Paper proves linked simplices in odd dimensions and disjoint tuples in even dimensions.
problem Proving linked simplices and disjoint tuples in various dimensions.
method Algebraic proofs based on Radon theorem.
result Linear Conway--Gordon--Sachs and van Kampen--Flores theorems proved.
A new approach to Morse theory using folded ribbon trees.
problem Applying Morse theory on symmetric products of surfaces.
method Introducing an A-infinity category with objects as κ-tuples of Morse functions, and showing conditions for the endomorphism to be a Hecke algebra.
result The endomorphism of a specific type of κ-tuple of Morse functions on T*R^2 is the Hecke algebra associated to the symmetric group.
A g-tuple of disjoint, linearly independent circles in a Riemann surface of genus g determines a `Heegaard torus' in its g-fold symmetric product. Changing the circles by a handleslide produces a new torus. It is proved that, for symplectic forms with certain properties, these two tori are Hamiltonian-isotopic La…
A new theorem eliminates higher-multiplicity intersections in manifold topology.
problem Eliminating intersections in manifold topology, especially in higher dimensions.
method Proved and applied the r-fold Whitney trick for proper maps from disjoint unions of disks to d-dimensional balls. result A continuous map can be modified to avoid intersections in higher dimensions.
Shows uniqueness of irreducible generating tuples for Fuchsian groups.
problem Identifying irreducible generating tuples in Fuchsian groups.
method Variation of ideas from \cite{W2} to show uniqueness of almost orbifold covers with rigid generating tuples.
result Irreducible generating tuples are unique up to equivalence and are irreducible.
Unified framework for N-tuples learning improves weakly supervised tasks.
problem Reducing annotation burden in supervised learning.
method Empirical risk minimization framework integrating pointwise unlabeled data.
result Framework improves generalization across various N-tuples learning tasks.
Develops n-tuple principal bundles from Lie group actions.
problem No specific problem stated; focuses on concept development.
method Compatibility condition for Lie group action on groupoid.
result Concept of n-tuple principal bundles established.
As previously known, all 3-manifolds of genus two can be represented by edge-coloured graphs uniquely defined by 6-tuples of integers satisfying simple conditions. The present paper describes an ``elementary transformation'' on these 6-tuples which changes the associated graph but does not change the represented manifo…
Study Gromov-Hausdorff convergence of metric pairs and tuples.
problem Understanding convergence in metric spaces.
method Prove equivalence of definitions, embedding, completeness, and compactness theorems.
result Relative version of Fukaya's theorem and finiteness theorem for stratified spaces.
Paper establishes equivalence between algebraic and functorial QFTs.
problem Challenges in globally hyperbolic Lorentzian bordisms and their impact on field theories.
method Introduces a new concept of bordisms and pseudo-operads to address challenges, defining FQFTs as pseudo-multifunctors.
result Equivalence theorem between globally hyperbolic Lorentzian FQFTs and AQFTs.
The paper generalizes Nielsen equivalence to 2-orbifolds.
problem Proving Nielsen equivalence for closed 2-orbifold groups.
method Proving that generating tuples of orbifold fundamental groups are represented by almost orbifold coverings.
result Generalization of Louder's Theorem to closed 2-orbifolds.
InfoTuple efficiently selects larger tuple queries for ranking multiple objects, improving efficiency and consistency.
problem Efficiently selecting and ranking multiple objects for similarity learning.
method Adaptive selection method using mutual information maximization.
result InfoTuple outperforms state-of-the-art methods on synthetic and human response datasets.
Study Morse theory on loop spaces and Hecke algebras.
problem Morse theory applied to loop spaces and Hecke algebras.
method Defined a Morse-type A∞-algebra and showed equivalence to Heegaard Floer algebras. result Equivalence of based multiloop A∞-algebra to wrapped higher-dimensional Heegaard Floer algebras. The paper analyzes orbits of integer tuples using braid diagrams.
problem Determining orbits of integer tuples under braid diagram actions.
method Monoid action of braid diagrams on integer tuples.
result Orbits of integer tuples under up-down action of braid diagrams.
Computed A-polynomial of twisted Whitehead links.
problem Calculating the A-polynomial for a specific class of links.
method Direct computation of the A-polynomial using mathematical techniques.
result Determined canonical components and volume formula for twisted Whitehead links.
DSRGAN learns independent structure and rendering without tuple supervision.
problem Learning disentangled representation for natural image generation without tuple supervision.
method Introducing an auxiliary domain with a common underlying-structure space, and designing a parallel generative network with a common Progressive Rendering Architecture.
result DSRGAN significantly outperforms state-of-the-art methods in disentanglability.
BHLR predicts hyperlink weights from data vectors using symmetric similarity functions and Bregman divergence.
problem Predicting hyperlink weights from data vectors in a general framework.
method BHLR learns a symmetric similarity function to minimize Bregman-divergence between hyperlink weights and estimated similarities.
result BHLR is statistically consistent and computationally tractable, providing theoretical guarantees for various methods.
Develops slope detection for 3-manifolds with torus boundaries.
problem Determining slopes on the boundary of 3-manifolds with torus boundaries.
method Introduces order-detection and representation-detection of slopes, proving their equivalence.
result Shows how slopes' behavior changes with cabling, improving previous results.
This work extends GNNs to handle multiple graphs with non-commuting operators, proving transferability.
problem Handling multiple graphs with non-commuting operators in graph neural networks.
method Developed a mathematical theory for graph-tuple neural networks (GtNNs) with non-commuting non-expansive operators.
result Proved universal transferability of GtNNs, ensuring no non-transferable energy under convergence.
Optimal sample complexity for contrastive learning of distances.
problem Minimum labeled tuples needed for high accuracy in learning distances.
method Analyzes sample complexity in various distance settings, proving tight bounds.
result Almost optimal bound on sample complexity for learning ℓp distances. Researchers solve the realization of Jordan-Kronecker invariants in Lie algebras.
problem Identifying which Jordan-Kronecker invariants can be realized by Lie algebras.
method Analyzing the Kronecker and Jordan cases, proving impossibility for certain invariants, and describing realizability for others.
result Complete solution for Jordan and Kronecker cases, partial answers for others.
The paper provides an algorithm to create curves touching a smooth cubic at specific intersection points.
problem Creating curves that touch a smooth cubic at specific intersection points.
method Algorithm based on divisions and Zariski tuples to produce n-contact curves. result An algorithm to generate n-contact curves to a smooth cubic. We show that for every n≥2 there exists a torsion-free one-ended word-hyperbolic group G of rank n admitting generating n-tuples (a1,…,an) and (b1,…,bn) such that the (2n−1)-tuples $$(a_1,\ldots ,a_n, \underbrace{1,\ldots ,1}_{n-1 \text{times}})\hbox{ and }(b_1,\ldots, b_n, \underbrace{…
The article studies Hamiltonian flows on surface group representations induced by invariant multi-functions.
problem Hamiltonian flows on surface group representations induced by invariant multi-functions.
method Introducing subsurface deformation and proving Poisson commutativity of induced invariant multi-functions.
result Hamiltonian flows on character varieties are of subsurface deformation type and Poisson commute if supporting subsurfaces are disjoint.
Study meromorphic k-differentials on Riemann surfaces, focusing on their singularities.
problem Characterize the local invariants of meromorphic k-differentials on Riemann surfaces.
method Analyzing orders of zeroes and poles, and k-residues at poles, for different genera.
result For genus g ≥ 2, every expected tuple of k-residues appears as actual residues.
Let G be a group given by the presentation [<a_1,...,a_k,b_1,... b_k\,| a_i=u_i(\bar b), b_i=v_i(\bar a) \hbox{for} 1\le i\le k>,] where k≥2 and where the ui∈F(b1,...,bk) and wi∈F(a1,...,ak) are random words. Generically such a group is a small cancellation group and it is clear that $(a_1,...,…
Improved analysis for extreme multi-class CRL with better sample complexity.
problem Theoretical sample complexity of CRL in extreme multi-class settings is poorly understood.
method Improved U-Statistics estimator to capture class concentration, proving O(k) sample complexity. result Sample complexity is O(k) for extreme multi-class learning, independent of class distribution. The paper introduces Absolute Shapley Value to handle negative contributions in machine learning model training.
problem Negative marginal contributions in machine learning model training.
method Investigates three philosophies: Original Shapley Value, Zero Shapley Value, and Absolute Shapley Value.
result Absolute Shapley Value significantly outperforms other definitions in evaluating data importance.
Curves in Lagrange Grassmannians naturally appear when one studies intrinsically "the Jacobi equations for extremals", associated with control systems and geometric structures. In this way one reduces the problem of construction of the curvature-type invariants for these objects to the much more concrete problem of fin…
A new framework for playing and learning board games.
problem Tackling the tedious and repetitive aspects of coding for board game AI.
method Developed a generic TD(λ)-n-tuple agent for arbitrary board games. result TD(λ)-n-tuple outperforms other generic agents on various games. A Heegaard splitting of a closed, orientable three-manifold satisfies the disjoint curve property if the splitting surface contains an essential simple closed curve and each handlebody contains an essential disk disjoint from this curve [Thompson, 1999]. A splitting is full if it does not have the disjoint curve proper…
The study connects polygon areas and projective structures in 3D space.
problem Relating polygon areas and projective structures in 3D space.
method Investigates positive tuples of complete flags in R^3 and their associated polygons in RP^2.
result Establishes a relationship between Holmes-Thompson area and projective structures.
Approximates cycles in planar and bounded-genus graphs.
problem Finding many disjoint cycles in planar and bounded-genus graphs.
method Constant-factor approximation algorithms for vertex-disjoint and edge-disjoint cycles.
result First algorithms for vertex-disjoint paths in fully planar and bounded-genus instances.
New compression scheme improves mini-batch ML efficiency.
problem Improving efficiency of mini-batch stochastic gradient descent.
method Tuple-oriented compression tailored for mini-batch stochastic gradient descent.
result Substantial compression ratios and runtime reductions for mini-batch ML.
Paper constructs graphs with girth four and distinct properties.
problem Characterize Ricci-flat graphs with specific girth.
method Constructs and characterizes graphs with girth four, focusing on edge-disjoint and vertex-disjoint 4-cycles.
result Characterizes all Ricci-flat graphs of girth four with vertex-disjoint 4-cycles.
We show that all finite-dimensional resolvable generalized manifolds with the piecewise disjoint arc-disk property are codimension one manifold factors. We then show how the piecewise disjoint arc-disk property and other general position properties that detect codimension one manifold factors are related. We also note …
Let (G) be a connected compact non-abelian Lie-group and (T) a maximal torus of (G). A torus manifold with (G)-action is defined to be a smooth connected closed oriented manifold of dimension (2\dim T) with an almost effective action of (G) such that (M^T\neq \emptyset). We show that if there is a torus manifold (M) wi…
The paper explores isotopic triples of triangles in 3D space.
problem Determining if triples of triangles are combinatorially isotopic.
method Continuous motion of triangles with disjoint outlines, algorithmic checks, and elementary proofs.
result Different types of triples of disjoint triangles are not isotopic.
Δ-UQ uses anchoring to estimate uncertainty in models.
problem Estimating uncertainty in predictive models.
method Anchoring input into a tuple for uncertainty estimation.
result Δ-UQ outperforms baselines in various use-cases.
We present a new property, the Disjoint Path Concordances Property, of an ENR homology manifold X which precisely characterizes when X times R has the Disjoint Disks Property. As a consequence, X times R is a manifold if and only if X is resolvable and it possesses this Disjoint Path Concordances Property.
Simple proof shows uncountable Möbius bands are impossible in space.
problem Proving the impossibility of uncountably many disjoint Möbius bands in 3D space.
method Algebraic proof
result Proves uncountable Möbius bands are impossible in R3 and higher dimensions. Paper analyzes CRL generalization under non-i.i.d. settings, providing bounds for practical data reuse.
problem Limited theoretical understanding of CRL generalization under non-i.i.d. data conditions.
method Inspired by U-statistics, derives generalization bounds for CRL under non-i.i.d. settings.
result Required number of samples scales logarithmically with class covering number.
A new classification method using disjoint centroids and normalized distance.
problem Improving classification accuracy and feature selection.
method Nearest disjoint centroid classifier with normalized distance.
result Our method outperforms other classifiers in terms of misclassification rates and feature usage.
Classifies 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
problem Classifying 3-manifolds from simplified (2,0)-trisections of 4-manifolds.
method Classifies vertical 3-manifolds as preimages of arcs on the plane for simplified (2,0)-trisection maps.
result Each 6-tuple of vertical 3-manifolds determines the source 4-manifold uniquely up to orientation reversing diffeomorphisms.
Detects synchronized behavior in streaming data.
problem Tracking synchronized behavior in time-stamped tuples.
method AugSplicing algorithm for streaming dense block detection.
result Effective and robust in detecting anomalous behavior.
Extends geostatistical simulation method to handle multiple variables and large grids.
problem Scalability and handling of multiple variables in geostatistical simulation.
method Uses Sinkhorn optimal transport with sparse matcher and FFT-MA Gaussian backbone.
result MST-Direct reproduces joint distribution with zero histogram error and accurately preserves spatial correlation.
A mathematical isomorphism connects Floer homology to DAHA representations.
problem Connecting Floer homology to DAHA representations.
method Isomorphism between Floer homology and DAHA module structure.
result Establishes equivalence between DAHA polynomial representation and Floer homology module structure.
New method shows disjoint set-theoretic subsolutions remain so for longer.
problem Ensuring disjointness of set-theoretic subsolutions under mean curvature flow.
method Set-theoretic approach to mean curvature flow on Riemannian manifolds.
result Disjoint subsolutions remain disjoint for longer periods if one is compact.