arXiv research

CuMPerLay vectorizes CMP for deep learning, improving image analysis.

Unified framework connects deformation theory and derived categories for multiparameter persistence.

New method for analyzing multiparameter persistence modules from smooth functions.

Stable density-based clustering via multiparameter persistence.

Extends Morse-Forman theory to vector-valued functions for multiparameter persistence.

Paper introduces stable vectorization for multiparameter PH using signed barcodes.

Develops 2-categorical methods for multi-parameter persistence.

Paper stabilizes persistent homology rank functions for statistical inference.

We establish multiparameter resolvent trace expansions for elliptic boundary value problems, polyhomogeneous both in the resolvent and the auxiliary parameter. The present analysis is rooted in the joint project with Matthias Lesch on multiparameter resolvent trace expansions on revolution surfaces with applications to…

The paper uses MDM theory to analyze multifiltering functions on simplicial complexes.

We prove the existence of multiparameter isospectral deformations of metrics on SO(n) $(n = 9$ and $n\geq 11)$, SU(n) $(n\geq 8)$, and $Sp(n)$ $(n\geq 4)$. For these examples, we follow a metric construction developed by Schueth who had given one-parameter families of isospectral metrics on orthogonal and unitary group…

A novel topological method analyzes fMRI data over time.

The central problem of strip theory is the calculation of potential flowaround 2D sections. One particular method of solutions to this problem is conformal mapping of the body section to the unit circle over which a solution of potential flow is available. Here, a new multiparameter conformal mapping method is presente…

We present Natural Gradient Boosting (NGBoost), an algorithm for generic probabilistic prediction via gradient boosting. Typical regression models return a point estimate, conditional on covariates, but probabilistic regression models output a full probability distribution over the outcome space, conditional on the cov…

Random complexes can be embedded linearly if certain conditions on parameters are met.

This study explores how choosing noninformative priors affects Thompson Sampling in multiparameter bandit models.

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the cla…

Given a Coxeter system $(W,S)$ and a positive real multiparameter $\bq$, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $Σ$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W,S)$ and the multiparameter $q$. The…

{\em Riemannian cubics} are curves in a manifold $M$ that satisfy a variational condition appropriate for interpolation problems. When $M$ is the rotation group SO(3), Riemannian cubics are track-summands of {\em Riemannian cubic splines}, used for motion planning of rigid bodies. Partial integrability results are know…

Sub-Riemannian cubics are a generalisation of Riemannian cubics to a sub-Riemannian manifold. Cubics are curves which minimise the integral of the norm squared of the covariant acceleration. Sub-Riemannian cubics are cubics which are restricted to move in a horizontal subspace of the tangent space. When the sub-Riemann…

The cubic lattice stick index of a knot type is the least number of sticks necessary to construct the knot type in the 3-dimensional cubic lattice. We present the cubic lattice stick index of various knots and links, including all (p,p+1)-torus knots, and show how composing and taking satellites can be used to obtain t…

This paper refines homotopy theory for cubical sets and uniform spaces.

According to our previous results, the conjugacy class of the involution induced by the complex conjugation in the homology of a real non-singular cubic fourfold determines the fourfold up to projective equivalence and deformation. Here, we show how to eliminate the projective equivalence and to obtain a pure deformati…

The paper studies conformally flat cubic metrics with isotropic curvature, finding they must be Minkowski.

Cubic fourfolds have K-stability and admit Kähler-Einstein metrics.

The study shows that certain cubical presentations lead to aspherical spaces.

New cubic forms linked to η-invariants and mod 2 indices.

It is shown that there exist non-singular cubic surfaces in CP^3 containing 5 twistor lines. This is the maximum number of twistor fibres that a non-singular cubic can contain. Cubic surfaces in CP^3 with 5 twistor lines are classified up to transformations preserving the conformal structure of S^4.

We study global log canonical thresholds of cubic surfaces with canonical singularities, and we prove the existence of a Kahler-Einstein metric on two singular cubic surfaces.

Solves infinite family of cubic polynomial problems.

Consider the family of smooth cubic surfaces which can be realized as threefold-branched covers of $\mathbb{P}^{2}$, with branch locus equal to a smooth cubic curve. This family is parametrized by the space $\mathcal{U}_{3}$ of smooth cubic curves in $\mathbb{P}^{2}$ and each surface is equipped with a $\mathbb{Z}/3\ma…

Motivated by applications in computational anatomy, we consider a second-order problem in the calculus of variations on object manifolds that are acted upon by Lie groups of smooth invertible transformations. This problem leads to solution curves known as Riemannian cubics on object manifolds that are endowed with norm…

A natural family of affine cubic surfaces arises from SL(2)-characters of the 4-holed sphere and the 1-holed torus. The ideal locus is a tritangent plane which is generic in the sense that the cubic curve at infinity consists of three lines pairwise intersecting in three double points. We show that every affine cubic s…

Classifies hexagonal circular 3-webs with cubic polar curves.

We show that, for Finsler spaces with cubic metric, Landsberg spaces are Berwaldian. Also, for decomposable metrics, we determine specific conditions for a space with cubic metric to be of Berwald type, thus refining the result in [6].

Approximating complex curves with simple parametric curves is widely used in CAGD, CG, and CNC. This paper presents an algorithm to compute a certified approximation to a given parametric space curve with cubic B-spline curves. By certified, we mean that the approximation can approximate the given curve to any given pr…

We give a generalization of the well-known result of E. Cartan on isoparametric cubics by showing that a homogeneous cubic polynomial solution of the eiconal equation $|\nabla f|^2=9|x|^4$ must be rotationally equivalent to either $x_n^3-3x_n(x_1^2+...+x_{n-1}^2)$, or to one of four exceptional Cartan cubic polynomials…

Study on choosing points on cubic curves, answering some questions about their flexibility.

Researchers classify and visualize 5-cube cubical surfaces.

No nontrivial automorphisms for cubic surfaces moduli space.

Study of algebraic dynamics on Markov cubics in tropical geometry.

Origami solves real cubic equations, revealing a specific curve.

Approaches for approximating persistent homology for large datasets.

Persistence landscapes map persistence diagrams into a function space, which may often be taken to be a Banach space or even a Hilbert space. In the latter case, it is a feature map and there is an associated kernel. The main advantage of this summary is that it allows one to apply tools from statistics and machine lea…

Constructs independent bases for cubic curve families using Hessian structures.

We establish a quadratic identity for the Yamada polynomial of ribbon cubic graphs in 3-space, extending the Tutte golden identity for planar cubic graphs. An application is given to the structure of the flow polynomial of cubic graphs at zero. The golden identity for the flow polynomial is conjectured to characterize …

The paper studies 8D manifolds with a specific tensor field called a cubic discriminant.

We consider non-degenerate centro-affine hypersurface immersions in R^n whose cubic form is parallel with respect to the Levi-Civita connection of the affine metric. There exists a bijective correspondence between homothetic families of proper affine hyperspheres with center in the origin and with parallel cubic form, …