Study eigenfunctions of a singular quasi-Laplacian on a non-complete metric.
problem Eigenfunctions of a singular quasi-Laplacian on a non-complete metric.
method Analyzing eigenfunctions of the quasi-Laplacian and drifted Laplacian.
result Non-constant eigenfunctions of the quasi-Laplacian and drifted Laplacian are discontinuous at infinity.
3-manifolds transform into figure-eight knot complements through complex hyperbolic deformations.
problem Understanding transformations of 3-manifolds into figure-eight knot complements.
method Deforming representations of complex hyperbolic triangle groups.
result The quotient space is always the figure-eight knot complement.
Study shows translators can have non-removable singularities at infinity but eventually converge to unique planes.
problem Understanding singularities and convergence of translators at infinity.
method Global analysis of quasilinear soliton equations, sharp non-standard elliptic decay estimates, and potential theory.
result Finite entropy, finite genus translators converge to uniquely determined planes at infinity.
New algorithm tackles optimization problems with discontinuous gradients in finance and insurance.
problem Optimization problems with discontinuous stochastic gradients in finance and insurance.
method Langevin dynamics based algorithm e-THεO POULA. result Non-asymptotic error bounds and expected excess risk estimates for e-THεO POULA. 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.
Study bandit problems with BMO functions, achieving poly-log δ-regret.
problem Bandit problems with discontinuous, unbounded functions.
method Developed a toolset for BMO bandits and an algorithm achieving poly-log δ-regret. result Achieved poly-log δ-regret against an optimal arm after removing a δ-sized portion of the arm space. 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…
Compactifies geodesic flows on hyperbolic surfaces, revealing attractive circles at infinity.
problem Geodesic flows on non-compact hyperbolic surfaces without cusps.
method Constructs a geometrical compactification using one-dimensional distributions tangent to stable and unstable horocycles.
result Existence of attractive circles at infinity in the compactified flow.
New non-rigid discrete groups found in hyperbolic spaces.
problem Uniqueness of conformal or spherical CR structures on spheres.
method Nilpotent Sierpiński carpet and stretching to construct non-rigid groups.
result Discrete hyperbolic groups can have non-rigid deformations.
By using Thurston's bending construction we obtain a sequence of faithful discrete representations ρ_n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic space H^4. The algebraic limit of ρ_n contains a finitely generated subgroup F whos…
New groups discovered with unique properties in a specific space.
problem Finding new discrete subgroups with special properties in a mathematical space.
method Proved by showing groups play ping-pong on cones, related to crooked surfaces.
result Infinite family of discrete subgroups with remarkable properties in Sp4(R). 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.
This work models overnight rates with jumps and discontinuities, extending classical short-rate models.
problem Capturing the jump behavior and discontinuities in overnight rates for accurate modeling.
method Developed a term structure modeling framework based on overnight rates, accommodating stochastic discontinuities.
result Simple specifications can capture the jump behavior of overnight rates, and explicit valuation formulas are provided.
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.
Extended CIR process with jumps at fixed dates for modeling overnight rates.
problem Modeling overnight rates with jumps at predetermined dates.
method Formal definition and existence proof of a CIR process with stochastic discontinuities.
result Extended CIR process inherits affine property and non-negativity.
Bayesian emulator tackles models with discontinuities efficiently.
problem Uncertainty quantification of models with multiple discontinuities.
method TENSE framework, designed correlation structures, single emulator object.
result Efficient emulation of complex models with multiple discontinuities.
Moebius rigidity proven for negatively curved surfaces without cocompactness.
problem Proving rigidity for Moebius maps on negatively curved surfaces.
method Analyzing boundary homeomorphisms between surfaces with pinched negative curvature.
result Moebius homeomorphisms between boundaries of negatively curved surfaces extend to isometries.
Estimates discontinuous optimal transport maps between a discrete and continuous distribution.
problem Estimating discontinuous optimal transport maps between a discrete and continuous distribution.
method Entropic optimal transport estimator, computationally efficient.
result The estimator converges at the minimax-optimal rate n−1/2 in the semi-discrete setting. If Γ is a discrete subgroup of PSL(3,C), it is determined the equicontinuity region Eq(Γ) of the natural action of Γ on PC2. It is also proved that the action restricted to Eq(Γ) is discontinuous, and Eq(Γ) agrees with the discontinuity set in the sense of Kulkarni whenever the limit s…
In the framework of Lorentzian warped products, we study the Friedmann-Robertson-Walker cosmological model to investigate non-smooth curvatures associated with multiple discontinuities involved in the evolution of the universe. In particular we analyze non-smooth features of the spatially flat Friedmann-Robertson-Walke…
Hikami observed a discontinuity in a WRT invariant at roots of unity.
problem Discontinuity of a WRT invariant at roots of unity.
method Using Bailey's lemma and false theta functions.
result Generalized Hikami's observation about the discontinuity.
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.
A new definition of umbilic points at infinity for polynomial surfaces.
problem Defining umbilic points at infinity for homogeneous polynomial graphs.
method Proposed a stronger definition than Toponogov's, proving all are isolated and pairs, with geometric interpretation.
result All umbilic points at infinity are isolated and occur in pairs, being zeroes of the projective extension of the third fundamental form.
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.
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.
We consider properly discontinuous, isometric, convex cocompact actions of surface groups on a CAT(-1) space. We show that the limit set of such an action, equipped with the canonical visual metric, is a (weak) quasicircle in the sense of Falconer and Marsh. It follows that the visual metrics on such limit sets are cla…
The paper proves mapping class groups of closed surfaces are simply connected at infinity.
problem Understanding connectivity at infinity for mapping class groups of surfaces.
method Proved a general simply connected at infinity result for finitely presented groups.
result All mapping class groups of closed surfaces of genus ≥ 3 are simply connected at infinity.
We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m at least 3, we construct an infinite family of pairs of m-manifolds on whic…
Paper uses Carleman estimates to study harmonic functions on surfaces at infinity.
problem Unique continuation of harmonic functions on surfaces at infinity.
method Develops Carleman estimates for surfaces in Euclidean space at infinity.
result Obtains unique continuation property for harmonic functions.
Study on bounded solutions of parabolic and elliptic equations on noncompact Riemannian manifolds with conditions at infinity.
problem Existence and uniqueness of bounded solutions for equations with unbounded coefficients.
method Investigation of solutions satisfying prescribed conditions at infinity, considering large-time behavior.
result Established existence of solutions for parabolic and elliptic equations on noncompact Riemannian manifolds.
Convex optimization method identifies LTV models with smooth or discontinuous dynamics.
problem Identifying LTV dynamical models with smooth or discontinuous time evolution.
method Convex optimization to design cost functions promoting either continuous or discontinuous changes in model coefficients.
result Demonstrates identification of LTV models with either continuous or discontinuous dynamics.
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…
The paper examines mass aspects at future null infinity and limits of quasilocal mass.
problem Understanding mass aspects and limits of quasilocal mass at future null infinity.
method Review and extension of Bondi mass and mass loss formula in Bondi-Sachs coordinate system.
result New results about the limit of quasilocal mass of unit spheres at null infinity.
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 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.
Generalizes double backpropagation for neural networks in Hilbert space.
problem Lack of a general description of derivatives in neural network loss functions.
method Develops a Hilbert space framework to cover various special cases of double backpropagation.
result Optimized backpropagation rules for real-world scenarios, reducing calculations by up to 33%.
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…
The paper defines curvature at infinity for flat manifolds.
problem Defining curvature at the boundary of flat manifolds.
method Constructing coordinates at infinity for asymptotically flat ends.
result A Weyl tensor and renormalized volume defined at infinity.
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.
A new model for the stock market price analysis is proposed. It is suggested to look at price as an everywhere discontinuous function of time of bounded variation.
The paper studies the structure at infinity of shrinking Ricci solitons with bounded curvature.
problem Understanding the structure at infinity of shrinking Ricci solitons with bounded curvature.
method Analyzes the limit behavior of complete gradient shrinking Ricci solitons and applies results to four-dimensional cases.
result The end of a shrinking Ricci soliton is asymptotic to a round cylinder at infinity.
We study the geometry at infinity of expanding gradient Ricci solitons of dimension greater than two with finite asymptotic curvature ratio without curvature sign assumptions. We mainly prove that they have a cone structure at infinity.
Let F be R or C, d the dimension of F over R. Denote by P(F) either the affine plane A(F) or the hyperbolic plane H(F) over F. An arrangement L of k lines in P(F) (pairwise non-parallel in the hyperbolic case) has a link at infinity K(L) comprising k unknotted (d-1)-spheres in the (2d-1)-sphere, whose topology reflects…
Computes quasi-local mass at null infinity using Bondi-Sachs coordinates.
problem Global properties of quasi-local mass at null infinity.
method Evaluation of Wang-Yau quasi-local mass on unit spheres in Bondi-Sachs coordinates.
result Quasi-local mass is related to the news function in Bondi-Sachs coordinates.
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.
Study asymptotic behavior of Weingarten surfaces at infinity.
problem Understanding the behavior of Weingarten surfaces at infinity.
method Derive asymptotic expansion and solve Dirichlet problem.
result Established maximum principle and solved Dirichlet problem.
A new model for the stock market price analysis is proposed. It is suggested to look at price as an everywhere discontinuous function of time of bounded variation.