Paper connects knot theory with cluster algebra via Alexander polynomials.
problem Understanding Alexander polynomials for 2-bridge knots.
method Use of cluster variables and ancestral triangles.
result Alexander polynomials are specializations of cluster variables.
This paper generalizes knot polynomials to include the HOMFLY polynomial.
problem Generalizing knot polynomials to include the HOMFLY polynomial.
method Using path posets to directly generalize the construction of the Jones and Alexander polynomials to the HOMFLY polynomial.
result The HOMFLY polynomial is obtained by specializing path posets.
Efficiently searches ancestral graphs using multivariate information.
problem Discovering causal relationships in graphs with latent variables.
method Greedy search-and-score algorithm with two-step approach.
result Outperforms existing methods on benchmark datasets.
Ancestral graph models, introduced by Richardson and Spirtes (2002), generalize both Markov random fields and Bayesian networks to a class of graphs with a global Markov property that is closed under conditioning and marginalization. By design, ancestral graphs encode precisely the conditional independence structures t…
New method for ancestral inference in branching processes with random environments.
problem Determining ancestor distribution parameters in branching processes with random environments.
method Generalized method of moments for ancestral inference.
result Limiting distribution of ancestor and offspring estimators decouple and converge to independent Gaussian variables under certain conditions.
New algorithm identifies causal relationships from graphs, even with selection bias.
problem Identifying causal relationships from graphs with selection bias.
method Developed a measure-theoretic version of Pearl's causal calculus and a sound, complete identification algorithm.
result General measure-theoretic version of causal calculus allows for identification of causal relationships under selection bias.
Study restricts causal graphs with expert knowledge.
problem Restricting causal graphs to include expert orientation knowledge.
method Prove properties, present new orientation rules, develop algorithms.
result Shows how to uniquely represent restricted essential ancestral graphs.
Paper discovers valid IVs from data without domain knowledge.
problem Inferring causal effects from observational data with latent confounders.
method Data-driven algorithm based on partial ancestral graphs (PAGs).
result Discovering valid IVs leads to accurate causal effect estimation.
An approach for learning ancestral causal relationships in high dimensions, validated on human genome-wide data.
problem Learning ancestral causal relationships in high-dimensional biological data.
method Supervised learning approach with discrete indicators treated as labels, scalable to large problems.
result The approach is highly effective and scalable to the human genome-wide setting, robust to perturbations of input information.
Many biological characteristics of evolutionary interest are not scalar variables but continuous functions. Here we use phylogenetic Gaussian process regression to model the evolution of simulated function-valued traits. Given function-valued data only from the tips of an evolutionary tree and utilising independent pri…
In this paper, we study classes of graphs with three types of edges that capture the modified independence structure of a directed acyclic graph (DAG) after marginalisation over unobserved variables and conditioning on selection variables using the m-separation criterion. These include MC, summary, and ancestral grap…
Improved method for unbiased causal discovery in presence of unobserved confounding.
problem Unbiased data synthesis for causal discovery algorithms in the presence of unobserved confounding.
method Explicit block-hierarchical ancestral sampling to address limitations of implicit parameterization.
result Our approach fully covers the space of causal models, including those generated by implicit parameterization.
AGFN improves causal discovery by integrating expert feedback and handling latent confounding.
problem Inaccurate causal discovery due to unreliable expert knowledge and latent confounding.
method Ancestral GFlowNet (AGFN) is a reinforcement learning algorithm that iteratively refines a policy based on noisy expert feedback to infer ancestral graphs.
result AGFN converges to the true ancestral graph given accurate expert responses and outperforms baselines in structural Hamming distance and Bayesian Information Criterion.
New RL approach builds short ancestral recombination graphs.
problem Building short ancestral recombination graphs (ARGs).
method Reinforcement Learning applied to genetic sequences.
result RL can build ARGs as short as heuristic algorithms.
A new algorithm learns MAGs from data more efficiently using entropy.
problem Learning MAGs from data is unstable and computationally expensive.
method Uses entropy estimation and refined Markov property to score MAGs.
result Algorithm is polynomial in number of nodes and outperforms existing methods.
CCHM algorithm learns BN structure with latent variables, improving causal effect measurement.
problem Latent variables cause spurious relationships in BN structure learning.
method Hybrid approach combining constraint-based and score-based learning, incorporating do-calculus.
result CCHM outperforms state-of-the-art in reconstructing true BN structure.
The paper develops methods to bound causal effects using Partial Ancestral Graphs.
problem Bounding causal effects from observational data when true causal diagrams are unknown.
method Proposes a method using Partial Ancestral Graphs to derive bounds on causal effects from observational data.
result Demonstrates the effectiveness of the method with synthetic and real data examples.
New algorithm groups variables by ancestral relationships to improve causal graph estimation accuracy.
problem Difficulty in estimating causal graphs with small sample sizes relative to variables.
method CAG algorithm groups variables based on ancestral relationships, reducing complexity and improving accuracy.
result CAG outperforms existing methods in estimation accuracy and computation time.
New method discovers causal relationships in confounded systems.
problem Discovering causal relationships in systems with unmeasured confounding variables.
method Differentiable algebraic constraints for continuous optimization of ADMGs.
result Effective method for causal discovery in confounded linear systems.
ASCEND discovers causal relationships in multi-omics data by leveraging known hierarchical structure.
problem Causal inference in high-dimensional multi-omics data, especially when ignoring the hierarchical structure.
method Two-tiered divide-and-conquer strategy with ancestral conditioning sets.
result Achieves polynomial-time complexity and accurately recovers ancestral relationships.
We prove that the criterion for Markov equivalence provided by Zhao et al. (2005) may involve a set of features of a graph that is exponential in the number of vertices.
Napoleonic triangles don't exist in hyperbolic geometry.
problem The existence of Napoleonic triangles in hyperbolic geometry.
method Analyzing the construction of equilateral triangles on hyperbolic triangles.
result Hyperbolic triangles do not form Napoleonic triangles, except equilateral ones.
New bounds on inscribed triangles in arbitrary planar domains.
problem Finding inscribed triangles in arbitrary planar domains with specific angle constraints.
method Proving the existence of uniformly fat triangles and not-too-fat triangles in bounded open sets.
result Existence of a maximal number Θ (between 0 and 60) for inscribed triangles with angles ≥ Θ degrees.
Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.
problem Finding the minimum of the spectral determinant on isosceles triangles.
method Analyzing the determinant of the Laplacian on Euclidean isosceles triangle envelopes of fixed area.
result Equilateral triangle envelope minimizes the determinant of the Laplacian.
Paper calculates eigenvalues of a specific triangle on a sphere.
problem Computing eigenvalues of a specific triangle on a sphere.
method Computed first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2) on the sphere.
result Computed the first two Dirichlet eigenvalues and eigenfunctions of the equilateral Schwarz triangle (3/2 3/2 3/2).
New method shows any triangle group generating pair is related to special coverings.
problem Understanding generating pairs of triangle groups.
method Special almost orbifold coverings.
result Any generating pair of a triangle group is represented by a special covering.
Shorter sides in geodesic triangles in hyperbolic plane.
problem Properties of geodesic triangles in hyperbolic surfaces.
method Analyzing lifts of a closed geodesic in hyperbolic 2-space.
result Sides of triangles formed by geodesics are shorter than the geodesic itself.
Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups t…
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.
Defines band maps in unoriented link Floer homology forming a skein exact triangle.
problem Understanding unoriented link Floer homology through band maps.
method Defines and analyzes band maps in unoriented link Floer homology.
result Band maps form an unoriented skein exact triangle.
Proves a new skein exact triangle for real monopole Floer homology.
problem None explicitly stated; focuses on proving a new mathematical structure.
method Introduces a new exact triangle for real monopole Floer homology.
result Proves an unoriented skein exact triangle for real monopole Floer homology.
New surgery exact triangles in Heegaard Floer homology for rational slopes.
problem Constructing new surgery exact triangles in Heegaard Floer homology.
method Combining combinatorial triangle and quadrilateral counting in genus 1 Heegaard diagrams.
result Solving the combinatorial problem for rational slopes, including tricky cases.
New causal models for growing networks avoid node deletion constraints.
problem Statistical models based on node exchangeability are not suitable for growing networks.
method Enumerated and partitioned causal directed acyclic graph (DAG) models over pairs of nodes.
result Simple model exhibits flexible power-law degree distributions and emergent phase transitions.
We answer the question "Does the Y-triangle move preserve intrinsic knottedness?" in the negative by giving an example of a graph that is obtained from the intrinsically knotted graph K_7 by triangle-Y and Y-triangle moves but is not intrinsically knotted.
Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.
problem Understanding eigenvalues and gaps in spherical triangles.
method Explicit computation of Dirichlet eigenvalues and eigenfunctions for spherical lunes and triangles.
result Fundamental gap of spherical triangles increases as the angle of the lune decreases.
We show that the triangle with angles Pi/12, Pi/3 and 7*Pi/12 has the lattice property and compute this triangle's Veech group.
We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in X. We construct an example of a complete path metr…
New method proves mateability of triangle groups with Blaschke products.
problem Proving mateability of triangle groups with Blaschke products.
method Associating two piecewise analytic circle maps to the triangle group, mating these with Blaschke products, and constructing a common lift.
result Proves mateability of all cusped triangle groups with suitable Blaschke products.
In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the con…
Study of complex tori using twistor triangles and algebraic representations.
problem Understanding the geometry of complex tori through twistor triangles.
method Using representation theory of algebras to analyze the period domain of complex tori.
result Introduced pseudometric invariants to distinguish triangles up to G-equivalence. Classifies complex hyperbolic triangle groups by types.
problem Classifying complex hyperbolic triangle groups.
method By types defined by the ellipticity of two short words.
result Improves Schwartz conjecture.
In Lorentzian geometry, limited definition of angles restricts the use of angle bisectors in study of triangles. This paper redefines angle bisectors so that they can be used to study attributes of triangles. Using the new definition, this paper investigates the existence of the incenter and the isogonal conjugate of a…
A formula for Rademacher symbols in triangle groups is provided.
problem No specific problem stated; focuses on a mathematical formula.
method Presentation of an explicit formula for Rademacher symbols.
result Generalizes Ghys' proof of modular knot linking numbers.
Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.
problem Understanding the stabilizers of complex hyperbolic triangle groups.
method Explicit generators and signatures of stabilizers computed for each group orbit of mirrors.
result Explicit generators and signatures of stabilizers for some triangle groups.
Triangle groups show rigidity in hyperbolic spaces.
problem Local rigidity of triangle groups generated by reflections.
method Geometric representation and diagonal embeddings in PGL(2,R) and PSp±(2n,R). result Triangle groups are locally rigid in hyperbolic spaces.
Criterion for stopping conjugacy class enumeration in triangle groups.
problem Enumerating all conjugacy classes in cocompact triangle groups.
method Encoding by P. Dehornoy and T. Pinsky; stopping criterion based on geometric length.
result Stopping criterion for the generation of conjugacy classes in cocompact triangle groups.
New skein exact triangles for link Floer homology.
problem Understanding link Floer homology through skein relations.
method Construction of skein triples for rational tangles.
result Established a framework for potential further skein exact triangles.
New theorem disproves Angle Defect for super triangles.
problem Angle Defect Theorem for N=1 super hyperbolic geometry.
method Action of OSp(1|2) on real super Minkowski space and brute-force computation.
result Disproves Angle Defect Theorem and provides novel additive function.