New boundary treatment improves accuracy for complex PDEs.
problem Order reduction in high-order IMEX schemes for multidimensional PDEs.
method Novel boundary treatment algorithms for Cartesian meshes, treating implicit-explicit stages similarly to interior points.
result Recovery of designed order of convergence by numerical verification.
Let X be a CAT(0) space, and G a discrete cyclic group of isometries of X. We investigate the domain of discontinuity for the action of G on the boundary ∂X.
Motivated by the work of McCarthy and Papadopoulos for subgroups of mapping class groups, we construct domains of proper discontinuity in the compactified Outer space and in the projectivized space of geodesic currents for any "sufficiently large" subgroup of Out(FN) (that is, a subgroup containing a hyperbolic iwip…
Develops LF-PPL for non-differentiable models with automatic boundary checks.
problem Handling non-differentiable models in probabilistic programming.
method Introduces LF-PPL with automatic boundary checks and a formalism ensuring measure zero discontinuities.
result Demonstrates efficient inference for non-differentiable models using DHMC.
Cut-DeepONet handles discontinuities and sharp transitions in neural operators.
problem Neural operators struggle with discontinuities and sharp transitions in PDEs.
method Two-stage training framework that explicitly models discontinuities via a lifting strategy and input-dependent discontinuity prediction.
result Cut-DeepONet outperforms state-of-the-art methods on benchmark PDEs with low-resolution datasets.
Study proves rigid spectral properties of planets with metric discontinuities.
problem Establishing spectral rigidity for spherically symmetric planets with discontinuities.
method Novel trace formula applied to two wave types in spherically symmetric manifolds with boundary and interior interfaces.
result Spectral rigidity of spherically symmetric planets with discontinuities is proven.
The paper analyzes Variable Annuities with surrender charges, providing a pricing formula and optimal exercise boundary.
problem Analyzing Variable Annuities with surrender charges and early termination rights.
method Formulated as an optimal stopping problem with a discontinuous payoff, non-monotonic optimal stopping boundaries are proven continuous and regular.
result A rigorous pricing formula and optimal exercise boundary for surrender options are derived.
We show that the twisted signature invariants of boundary link concordance derived from unitary representations of the free group are actually ordinary link concordance invariants. We also show how the discontinuity locus of this signature function is determined by Seifert matrices of the link.
New model considers spontaneous curvature for lipid bilayer membranes, including discontinuities.
problem Modeling discontinuities at interfaces in lipid bilayer membranes.
method Introduced a family of energies for smooth surfaces and phase fields, derived a sharp interface limit.
result Theoretical result extends classical model by assigning bending energy to tangential discontinuities.
The paper analyzes optimal retirement strategies in a market with habit persistence and jump diffusion, finding discontinuous investment strategies.
problem Optimal retirement decision in a market with habit persistence and jump diffusion.
method Habit reduction method and duality approach to solve the dual problem using a C1 version of Itô's formula. result Discontinuous investment strategies are possible when the so-called ``de facto wealth'' exceeds a critical proportion of wage.
Study nearly geodesic immersions in symmetric spaces, leading to explicit fibrations.
problem Understanding geodesic immersions in higher rank symmetric spaces.
method Define and analyze nearly geodesic immersions with a bound on fundamental form.
result Explicit fibrations of domains of discontinuity for certain representations.
An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual…
RLGP model improves robustness and accuracy for discontinuous response surfaces.
problem Challenges in modeling abrupt jumps and discontinuities in response surfaces.
method Integrates adaptive nearest-neighbor selection with robustification mechanism.
result Consistently delivers high predictive accuracy and robustness in higher dimensions.
CAT(0) spaces can split into products if their boundary has certain properties.
problem Detecting product splittings in CAT(0) spaces.
method Analyzing the boundary properties of CAT(0) spaces and their group actions.
result Conditions for a CAT(0) space to contain a quasi-dense, closed convex subspace that splits as a product.
Proposes a new adaptive design strategy for discontinuous regression functions.
problem Designing input variables for discontinuous regression functions.
method Sequential adaptive design strategy using statistical properties.
result Effective design points selection for discontinuous regression functions.
The paper prices options using a novel finite element method.
problem Pricing European and American options under the Heston model.
method Discontinuous Galerkin finite element method (dGFEM) with interior penalty and Rannacher smoothing.
result Efficient and accurate pricing of options, demonstrated through comparisons and experiments.
Study optimal stopping for variable annuity contracts with discontinuous rewards.
problem Optimal timing to surrender a variable annuity contract with guaranteed minimum benefit.
method Analytical study of an optimal stopping problem with a discontinuous reward function, considering general fee and surrender charge functions.
result Characterization of the surrender region and its interrelation with fee and surrender charge functions.
We show here that the Nielsen core of the bumping set of the domain of discontinuity of a Kleinian group Γ is the boundary of the characteristic submanifold of the associated 3-manifold with boundary. Some examples of interesting characteristic submanifolds are given. We also give a construction of the characteristic…
Every oriented 4-manifold admits a folded symplectic structure, which in turn determines a homotopy class of compatible almost complex structures that are discontinuous across the folding hypersurface ("fold") in a controlled fashion. We define folded holomorphic maps, i.e. pseudo-holomorphic maps that are discontinuou…
We survey recent work on the dynamics of the outer automorphism group of a word hyperbolic group on spaces of (conjugacy classes of) representations ofthe group into a semi-simple Lie group G. All these results are motivated by the fact that the mapping class group of a closed surface acts properly discontinuously on t…
Study geodesic currents on surfaces, proving new properties of character varieties.
problem Characterizing discontinuity sets in Hitchin and maximal character varieties.
method Decompose geodesic currents into measured laminations or positive systole components.
result Bi-Lipschitz equivalence of intersection functions to length functions for currents with positive systole.
Model for corporate bond pricing with credit rating migration, solving a double free boundary problem.
problem Corporate bond pricing with credit rating migration risks.
method Established a pricing model as a double free boundary problem, proving existence, uniqueness, and regularity of the solution.
result Two free boundaries are shown to be smooth and converge to a traveling wave solution as time goes to infinity.
The Tits alternative applies to groups acting on specific CAT(0) spaces.
problem Understanding the structure of groups acting on CAT(0) spaces.
method Proving the Tits alternative for groups acting on visibility CAT(0) spaces with bounded packing property.
result Groups either almost nilpotent or contain a free nonabelian subgroup of rank 2.
Study Dirac operators on finite warped cylinders with gauge fields.
problem Characterize spectral flow on finite warped cylinders with gauge fields.
method Identify endpoint operators, derive determinant characterization, introduce regularized APS conditions.
result Regularized APS conditions admit a spectral-flow framework, matching zero-mode sets.
Estimates funding impact from an algorithmic relief rule, finding little effect on hospital activities.
problem Evaluating the impact of algorithmic policy decisions.
method Developed a treatment-effect estimator using algorithmic decisions as instruments.
result Funding from an algorithmic relief rule had little effect on COVID-19-related hospital activities.
Paper unifies three invariants for flat bundles over surfaces with boundary.
problem Measuring flat bundles over surfaces with boundary.
method Extending Atiyah-Patodi-Singer rho invariant to discontinuous class functions.
result Unified signature, Toledo, and rho invariants.
Proves nonemptyness of domains for specific group actions.
problem Nonemptyness of domains of proper discontinuity for Anosov groups of affine Lorentzian transformations.
method Proof of nonemptyness of domains of proper discontinuity.
result Proves nonemptyness of domains for Anosov groups of affine Lorentzian transformations.
Let M be a hyperbolizable, nontrivial compression body without toroidal boundary components. In this paper, we characterize which discrete and faithful representations of the fundamental group of M into PSL(2,C) are separable-stable. The set of separable-stable representations forms a domain of discontinuity for the ac…
We give a geometric interpretation of the maximal Satake compactification of symmetric spaces X=G/K of noncompact type, showing that it arises by attaching the horofunction boundary for a suitable G-invariant Finsler metric on X. As an application, we establish the existence of natural bordifications, as orbifold…
Let X be a proper CAT(0) space and let G be a cocompact group of isometries of X which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the contracting boundary. The contracting boundary imitates the Gromov boundary for $δ…
We construct compactifications of Riemannian locally symmetric spaces arising as quotients by Anosov representations. These compactifications are modeled on generalized Satake compactifications and, in certain cases, on maximal Satake compactifications. We deduce that these Riemannian locally symmetric spaces are topol…
The present paper is devoted to the joint motion of two immiscible incompressible liquids in porous media. The liquids have different densities and initially separated by a surface of strong discontinuity (free boundary). We discuss the results of numerical simulations for exact free boundary problems on the microscopi…
DGNet solves complex dynamical systems with neural networks and constraints.
problem Real-time accurate solutions for large-scale complex systems.
method Model-constrained discontinuous Galerkin Network (DGNet) for compressible Euler equations.
result DGNet achieves out-of-distribution generalization and improved stability.
We consider the deformation of a discontinuous group acting on the Euclidean space by affine transformations. A distinguished feature here is that even a `small' deformation of a discrete subgroup may destroy proper discontinuity of its action. In order to understand the local structure of the deformation space of disc…
Study proves existence of equilibrium in incomplete economies with discontinuous volatility.
problem Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
method Established existence of solution for Markovian quadratic BSDEs with discontinuous generators using unique continuation and backward uniqueness.
result Existence of incomplete Radner equilibrium with nondegenerate endogenous volatility.
New domains of discontinuity found for Anosov representations.
problem Understanding Anosov representations acting on homogeneous spaces.
method Constructing open domains of discontinuity for Anosov representations acting on specific homogeneous spaces.
result Describes the largest possible open domains of discontinuity for Zariski dense Anosov representations.
As early as 1972, Penrose - in a purely formal way - introduced a "discontinuous coordinate transformation", which relates a continuous representation of the metric of impulsive pp-waves to a discontinuous one. On the basis of the invertibility concept for generalized functions developed recently by the first author, w…
Proves the bending map is proper for hyperbolic 3-manifolds.
problem Properness of the bending map in hyperbolic 3-manifolds.
method Analyzes geometric properties and isotopy classes of homeomorphisms.
result Proving the bending map is proper for hyperbolic 3-manifolds.
A math surface has varying perimeter rules.
problem Finding a surface with a non-continuous isoperimetric profile.
method Constructed a 2D Riemannian manifold.
result Discovered a surface with a discontinuous isoperimetric profile.
Study the discontinuity of functions not embeddable in Euclidean space.
problem Understanding discontinuity of non-embeddable functions.
method Define a modulus of discontinuity and establish lower bounds.
result Quantified nonembeddability results and topological Tverberg theorem.
Bayesian nonparametric discontinuity design improves causal inference without randomization.
problem Causal inference in non-randomized settings with additional assumptions.
method Bayesian model comparison and Gaussian process regression.
result BNDD can detect discontinuities of any order and spectral features.
Characterizes discontinuous motions of limit sets in quasi-Fuchsian groups.
problem Understanding discontinuous motions of limit sets in quasi-Fuchsian groups.
method Characterization of discontinuous motions through Cannon-Thurston maps for algebraically convergent groups.
result Completely characterizes discontinuous motions of limit sets.
Study new symmetries in non-symmetric spaces and discontinuous groups.
problem Analyze symmetries in non-symmetric homogeneous spaces and discontinuous groups.
method Investigate discrete series, discontinuous groups, and analysis on pseudo-Riemannian spaces.
result New insights into symmetries of non-symmetric homogeneous spaces and discontinuous groups.
Improved method for estimating derivatives of discontinuous functions using stochastic algorithmic differentiation and regression.
problem High Monte-Carlo error in finite difference approximation of discontinuous functions.
method Combining stochastic algorithmic differentiation and regression to estimate derivative of expectations of discontinuous functions.
result Reduction in Monte-Carlo error through decoupling integration of Dirac delta and conditional expectation.
A new method for large-scale GP regression using seamless patching of local models.
problem Boundary discontinuity issue in local partitioned GP models.
method Partitioning input domain into local regions, applying different local GP models, and ensuring seamless patching with pseudo-observations.
result Mitigates boundary discontinuity problem, leading to improved accuracy and predictive uncertainty.
New subgroup properties proven based on boundary compactness.
problem Characterizing acylindrically hyperbolic subgroups.
method Analyzing the contracting boundary and action on space of triples.
result Subgroups with certain boundary properties are acylindrically hyperbolic.
Paper finds surface groups can deform in reductive symmetric spaces.
problem Finding deformations of discontinuous groups in reductive symmetric spaces.
method Analyzing Zariski dense surface subgroups and their deformations.
result Surface groups of high genus can deform in reductive symmetric spaces.
A new method for unsupervised disentanglement using axis-aligned cliffs.
problem Unsupervised disentanglement of latent factors under nonlinear maps.
method Encouraging axis-aligned discontinuities (cliffs) in the estimated density of factors.
result Cliff method outperforms baselines on disentanglement benchmarks.