Researchers link vertex algebras to non-Kähler solutions of the Hull-Strominger system.
problem Constructing representations of vertex algebras from non-Kähler solutions of the Hull-Strominger system.
method Embedding the N=2 superconformal vertex algebra in the chiral de Rham complex of a string Courant algebroid, with a condition on the Hermitian-Yang-Mills connection.
result Any solution of the Hull-Strominger system satisfying the Hermitian-Yang-Mills condition has an associated N=2 embedding.
We introduce the notion of a ``projective hull'' for subsets of complex projective varieties, parallel to the idea of the polynomial hull in affine varieties. With this concept, a generalization of J. Wermer's classical theorem on the hull of a curve in Cn is established in the projective setting. The projective hul…
We compute the algebraic hull of the Kontsevich-Zorich cocycle over any GL^+_2(R) invariant subvariety of the Hodge bundle, and derive from this finiteness results on such subvarieties.
We show how to construct the nonstandard hull of certain infinite-dimensional Lie algebras in order to generalize a theorem of Pestov on the enlargeability of Banach-Lie algebras. In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to ensure that the induced functio…
We represent the coordinate ring of algebraic hulls (which are generalizations of the Malcev completions of nilpotent groups for solvable groups) of solvmanifolds G/Γ by using Miller's exponential iterated integrals (which are extensions of Chen's iterated integrals) of invariant differential forms.
This paper defines maximal measurable cocycles for surface groups into Hermitian Lie groups and studies their algebraic hulls.
problem Understanding maximal measurable cocycles for surface groups into Hermitian Lie groups.
method Introducing the notion of maximal measurable cocycles, defining Toledo invariant, and studying the algebraic hulls.
result The algebraic hull of a maximal cocycle is reductive and its centralizer is compact.
The paper is motivated by the study of graded representations of Takiff algebras, cominuscule parabolics, and their generalizations. We study certain special subsets of the set of weights (and of their convex hull) of the generalized Verma modules (or GVM's) of a semisimple Lie algebra $\lie g$. In particular, we exten…
We prove that if K is a compact subset of an affine variety O = P^n - D (where D is a projective hypersuface), and if K is a compact subset of a closed analytic subvariety V \subset O, then the projective hull K^ of K has the property that K^ \cap O is contained in V. If V is smooth and 1-dimensional, then K^ \cap O …
We consider finite deformations of the Hull--Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a third-order Maurer--Cartan equation that controls the moduli. The resulting…
We present a nonstandard hull construction for locally uniform groups in a spirit similar to Luxembourg's construction of the nonstandard hull of a uniform space. Our nonstandard hull is a local group rather than a global group. We investigate how this construction varies as one changes the family of pseudometrics used…
We prove a generalization of Livsic's Theorem on the vanishing of the cohomology of certain types of dynamical systems. As a consequence, we strengthen a result due to Zimmer concerning algebraic hulls of Anosov actions of semisimple Lie groups. Combining this with Topological Superrigidity, we find a Holder geometric …
The paper characterizes sets with infinite hyperbolic convex hull volume.
problem Characterizing sets with infinite hyperbolic convex hull volume.
method Geometric conditions and self-similar sets.
result Characterizes continua and planar self-similar sets with infinite hyperbolic convex hull volume.
The n-th hull of a union of curves in R^3 is the set of points with the property: Any plane passing through the point intersects the curves at least 2n times. The hull number u(L) of a link L is defined as the minimum number of non-empty hulls a representative of L can have. We show that the hull numbers of torus links…
The main result of this paper is a characterization of the minimal surface hull of a compact set K in R3 by sequences of conformal minimal discs whose boundaries converge to K in the measure theoretic sense, and also by 2-dimensional minimal currents which are limits of Green currents supported by conf…
Designing and modifying complex hull forms for optimal vessel performances have been a major challenge for naval architects. In the present study, Principal Component Analysis (PCA) is introduced to compress the geometric representation of a group of existing vessels, and the resulting principal scores are manipulated …
Classifies 6D homogeneous spaces with holomorphically trivial canonical bundle.
problem Classifying 6D homogeneous spaces with specific geometric properties.
method Analyzing Lie algebras and homogeneous spaces, studying connections and instantons.
result Compact non-Kähler homogeneous spaces are unique in their solutions to the Hull-Strominger system.
Develops harmonic metrics for Hull-Strominger system stability.
problem Existence of solutions to the Hull-Strominger system with balanced class.
method Uses non-Hermitian Yang-Mills connections and holomorphic Courant algebroids, introduces harmonic metrics.
result Expected existence of a numerical stability condition for generic families of solutions.
Let X be a complex manifold and c a simple closed curve in X. We address the question: What conditions on c ensure the existence of a 1-dimensional complex subvariety V with boundary c in X. When X = C^n, an answer to this question involves the polynomial hull of gamma. When X = P^n, complex projective space, the proje…
Study analyzes correlation structure in two-factor Hull-White model for XVA calculations.
problem Capturing the correlation structure in two-factor Hull-White model for accurate XVA calculations.
method Combination of approximation formula and Monte-Carlo simulation to investigate correlation structure.
result Hull-White model effectively captures de-correlation of the yield curve under specific parameter conditions.
Study shows non-compact convex hulls in certain metric spaces.
problem Compactness of convex hulls in weakly non-positive curvature spaces.
method Introduced a conical geodesic bicombing and used it to construct a counterexample.
result Existence of a metric space with a finite subset whose convex hull is not compact.
Deep learning models generalize by extending decision boundaries outside the convex hull of training data.
problem Understanding how deep learning models generalize beyond their training data.
method Investigation of decision boundaries inside and outside the convex hull of training sets, using various neural network architectures and training regimes.
result Over-parameterization is necessary for deep learning models to extend decision boundaries outside the convex hull of their training data.
The paper transforms a convex hull into a concave surface around a point cloud.
problem Creating a concave surface that encloses all points in a point cloud.
method Iterative facet replacement and expansion of the convex hull.
result A method to evolve a convex hull into a concave surface that fits the point cloud.
The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is a ratio …
Study characterizes hulls and capacities on Riemannian manifolds, proving isoperimetric inequalities.
problem Characterizing hulls and capacities on Riemannian manifolds.
method Investigates strictly outward minimising hulls and uses p-capacities to recover their areas.
result Sharp isoperimetric inequality on complete noncompact manifolds with nonnegative Ricci curvature.
The convex hull of a set K in space consists of points which are, in a certain sense, "surrounded" by K. When K is a closed curve, we define its higher hulls, consisting of points which are "multiply surrounded" by the curve. Our main theorem shows that if a curve is knotted then it has a nonempty second hull. This pro…
New proofs given for space curves with totally positive torsion.
problem Description of convex hulls of space curves with totally positive torsion.
method New proofs of parametric representation, surface area, and volume formulas.
result Recovery of formulas for convex hull's surface area and volume.
Estimates convex hulls of smooth function images with error bounds.
problem Estimating the convex hull of the image of a smooth boundary set.
method Using submersion properties and sampling inputs, derive bounds on Hausdorff distance.
result New tighter and more general error bounds for geometric inference.
Study finds knots with ideal length need not have smallest volume.
problem Tackles the conjecture that ideal knot length equals smallest volume.
method Measures convex hull volume of knots during length annealing.
result Identifies knots with non-ideal global minimum volume.
In this paper we present short algebraic proofs of the Linear Conway--Gordon--Sachs and the Linear van Kampen--Flores theorems in the spirit of the Radon theorem on convex hulls. {\bf Theorem.} {\it Take any n+3 general position points in Rn. If n is odd, then there are two linked (n+1)/2-simplices wi…
Study Hull-Strominger system and Anomaly flow on specific solvmanifolds.
problem Characterize invariant solutions to Hull-Strominger system and investigate flow of invariant metrics.
method Characterization of invariant solutions using Gauduchon connections, investigation of Anomaly flow, and proof of flow immortality under certain conditions.
result Anomaly flow reduces to a special form and always converges to a Kähler metric when slope parameter is zero.
New solutions found for complex structures on specific manifolds.
problem Constructing smooth solutions to the Hull-Strominger system.
method Using fibrations over K3 orbisurfaces.
result Proved existence of solutions for certain manifolds.
Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.
problem Understanding properties of convex hulls of coadjoint orbits of compact groups.
method Introduce partial convex hulls and use them to define numerical invariants.
result Orbits with new invariants form rational convex polyhedral cones related to Littlewood-Richardson cones.
For a convex curve in an even-dimensional affine space we introduce a series of convex domains (called Young hulls), describe their structure and give a formulas fo the volume of the biggest of these domains. This paper is an attempt to generalize the classical isoperimetric inequality for the volume of the convex hull…
We study a hybrid tree-finite difference method which permits to obtain efficient and accurate European and American option prices in the Heston Hull-White and Heston Hull-White2d models. Moreover, as a by-product, we provide a new simulation scheme to be used for Monte Carlo evaluations. Numerical results show the rel…
Optimal algorithm finds if point is in convex hull of distributions.
problem Determining if a point is inside the convex hull of means of multiple distributions.
method Thompson-CHM algorithm with modular design of stopping and sampling rules.
result First asymptotically optimal algorithm for CHM problem in one dimension.
We obtain an upper bound for the volume of the convex hull of a simple closed Frenet curve with exactly four vertices, i.e., four points of vanishing torsion, and lying on the boundary of its convex hull. Moreover, we show that the upper bound is attained when the curve intersects every plane in at most four points, a …
New flow defined to solve Hull-Strominger system, with estimates and convergence results.
problem Constructing solutions to the Hull-Strominger system of equations.
method Introducing a natural extension of pluriclosed flow and using string algebroids and higher gauge theory.
result Proves global existence and convergence of the flow on special backgrounds.
Optimum in Convex Hulls (OCH) generalizes clinical trial results to broader populations.
problem Clinical trials exclude confounding but limit recruitment; observational data are more inclusive but suffer from confounding.
method OCH uses convex hulls of conditional expectations or densities to approximate the true treatment effect from both observational and trial data.
result OCH estimates the treatment effect with state-of-the-art accuracy in terms of both expectations and densities.
A wide range of fundamental machine learning tasks that are addressed by the maximum a posteriori estimation can be reduced to a general minimum conical hull problem. The best-known solution to tackle general minimum conical hull problems is the divide-and-conquer anchoring learning scheme (DCA), whose runtime complexi…
For actions with a dense orbit of a connected noncompact simple Lie group G, we obtain some global rigidity results when the actions preserve certain geometric structures. In particular, we prove that for a G-action to be equivalent to one on a space of the form (G×K\H)/Γ, it is necessary and suff…
Estimate collapsibility of causal effects in CPDAGs via strong d-convex hulls.
problem Estimate causal effects in CPDAGs.
method Use strong d-convex hulls to characterize minimal collapsible sets.
result Efficient algorithm for obtaining collapsible sets in DAGs and CPDAGs.
Automorphisms and subdivisions of Helly graphs are studied, leading to explicit models and rational translation lengths.
problem Understanding automorphisms and subdivisions of Helly graphs.
method Simple fine simplicial subdivisions and explicit simplicial models of the injective hull.
result Any automorphism of a Helly graph is either elliptic or hyperbolic, with rational translation lengths.
We use twistor theory to identify the harmonic hull of an arbitrary connected open subset U of R^{2m} for m at least 2. It is the natural domain of analytic continuation in C^{2m} for harmonic functions on U.
New obstruction found for Hull-Strominger system solutions.
problem Existence of solutions to the Hull-Strominger system.
method Construction of Hermitian-Einstein metric on string algebroid.
result Definition of Futaki invariants obstructing solutions.
Sketching algorithm finds closest point on convex hull efficiently.
problem Finding the closest point on a convex hull of large datasets.
method Sketching procedure to exploit data structure, gradient project method.
result Faster solution than standard optimization algorithms.
GraphHull models networks with clear multi-scale explanations of community structure.
problem Lack of self-explainable models in graph machine learning.
method Two-level convex hulls with global archetypes and local prototypes.
result GraphHull models networks with clear multi-scale explanations.
The main result is a direct proof of the implication (LVKFk,3)⇒(LT3k−1,3) below. Consider the following statements: (LVKF1,3) From any 11 points in R3 one can choose 3 pairwise disjoint triples whose convex hulls have a common point. (LVKFk,3) From any 6k+5 points in $ \m…
We construct new smooth solutions to the Hull-Strominger system, showing that the Fu-Yau solution on torus bundles over K3 surfaces can be generalized to torus bundles over K3 orbifolds. In particular, we prove that, for 13≤k≤22 and 14≤r≤22, the smooth manifolds S1×♯k(S2×S3)…