New concept (p,m)-width realized as minimal hypersurface volume.
problem Existence of minimal hypersurfaces in compact manifolds.
method Introduced (p,m)-width and proved its realization as minimal hypersurface volume. result The (p,m)-width can be realized as the volume of minimal hypersurfaces. Improved multi-group learning with group-realizable concepts.
problem Enhancing multi-group learning efficiency.
method Empirical risk minimization over group-realizable concepts.
result Improved sample complexity in group-realizable settings.
The paper constructs minimal realizations of signed Gauss paragraphs using graph theory.
problem Constructing minimal realizations of signed Gauss paragraphs on surfaces.
method Theory of embedded graphs on oriented and compact PL-surfaces, intersection pairing of immersed PL-normal curves.
result The genus of the ambient surface can be a function of the maximum number of Carter's circles.
A realization of a virtual link diagram is obtained by choosing over/under markings for each virtual crossing. Any realization can also be obtained from some representation of the virtual link. (A representation of a virtual link is a link diagram on an oriented 2-dimensional surface.) We prove that if a minimal genus …
We show that every area-minimizing hypercone and every oriented Lawlor cone in [Law91] can be realized as a tangent cone at a point of some homologically area-minimizing singular compact submanifold. In particular this generalizes the result of N. Smale [Sma99].
New superbridge index calculations for knots with odd edges.
problem Computing superbridge index of knots.
method Polygonal realizations with odd edges and linear programming.
result Exact superbridge index of many new knots, including 9- and 12-crossing knots.
The article improves and extends a heuristic for realizing surfaces with few vertices.
problem Realizing triangulated orientable and non-orientable surfaces with minimal vertices.
method Polyhedral embeddings and immersions of triangulated 2-manifolds with few vertices.
result Obtained numerous symmetric realizations of non-orientable surfaces with minimal vertices.
Optimal Euclidean structure minimizes energy in weighted toroidal graphs.
problem Finding the optimal Euclidean structure for weighted toroidal graphs.
method Minimizing Dirichlet energy over all possible Euclidean structures and realizations within a fixed homotopy class.
result The optimal Euclidean structure induces a weighted Delaunay decomposition.
Discuss Alan Schoen's I-WP minimal surface with geometric realizations.
problem Characterize and visualize Alan Schoen's I-WP minimal surface.
method Examine various geometric realizations of the I-WP surface and analyze the associate family.
result The associate family of the I-WP surface contains six congruent surfaces at specific angles.
The paper proves the existence of infinitely many minimal hypersurfaces in higher-dimensional manifolds.
problem Finding minimal hypersurfaces in higher-dimensional closed manifolds.
method Generic metrics and Baire sense arguments.
result Infinitely many singular minimal hypersurfaces are found in closed manifolds with optimal regularity.
Riemannian manifolds can be realized as leaf spaces of matchbox manifolds.
problem Realizing Riemannian manifolds as leaf spaces of matchbox manifolds.
method Graph coloring techniques to prove realization of manifolds as leaves.
result Any repetitive Riemannian manifold of bounded geometry can be realized as a leaf of a minimal Riemannian matchbox manifold without holonomy.
Computes minimal dilatation for Thurston maps on surfaces.
problem Finding the minimal dilatation of Thurston maps on surfaces.
method Explicit computation using spectral radius in a congruence subgroup of PSL2(Z).
result Explicitly computes minimal dilatation for Thurston maps.
New algorithm SELECT minimizes satisficing regret in bandits.
problem Minimizing regret in bandit optimization with satisficing arms.
method SELECT algorithm for satisficing regret minimization.
result SELECT achieves constant expected satisficing regret.
We give a combinatorial characterization of generic minimally rigid reflection frameworks. The main new idea is to study a pair of direction networks on the same graph such that one admits faithful realizations and the other has only collapsed realizations. In terms of infinitesimal rigidity, realizations of the former…
We define generalized currents associated with immersions of abstract oriented solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geo…
Framework synthesizes geological images minimizing patch distribution discrepancy.
problem Synthesizing realistic geological images from a single exemplar.
method Uses kernel discrepancies and generative neural networks for efficient synthesis.
result Synthesized images match visual patterns and spatial statistics of the exemplar.
Paper tackles MLR prediction error without assuming realizable models.
problem Prediction error in mixture of linear regressions without realizable assumptions.
method Developed algorithms for list-decoding MLR predictions and minimized empirical risk.
result Alternating minimization algorithm finds best fit lines in non-realizable settings.
The study proves properties of metrics and their conformal classes on specific manifolds.
problem Understanding metrics and their conformal classes on certain manifolds.
method Combining Simons' gap theorem with minimal isometric embeddings and coherent embeddings of standard Einstein metrics.
result The conformal classes of the product metrics and projective spaces realize the sigma invariant uniquely.
Boosts weak online learners to strong ones with sublinear regret.
problem Online learning agnostic setting without strong guarantees.
method Reduction to online convex optimization, boosting via marginally-better-than-trivial regret guarantees.
result First agnostic online boosting algorithm with sublinear regret.
This paper classifies knots with maximal exceptional surgeries on the minimally twisted 5-chain link.
problem Identifying knots with maximal exceptional surgeries on the minimally twisted 5-chain link.
method Enumerating all hyperbolic knots with maximal distance between exceptional surgeries.
result Examples of knots with maximal distance between exceptional surgeries are not obtained by filling the Berge manifold.
Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.
problem Finding optimal discrete harmonic maps between hyperbolic surfaces.
method Minimizing Dirichlet energy over all possible hyperbolic structures and realizations within a fixed homotopy class.
result At the optimal hyperbolic structure, the discrete harmonic map and edge weights are induced from a weighted Delaunay decomposition.
The paper establishes a minimal state-space realization for VAR models using Kalman's theorem.
problem Finding a minimal state-space realization for Vector Autoregressive Models (VARX).
method Introducing AR-state-space realization and applying Kalman's theorem to VAR models.
result Each VARX model has a minimal AR-state-space realization with specific matrix properties.
New method for distributional off-policy evaluation using Bellman residual minimization.
problem Learning return distribution from offline data generated by a different policy.
method Energy Bellman Residual Minimizer (EBRM) method.
result Established finite-sample error bound for EBRM estimator.
Abstract framework for two meromorphic forms on punctured surfaces.
problem Developing a framework for two meromorphic forms on punctured Riemann surfaces.
method Abstract framework with Teichmüller regularity, degeneration detection, and pushability.
result Existence of a surface carrying two meromorphic differentials realizing any prescribed restricted pair.
Study realizes symplectic algebras and homotopy types on manifolds.
problem Realizing symplectic algebras and homotopy types on manifolds.
method Addressing questions on realizability of symplectic algebras and rational homotopy types by closed symplectic manifolds.
result Realization of symplectic algebras and homotopy types in various dimensions.
New knots found with Seifert genus not matching minimal genus Seifert surfaces.
problem Discrepancy between Seifert genus and minimal genus Seifert surfaces.
method Constructed knots with specific genus and handle numbers to demonstrate the discrepancy.
result Found knots where Seifert genus is not realized by minimal genus Seifert surfaces.
We realize a given (monic) Alexander polynomial by a (fibered) hyperbolic arborescent knot and link of any number of components, and by infinitely many such links of at least 4 components. As a consequence, a Mahler measure minimizing polynomial, if it exists, is realized as the Alexander polynomial of a fibered hyperb…
New active learning framework for multiclass classification beyond realizability assumption.
problem Active learning in non-realizable settings with convex model classes.
method Surrogate risk minimization, epoch-based fitting, aggregation of models.
result Achieves label and sample complexity comparable to prior work in non-realizable settings.
We develop Teichmuller theoretical methods to construct new minimal surfaces in $\BE^3$ by adding handles and planar ends to existing minimal surfaces in $\BE^3$. We exhibit this method on an interesting class of minimal surfaces which are likely to be embedded, and have a low degree Gaußmap for their genus; the (Weier…
Given a one-dimensional homology class in a lens space, a question related to the Berge conjecture on lens space surgeries is to determine all knots realizing the minimal rational genus of all knots in this homology class. It is known that simple knots are rational genus minimizers. In this paper, we construct many non…
Paper finds shortest geodesic paths on hyperbolic surfaces.
problem Finding the shortest geodesic paths on hyperbolic surfaces.
method Analyzes genus g hyperbolic surfaces to find minimal length geodesics.
result Minimal geodesic length is realized by a specific polygon.
We study configurations of immersed curves in surfaces and surfaces in 3-manifolds. Among other results, we show that primitive curves have only finitely many configurations which minimize the number of double points. We give examples of minimal configurations not realized by geodesics in any hyperbolic metric.
Finite group symmetries on surfaces embedded in 3-torus are bounded.
problem Bounding the maximum symmetries of surfaces in 3-torus.
method Analyzing group actions and embeddings of surfaces into 3-torus.
result Upper bound of symmetries and surfaces achieving this bound.
This paper studies topological properties of line arrangements in complex projective plane.
problem Understanding the topology of line arrangements in complex projective plane.
method Established foundational results using homological methods to compute cohomology rings and studied homotopy types.
result Complement of a symplectic line arrangement has the homotopy type of a minimal CW complex.
Study finds bound on energy of minimal spheres on complex manifolds.
problem Finding bounds on energy of minimal spheres on complex manifolds.
method Proving existence of harmonic spheres with Morse index bound one.
result Sum of energies of minimal spheres realizes a geometric invariant width.
Characterizes concept classes for optimistic online learning.
problem Understanding minimal assumptions for online learnability.
method Investigates two questions about concept classes' learnability.
result Characterizes all concept classes for optimistically universal online learnability.
We study manifolds with split-complex structure and apply some general results to the study of Lorentz surfaces. In particular, we apply our results to timelike minimal immersions. The conformal realization of these surfaces is obtained using a representation based on loop groups. The classical Weierstrass representati…
To any generic curve in an oriented surface there corresponds an oriented chord diagram, and any oriented chord diagram may be realized by a curve in some oriented surface. The genus of an oriented chord diagram is the minimal genus of an oriented surface in which it may be realized. Let g_n denote the expected genus o…
The study constructs a hyperbolic orbifold to show how certain Salem numbers can be realized geometrically.
problem Constructing a geometric model for Salem numbers of degree 4.
method Constructing an arithmetic hyperbolic 6-orbifold and proving its properties.
result Any square-rootable Salem number of degree at most 4 can be realized as the exponential of a closed geodesic length in the constructed orbifold.
We define generalized currents associated with immersions of abstract solenoids with a transversal measure. We realize geometrically the full real homology of a compact manifold with these generalized currents, and more precisely with immersions of minimal uniquely ergodic solenoids. This makes precise and geometric De…
The paper introduces a dynamic MVP model using high-frequency financial data.
problem Capturing the dynamics of minimum variance portfolio weights in financial markets.
method Imposes autoregressive structure on MVP processes and uses CLIME and LASSO for estimation.
result Proposes DR-MVP model with established asymptotic properties.
The paper studies families of curves on surfaces that realize all types of pants decompositions.
problem Finding the minimal size of families of curves on surfaces that realize all types of pants decompositions.
method Investigates exponential and superlinear bounds for surfaces without punctures, and provides bounds for surfaces with punctures.
result Provides bounds for the minimal size of families of curves on surfaces with and without punctures.
The paper generalizes a theorem about metrics realizing maxima of the first non-zero Laplace eigenvalue.
problem Maxima of the first non-zero Laplace eigenvalue and their induced metrics.
method Generalization of a theorem to metrics with conical singularities.
result Properties of metrics induced from S2 differ significantly from those induced from Sm with m>2. We prove that the Whitehead link complement and the (-2, 3, 8) pretzel link complement are the minimal volume orientable hyperbolic 3-manifolds with two cusps, with volume 3.66... = 4 x Catalan's constant. We use topological arguments to establish the existence of an essential surface which provides a lower bound on vo…
Construct locally minimizing (1,2)-clusters with prescribed asymptotic geometry.
problem Minimizing clusters with prescribed asymptotic geometry.
method Develop a refined construction using the Hardt-Simon foliation.
result Produce a countably infinite family of distinct locally minimizing clusters asymptotic to a singular area-minimizing hypercone.
Infinite G-invariant minimal hypersurfaces found in Riemannian manifolds.
problem Finding minimal hypersurfaces in manifolds with group actions.
method New algorithm using multi-stage maximal cuttings.
result Each G-homology class admits infinitely many distinct realizations by embedded minimal G-hypersurfaces. We collect some examples showing that some Vassiliev invariants are not obtainable from the HOMFLY and Kauffman polynomials in the real sense, namely, that they distinguish knots not distinguishable by the HOMFLY and/or Kauffman polynomial.
Study topological constraints for minimal hyperbolic surface laminations.
problem Understanding which topological surfaces can be leaves in minimal hyperbolic surface laminations.
method Analyzes topological obstructions and embeddings of Cantor tree-like surfaces.
result All possible topological types of leaves can be simultaneously embedded in a lamination.