We construct a finitely presented group with non-quadratic Dehn function majorizable by a quadratic function on arbitrary long intervals.
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The paper classifies biharmonic quadratic maps between spheres, proving their energy density properties.
problem Classifying non-harmonic biharmonic quadratic forms between spheres.
method Proving non-harmonic biharmonic quadratic forms have constant energy density and classifying them.
result Non-harmonic biharmonic quadratic forms have constant energy density .
Recently, deep learning has achieved huge successes in many important applications. In our previous studies, we proposed quadratic/second-order neurons and deep quadratic neural networks. In a quadratic neuron, the inner product of a vector of data and the corresponding weights in a conventional neuron is replaced with…
Quadratic metric comparisonsmath.DG
The study examines how quadratic inequalities affect distances in length spaces.
problem Effects of quadratic inequalities on distances in length spaces.
method Analyzes quadratic inequalities on distances between points in quadruples.
result Quadratic inequalities significantly alter distances in length spaces.
We consider a proximal operator given by a quadratic function subject to bound constraints and give an optimization algorithm using the alternating direction method of multipliers (ADMM). The algorithm is particularly efficient to solve a collection of proximal operators that share the same quadratic form, or if the qu…
Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has r…
Quadratic Hessian equationmath.AP
Abstract: Survey on quadratic Hessian equations, their properties, and open problems.
problem Understanding quadratic Hessian equations and their solutions.
method Survey and review of existing research.
result Survey of entire solutions, viscosity solutions, and Hessian estimates.
This paper classifies quadratic form parameters over integers and computes their Witt groups.
problem Classifying quadratic form parameters over integers and computing their Witt groups.
method Study of quadratic forms and extended quadratic forms over the integers, defining and comparing different definitions of extended quadratic forms.
result Classification of all quadratic form parameters over the integers and computation of their Witt groups.
Inspired by complexity and diversity of biological neurons, our group proposed quadratic neurons by replacing the inner product in current artificial neurons with a quadratic operation on input data, thereby enhancing the capability of an individual neuron. Along this direction, we are motivated to evaluate the power o…
Classifies extended Abelian Chern-Simons theories using quadratic modules.
problem Classifying extended Abelian Chern-Simons theories.
method Using quadratic modules to classify theories.
result Finite quadratic modules classify extended Abelian Chern-Simons theories.
Finite intersection numbers between horizontal foliations of quadratic differentials.
problem Intersection properties of horizontal foliations in quadratic differentials.
method Joint continuity of intersection number in -norm.
result Intersection number is finite and jointly continuous.
Python package for projecting onto quadratic hypersurfaces.
problem Projections onto non-cylindrical central quadratic hypersurfaces.
method User-friendly Python package with documentation.
result Efficiently projects points onto quadratic hypersurfaces.
The Samuelson condition is not satisfied by tangent lines of quadratic curves.
problem Area condition for Lagrangian 2-web
method Show that the Samuelson condition is not satisfied
result The Samuelson condition is not satisfied by tangent lines of quadratic curves.
Study on pseudo-Hermitian quadratic nilpotent Lie algebras with methods and classifications.
problem Characterizing and classifying pseudo-Hermitian quadratic nilpotent Lie algebras.
method Construction methods and double extension by planes.
result Complete classification of nilpotent quadratic Lie algebras and pseudo-Hermitian metrics up to dimension 8.
Unique minimal surfaces near quadratic cones are identified.
problem Identifying minimal surfaces near quadratic cones.
method Analyzing minimal hypersurfaces inside the unit ball with perturbed boundary conditions.
result Minimal surfaces are uniquely determined by their boundary conditions.
Paper proves flat metrics from holomorphic quadratic differentials can be identified by length spectrum.
problem Identifying flat metrics from holomorphic quadratic differentials.
method Proved using length spectrum on closed oriented surfaces.
result Flat metrics from holomorphic quadratic differentials can be distinguished by their length spectrum.
Study on smoothness of special algebra types.
problem Differential smoothness of bi-quadratic algebras with PBW basis.
method Investigation of algebra properties.
result Results on differential smoothness.
Study automorphism groups of Inoue surfaces using quadratic number fields.
problem Understanding automorphism groups of Inoue surfaces.
method Construction and description of automorphism groups using quadratic number fields.
result Automorphism groups of Inoue surfaces described in terms of quadratic number fields.
A Finsler space is called Ricci-quadratic if its Ricci curvature is quadratic in . It is called a Berwald space if its Chern connection defines a linear connection directly on the underlying manifold . In this article, we prove that a homogeneous Randers space is Ricci-quadratic if and only if it is of…
Quadratic models explain neural network behavior during training.
problem Understanding neural network dynamics during training with large learning rates.
method Developed and tested Neural Quadratic Models.
result Neural Quadratic Models exhibit the 'catapult phase' similar to neural networks.
Proves transitivity of a specific class of quadratic polynomials.
problem Transitivity of pure Hurwitz classes of post-critically finite quadratic polynomials.
method Uses mapping classes of the sphere with finitely many marked points.
result Establishes transitivity for pure Hurwitz classes of post-critically finite quadratic polynomials.
We consider the problem of solving a large-scale Quadratically Constrained Quadratic Program. Such problems occur naturally in many scientific and web applications. Although there are efficient methods which tackle this problem, they are mostly not scalable. In this paper, we develop a method that transforms the quadra…
QENDy learns quadratic dynamics from nonlinear systems data.
problem Identifying governing equations of highly nonlinear dynamical systems.
method QENDy embeds nonlinear dynamics into a quadratic feature space, requiring trajectory data and preselected basis functions.
result QENDy accurately identifies quadratic dynamics and outperforms SINDy and deep learning methods.
We study quadratic Lie algebras over a field K of null characteristic which admit, at the same time, a symplectic structure. We see that if K is algebraically closed every such Lie algebra may be constructed as the T*-extension of a nilpotent algebra admitting an invertiblederivation and also as the double extension of…
Quadratic bounds found for graph dimensions.
problem Understanding dimensions of arc and disk graphs.
method Quadratic upper bounds calculation.
result Asymptotic dimensions of arc and disk graphs have been bounded.
Deep Quadratic Hedgingq-fin.CP
Deep learning solves high-dimensional quadratic hedging problems.
problem High-dimensional incomplete markets with mean-variance and local risk minimization.
method Deep learning-based BSDE solver for optimal hedging strategies.
result High-dimensional quadratic hedging is efficiently computed with deep learning.
A correspondence between different -type structures on a compact surface and quadratic (linear) forms on its homology is constructed. Addition of structures is defined and expressed in terms of these quadratic forms.
The study characterizes infinite Riemann surfaces and their foliations using quadratic differentials.
problem Characterizing infinite Riemann surfaces and their foliations using quadratic differentials.
method Extending Hubbard-Masur theorem to infinite surfaces and analyzing Jenkins-Strebel differentials.
result Density of Jenkins-Strebel differentials and extension of Kerckhoff's formula for Teichmüller metric.
Ricci solitons as critical points of quadratic curvature functionals
problem Einstein metrics and Ricci solitons as critical points of quadratic Riemannian functionals
method Study of Ricci solitons as critical points of a special quadratic curvature functional
result Ricci solitons are non-Einstein critical points of these functionals
Quadratic points of a surface in the projective 3-space are the points which can be exceptionally well approximated by a quadric. They are also singularities of a 3-web in the elliptic part and of a line field in the hyperbolic part of the surface. We show that generically the index of the 3-web at a quadratic point is…
The volumes of strata of Abelian or quadratic differentials play an important role in the study of dynamics on flat surfaces, related to dynamics in polygonal billiards. This article reviews all known ways to compute volumes in the quadratic case and provides explicit values of volumes of the strata of meromorphic quad…
We provide explicit solutions of certain forward-backward stochastic differential equations (FBSDEs) with quadratic growth. These particular FBSDEs are associated with quadratic term structure models of interest rates and characterize the zero-coupon bond price. The results of this paper are naturally related to simila…
Researchers found all special metrics in 4D for certain curvature functionals.
problem Identifying special metrics in 4D for quadratic curvature functionals.
method Determined all homogeneous metrics that are critical for quadratic curvature functionals.
result All homogeneous metrics in 4D for some quadratic curvature functionals have been identified.
New stability conditions identified from quadratic differentials on surfaces.
problem Identifying stability conditions from quadratic differentials.
method Comparison of exchange graphs from tilting hearts and flipping mixed angulations.
result Spaces of stability conditions identified with moduli spaces of quadratic differentials.
We derive caplet volatilities for quadratic models, providing an asymptotic approximation.
problem Calculating caplet volatilities for quadratic term-structure models.
method Asymptotic approximation for caplet volatilities under quadratic models.
result Asymptotic accuracy of the derived caplet volatilities.
Investigates smoothness of specific algebra structures.
problem Smoothness of bi-quadratic algebras on three generators.
method Analyzes differential smoothness with PBW basis.
result Characterizes conditions for smoothness.
To determine the Lie groups that admit a flat (eventually complete) left invariant semi-Riemannian metric is an open and difficult problem. The main aim of this paper is the study of the flatness of left invariant semi Riemannian metrics on quadratic Lie groups i.e. Lie groups endowed with a bi-invariant semi Riemannia…
New geometric Joyce structures on moduli spaces of quadratic differentials.
problem Constructing Joyce structures on moduli spaces of quadratic differentials.
method Isomonodromic deformations of second-order linear ODEs with rational potential.
result Construction of Joyce structures on moduli spaces of quadratic differentials.
Study identifies specific subvarieties in translation surfaces with quadratic field.
problem Characterizing invariant subvarieties in translation surfaces with quadratic field.
method Analyzing algebraically primitive subvarieties in strata of translation surfaces.
result Only specific subvarieties identified: decagon, Weierstrass curves, etc.
We prove a uniform estimate, valid for every closed Riemann surface of genus at least two, that bounds the distance of any quadratic differential to the finite dimensional space of holomorphic quadratic differentials in terms of its antiholomorphic derivative.
Novel link classification connects quadratic forms and knot theory.
problem Classifying isotopy classes of links in 3D space.
method Established a correspondence between quadratic forms and isotopy classes of links.
result Class numbers of quadratic number fields measure link distinguishability.
New method uses binary quadratic forms to classify Seifert surfaces in 4-ball.
problem Classifying non-isotopic Seifert surfaces in 4-ball.
method Composition of binary quadratic forms and number-theoretic approach.
result Established a new connection between Bhargava cube and Gauss composition.
This paper shows how to create quadratic differentials with any given singularities.
problem Creating quadratic differentials with prescribed singularities.
method Using the flat metric induced by the differentials, the authors classify and construct quadratic differentials with specific singularities.
result Every pattern of local invariants can be obtained by a quadratic differential on some Riemann surface, with exceptions in genera zero and one.
Abstract compares two norms in holomorphic quadratic differentials.
problem Comparing two norms in holomorphic quadratic differentials.
method Comparison between Avila-Gouëzel-Yoccoz norm and Teichmüller norm.
result Comparison of two norms in holomorphic quadratic differentials.
Quadratic Killing tensors on Lie groups are always decomposable.
problem Characterize Killing tensors on Lie groups.
method Analyzing the algebraic structure of Killing tensors on Lie groups.
result Quadratic Killing tensors on compact Lie groups are decomposable.
New rigidity results for critical metrics of a quadratic curvature functional.
problem Proving uniqueness of critical metrics for a specific curvature functional.
method Analyzing complete, possibly non-compact, critical metrics of the quadratic curvature functional.
result Critical metrics with finite energy are scalar flat (global minima) for dimensions n≥10.
Configurations of rigid collections of saddle connections are connected component invariants for strata of the moduli space of quadratic differentials. They have been classified for strata of Abelian differentials by Eskin, Masur and Zorich. Similar work for strata of quadratic differentials has been done in Masur and …
We investigate classification results for general quadratic functions on torsion abelian groups. Unlike the previously studied situations, general quadratic functions are allowed to be inhomogeneous or degenerate. We study the discriminant construction which assigns, to an integral lattice with a distinguished characte…