifolds.
method Homogeneity degrees and weight vector field.
result Proofs of homogeneous Poincaré Lemma, Frobenius Theorem, and Darboux Theorem.
The paper studies weak singular Hermite-Einstein structures on homogeneous vector bundles.
problem Existence of weak singular Hermite-Einstein structures on homogeneous holomorphic vector bundles.
method Using Cartan's highest weight theory, the paper establishes an algebraic criterion for topological splitting and decouples the prescribed mean curvature equation.
result A sufficient algebraic condition for realizing an -function as the mean curvature of a singular Hermitian structure on an irreducible homogeneous bundle.
The study examines the independence of GKM manifolds and symmetric spaces.
problem Understanding the independence of isotropy weights in GKM manifolds.
method Using weighted graphs and properties of symmetric spaces, the study analyzes the independence of isotropy weights.
result The maximal independence of is , , or , corresponding to symmetric spaces of rank .
New algorithm detects communities in weighted networks, improving on binary ones.
problem Few methods exist for detecting communities in weighted networks.
method Pseudo-likelihood approach for weighted stochastic block model.
result The method is consistent and works well for both homogeneous and heterogeneous networks.
Directional Convergence Near Small Initializations and Saddles in Two-Homogeneous Neural Networkscs.LG
The paper studies neural networks' convergence near origin and saddle points.
problem Directional convergence of neural networks near small initializations and saddle points.
method Gradient flow dynamics analysis of two-homogeneous neural networks.
result Neural networks' weights approximately converge in direction to KKT points for small initializations.
Gradient descent on normalized networks reveals sparsity preferences.
problem Understanding the inductive bias of gradient descent on normalized neural nets.
method Analysis of gradient descent on weight-normalized smooth homogeneous neural nets, focusing on SWN and EWN.
result EWN causes weights to be updated in a way that prefers asymptotic relative sparsity.
Early training of deep neural networks leads to small, directionally converging weights.
problem Training dynamics of deep homogeneous neural networks with small initializations.
method Gradient flow analysis and study of KKT points for neural correlation function.
result Weights converge in direction to KKT points during early training stages.
The paper studies graded manifolds and their functorial relationship.
problem Understanding the functor between two categories of graded manifolds.
method Examines polynomial filtrations and homogeneity structures, applying the Batchelor-Gawedzki theorem and Borel-Whitney theorem.
result The functor is full and surjective on objects between the categories of graded vector bundles and manifolds.
The paper examines bi-Lipschitz triviality of function germs on singular varieties.
problem Analyzing the bi-Lipschitz triviality of deformations of function germs on singular varieties.
method Introducing strongly rational -bi-Lipschitz trivial families and providing an infinitesimal criterion for bi-Lipschitz triviality.
result Bi-Lipschitz triviality of deformations of on when and are homogeneous of the same degree.
The paper validates Stokes' theorem for differential subcomplexes in positively graded Lie groups.
problem Validating Stokes' theorem for differential subcomplexes in positively graded Lie groups.
method Introducing geometric conditions and spectral complexes to recover Stokes' theorem on locally smooth intrinsic graphs.
result Stokes' theorem holds for Rumin complex and new spectral complexes on Carnot groups.
V1 cortex reconstructs images as Poisson equation solutions with varying weights.
problem Reconstructing images from V1 cortical cell receptive profiles.
method Solves a heterogeneous Poisson equation with varying weights representing neural connectivity.
result Reconstructions converge to homogeneous solutions using homogenization techniques.
New capacity measure for deep ReLU networks derived from weight norms.
problem Identifying a suitable capacity measure for deep ReLU networks.
method Generalization of a recently proposed sampling argument to demonstrate the existence of sparse approximants of positive homogeneous networks.
result Bounding generalization error in multi-class classification using covering number bounds.
New Stein fillings found for non-weighted homogeneous singularities.
problem Finding Stein fillings for certain non-weighted homogeneous singularities.
method Using spinal open books and nearly Lefschetz fibrations.
result Stein rational homology disk fillings for non-weighted homogeneous singularities.
This paper analyzes convergence of large-scale Transformers with weight decay.
problem Understanding optimization guarantees in large-scale Transformer training.
method Construct mean-field limit, show gradient flow convergence to PDE, demonstrate global minimum consistency.
result Gradient flow reaches global minimum in large-scale Transformers with small weight decay.
Direct learning framework for integrating multi-source causal data.
problem Conditional average treatment effects inference from heterogeneous data.
method Direct learning framework, double robustness, causal information-aware weighting function.
result Effective causal data fusion in both homogeneous and heterogeneous scenarios.
On geometric properties of the functors of positively homogenous and semiadditive functionalsmath.GN
In this paper we investigate the functors of OH of positively homogenous functionals and OS of semiadditive functionals. We show that OH(X) is AR if and only if X is openly generated, and OS(X) is AR if and only if X is an openly generated compactum of weight less than . Also, we investigate the multiplication map…
Low regularity spacetimes split into simpler structures.
problem Proving splitting theorem for metrics and weights.
method Combining elliptic techniques and line-adapted curves.
result Extends Lorentzian splitting theorem to settings.
Characterizes Schrödinger operator boundedness on weighted Riemannian manifolds.
problem Classifying functions for bounded Schrödinger operator .
method Investigates weighted -boundedness of Hodge projector.
result Characterizes function for Schrödinger operator boundedness.
Positivity of mild solution to stochastic evolution equations with an application to forward ratesmath.AP
Maximum principle proves positivity of forward rates in stochastic models.
problem Proving positivity of forward rates in stochastic models.
method Maximum principle for mild solutions to SPDEs with Lipschitz coefficients and Wiener noise.
result Sufficient conditions for positivity of forward rates in the Heath-Jarrow-Morton model.
Homogenized Transformersmath.PR
Study on deep multi-head self-attention dynamics, proving homogenized limits under specific scalings.
problem Understanding the behavior of deep multi-head self-attention models as depth increases.
method Random model of deep multi-head self-attention, viewing depth as time, and analyzing the residual stream as a particle system.
result Homogenized limit of the dynamics, leading to deterministic or stochastic behavior depending on scaling, with implications for representation collapse.
We study the problem of the existence and the holomorphicity of the Monge-Ampère foliation associated to a plurisubharmonic solutions of the complex homogeneous Monge-Ampère equation even at points of arbitrary degeneracy. We obtain good results for real analytic unbounded solutions. As a consequence we also provide a …
The paper characterizes symplectic fillings of Seifert 3-manifolds using rational blowdowns.
problem Understanding symplectic fillings of Seifert 3-manifolds.
method Rational blowdown surgery and minimal symplectic fillings.
result A necessary and sufficient condition for minimal symplectic fillings to be obtained by rational blowdowns.
We define and make initial study of Lie groupoids equipped with a compatible homogeneity (or graded bundle) structure, such objects we will refer to as weighted Lie groupoids. One can think of weighted Lie groupoids as graded manifolds in the category of Lie groupoids. This is a very rich geometrical theory with numero…
We prove regularity results up to the boundary for time independent generalized Maxwell equations on Riemannian manifolds with boundary using the calculus of alternating differential forms. We discuss homogeneous and inhomogeneous boundary data and show 'polynomially weighted' regularity in exterior domains as well.
Toeplitz operators with quasi-radial quasi-homogeneous symbols and bundles of Lagrangian framesmath.OA
We prove that the quasi-homogenous symbols on the projective space yield commutative algebras of Toeplitz operators on all weighted Bergman spaces, thus extending to this compact case known results for the unit ball . These algebras are Banach but not . We prove the existen…
MANA-Net improves market predictions by dynamically weighting news sentiments.
problem Aggregated Sentiment Homogenization in financial news data.
method Dynamic market-news attention mechanism to aggregate sentiments.
result MANA-Net outperforms recent market prediction methods by 1.1% Profit & Loss and 0.252 daily Sharpe ratio.
Homogeneous compatible almost complex structures on symplectic manifolds are studied, focusing on those which are special, meaning that their Chern-Ricci form is a multiple of the symplectic form. Non Chern-Ricci flat ones are proven to be covered by co-adjoint orbits. Conversely, compact isotropy co-adjoint orbits of …
In this paper we study the cobordism of algebraic knots associated with weighted homogeneous polynomials, and in particular Brieskorn polynomials. Under some assumptions we prove that the associated algebraic knots are cobordant if and only if the Brieskorn polynomials have the same exponents.
Defines magnitude for length spaces with measures, agreeing with finite spaces' magnitude.
problem Defining magnitude for non-finite metric spaces with measures.
method Integrals over geodesics, using counting and weight measures.
result Magnitude agrees with finite spaces' magnitude and volume under specific conditions.
We introduce W-spin structures on a Riemann surface and give a precise definition to the corresponding W-spin equations for any quasi-homogeneous polynomial W. Then, we construct examples of nonzero solutions of spin equations in the presence of Ramond marked points. The main result of the paper is a compactness theore…
Detecting spatial homogeneity in the world trade web with Detrended Fluctuation Analysisphysics.data-an
In a spatially embedded network, that is a network where nodes can be uniquely determined in a system of coordinates, links' weights might be affected by metric distances coupling every pair of nodes (dyads). In order to assess to what extent metric distances affect relationships (link's weights) in a spatially embedde…
We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of . Our result applies to…
We consider a connected symplectic manifold acted on properly and in a Hamiltonian fashion by a connected Lie group . Inspired to the recent paper \cite{gb2}, see also \cite{ch} and \cite{pacini}, we study Lagrangian orbits of Hamiltonian actions. The dimension of the moduli space of the Lagrangian orbits is giv…
We study the problem of identifying different behaviors occurring in different parts of a large heterogenous network. We zoom in to the network using lenses of different sizes to capture the local structure of the network. These network signatures are then weighted to provide a set of predicted labels for every node. W…