arXiv research

This paper analyzes the Bochner formula for Riemannian flows and derives eigenvalue estimates.

We construct a cubical CW-complex CK(M^3) whose rational cohomology algebra contains Vassiliev invariants of knots in the 3-manifold M^3. We construct \bar{CK}(R^3) by attaching cells to CK(R^3) for every degenerate 1-singular and 2-singular knot, and we show that π_1(\bar{CK}(R^3))=1 and π_2(\bar{CK}(R^3))=Z. We give …

CK simplifies ML model deployment and reproducibility with open APIs and DevOps.

We consider canonical metrics on Fano manifolds. First we introduce a norm-type functional on Fano manifolds, which has Kahler-Einstein or Kahler-Ricci soliton as its critical point and the Kahler-Ricci flow can be viewed as its (reduced) gradient flow. We then obtain a natural lower bound of this functional. As an app…

Study of eigenvalues in nonlinear kernels for classification of separable data.

Study eigenvalue distributions of neural kernels for linear-width networks.

Study eigenvalues and eigenvectors in neural networks, focusing on signal propagation.

Final revision. To appear in the Journal of Differential Geometry. This paper studies knots that are transversal to the standard contact structure in $\reals^3$, bringing techniques from topological knot theory to bear on their transversal classification. We say that a transversal knot type $\cTK$ is {\it transversally…

DEQs and explicit networks are nearly equivalent for Gaussian mixtures.

Researchers derived Kauffman bracket polynomial for Celtic link shadows using two methods.

A hypercomplex manifold M is a manifold with a triple I,J,K of complex structure operators satisfying quaternionic relations. For each quaternion L=aI +bJ+cK, L^2=-1, L is also a complex structure operator on M, called an induced complex structure. We are studying compact complex subvarieties of (M,L), when L is a gene…

Analyzes neural networks using spectral perspectives.

The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and with a metric of any signature, either Riemannian (definite positive) or Lorentzian (in…

We study spectral behavior of the complex Laplacian on forms with values in the $k^{\text{th}}$ tensor power of a holomorphic line bundle over a smoothly bounded domain with degenerated boundary in a complex manifold. In particular, we prove that in the two dimensional case, a pseudoconvex domain is of finite type if a…

Given (M, g0) we consider the problem -ε^2Delta_{g0+h}u + u = (u+)^{p-1} with (ε, h) \in (0, ε0) \times Bρ. Here Bρ is a ball centered at 0 with radius ρ in the Banach space of all Ck symmetric covariant 2-tensors on M. Using the Poincaré polynomial of M, we give an estimate on the number of nonconstant solutions with …

Twistor space of hypercomplex manifolds is never Moishezon.

The twisting technique creates infinite links.

Study finds new minimal surfaces in Schwarzschild space.

Study quasi-Einstein metrics on real hypersurfaces of complex space forms.

We develop an algebraic representation for (1,1)-knots using the mapping class group of the twice punctured torus MCG(T,2). We prove that every (1,1)-knot in a lens space L(p,q) can be represented by the composition of an element of a certain rank two free subgroup of MCG(T,2) with a standard element only depending on …

The paper studies neural networks with wide layers and finds a deformed semicircle law.

Paper analyzes learning dynamics in quasi-periodic environments, showing consistent solutions.

Paper shows how a Lie superalgebra can be realized using matrices.

Proposes a linear model for facial action recognition without requiring large datasets.

New insights into query complexity for Nash equilibrium learning.

Study estimates squared error in high-dimensional binary regression, revealing phase transitions and structural properties.

New algorithm reduces online learning regret in uninformed Markov games.

New class of metric f-manifolds introduced.

Characterizes and describes selfsimilar Hessian manifolds with homothetic vector fields.

Extends Tian theorem to Vaisman manifolds for approximations.

The paper defines trans-para-Sasakian manifolds and explores their geometric properties.

In 1972, K. Kenmotsu studied a class of almost contact Riemannian manifolds. Later, such a manifold was called a Kenmotsu manifold. This paper, we studied Kenmotsu manifolds with $(2n+s)$-dimensional $s-$contact metric manifold and this manifold, we have called generalized Kenmotsu manifolds. Necessary and sufficient c…

Study on a new type of manifolds that generalize almost C-manifolds.

New class of complex manifolds defined, properties studied.

Stabilized convex symplectic manifolds are equivalent to flexible Weinstein manifolds.

New criteria for Sasakian manifolds and classification of nearly cosymplectic manifolds.

Smoothly embed 3-manifolds in 5-manifolds, simplifying topological to smooth.

Determines the type of 3-manifolds from their covers.

The study identifies criteria for 3-manifolds to be boundaries of exotic 4-manifolds.

The study provides a structure theorem for a new class of noncompact 3-manifolds.

Compact Vaisman manifolds split into simpler parts.

Condition for intersection of real flag manifolds in complex flag manifold.

The paper studies extended quasi-Einstein manifolds with special geometric properties and solitons.

The paper explores toric Vaisman manifolds and their connections to Sasaki and Kähler geometry.

Conditions for flat manifolds as cusp cross-sections in arithmetic hyperbolic manifolds.

Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.

Study on 3D manifolds with specific tensor structures and their properties.

New manifold type PNDP-manifold defined with Einstein warped product structure.