Add expert knowledge to resolve ambiguities in ANN models.
problem Non-unique parameter fitting in material science.
method Augment a black-box ANN model with expert knowledge at two levels.
result Expert knowledge resolves ambiguities in parameter space.
We describe a method for constructing Teichmüller geodesics where the vertical measured foliation ν is minimal but is not uniquely ergodic and where we have a good understanding of the behavior of the Teichmüller geodesic. The construction depends on various parameters, and we show that one can adjust the parameters …
Paper shows non-uniqueness of solutions to a specific Minkowski problem.
problem Investigating non-uniqueness in the dual Lp-Minkowski problem for p<0<q. method Proved existence of solutions and constructed an example.
result Non-uniqueness of solutions to the dual Lp-Minkowski problem. A new principle minimizes residual and introduces momentum to improve PDE solution dynamics.
problem Ill-conditioning in Dirac-Frenkel residual minimization leads to non-unique parameter dynamics.
method Introduces a history variable (momentum) to select better-conditioned parameter velocities, preserving residual minimization while promoting smooth parameter evolutions.
result The approach leads to increased robustness in singular and near-singular PDE solution regimes.
Study Born-Infeld solitons and solve Björling problem for them.
problem Existence and non-uniqueness of solutions to the Björling problem for Born-Infeld solitons.
method Two approaches: treating as time-like minimal surfaces or using Barbashov-Chernikov representation.
result Solution to Björling problem may not be unique.
Non-uniqueness found in option valuation for certain α values.
problem Non-uniqueness in the value of call options for specific α values.
method Mathematical theory of degenerate parabolic equations and boundary conditions.
result Non-uniqueness explained by initial data outside Täcklind class and absence of boundary condition at infinity.
In this technical note, we adapt an idea of Gabai to construct non-uniquely ergodic, non-geometric, arational trees.
Unique 3-balls in S^4 can be non-isotopic.
problem Existence and uniqueness of spanning 3-balls in S^4.
method Introducing barbell diffeomorphisms, implantations, and twistings; 2-parameter calculus of embeddings; framed cobordism method.
result Non-uniqueness of spanning 3-balls in S^4.
Non-unique option pricing in Heston model analyzed mathematically.
problem Non-uniqueness of call option prices in the Heston model.
method Analysis of degenerate parabolic equations in the context of option pricing.
result Construction of a new example demonstrating the accuracy of a uniqueness theorem.
Wave maps can have multiple bubbling solutions at blow-up points.
problem Non-uniqueness of bubbling solutions in wave maps.
method Example construction of multiple bubbling solutions.
result First known example of non-uniqueness of bubbling for dispersive equations.
Study shows non-uniqueness and failure of compactness in constant curvature equations.
problem Non-uniqueness and failure of compactness in constant curvature equations.
method Warped product manifold construction and smooth counterexample creation.
result Compactness of solutions fails for dimensions ≥ 62.
New 1-parameter family of ovals identified in 4d Ricci flow classification.
problem Classifying κ-solutions in 4d Ricci flow. method Introducing conjectures and constructing new examples.
result Established canonical neighborhood theorem for 4d Ricci flow.
The study examines uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
problem Uniqueness and non-uniqueness of minimal surfaces in hyperbolic space.
method Analyzes criteria for uniqueness and constructs examples of non-uniqueness.
result Uniqueness of minimal surfaces is equivalent to uniqueness in a smaller class of stable minimal disks.
We construct Weil-Petersson (WP) geodesic rays with minimal filling non-uniquely ergodic ending lamination which are recurrent to a compact subset of the moduli space of Riemann surfaces. This construction shows that an analogue of the Masur's criterion for Teichmüller geodesics does not hold for WP geodesics.
Researchers construct translators asymptotic to self-shrinkers, proving non-uniqueness and fattening.
problem Constructing translators with prescribed ends and understanding their asymptotic behavior.
method Constructing families of complete translators polynomially asymptotic to self-shrinkers, proving non-uniqueness and fattening.
result Non-uniqueness and fattening of translators, with at least two translators asymptotic to each other at an exponential rate.
The paper solves problems related to curvature on a 3-sphere.
problem Prescribing positive cross curvature on the three-dimensional sphere.
method Existence results and a non-uniqueness example.
result Disproved a conjecture of Hamilton's about uniqueness.
Paper explores non-uniqueness and uniqueness class for wave equations on graphs.
problem Non-uniqueness of solutions to wave equations on infinite graphs.
method Analyticity of solutions in the uniqueness class, extension to a wide class of linear evolution equations.
result Sharp uniqueness class for solutions of wave equations on graphs.
Barren plateaus are not an average-case phenomenon, but a highly non-unique problem.
problem Avoiding barren plateaus in neural network training
method First-moment framework for initialization strategies
result Many families of inequivalent initialization strategies can avoid concentration
The paper assesses conditions for the uniqueness of k-means clustering.
problem Conditions for the uniqueness of k-means clustering.
method Analyzes the choice of k and provides necessary and sufficient conditions for uniqueness.
result Determines the asymptotic distribution of the within cluster sum of squares (WCSS) and provides a bootstrap test for uniqueness.
Study shows non-uniqueness of Brakke flow near flat singular points.
problem Exploring instability of minimal surfaces at flat singular points.
method Analyzes the behavior of stationary varifolds and their blow-ups.
result Proves existence of non-constant Brakke flow near flat singular points.
Percolation study in non-hyperbolic groups proves non-uniqueness phase.
problem Percolation in acylindrically hyperbolic groups.
method Analyzing Bernoulli bond percolation on Cayley graphs of groups.
result Non-uniqueness phase in percolation on Cayley graphs of acylindrically hyperbolic groups.
Curve shortening flow is not unique on certain metrics.
problem Non-uniqueness of curve shortening flow on specific metrics.
method Formulated a uniqueness conjecture and constructed a non-static solution.
result Curve shortening flow is not unique on a non-flat metric on the plane.
Nonnegative matrix factorization (NMF) is a popular dimension reduction technique that produces interpretable decomposition of the data into parts. However, this decompostion is not generally identifiable (even up to permutation and scaling). While other studies have provide criteria under which NMF is identifiable, we…
We consider the issue of solution uniqueness for portfolio optimization problem and its inverse for asset returns with a finite number of possible scenarios. The risk is assessed by deviation measures introduced by [Rockafellar et al., Mathematical Programming, Ser. B, 108 (2006), pp. 515-540] instead of variance as in…
New method for estimating value of optimal policies in uncertain scenarios.
problem Inference for optimal policies when they are non-unique or nearly deterministic.
method Semiparametric efficiency bound, uniformly weighted estimator, NSAVE method.
result Proposes NSAVE method for robust inference in uncertain optimal policies.
New non-canonical flows found via parabolic Allen-Cahn equations.
problem Existence of non-canonical mean curvature flows inside fattening regions.
method Construction of non-canonical flows as limits of parabolic ε-Allen-Cahn solutions.
result First examples of non-outermost, non-canonical integral Brakke motions.
We show, in this note, that on any symplectic supermanifold, even or odd, there exist an infinite dimensional affine space of symmetric connections, compatible to the symplectic form.
Study finds existence and non-uniqueness of cone spherical metrics on compact Riemann surfaces.
problem Existence and non-uniqueness of cone spherical metrics with prescribed singularities.
method Utilizing polystable extensions of line bundles, the study establishes three primary results concerning these metrics.
result Existence of multiple irreducible and reducible cone spherical metrics for certain effective divisors.
Paper analyzes adaptive ISTA with MAD for LASSO problem.
problem Finding solutions to LASSO problems without tuning λ. method Adaptive ISTA with median absolute deviation (MAD) for estimating noise level.
result Local linear convergence and global convergence of the algorithm.
Study on Seifert fibered spherical 3-orbifolds, determining their unique fibrations.
problem Analyzing the non-uniqueness of Seifert fibrations in spherical 3-orbifolds.
method Examined closed spherical Seifert three-orbifolds, determining the number and describing algorithms for equivalence.
result Determined the number of inequivalent fibrations for any closed spherical Seifert three-orbifold.
New findings show non-uniqueness in Ricci flow solutions for dimensions n≥5.
problem Non-uniqueness in Ricci flow solutions for dimensions n≥5.
method Exhibited a family of asymptotically conical gradient shrinking solitons with non-unique forward continuations.
result Non-uniqueness of Ricci flow solutions for dimensions n≥5.
Proves existence of multi-phase flows from arbitrary initial data.
problem Non-uniqueness issue in Brakke flows.
method Global existence proof for multi-phase mean curvature flow.
result Validates explicit identity for evolving grain volumes.
The paper explores uniqueness and non-uniqueness of spacetime extensions in general relativity.
problem Investigating the uniqueness and non-uniqueness of spacetime extensions in general relativity.
method Analyzes the extension of globally hyperbolic Lorentzian manifolds with a focus on low regularities.
result Local uniqueness of anchored extensions for certain regularity classes of extensions.
The paper characterizes conditions for convergence of RL methods with linear approximations.
problem Characterizing non-uniqueness issues for reinforcement learning algorithms with linear function approximation.
method Proves a condition on features that determines convergence or non-uniqueness of natural algorithms.
result Natural algorithms converge to the correct solution if and only if value functions in the approximation space satisfy a certain shape.
We apply various topological methods to distinguish connected components of moduli spaces of complete Riemannian metrics of nonnegative sectional curvature on open manifolds. The new geometric ingredient is that souls of nearby nonnegatively curved metrics are ambiently isotopic.
The conformal method has been effective for parametrizing solutions to the Einstein constraint equations on closed 3-manifolds. However, it is still not well-understood; for example, existence of solutions to the conformal equations for zero or negative Yamabe metrics is still unknown without the so-called ``CMC'' or `…
After giving a general introduction to the main known results on the anisotropic Calder{ó}n problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5, 6] for the anisotropic Calder{ó}n problem at fixed frequency, in dimension n $…
In this review paper we give a geometrical formulation of the field equations in the Lagrangian and Hamiltonian formalisms of classical field theories (of first order) in terms of multivector fields. This formulation enables us to discuss the existence and non-uniqueness of solutions, as well as their integrability.
Statistical framework improves LLM chatbot ranking.
problem Improving evaluation of LLM-based chatbots through pairwise comparisons.
method Factored tie model, covariance modeling, and parameter constraints.
result Substantial improvements in modeling pairwise comparison data.
Study well-posedness of SPDE on Riemannian manifolds with rough initial conditions.
problem Well-posedness of parabolic Anderson model on Riemannian manifolds with rough initial conditions.
method Construct intrinsic Gaussian noises, explore global geometry, use Feynman-Kac formula.
result Show well-posedness with non-positive curvature and conditions on α. The Volterra square-root process shows non-uniqueness of limiting distributions and regularity of its law.
problem Non-uniqueness of limiting distributions in the Volterra square-root process.
method Establishing existence of limiting distributions using integrability of the Volterra convolution kernel and exponential-affine transformation.
result The limiting distributions of the Volterra square-root process depend on the initial state and belong to weighted Besov spaces.
We prove that, given ∣H∣<1, a generic simple closed curve embedded in the asymptotic boundary of H3 (with respect to the supremum metric) bounds more than one complete surface embedded in H3 which has constant mean curvature H. We remark that this is not true for the space of simple closed $…
We consider a class of finite Markov moment problems with arbitrary number of positive and negative branches. We show criteria for the existence and uniqueness of solutions, and we characterize in detail the non-unique solution families. Moreover, we present a constructive algorithm to solve the moment problems numeric…
Study finds non-uniqueness in sphere metrics with constant fractional curvature.
problem Non-uniqueness of metrics with constant positive fractional curvature on spheres.
method Bifurcation techniques applied to non-local equations with critical non-linearity.
result Non-uniqueness results for complete metrics on Sn∖Sk. Unified framework for ranking-and-selection with multiple correct answers and non-answerable estimates
problem Fixed-precision ranking-and-selection in structured settings with non-unique answers and non-answerable estimates
method Unified framework based on answer-wise acceptance sets, restricted generalized likelihood ratio stopping, and answer-pitfall decomposition
result Unified recipe performs well across a broad range of pure-exploration problems
New method designs multilayer nanoparticles using AI.
problem Difficult to design multilayer nanoparticles by trial and error.
method Combines genetic algorithm and neural network for inverse design.
result Successfully designs multilayer nanoparticles efficiently.
We show that non-uniqueness of the Leray-Hopf solutions of the Navier--Stokes equation on the hyperbolic plane observed in arXiv:1006.2819 is a consequence of the Hodge decomposition. We show that this phenomenon does not occur on the hyperbolic spaces of higher dimension. We also describe the corresponding general Ham…
New Teichmüller geodesic rays found with unique foliations.
problem Capturing generic directions in Teichmüller space.
method Using Chaika-Masur-Wolf and Durham-Zalloum work.
result First sublinearly-Morse geodesic rays with minimal non-uniquely ergodic foliations.