arXiv research

Constructs unique bases for CY varieties over valued fields.

Discrete Lagrange problems solved with Lie group constraints.

This work develops discrete Gaussian models for vector-valued data on triangular meshes.

New finite element method for complex forms in any dimension.

We propose an original model for inferring team strengths using a Markov Random Field, which can be used to generate historical estimates of the offensive and defensive strengths of a team over time. This model was designed to be applied to sports such as soccer or hockey, in which contest outcomes take value in a limi…

New method speeds up sampling of Markov random fields.

The paper introduces novel Gaussian process models for vector-valued signals on manifolds.

High order discretization schemes of SDEs by using free Lie algebra valued random variables are introduced by Kusuoka, Lyons-Victoir, Ninomiya-Victoir and Ninomiya-Ninomiya. These schemes are called KLNV methods. They involve solving the flows of vector fields associated with SDEs and it is usually done by numerical me…

Paper studies gradient fields from discrete Morse functions for watershed-cut computation.

New method for mesh denoising using TGV of normal vector field.

The study identifies all possible vector field structures on specific 2D shapes.

We give another definition of two-dimensional extended homotopy field theories (E-HFTs) with aspherical targets and classify them. When the target of E-HFT is chosen to be a $K(G,1)$-space, we classify E-HFTs taking values in the symmetric monoidal bicategory of algebras, bimodules, and bimodule maps by certain Frobeni…

In an earlier work we identified the types and numbers of static equilibrium points of solids arising from fine, equidistant $n$-discretrizations of smooth, convex surfaces. We showed that such discretizations carry equilibrium points on two scales: the local scale corresponds to the discretization, the global scale to…

The paper explores heat flow and constants on graphs, proving properties and proposing new concepts.

A learning algorithm optimizes beamforming for holographic transceivers in far-field communication.

A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra)…

We present a theory and applications of discrete exterior calculus on simplicial complexes of arbitrary finite dimension. This can be thought of as calculus on a discrete space. Our theory includes not only discrete differential forms but also discrete vector fields and the operators acting on these objects. This allow…

New methods merge discrete gradient fields from patches to correct errors.

The study examines discrete subgroups of PSL2 over non-archimedean fields.

In this technical note we give a purely geometric understanding of discrete torsion, as an analogue of orbifold Wilson lines for two-form tensor field potentials. In order to introduce discrete torsion in this context, we describe gerbes and the description of certain type II supergravity tensor field potentials as con…

Defines discrete differential geometry concepts in homotopy type theory.

Simplicial versions of topological abelian gauge theories are constructed which reproduce the continuum expressions for the partition function and Wilson expectation value of linked loops, expressible in terms of R-torsion and linking numbers respectively. The new feature which makes this possible is the introduction o…

Paper establishes NE existence and efficient algorithms for weakly monotone GMFGs.

Study shows financial value of weak information converges in discrete vs continuous markets.

A line field on a manifold is a smooth map which assigns a tangent line to all but a finite number of points of the manifold. As such, it can be seen as a generalization of vector fields. They model a number of geometric and physical properties, e.g. the principal curvature directions dynamics on surfaces or the stress…

Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.

A new discrete calculus for bundle-valued forms is proposed and validated.

We analyze quantum Yang-Mills theory on $\mathbb{R}^2$ using a novel discretization method based on an algebraic analogue of stochastic calculus. Such an analogue involves working with "Gaussian" free fields whose covariance matrix is indefinite rather than positive definite. Specifically, we work with Lie-algebra valu…

The paper develops algorithms and topological invariants for distinguishing dynamic systems.

This paper presents a geometric-variational approach to continuous and discrete {\it second-order} field theories following the methodology of \cite{MPS}. Staying entirely in the Lagrangian framework and letting $Y$ denote the configuration fiber bundle, we show that both the multisymplectic structure on $J^3Y$ as well…

We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on spaces $X$ in the following cases: 1) $G$ acts freely, properly discontinuously by isometries on a CAT(-1) space $X$, 2) $G$ is a lattice in a higher rank Lie group, acting on a symmetric space $X$, 3) $G$ is t…

We adapt a Markov Random Field learning algorithm for continuous variables.

Diffeomorphic Time Warping (DiffTW) is a novel method for time series classification that learns a diffeomorphic mapping between time series.

Given a triangulated region in the complex plane, a discrete vector field $Y$ assigns a vector $Y_i\in \mathbb{C}$ to every vertex. We call such a vector field holomorphic if it defines an infinitesimal deformation of the triangulation that preserves length cross ratios. We show that each holomorphic vector field can b…

A method for calculating multi-portfolio time consistent multivariate risk measures in discrete time is presented. Market models for $d$ assets with transaction costs or illiquidity and possible trading constraints are considered on a finite probability space. The set of capital requirements at each time and state is c…

Paper proves a discrete Schwarz-Pick lemma for generalized circle packings.

Study on 2-valued dynamics on complex plane, showing some dynamics can't be group actions.

Estimating causal models from observational data is a crucial task in data analysis. For continuous-valued data, Shimizu et al. have proposed a linear acyclic non-Gaussian model to understand the data generating process, and have shown that their model is identifiable when the number of data is sufficiently large. Howe…

Develops combinatorial theory of vector bundles on simplicial complexes.

The paper proves the existence of a complete holomorphic vector field on a complex manifold with a Kähler-Einstein metric.

A Neural Network (NN) based numerical method is formulated and implemented for solving Boundary Value Problems (BVPs) and numerical results are presented to validate this method by solving Laplace equation with Dirichlet boundary condition and Poisson's equation with mixed boundary conditions. The principal advantage o…

A new method splits surface flow discretizations into streamfunctions and harmonic fields.

In this study, we propose a new definition of multivariate conditional value-at-risk (MCVaR) as a set of vectors for discrete probability spaces. We explore the properties of the vector-valued MCVaR (VMCVaR) and show the advantages of VMCVaR over the existing definitions given for continuous random variables when adapt…

Summarizes quantum field theories with discrete symmetry, classifying representations and anomalies.

Study Transformer layers under cross-entropy training using mean field control.

Study optimizes portfolio liquidation strategies with complex market impacts.

We examine the effect of clamping variables for approximate inference in undirected graphical models with pairwise relationships and discrete variables. For any number of variable labels, we demonstrate that clamping and summing approximate sub-partition functions can lead only to a decrease in the partition function e…

RFM uses tangent vector fields to match data on manifolds, analyzing TV convergence for Euler discretization.