Directed graphs can be intrinsically knotted and 4-linked.
problem Intrinsic linking and knotting in directed graphs.
method Construction of examples and operations (consistent edge contraction, H-cyclic subcontraction).
result Directed graphs can have consistently oriented knotted cycles and intrinsically 3- and 4-linked structures.
EdgePool improves GNN performance by pooling edges, not nodes.
problem Lack of effective graph pooling methods in GNNs.
method Edge contraction pooling approach.
result EdgePool outperforms alternative pooling methods.
Grid homology properties for MOY graphs studied.
problem Defining and studying properties of grid homology for MOY graphs.
method Defined grid homology from Harvey and O'Donnol's work. Studied properties using oriented skein relation, edge contraction, and parallel edge unification.
result Properties of grid homology for MOY graphs were studied and defined.
A triangulation of a surface with fixed topological type is called irreducible if no edge can be contracted to a vertex while remaining in the category of simplicial complexes and preserving the topology of the surface. A complete list of combinatorial structures of irreducible triangulations is made by hand for the on…
The study explores convex unions and completions in simplicial pseudomanifolds, revealing unexpected behavior.
problem Understanding the behavior of convex unions in simplicial pseudomanifolds.
method Generalization to simplicial pseudomanifolds, considering PL homeomorphisms and edge subdivisions.
result Unexpected behavior in convex unions and completions, including empty contraction spaces and large/small contraction spaces.
New graph shows edge deletion/contraction doesn't always result in intrinsically linked graphs.
problem Edge operations in intrinsically knotted graphs don't always produce intrinsically linked graphs.
method Presented a new intrinsically knotted graph.
result Edge operations in intrinsically knotted graphs don't always result in intrinsically linked graphs.
Two algorithms learn Gaussian graphical models from Glauber dynamics trajectories.
problem Learning Gaussian graphical models from dependent data.
method Two complementary approaches: local edge-testing and burn-in/thinning reduction.
result Both approaches provide finite-sample recovery guarantees and empirical comparisons.
Paper determines Assouad-Nagata dimension for all minor-closed metrics.
problem Understanding the Assouad-Nagata dimension of minor-closed metrics.
method Using edge-weighted graphs and edge-deletion/contraction to model minor-closed metrics, determining their Assouad-Nagata dimension.
result Determined the Assouad-Nagata dimension for every minor-closed metric.
We present a necessary and sufficient condition for existence of a contractible Hamiltonian Cycle in the edge graph of equivelar maps on surfaces. We also present an algorithm to construct such cycles. This is further generalized and shown to hold for more general maps.
We present a necessary and sufficient condition for existence of a contractible, non-separating and noncontractible separating Hamiltonian cycle in the edge graph of polyhedral maps on surfaces. In particular, we show the existence of contractible Hamiltonian cycle in equivelar triangulated maps. We also present an alg…
The study examines six variations of apex graphs and their planar properties.
problem Characterizing apex, edge apex, and contraction apex graphs.
method Defined and analyzed six variations of apex graphs, using the Graph Minor Theorem and determining obstruction graphs.
result Found at least 36, 55, and 82 obstruction graphs for apex, edge apex, and contraction apex respectively.
Bayesian KANs achieve near-minimax posterior contraction rates in anisotropic Besov spaces.
problem Statistical foundation for Bayesian Kolmogorov-Arnold networks in anisotropic Besov spaces.
method Sparse Bayesian KANs with spike-and-slab priors, hyperprior on model size, and approximation complexity bounds.
result Posterior contraction rates depend on intrinsic anisotropic smoothness and effective dimension of the compositional structure.
This paper introduces a new method to price long-dated insurance contracts.
problem Pricing of long-dated, insurance-type contracts is complex and inconsistent.
method Loading pricing combines theoretically minimal and formally risk-neutral prices.
result Loading degree is constant for minimally fluctuating contracts and is a key characteristic.
Kakimizu complex of a knot is a flag simplicial complex whose vertices correspond to minimal genus Seifert surfaces and edges to disjoint pairs of such surfaces. We discuss a general setting in which one can define a similar complex. We prove that this complex is contractible, which was conjectured by Kakimizu. More ge…
Researchers prove conjecture about contractible subcomplexes in noncrossing partition link.
problem Understanding contractibility of subcomplexes in the noncrossing partition link.
method Combining contractibility of flag complexes' stars with noncrossing hypertrees theory.
result Proved conjecture about contractible subcomplexes in the noncrossing partition link.
Regular languages describe contracting geodesics in groups.
problem Characterizing groups with infinite contracting geodesics.
method Analyzing geodesics in Cayley graphs with a contracting property.
result Groups with infinite contracting geodesics are either virtually Z or acylindrically hyperbolic.
Proposes a Bayesian approach for automatic node selection in sparse neural networks.
problem Reduces structural complexity and computational speedup in large-scale predictive models.
method Uses spike-and-slab Gaussian priors and variational Bayes approach for node selection.
result Establishes variational posterior consistency and optimal contraction rates for sparse networks.
Characterizes Forman curvature bounds and proves curvature equivalence.
problem Characterize Forman curvature bounds and prove curvature equivalence.
method Contractivity of the Hodge Laplacian semigroup, translation between 2-cells and transport plans.
result Ollivier and Forman curvature coincide on edges when maximizing Forman curvature.
We extend average edge order results to normal 3-pseudomanifolds.
problem Determining the average edge order of normal 3-pseudomanifolds.
method Extending previous results on 3-manifolds to 3-pseudomanifolds with singularities.
result For a normal 3-pseudomanifold K, μ0(K)≥730, with equality if and only if K is a specific triangulation of RP2. The paper examines how edge subdivisions affect the vanishing of L2-homology in Coxeter groups.
problem The vanishing of L2-homology in Coxeter groups under edge subdivisions. method Investigates conditions for the vanishing of L2-homology to be preserved under edge subdivisions of flag triangulations. result Conditions are given to preserve the vanishing of L2-homology under edge subdivisions, and counterexamples are constructed for a torsion growth analogue of Singer's conjecture. This paper explores how insurance contracts can be traded in financial markets.
problem The exclusion of arbitrage in insurance contracts due to their non-tradability.
method Defining strategies on insurance portfolios and combining them with financial trading strategies.
result The existence of an insurance-finance-consistent probability, leading to the expected discounted cash-flows.
The paper proves ML estimators are strongly consistent for identifying edge weights in BAR models.
problem Identifying edge weights in Bernoulli Autoregressive (BAR) models.
method Maximum Likelihood (ML) estimation for two variants of BAR models.
result ML estimators are strongly consistent for edge weight identification.
Proves equivalence of two types of boundaries in metric spaces.
problem Proving equivalence of two types of boundaries in metric spaces.
method Analyzes and compares contracting and κ-Morse boundaries. result Proves equivalence of 1-Morse boundary and contracting boundary as topological spaces.
If X is a compact set, a {\it topological contraction} is a self-embedding f such that the intersection of the successive images fk(X), k>0, consists of one point. In dimension 3, we prove that there are smooth topological contractions of the handlebodies of genus ≥2 whose image is essential. Our proof i…
Study financial contracts pricing in markets with nonproportional costs and constraints.
problem Financial contract pricing in markets with nonproportional transaction costs and portfolio constraints.
method Direct and dual characterization of market-consistent prices with acceptable risk thresholds.
result Extension of the Fundamental Theorem of Asset Pricing to include good deals and scalable good deals.
The study characterizes homology 4-manifolds with g2≤5 combinatorially.
problem Characterizing homology 4-manifolds with specific g2 values. method Combinatorial approach using various operations on triangulated 4-spheres.
result Homology 4-manifolds with g2≤5 are triangulated spheres and can be derived from 4-spheres with g2≤2. Filling length measures the length of the contracting closed loops in a null-homotopy. The filling length function of Gromov for a finitely presented group measures the filling length as a function of length of edge-loops in the Cayley 2-complex. We give a bound on the filling length function in terms of the log of an …
This paper improves a result on homology concordance in contractible manifolds and two bridge links.
problem Improving the understanding of homology concordance in contractible manifolds and two bridge links.
method Using a family of knots obtained by blowing down a component of a two-bridge link, the paper constructs examples and classifies knot Floer homology.
result There exists a family generating a Z∞-summand in the quotient group, and a Z-count of such families is provided. In this paper Legendrian graphs in (R3,ξst) are considered modulo Legendrian isotopy and edge contraction. To a Legendrian graph we associate a (generalized) rectangular diagram --- a purely combinatorial object. Moves of rectangular diagrams are introduced so that equivalence classes of Legendr…
The study characterizes 3-pseudomanifolds with up to two singularities.
problem Characterizing face-number-related invariants of normal 3-pseudomanifolds with up to two singularities.
method Proves properties of normal 3-pseudomanifolds using specific operations and upper bounds.
result Proves that normal 3-pseudomanifolds with up to two singularities are constructed from boundary complexes of 4-simplices.
Edge augmentation connects disconnected graphs by elevating eigenvalues.
problem Connecting disconnected subgraphs in graphs with zero eigenvalues.
method Elevating zero eigenvalues of graph's spectrum to connect subgraphs.
result The algorithm consistently connects graph components, achieving >50% inter-community edges.
Approach for recovering shared structure from multiple networks with unknown noise.
problem Recovering shared structure from multiple networks with unknown edge distributions.
method Exploits shared mean structure to denoise edge-level measurements and estimate population-level parameters.
result Established a finite-sample concentration inequality for low-rank eigenvalue truncation of a random weighted adjacency matrix.
We analyze conditional optimization problems arising in discrete time Principal-Agent problems of delegated portfolio optimization with linear contracts. Applying tools from Conditional Analysis we show that some results known in the literature for very specific instances of the problem carry over to translation invari…
New method enhances graph neural networks using contractions and hourglass persistence.
problem Limitations of traditional persistent homology in graph neural networks.
method Hourglass Persistence, Contraction Homology, contractions as a topological operation.
result Hourglass Persistence boosts expressivity, learnability, and stability in graph representation learning.
Paper provides statistical guarantees for GNNs in link prediction.
problem Link prediction accuracy in graph neural networks.
method Proposes a linear GNN architecture (LG-GNN) and derives statistical guarantees.
result LG-GNN produces consistent estimators for edge probabilities and has better detection of high-probability edges.
Study on network-valued processes with asynchronous updates, proving consistency in community and changepoint estimation.
problem Understanding the behavior of network-valued stochastic processes with asynchronous updates.
method Analysis of concentration properties of aggregated adjacency and Laplacian matrices for lazy network-valued stochastic processes.
result Demonstrates consistency of estimators in community and changepoint estimation problems.
Region detection in Gaussian Markov fields with limited samples.
problem Consistent graph recovery in sample deficient scenarios.
method Partitioning the graph into spatial regions with similar edge parameters and regular boundaries, developing new sample complexity bounds, and introducing an efficient region growing algorithm.
result A bounded number of samples can be sufficient for consistent region recovery.
Foresight Arena benchmarks AI forecasting on real-world markets, isolating predictive edge.
problem Evaluating AI forecasting ability in real-world markets is challenging due to overfitting, centralized trust, and conflated metrics.
method Permissionless, on-chain benchmark using probabilistic forecasts, commit-reveal protocol, and smart contracts.
result Demonstrates the need for 350 predictions to reliably distinguish agents of different skill levels.
Paper proposes new methods for improving interatomic potentials.
problem Limitations of conventional SO(2) Linear architectures in MLIPs.
method Direct Cartesian construction, recursive Clebsch-Gordan construction, Edge Complex Product Basis, Radial Rotary Complex Attention.
result TECE-OAM-RRA-1.0 achieves SOTA performance on Matbench Discovery.
New method solves group synchronization with cycle-edge message passing.
problem Solving group synchronization with adversarial or uniform corruption and small noise.
method Cycle-edge message passing procedure using cycle consistency information.
result Exact recovery and linear convergence guarantees under adversarial corruption.
Study on symmetric automorphisms of RAAGs, proving finiteness properties and contractibility.
problem Finiteness properties and contractibility of symmetric automorphisms of RAAGs.
method Definition of symmetric automorphism group, construction of symmetric Outer space, proof of contractibility.
result Finiteness properties and contractibility results for symmetric automorphisms of RAAGs.
We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a number of known results. New results include a combinatorial characterization of c…
Bayesian SSR on graphs improves regression with noisy labels.
problem Estimating function values on graphs from noisy labeled data.
method Bayesian approach using graph Laplacian and Gaussian prior.
result Rates of contraction of posterior measure around ground truth.
This paper tackles graph translation challenges by predicting both node and edge attributes simultaneously.
problem Challenges in predicting both node and edge attributes in graph translation, especially in interactive, iterative, and asynchronous processes.
method Developed a novel framework integrating both node and edge translations seamlessly, using spectral graph regularization to maintain consistency.
result Demonstrated the effectiveness of the proposed method on both synthetic and real-world application data.
In this paper we use Bernstein and Chebyshev polynomials to approximate the price of some basket options under a bivariate Black-Scholes model. The method consists in expanding the price of a univariate related contract after conditioning on the remaining underlying assets and calculating the mixed exponential-power mo…
ARGEW improves node embeddings for weighted homophilous graphs by emphasizing strong edge weights.
problem Lack of accurate node embeddings for weighted homophilous graphs.
method ARGEW (Augmentation of Random walks by Graph Edge Weights) augments random walks by emphasizing nodes with larger edge weights.
result ARGEW produces embeddings where node pairs with strong edge weights have closer embeddings.
New method recovers graph latent positions under edge differential privacy.
problem Recovering latent graph information from privatized graphs.
method Applying geometric insights to adjust statistical inference for privatized graphs.
result Achieves consistent recovery of latent positions under local edge differential privacy constraints.
Paper proposes using blockchain for trustable machine learning with streaming layer and synthetic data.
problem Machine learning results are not fully trusted due to mutable data and difficulty in automation.
method Blockchain technology for immutable data storage and smart contracts for automation. Server, streaming, and smart contract implementations.
result A compact binary model format for streaming layer and synthetic data generation for limited training data.