arXiv research

The paper constructs a multi-valued inverse of quasiregular maps and develops pull-back theory for differential forms.

We construct several types of multi-valued solutions to the Monge-Ampere equation in higher dimensions.

Extends Campanato theory to multi-valued functions for geometric variational problems.

XTNet estimates complex cross-treatment effects in multi-category, multi-valued settings.

We apply the technique of integrable extensions to the symmetry pseudo-group of the dKP-hyper CR interpolating equation. This allows us to find a covering for this equation and to construct multi-valued Einstein-Weyl structures.

In this paper we prove that an embedded and simply connected constant mean curvature surface with curvature large at a point contains a multi-valued graph around that point on the scale of $|A|^2$, where $|A|^2$ is the norm squared of the second fundamental form. This generalizes Colding and Minicozzi's result for mini…

This paper analyzes meta-learners for estimating multi-valued treatment effects.

New method estimates causal effects with multi-valued, time-varying treatments.

New category theory for complex projective plane sections.

In this article, we study local holomorphic isometric embeddings from ${\BB}^n$ into ${\BB}^{N_1}\times... \times{\BB}^{N_m}$ with respect to the normalized Bergman metrics up to conformal factors. Assume that each conformal factor is smooth Nash algebraic. Then each component of the map is a multi-valued holomorphic m…

DiffOPF solves multi-valued OPF problems by sampling from system history.

In the early 1980's Almgren developed a theory of Dirichlet energy minimizing multi-valued functions, proving that the Hausdorff dimension of the singular set (including branch points) of such a function is at most $(n-2),$ where $n$ is the dimension of its domain. Almgren used this result in an essential way to show t…

VOWEL trains WTA-SNNs for multi-valued events, overcoming resource limitations.

This paper is the second in a series where we attempt to give a complete description of the space of all embedded minimal surfaces of fixed genus in a fixed (but arbitrary) closed 3-manifold. The key for understanding such surfaces is to understand the local structure in a ball and in particular the structure of an emb…

This is the second paper of a series of three on the regularity of higher codimension area minimizing integral currents. Here we perform the second main step in the analysis of the singularities, namely the construction of a center manifold, i.e. an approximate average of the sheets of an almost flat area minimizing cu…

New method constructs multi-monopoles on mapping tori.

The Chekanov theorem generalizes the classic Lyusternik-Shnirel'man and Morse theorems concerning critical points of a smooth function on a closed manifold. A Legendrian submanifold Λof space of 1-jets of the functions on a manifold M defines a multi-valued function whose graph is the projection of Λin J^0 M = M x R. T…

After appropriate normalizations an embedded disk whose second fundamental form has large norm contains a multi-valued graph, provided the L^P norm of the mean curvature is sufficiently small. This generalizes to non-minimal surfaces a well known result of Colding and Minicozzi.

For multi-valued functions---such as when the conditional distribution on targets given the inputs is multi-modal---standard regression approaches are not always desirable because they provide the conditional mean. Modal regression algorithms address this issue by instead finding the conditional mode(s). Most, however,…

A conformal metric $g$ with constant curvature one and finite conical singularities on a compact Riemann surface $Σ$ can be thought of as the pullback of the standard metric on the 2-sphere by a multi-valued locally univalent meromorphic function $f$ on $Σ\backslash \{{\rm singularities}\}$, called the {\it developing …

The paper introduces a statistical distance matrix for better feature representation and clustering.

In the 1980's, Almgren developed a theory of multi-valued Dirichlet energy minimizing functions on $n$ dimensional domains and used it, in an essential way, to bound the Hausdorff dimension of the singular sets of area minimizing rectifiable currents of dimension $n$ and codimension $\geq 2$. Recent work of the second …

This paper analyzes SHAP values using Fourier expansions for model interpretability.

Study on a weighted Suita conjecture for higher derivatives and their geometric properties.

New solutions to SU(n+1) Toda system found on compact Riemann surfaces with cone singularities.

In this paper, we consider multi-valued graphs with a prescribed real analytic interface that minimize the Dirichlet energy. Such objects arise as a linearized model of area minimizing currents with real analytic boundaries and our main result is that their singular set is discrete in 2 dimensions. This confirms (and p…

We analyze a notion of multiple valued sections of a vector bundle over an abstract smooth Riemannian manifold, which was suggested by W. Allard in the unpublished note "Some useful techniques for dealing with multiple valued functions" and generalizes Almgren's $Q$-valued functions. We study some relevant properties o…

We prove an Alexandrov type theorem for a quotient space of $\mathbb H^2\times \mathbb R$. More precisely we classify the compact embedded surfaces with constant mean curvature in the quotient of $\mathbb H^2\times \mathbb R$ by a subgroup of isometries generated by a parabolic translation along horocycles of $\mathbb …

Alternative proof and extension of curvature estimates for minimal immersions.

Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.

In this paper, we propose an active learning method for an inverse problem that aims to find an input that achieves a desired structured-output. The proposed method provides new acquisition functions for minimizing the error between the desired structured-output and the prediction of a Gaussian process model, by effect…

A clustering may be considered as fair on pre-specified sensitive attributes if the proportions of sensitive attribute groups in each cluster reflect that in the dataset. In this paper, we consider the task of fair clustering for scenarios involving multiple multi-valued or numeric sensitive attributes. We propose a fa…

Optimal LDP mechanisms reduce data unfairness in classification.

Proves regularity for stable varifolds near cones, expanding previous work.

The paper uses neural networks to estimate treatment effects even with many confounders.

Paper generalizes wrinkled embedding concept to jet spaces.

We consider a compact manifold of dimension greater than 2 and a differential form of degree one which is closed but non-exact. This form, viewed as a multi-valued function has a gradient vector field with respect to any Riemannian metric. After S. Novikov's work and a complement by J.-C. Sikorav, under some genericity…

Paper finds local normal forms for wavefronts in flat coordinates.

Proves regularity for stable varifolds near specific cones.

The paper explores how topology affects the solvability of first-order differential equations.

Many signals on Cartesian product graphs appear in the real world, such as digital images, sensor observation time series, and movie ratings on Netflix. These signals are "multi-dimensional" and have directional characteristics along each factor graph. However, the existing graph Fourier transform does not distinguish …

Paper introduces r-DEP classifier for binary classification tasks.

Paper tackles optimal policy learning with observational data in multi-action scenarios.

While it has become common to perform automated translations on natural language, performing translations between different representations of mathematical formulae has thus far not been possible. We implemented the first translator for mathematical formulae based on recursive neural networks. We chose recursive neural…

Proves branch set dimension for stationary varifolds with ε-regularity.

The study examines stationary integral varifolds near multiplicity 2 planes, proving regularity under specific conditions.

The article explores the mapping class group using unicellular maps and provides filtrations.

Constructs a moment map flow for isotropic maps on surfaces.