This paper studies constrained polygonal linkages and their configuration spaces.
problem Understanding the configuration spaces of constrained polygonal linkages.
method The paper uses Bott-Morse functions and critical point analysis to study the configuration spaces.
result The oriented area is a Bott-Morse function with computed indices.
Oriented area function is a perfect Morse function for polygonal linkages.
problem Understanding the topology of polygonal linkages.
method Generalization of oriented area function as a Morse function.
result Cyclic equilateral polygons are independent generators of configuration space's homology.
The geometric, topological, and symplectic properties of moduli spaces (spaces of configurations modulo rotations and translations) of polygonal linkages have been studied by Kapovich, Millson, and Kamiyama, et. al. One can form a polygonal linkage by taking two free linkages and identifying initial and terminal vertic…
We prove diffeomorphisms of polygonal linkage moduli spaces to Euclidean spaces.
problem Moduli spaces of self-avoiding polygonal linkages and configurations.
method Construct Lyapunov-Reeb functions to show diffeomorphisms.
result Moduli spaces are diffeomorphic to Euclidean spaces.
The oriented area function A is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the Morse index of a critical point. However, for planar polygons, the function A i…
Moduli spaces of planar polygonal linkages admit a cell structure which can be realized as a surgery on the permutohedron. We present a 3D visualization of the result of the surgery for all types of non-degenerate pentagonal linkages.
Starting by a simple game Q as a combinatorial data, we build up a cell complex M(Q), whose construction resembles combinatorics of the permutohedron. The cell complex proves to be a combinatorial manifold; we call it the \textit{ simple game induced manifold.} By some motivations coming from polygonal linkages, w…
Study spider mechanism configuration spaces using squared distance function.
problem Understand configuration spaces of spider mechanisms.
method Use Morse theory of squared distance function from body to fixed point.
result List and describe critical manifolds of squared distance function as products of polygon spaces.
We study the symplectic geometry of the moduli space of closed n-gons with fixed side-lengths in hyperbolic 3-space. We prove that these moduli spaces have a symplectic structure coming from Poisson Lie theory. We construct completely integrable systems on these moduli spaces by bending n-gons along their diagonals. Th…
In 1985 Kevin Walker in his study of topology of polygon spaces raised an interesting conjecture in the spirit of the well-known question "Can you hear the shape of a drum?" of Marc Kac. Roughly, Walker's conjecture asks if one can recover relative lengths of the bars of a linkage from intrinsic algebraic properties of…
We give a proof of a Conjecture of Walker which states that one can recover the lengths of the bars of a circular linkage from the cohomology ring of the configuration space. For a large class of length vectors, this has been shown by Farber, Hausmann and Schuetz. In the remaining cases, we use Morse theory and the fun…
A new clustering method versatile linkage improves on existing strategies.
problem Improving agglomerative hierarchical clustering methods.
method Introducing versatile linkage, a family of clustering strategies based on generalized means.
result Versatile linkage is space-conserving compared to existing methods.
New method for estimating firm linkages using CVLs and QCML.
problem Estimating firm linkages for profitable trading strategies.
method Characteristic Vector Linkages (CVLs) and Quantum Cognition Machine Learning (QCML).
result QCML similarity outperforms Euclidean similarity in constructing profitable trading strategies.
Minimax linkage improves clustering interpretability by minimizing maximum distance to prototypes.
problem Improving clustering interpretability and performance.
method Uses distances to prototypes for cluster formation, evaluated on multiple metrics.
result Minimax linkage often produces the smallest maximum minimax radius, but not always best across all metrics.
Paper introduces a method for supervised hierarchical clustering with Exponential Linkage.
problem Discrepancy between training and clustering objectives in supervised clustering.
method Tightly couples supervised training of dissimilarity function with hierarchical clustering, using Exponential Linkage.
result Joint training procedure consistently matches or outperforms other methods, improving dendrogram purity by up to 8 points.
This paper studies the configuration space of all possible positions of a linkage in R^n. For example, it shows that for every compact algebraic set, there is a linkage whose configuration space is analytically isomorphic to a finite number of copies of the algebraic set. If flexible edges are allowed, any compact set …
New closed linkage mechanisms with Möbius strip properties.
problem Designing closed linkage mechanisms with arbitrary number of hinges.
method Proposed a new family of closed linkage mechanisms with singular properties.
result These mechanisms can be considered as discrete Möbius strips.
Cryptocurrencies are becoming more linked in their returns and volatilities.
problem Understanding the increasing interconnectivity of cryptocurrencies.
method Examined market linkages using returns and volatilities, applied various methodologies.
result Significant increase in market linkages for both returns and volatilities.
Paper addresses limitations of traditional hierarchical clustering methods.
problem Traditional hierarchical clustering methods face limitations in binary trees and ultrametrics.
method Introduces the notion of a valid hierarchy and a two-step algorithm to construct a binary tree and prune it to enforce validity.
result Proposes a method to recover the finest valid hierarchy, which is not constrained to binary structures.
New model detects crime linkages from text, time, and space.
problem Detecting crime linkages from limited information.
method Spatio-temporal-textual Hawkes processes with text embeddings.
result Joint modeling of space, time, and text enhances crime linkage detection.
The study sets performance limits for record linkage using KL divergence.
problem Efficiently merging records in large, noisy databases to remove duplicates.
method Assesses performance bounds using Kullback-Leibler divergence in a Bayesian record linkage framework.
result Provides upper and lower bounds on misclassification probability.
New method extracts all reliable linkages in agglomerative clustering.
problem Finding optimal clustering solutions with adaptive and flexible criteria.
method Extracting all reliable linkages at each step for various criteria.
result Single linkage criterion yields minimum spanning tree.
Paper analyzes a three-loop linkage, showing it's overconstrained and shaky.
problem Analyzing a three-loop spatial linkage's degree of freedom and configuration space.
method Local analysis of differential degrees of freedom, computation of kinematic tangent cone, and c-space approximation.
result The linkage has a finite degree of freedom 3 and is locally a smooth manifold, making it shaky.
We prove realizability theorems for vector-valued polynomial mappings, real-algebraic sets and compact smooth manifolds by moduli spaces of planar linkages. We also establish a relation between universality theorems for moduli spaces of mechanical linkages and projective arrangements.
A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…
RLINK uses deep reinforcement learning to improve user identity linkage across social networks.
problem Recognizing the same user across different social networks.
method Converts user identity linkage into a sequence decision problem and uses deep reinforcement learning to optimize the linkage strategy.
result Achieves better performance than state-of-the-art methods in experiments on various datasets.
New configuration space accounts for spatial linkages and collisions.
problem Modeling spatial linkages considering collisions.
method Constructed completed and simplified configuration spaces.
result New configuration spaces account for linkages touching each other.
Study examines insurance sector linkages and systemic risk using dynamic spanning trees.
problem Interlinkages and systemic risk in the European insurance sector.
method Analysis of linkage dynamics and systemic risk using correlation networks, copulas, and minimum spanning trees.
result Minimum spanning trees describe linkage dynamics in the European insurance sector.
A linkage is a finite graph with lengths assigned to each edge. A planar realization is a map to the plane which preserves edge lengths. It can be thought of as a mechanical device formed from stiff rods and rotating joints. We look at the configuration space of all planar realizations of a linkage (following work of K…
Paper studies how discrete space curves with constant torsion deform to model linkage motions.
problem Modeling and understanding the motion of discrete space curves with constant torsion.
method Using semi-discrete mKdV equations to describe the motion of discrete space curves.
result The motion of discrete space curves is governed by semi-discrete mKdV equations.
A mechanical linkage is a mechanism made of rigid rods linked together by flexible joints, in which some vertices are fixed and others may move. The partial configuration space of a linkage is the set of all the possible positions of a subset of the vertices. We characterize the possible partial configuration spaces of…
Study explores financial market linkages between Japan and US markets.
problem Inconsistency in empirical studies regarding financial market causal linkages.
method Causal discovery methods including VAR-LiNGAM and LPCMCI with domain knowledge.
result VAR-LiNGAM reveals causal influences among financial markets, while LPCMCI identifies potential latent confounders.
New bounds improve linkage methods for clustering, distinguishing complete-link from single-link.
problem Improving bounds on linkage methods for clustering quality.
method Developed new bounds for complete-link and average-link methods in agglomeration clustering.
result Separated complete-link from single-link in terms of approximation for diameter.
The task of matching co-referent records is known among other names as rocord linkage. For large record-linkage problems, often there is little or no labeled data available, but unlabeled data shows a reasonable clear structure. For such problems, unsupervised or semi-supervised methods are preferable to supervised met…
Hierarchical clustering uses OWA operators to generalize linkage methods and avoid dendrogram inversions.
problem Avoiding unaesthetic inversions in hierarchical clustering dendrograms.
method OWA-based linkages combined with the Lance-Williams formula and conditions on weight generators.
result Conditions for weight generators to produce dendrograms without inversions.
LinkNBed learns entity and relationship representations across multiple graphs.
problem Jointly learn over multiple graphs and construct a unified graph.
method LinkNBed is a deep relational learning framework that learns entity and relationship representations across multiple graphs. It identifies entity linkage as a vital component and designs a novel objective to leverage it.
result Substantial improvements in link prediction and entity linkage over state-of-the-art relational learning approaches.
We study spaces of realisations of linkages (weighted graphs) whose underlying graph is a series parallel graph. In particular, we describe an algorithm for determining whether or not such spaces are connected.
New methods classify convex lattice polygons for affine dimers.
problem Not all convex lattice polygons are characteristic polygons of affine dimers.
method General constructions and algorithm for finding affine dimers with prescribed polygons.
result All lattice triangles, generalised parallelograms, and polygons of genus at most two admit an affine dimer.
Study maps interdependence of SDGs, finds complex, dynamic linkages.
problem Identify which SDGs promote progress and how quickly.
method Used a balanced panel of 114 countries from 2000 to 2024, applying two estimators to recover directed interaction network and measure dynamic linkages.
result 84 goal linkages survive false-discovery control, showing both synergies and trade-offs, with no single goal acting as a universal accelerator.
The pentagram map's limit point is related to infinitesimal perturbations of polygons.
problem Understanding the limit point of the pentagram map and its relation to polygon perturbations.
method Interpreting Glick's operator as the infinitesimal monodromy of a polygon.
result Glick's operator measures the extent to which a perturbed polygon does not close up.
The paper explores centroaffine geometry of polygons and their duals.
problem Understanding centroaffine dual pairs of spatial polygons.
method Defining centroaffine dual pairs and proving properties of polygon duals.
result Constant curvature polygons are dual to planar polygons.
Foliation of star-shaped polygons with fixed perimeter and area.
problem Characterizing star-shaped polygons with fixed perimeter and area.
method Analyzing families of star-shaped n-polygons in the Euclidean plane.
result Existence and properties of foliations on the space of star-shaped n-polygons.
The article studies polygon flows and their asymptotic behavior.
problem Investigating the behavior of polygon flows over time.
method Analogous to curve shortening flow, the article analyzes the β-polygon flow. result Regular polygons with five or more vertices are asymptotically stable.
Grinch efficiently clusters large datasets with complex structures.
problem Large-scale hierarchical clustering with complex linkage functions.
method Rotate and graft subroutines for efficient reconfiguration.
result Grinch guarantees accurate cluster trees for consistent models.
General area-preserving motion of polygonal curves is formulated as a system of ODEs. Solution polygonal curves belong to a prescribed polygonal class, which is similar to the admissible class used in the crystalline curvature flow. The ODEs are discretized implicitly in time keeping a given constant area speed while s…
The pentagram map preserves Poncelet polygons in convex cases.
problem Characterizing Poncelet polygons using the pentagram map.
method Theory of commuting difference operators, properties of real elliptic curves, and theta functions.
result A convex polygon is Poncelet if and only if it is projectively equivalent to its pentagram image.
Study finds finitely many non-congruent polygonal domains with same Steklov spectrum.
problem Inverse Steklov problem on convex polygons.
method Analysis of Steklov eigenvalues and isoperimetric bounds.
result For almost all convex polygonal domains, there exist at most finitely many non-congruent domains with the same Steklov spectrum.
Study of polygon spaces, characterizing critical points of area function.
problem Characterizing critical points of area function in polygon spaces.
method Geometric characterization of critical points and calculation of Morse indices.
result Generalization of isoperimetric theorems for polygons in the plane.