arXiv research

The spectral properties of p-forms on the fundamental domains of regular tesselations of the d-dimensional sphere are discussed. The degeneracies for all ranks, p, are organised into a double Poincare series which is explicitly determined. In the particular case of coexact forms of rank (d-1)/2, for odd d, it is shown …

Study on existence of $p$-Kähler structures on nilmanifolds with nilpotent complex structures.

Through the theory of Lie bi-algebroids and generalized complex structures, one could define a cohomology theory naturally associated to a holomorphic Poisson structure. It is known that it is the hypercohomology of a bi-complex such that one of the two operators is the classical $\overline{\partial}$-operator. Another…

RP-GFRFT unifies fractional order and rotation control for graph signals.

Formula derived for holomorphic Poisson blow-ups.

We derive a blow-up formula for the de Rham cohomology of a local system of complex vector spaces on a compact complex manifold. As an application, we obtain the blow-up invariance of $E_{1}$-degeneracy of the Hodge-de Rham spectral sequence associated to a local system of complex vector spaces.

The paper proves a logarithmic partial derivative lemma and applies it to several geometric problems.

We consider the problem of training input-output recurrent neural networks (RNN) for sequence labeling tasks. We propose a novel spectral approach for learning the network parameters. It is based on decomposition of the cross-moment tensor between the output and a non-linear transformation of the input, based on score …

Research on mixed polynomials, extending non-degeneracy concepts to complex variables.

Classifies patterns of symmetry breaking and vacuum degeneracy in scalar and gauge fields.

CP degeneracy affects tensor regression solutions, especially in high dimensions.

Note removes degeneracy in Kähler geometry estimates.

Paper proves spectral sequences of knot spaces are isomorphic over fields.

We prove a blow-up formula for Dolbeault cohomologies of compact complex manifolds by introducing relative Dolbeault cohomology. As corollaries, we present a uniform proof for bimeromorphic invariance of $(\bullet,0)$- and $(0,\bullet)$-Hodge numbers on a compact complex manifold, and obtain the equality for the number…

We construct a complete convergent normal form for a real hypersurface in $\CC{N},\,N\geq 2$ at generic Levi degeneracy. This seems to be the first convergent normal form for a Levi-degenerate hypersurface. In particular, we obtain, in the spirit of the work of Chern and Moser \cite{chern}, distinguished curves in the …

Study ratio-limit boundaries for random walks on hyperbolic groups.

The study examines determinantal point processes linked to a specific operator on Riemannian manifolds.

Study on Dirac operator spectrum on shrinking surfaces with cusps.

This research connects quantum spectra of flag bundles to prime factorization of integers.

Network degeneracy affects training performance, especially in deep networks.

TAMD prevents degeneracy in finite mixtures, offering strong guarantees but modest practical improvements.

We analyze relations between BPS degeneracies related to Labastida-Marino-Ooguri-Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the correspond…

New cylindrical solutions found for Grushin-type problem.

Geometric regularisation improves statistical models by avoiding degeneracy loci.

Random Matrix Theory explains loss surface Hessians in neural networks.

Study non-degeneracy of minimal hypersurfaces asymptotic to cones, proving Jacobi equation solvability.

Study families of Morse functions for manifolds with boundary.

Investigates metric degeneracies on symplectic leaves using a generalized gradient flow.

A method to automatically and symbolically detect and resolve degenerate parameter combinations from parameter-data pairs.

Latent variable models with hidden binary units appear in various applications. Learning such models, in particular in the presence of noise, is a challenging computational problem. In this paper we propose a novel spectral approach to this problem, based on the eigenvectors of both the second order moment matrix and t…

Study non-degenerate singular points of Poisson-Nijenhuis structures.

For regular particle filter algorithm or Sequential Monte Carlo (SMC) methods, the initial weights are traditionally dependent on the proposed distribution, the posterior distribution at the current timestamp in the sampled sequence, and the target is the posterior distribution of the previous timestamp. This is techni…

Study on automorphisms of complex $b^k$-manifolds, extending previous work.

Generalizing some results from R. Leung's thesis, we compute, in rational cohomology, the Poincare dual of the degeneracy locus of the family of Dirac operators parameterized by the moduli space of projectively anti-self-dual $\SO(3)$ connections. This is the first step in a program to derive a relation between the Don…

Proves existence of proper solutions for inverse mean curvature flow.

Tian's theorem connects Chern classes of bundles to random section zeros and degeneracy sets.

We prove a convergence result for a family of Yang-Mills connections over an elliptic $K3$ surface $M$ as the fibers collapse. In particular, assume $M$ is projective, admits a section, and has singular fibers of Kodaira type $I_1$ and type $II$. Let $Ξ_{t_k}$ be a sequence of $SU(n)$ connections on a principal $SU(n)$…

Unified Morse-Bott-Smale chain complex, resolves well-definedness issue.

We produce examples of generalized complex structures on manifolds by generalizing results from symplectic and complex geometry. We produce generalized complex structures on symplectic fibrations over a generalized complex base. We study in some detail different invariant generalized complex structures on compact Lie g…

In this paper, we study the interplay between modules and sub-objects in holomorphic Poisson geometry. In particular, we define a new notion of "residue" for a Poisson module, analogous to the Poincaré residue of a meromorphic volume form. Of particular interest is the interaction between the residues of the canonical …

Counterfactual learning is a natural scenario to improve web-based machine translation services by offline learning from feedback logged during user interactions. In order to avoid the risk of showing inferior translations to users, in such scenarios mostly exploration-free deterministic logging policies are in place. …

Study slopes in 3-manifolds, proving conjectures about knots.

A new method for neural network initialization using graph degeneracy.

The paper studies parallel surfaces of cuspidal cross caps and their degeneracy.

The paper examines how the angle between inputs in ReLU networks decreases with depth, impacting training.

We present a new method to solve certain $\bar{\partial}$-equations for logarithmic differential forms by using harmonic integral theory for currents on Kahler manifolds. The result can be considered as a $\bar{\partial}$-lemma for logarithmic forms. As applications, we generalize the result of Deligne about closedness…

Local bifurcation theory typically deals with the response of a degenerate but isolated equilibrium state or periodic orbit of a dynamical system to perturbations controlled by one or more independent parameters, and characteristically uses tools from singularity theory. There are many situations, however, in which the…

Unified interpretation of sub-Riemannian Gauss-Bonnet theorem for surfaces in 3D contact manifolds.