arXiv research

Vanishing result for cohomology leads to extension theorem for pluriharmonic functions.

We show that there is no analog of Kirszbraun's extension theorem for Almgren's multiple valued functions.

Kan extensions help in data science extrapolation and learning.

The paper solves conditions for extending circle-valued Morse functions.

Extends functions on symmetric spaces to analytic functions.

This research extends quasiplurisubharmonic functions on compact Kähler manifolds.

Paper proposes a loss extension for neural networks to improve OSR performance.

Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …

We prove that functions defined on a lattice in a finite dimensional torus with bounded finite differences can be smoothly extended to the whole torus, and relate the bounds on the extension's derivatives with bounds on the original function's finite differences.

New method for probabilistic modeling of integer submodular functions.

The paper explores uniqueness and non-uniqueness of spacetime extensions in general relativity.

Submodular extensions of an energy function can be used to efficiently compute approximate marginals via variational inference. The accuracy of the marginals depends crucially on the quality of the submodular extension. To identify the best possible extension, we show an equivalence between the submodular extensions of…

New risk measure extensions preserve key properties.

In this paper we study a sharp Hardy-Littlewood-Sobolev (HLS) type inequality with Riesz potential on bounded smooth domains. We obtain the inequality for a general bounded domain $Ω$ and show that if the extension constant for $Ω$ is strictly larger than the extension constant for the unit ball $B_1$ then extremal fun…

Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. In this form, t…

Paper extends Ohsawa-Takegoshi theorem to more general domains, proving removable singularities for plurisubharmonic functions.

Extends geometric decompositions to arbitrary meshes and forms.

One-parameter smooth families of circles in the complex plane with the following property are described: a function is polyanalytic if and only if it has meromorphic extension inside any circle from the family, with the only singularity-a pole at the center.

Introduces trace operator for quasi-plurisubharmonic functions on Kähler manifolds.

Local well-posedness proved for Bartnik static extension near Schwarzschild spheres.

Maximizes capacity of extensions with fixed boundary data.

Framework extends neural operators to handle functions outside training set.

We consider a global, nonlinear version of the Whitney extension problem for manifold-valued smooth functions on closed domains $C$, with non-smooth boundary, in possibly non-compact manifolds. Assuming $C$ is a submanifold with corners, or is compact and locally convex with rough boundary, we prove that the restrictio…

Study proves finiteness for distance functions on curved surfaces with controlled curvature.

We define the geometric complex associated to a Morse-Bott-Smale vector field, cf. [Austin-Braam, 1995], and its associated spectral sequence. We prove an extension of the Bismut-Zhang theorem to Morse-Bott-Smale functions. The proof is based on the Bismut-Zhang theorem for Morse-Smale functions, see [Bismut-Zhang, 199…

Improved bounds on acylindricity for right-angled Artin groups.

This is an extensive (published) survey on CR geometry, whose major themes are: formal analytic reflection principle; generic properties of Systems of (CR) vector fields; pairs of foliations and conjugate reflection identities; Sussmann's orbit theorem; local and global aspects of holomorphic extension of CR functions;…

The paper proves a function extension on Kähler manifolds.

Extends curve functions to geodesic currents with a simple criterion.

We consider the problem of extending functions φ:\to S^n to functions u:B^{n+1}\to S^n for n=2,3. We assume φto belong to the critical space W^{1,n} and we construct a W^{1,(n+1,\infty)}-controlled extension u. The Lorentz-Sobolev space W^{1,(n+1,\infty)} is optimal for such controlled extension. Then we use such resul…

The equivalence of principal bundles with transitive Lie groupoids due to Ehresmann is a well known result. A remarkable generalisation of this equivalence, due to Mackenzie, is the equivalence of principal bundle extensions with those transitive Lie groupoids over the total space of a principal bundle, which also admi…

Real analytic functions can be extended on manifolds with normal crossings.

In this note, we study the dynamics and associated zeta functions of conformally compact manifolds with variable negative sectional curvatures. We begin with a discussion of a larger class of manifolds known as convex co-compact manifolds with variable negative curvature. Applying results from dynamics on these spaces,…

Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-Itô, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes whe…

XGBoost is often presented as the algorithm that wins every ML competition. Surprisingly, this is true even though predictions are piecewise constant. This might be justified in high dimensional input spaces, but when the number of features is low, a piecewise linear model is likely to perform better. XGBoost was exten…

Continuing the previous work, we propose a further extension of the structure equation for a truncated CMC hierarchy by the non-commuting, truncated Virasoro algebra of non-local symmetries. Via a canonical dressing transformation, we first define a wave function for the CMC hierarchy. This leads to a pair of additiona…

Extends Kodaira Spencer to higher-dimensional Calabi-Yau manifolds.

We construct a group (an HNN extension of a free group) with polynomial isoperimetric function, linear isodiametric function and non-simply connected asymptotic cones.

Building on an idea laid out by Martelli--Sparks--Yau, we use the Duistermaat-Heckman localization formula and an extension of it to give rational and explicit expressions of the volume, the total transversal scalar curvature and the Einstein--Hilbert functional, seen as functionals on the Sasaki cone (Reeb cone). Stud…

Extends Lipschitz functions while preserving local constants.

simpcomp is an extension (a so called package) to GAP, the well known system for computational discrete algebra. The package enables the user to compute numerous properties of (abstract) simplicial complexes, provides functions to construct new complexes from existing ones and an extensive library of triangulations of …

We extend rectified flow to infinite-dimensional Hilbert space.

Employing Morse theory for the global control of monodromy and the method of analytic discs for local extension, we establish a version of the global Hartogs extension theorem in a singular setting: for every domain D of an (n-1)-complete normal complex space X of pure dimension n >= 2 and for every compact set K in D …

We extend neural networks with fractional and mixed activation functions for better function approximation.

In this paper, we obtain two extension theorems for cohomology classes and holomorphic sections defined on analytic subvarieties, which are defined as the supports of the quotient sheaves of multiplier ideal sheaves of quasi-plurisubharmonic functions with arbitrary singularities. The first result gives a positive answ…

Study of Kähler-Ricci flow on toric Fano varieties with preserved symplectic condition.

Krein's formula for conic Laplacians on compact Riemann surfaces

A novel online framework for analyzing multidimensional functional data.