Ribbon cobordism forms a partial order in 3-manifolds.
problem Understanding partial orders in 3-manifolds.
method Utilizing recent methods from Ian Agol's work on knot concordance.
result Ribbon rational homology cobordism forms a partial order.
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Ribbon cobordisms form a partial order on 3-manifolds.
problem Understanding the partial order structure of 3-manifolds.
method Building on Agol's proof for knots, we extend the partial order to 3-manifolds.
result Ribbon rational homology cobordism forms a partial order.
Prove strong ribbon concordance induces a partial order on links, certify minimality for a handful of knots, and find minimal ribbon minimal knots.
problem Prove strong ribbon concordance induces a partial order on links.
method Use results from knot Floer homology to certify minimality under the ribbon partial order.
result Certify minimality for a handful of knots and find minimal ribbon minimal knots.
Rational homology ribbon cobordism defines a partial order on 3-manifolds.
problem Classifying 3-manifolds based on their homology
method Proving a partial order on homeomorphism classes of 3-manifolds
result Closed, connected, oriented 3-manifolds are partially ordered by rational homology ribbon cobordism.
Differentiable relaxation for inferring partial orders from noisy linear data.
problem Inference of partial orders from linear data with noisy observations.
method Introducing a differentiable relaxation to model noisy linear extensions, replacing discontinuous precedence and feasibility with smooth surrogates.
result Smooth posterior that preserves partial-order semantics, supports gradient-based inference, and converges to hard likelihood.
The paper uses belief propagation to analyze rankings and partial orders from partial information.
problem Analyzing rankings and partial orders from incomplete data.
method Continuous spin system and belief propagation algorithm.
result Computes marginal distribution and approximates number of linear extensions.
Paper proposes methods to learn DAGs from partial orderings.
problem Learning DAGs from partial orderings is challenging.
method General estimation framework and efficient algorithms for low- and high-dimensional problems.
result Efficient estimation of DAGs from partial orderings is possible.
Algorithm improves reinforcement learning in MDPs with partial order policies.
problem Improving reinforcement learning in MDPs with partial order policies.
method Epoch-based reinforcement learning algorithm leveraging a partial order over policy class.
result Achieves an regret bound, independent of state and action space sizes.
In this paper, we introduce a partial order on neighborhood equivalence classes of maximally spread essential multibranched surfaces embedded in a 3-manifold. We show that if a maximally spread essential multibranched surface is atoroidal and acylindrical, then its equivalence class is minimal with respect to the parti…
It is well known that a countable group admits a left-invariant total order if and only if it acts faithfully on R by orientation preserving homeomorphisms. Such group actions are special cases of group actions on simply connected 1-manifolds, or equivalently, actions on oriented order trees. We characterize a class of…
Hierarchical Partial-Order Models for Ranking
problem Rank aggregation combining ordered lists
method Hierarchical partial-order models
result Bayesian inference for latent poset hierarchy
New statistical models for predicting ranked preferences from partial orders.
problem Statistical models overlook information in list length.
method Composite and augmented ranking models for joint modeling of partial orders and list lengths.
result Augmented ranking models best predict both length and preferences.
We present a new Markov chain Monte Carlo method for estimating posterior probabilities of structural features in Bayesian networks. The method draws samples from the posterior distribution of partial orders on the nodes; for each sampled partial order, the conditional probabilities of interest are computed exactly. We…
A novel transformer model improves classification of partially ordered sequences.
problem Classification of partially ordered sequences with uncertainty in timestamps.
method Developed a transformer-based model for partially ordered sequences, benchmarked against set models.
result Transformer-based model outperforms set models on three datasets.
New depth function for partial orders helps compare machine learning algorithms.
problem Comparing machine learning algorithms using non-standard data types.
method Adapted simplicial depth to partial orders, using ufg depth for comparison.
result Demonstrates promising variety of analysis approaches based on ufg methods.
As part of his celebrated Complex Frobenius Theorem, Nirenberg showed that given a smooth elliptic structure (on a smooth manifold), the manifold is locally diffeomorphic to an open subset of (for some and ) in such a way that the structure is locally the span of $\frac{\partial…
Sharp Hölder regularity found for complex Frobenius theorem coordinates.
problem Finding optimal Hölder-Zygmund regularity for complex Frobenius theorem coordinates.
method Analyzing necessary and sufficient conditions for coordinate charts achieving the theorem's structure.
result The optimal Hölder-Zygmund regularity for coordinate charts is shown to be .
Here we give a concrete description of the cork automorphism of the infinite order loose-cork , defined in \cite{a2}. It is obtained by concatenating the defining ribbon disk of in by an infinite order isotopy of the boundary knot.
Method preserves order in hierarchical clustering of ordered data.
problem Order preserving hierarchical clustering of directed acyclic graphs.
method Combination of classical hierarchical clustering and ultrametric fitting.
result Optimal clustering preserves both cluster quality and order.
We say that a link is an s-major of a link if any diagram of can be transformed into a diagram of by changing some crossings and smoothing some crossings. This relation is a partial ordering on the set of all prime alternating links. We determine this partial order for all prime alternating knot…
Framework benchmarks optimizers on multiple criteria.
problem Benchmarking optimizers across diverse test functions.
method Union-free generic depth function for partial orders/rankings.
result Identifies central and outlying rankings of optimizers.
We give a survey of some recent papers by the authors and Masaaki Wada relating the twisted Alexander polynomial with a partial order on the set of prime knots. We also give examples and pose open problems.
As in [5], we study holomorphic maps of positive degree between compact complex manifolds, and prove that any holomorphic map of degree one from a compact complex manifold to itself is biholomorphic. This conclusion confirms that under a mild restriction the holomorphic Gromov relation ">_" is indeed a partial order.
Knots can be ordered by ribbon concordance, solving a long-standing question.
problem Ordering knots by ribbon concordance.
method Representation varieties of knot groups to and relations induced by ribbon concordance.
result Ribbon concordance forms a partial ordering on the set of knots.
The natural partial ordering of the orbit types of the action of the group of local gauge transformations on the space of connections in space-time dimension d<=4 is investigated. For that purpose, a description of orbit types in terms of cohomology elements of space-time, derived earlier, is used. It is shown that, on…
A complete solution to the multiplier version of the inverse problem of the calculus of variations is given for a class of hyperbolic systems of second-order partial differential equations in two independent variables. The necessary and sufficient algebraic and differential conditions for the existence of a variational…
Inter-order relations between moments of a Student distribution, with an application to -quantilesmath.ST
The paper derives formulas for moments of a Student t distribution and applies them to quantify -quantiles.
problem Understanding the moments and quantiles of a Student t distribution.
method Developed formulas for partial and complete moments, and derived relationships between -quantiles.
result For a Student t distribution, the -quantile and -quantile coincide at any confidence level.
A partial order on the set of prime knots can be defined by the existence of an epimorphism between knot groups. We prove that all the prime knots with up to crossings are minimal. We also show that each fibered knot with the irreducible Alexander polynomial is minimal.
In this paper we use continued fractions to study a partial order on the set of 2-bridge knots derived from the work of Ohtsuki, Riley, and Sakuma. We establish necessary and sufficient conditions for any set of 2-bridge knots to have an upper bound with respect to the partial order. Moreover, given any 2-bridge knot K…
The paper proves tight fibered knots are minimal in a specific knot order.
problem Understanding the minimality of tight fibered knots.
method Proved ribbon concordance forms a partial order and used it to show tight fibered knots are minimal.
result All tight fibered knots are minimal in the ribbon concordance order.
Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the se…
Motivated by the relationship between orthogonal complex structures and spure spinors, we define twisted partially pure spinors in order to characterize spinorially subspaces of Euclidean space endowed with a complex structure.
We prove, for a class of contact manifolds, that the universal cover of the group of contact diffeomorphisms carries a natural partial order. It leads to a new viewpoint on geometry and dynamics of contactomorphisms. It gives rise to invariants of contactomorphisms which generalize the classical notion of the rotation …
Study immersions of punctured 4-manifolds for quantum automata applications.
problem Existence of immersions between specific 4-manifolds.
method Analyzing immersions of punctured 4-manifolds to establish a partial order.
result Established a partial order on closed 4-manifolds via immersions.
Classifies scalar second-order PDEs with low-dimensional symmetry groups.
problem Classifying differential equations with specific symmetry groups.
method Algebraic technique based on covariant form for constructing equations.
result Complete classification of quasi-linear scalar second-order PDEs with free symmetry groups of dimension ≤3.
In this paper, we consider an equivalence problem of second order partially differential equations (PDE) and a duality of the flat differential equation. For the equivalence problem, explicit form of invariants (curvatures) are given. We also investigate a duality associated with the flat equation using double fibratio…
Studies projective geometry and partial differential equations prolongation.
problem Understanding the prolongation of overdetermined geometric partial differential equations.
method Introduction to differential geometry and tractor calculus, study of prolongation of equations.
result Recovery of projective tractor and cotractor connections via partial differential equations prolongation.
Mixtures of ranking models have been widely used for heterogeneous preferences. However, learning a mixture model is highly nontrivial, especially when the dataset consists of partial orders. In such cases, the parameter of the model may not be even identifiable. In this paper, we focus on three popular structures of p…
We study an intrinsic distribution, called polar, on the space of -dimensional integral elements of the higher order contact structure on jet spaces. The main result establishes that this exterior differential system is the prolongation of a natural system of PDEs, named pasting conditions, on sections of the bundle…
A new method for state estimation on complex networks.
problem Reconstructing latent dynamics from multivariate time-series on topological cell complexes.
method Topology-aware state space framework derived from stochastic partial differential equations, with state evolution following heat-like topological diffusion.
result The proposed method successfully recovers latent states and topological structures in real-world networks.
In this note we establish some appropriate conditions for stochastic equality of two random variables/vectors which are ordered with respect to convex ordering or with respect to supermodular ordering. Multivariate extensions of this result are also considered.
We show that the notion of -hyperconvexity on oriented flag manifolds defines a partial cyclic order. Using the notion of interval given by this partial cyclic order, we construct Schottky groups and show that they correspond to images of positive representations in the sense of Fock and Goncharov. We construct poly…
Model optimizes trading strategy with unobservable toxicity.
problem Maximizing daily trading profit with unobservable toxicity.
method Formulated as a partially observable stochastic control problem, solved in two steps.
result P&L performance gap is negligible (0.01%) in all scenarios.
This paper studies an unsupervised deep learning-based numerical approach for solving partial differential equations (PDEs). The approach makes use of the deep neural network to approximate solutions of PDEs through the compositional construction and employs least-squares functionals as loss functions to determine para…
New algorithm minimizes expert selection regret in partial bandit feedback.
problem Minimizing expert selection regret in partial bandit feedback.
method Develops a sequential minimax optimal algorithm for a generalized partial monitoring setting.
result Second order regret bounds against a general expert selection sequence.
Recent work in learning ontologies (hierarchical and partially-ordered structures) has leveraged the intrinsic geometry of spaces of learned representations to make predictions that automatically obey complex structural constraints. We explore two extensions of one such model, the order-embedding model for hierarchical…
This paper compares rank aggregation methods for partial label ranking.
problem Handling partial label ranking with ties.
method Scoring-based and non-parametric probabilistic-based rank aggregation methods.
result Scoring-based variants consistently outperform the state-of-the-art method.
We propose an analytically tractable class of models for the dynamics of a limit order book, described through a stochastic partial differential equation (SPDE) with multiplicative noise for the order book centered at the mid-price, along with stochastic dynamics for the mid-price which is consistent with the order flo…