The paper develops algorithms and topological invariants for distinguishing dynamic systems.
problem Distinguishing the topological type of surfaces and functions in dynamic systems.
method Construction of algorithms and topological invariants using discrete topological structures.
result The development of discrete topological structures for topological equivalence of dynamic systems.
Study improves systemic risk assessment by considering local network environments.
problem Identifying systemic financial institutions using network metrics.
method Two-step procedure: 1) recover network communities, 2) regress vulnerability on topological measures at global, local, and aggregated levels.
result Local network metrics predict distress better than global metrics during financial crises.
Constructs algorithms to recognize and classify 2D surfaces.
problem Recognizing and classifying 2D surfaces in dynamic systems.
method Discrete topological structures and algorithms for simplicial and CW-complexes.
result Determines the topological type of 2-manifolds.
In this paper we try to establish a connection between a three-dimensional Lotka--Volterra dynamical system and two-dimensional topological surgery. There are many physical phenomena exhibiting two-dimensional topological surgery through a `hole drilling' process. By our connection, such phenomena may be modelled mathe…
Investigates integrable systems with linear periodic integral for e(3) Lie algebra.
problem Analyzes singularities and topological properties of integrable systems.
method Examines singularities of Liouville foliation, bifurcation diagram, transformations of Liouville tori, and isoenergy surfaces.
result Discovers topological properties of integrable systems with linear periodic integral.
The main purpose of this paper is to give a topological and symplectic classification of completely integrable Hamiltonian systems in terms of characteristic classes and other local and global invariants.
New framework for 3D spatial topology enumeration and identification.
problem Efficient navigation through complex engineering system topologies.
method Mathematical spatial graph theory to represent, enumerate, and identify unique topological classes.
result Identification of distinctive 3D topological classes for engineering systems.
Researchers create topologically protected knots in a realizable system.
problem Creating topologically protected vortex knots in experimentally realizable systems.
method Investigated non-Abelian vortices in tetrahedral order in spin-2 Bose--Einstein condensates and bent-core nematic liquid crystals.
result Discovered the first topologically protected knots in an experimentally realizable system.
We prove that a topological contact isotopy uniquely defines a topological contact Hamiltonian. Combined with previous results from [MS11], this generalizes the classical one-to-one correspondence between smooth contact isotopies and their generating smooth contact Hamiltonians and conformal factors to the group of top…
The paper studies bifurcations in discrete dynamical systems on manifolds.
problem Understanding bifurcations in discrete dynamical systems on manifolds.
method Topological techniques based on concentricity of manifolds.
result General result for attractors in n-dimensional manifolds.
New methods assess topological entanglement in periodic systems.
problem Assessing topological entanglement in systems with periodic boundary conditions.
method Introducing Periodic Jones polynomial and Cell Jones polynomial.
result Periodic Jones polynomial is a recurring factor of Jones polynomial of finite cutoffs.
Quantum phase diagrams for Chern topological insulators show jumps at critical loci.
problem Understanding phase transitions in Chern topological insulators.
method Mathematical formulation and explicit families of physical systems.
result Synthetic design of arbitrary Chern jumps in topological phases.
Causal fermion systems and Riemannian fermion systems are proposed as a framework for describing non-smooth geometries. In particular, this framework provides a setting for spinors on singular spaces. The underlying topological structures are introduced and analyzed. The connection to the spin condition in differential…
The paper uses topological concepts to analyze neural networks, revealing complex structure and dynamics.
problem Understanding the structure and dynamics of deep learning models.
method Topological dynamical systems, index theory, and computational homology.
result Neurons correspond to simplexes in a simplicial complex, and topological invariants can be computed.
The paper deals with the problem of reconstructing the topological structure of a network of dynamical systems. A distance function is defined in order to evaluate the "closeness" of two processes and a few useful mathematical properties are derived. Theoretical results to guarantee the correctness of the identificatio…
Topological method detects Hopf bifurcations from time series.
problem Detecting Hopf bifurcations in nonlinear systems from time series data.
method Persistent homology applied to Takens embedding for phase space reconstructions.
result A simple scalar topological functional identifies critical bifurcation points.
The abstract applies waist inequality to dynamical systems and entropy.
problem Understanding the relationship between waist inequality and dynamical systems.
method Applying waist inequality to entropy and mean dimension of dynamical systems.
result Maps between dynamical systems have positive conditional metric mean dimension under certain conditions.
A new approach uses circuit topology to study complex polymer interactions.
problem Understanding structural phase transitions in entangled polymer systems.
method Braided circuit topology framework for multiple-chain systems.
result Circuit topological motif fractions are effective order parameters for structural transitions.
Deep learning detects bifurcations in dynamical systems.
problem Predicting catastrophic changes in dynamical systems across sciences.
method Data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries.
result Extracts topologically invariant features to detect bifurcation boundaries in unseen systems.
AdaCGP learns dynamic graph topology from time series data, improving over existing methods.
problem Learning dynamic graph topology from time-varying signals, especially in real-time applications.
method AdaCGP is a sparsity-aware adaptive algorithm that recursively estimates the Graph Shift Operator (GSO) through variable splitting.
result AdaCGP outperforms state-of-the-art methods in GSO estimation, achieving improvements exceeding 83%.
Survey uses Milnor fibrations to classify first integrals of differential systems.
problem Classifying first integrals of differential systems using geometric-topological methods.
method Utilizing Milnor fibrations and connections with harmonic morphisms to provide topological and geometric descriptions.
result Geometric-topological classifications of first integrals for both isolated and non-isolated singularities.
Topologically protected vortex knots and links are proposed and proven.
problem Decaying of tangled vortex structures through local reconnections and strand crossings.
method Proposed and proven topologically protected vortex structures using non-Abelian topological vortices.
result Existence of topologically protected Q8-colored links and classification using the Q-invariant. Proves left-orderability of mapping class groups of infinite-type surfaces.
problem Left-orderability of mapping class groups of infinite-type surfaces.
method Inductive construction of a stable Alexander system and ideal arc systems.
result Proves left-orderability using carefully chosen exhaustion by finite-type subsurfaces.
The topological complexity TC(X) is a numerical homotopy invariant of a topological space X which is motivated by robotics and is similar in spirit to the classical Lusternik-Schnirelmann category of X. Given a mechanical system with configuration space X, the invariant TC(X) measures the complexity of all possible mot…
Study detects P-type bifurcations in single system realizations using unreliable kernel density estimates.
problem Detecting P-type bifurcations in signals with unreliable kernel density estimates.
method Create persistence diagrams from single system realization, statistically analyze resulting set, compare point process modeling methods.
result Subsampling outperforms other point process modeling methods in predicting P-type bifurcations.
This work generalizes a formula linking Seiberg-Witten prepotential and topological recursion.
problem Analyzing the relationship between Seiberg-Witten curves and topological recursion.
method Analytical approach using Seiberg-Witten family of curves.
result A generalized formula relating Seiberg-Witten prepotential to the genus zero part of topological recursion on a Seiberg-Witten curve.
New TC variant dTC better fits motion planning for some systems.
problem Improving motion planning for autonomous systems.
method Defined and computed new homotopy invariant dTC.
result dTC and dcat provide better motion planning solutions.
Study confirms Chern's conjecture on compact Hessian manifolds and classifies their topologies.
problem Global topological constraints and structural properties of compact Hessian manifolds.
method Novel fibration and splitting theorems, Chern's conjecture, Hitchin systems, Cheng-Yau solution.
result Topological classification of complete Hessian surfaces and closed orientable Hessian 3-manifolds.
Framework for joint inference of network topology and interaction types in heterogeneous systems.
problem Joint inference of network topology, multi-type interaction kernels, and latent type assignments in heterogeneous interacting particle systems.
method Three-stage approach: shared structure recovery, discrete interaction type identification, and matrix factorization.
result The method yields accurate reconstruction of underlying dynamics and is robust to noise.
The two main theorems of this paper provide a characterization of hyperbolic affine iterated function systems defined on Rm. Atsushi Kameyama (Distances on Topological Self-Similar Sets, Proceedings of Symposia in Pure Mathematics, Volume 72.1, 2004) asked the following fundamental question: given a topological self-si…
This paper introduces a new method for neural networks that doesn't need a global coordinate system.
problem The lack of a global coordinate system in neural networks limits their performance and explainability.
method Proposes a learnable topological layer that works in a general metric space (Hilbert space) without requiring a Euclidean space.
result The proposed method eliminates the need for a costly parametrization stage and achieves optimal network performance.
This work develops agents to learn generalizable policies for dynamic network environments.
problem Real-world network topologies change due to attackers, defenders, or system failures, leading to failures in adaptive ACD systems.
method Developing agents to learn generalizable policies across dynamic network environments.
result Agents can learn robust policies for dynamic network topologies and diverse attackers.
Arnold-Liouville systems cannot be bi-Hamiltonian generically.
problem The bi-Hamiltonian structure of Arnold-Liouville systems.
method Proving that a specific class of smooth functions is a meagre subset for the Fréchet topology, which implies Arnold-Liouville systems cannot be bi-Hamiltonian.
result Generically, Arnold-Liouville systems cannot be bi-Hamiltonian.
Determines the crossing number of polynomial curve systems on surfaces.
problem Calculating the crossing number of polynomial curve systems on surfaces.
method Determines the crossing number in terms of the genus for polynomial curve systems.
result High precision determination of crossing number for polynomial curve systems.
We investigate the special Kähler geometry of the base of the Hitchin integrable system in terms of spectral curves and topological recursion. The Taylor expansion of the special Kähler metric about any point in the base may be computed by integrating the g=0 Eynard-Orantin invariants of the corresponding spectral …
The paper studies the convergence of SAA for systemic risk measures.
problem Theoretical convergence of SAA for set-valued systemic risk measures.
method General theory and specific case study with mixed-integer programming formulations.
result Theoretical convergence results for SAA under Wijsman and Hausdorff topologies.
Topological entropy measures the number of distinguishable orbits in a dynamical system, thereby quantifying the complexity of chaotic dynamics. One approach to computing topological entropy in a two-dimensional space is to analyze the collective motion of an ensemble of system trajectories taking into account how traj…
This paper clarifies a global structure of Stokes-Dirac structures used for describing interconnected port-Hamiltonian systems defined on manifolds with non-trivial topology under consistent boundary condition.
Study uses TDA to improve OEE forecasting in manufacturing.
problem Volatility and nonlinearity in OEE time series data.
method Topological Data Analysis (TDA) to transform raw data into structured knowledge.
result Improves OEE forecasting accuracy by at least 17%.
The financial market is a complex dynamical system composed of a large variety of intricate relationships between several entities, such as banks, corporations and institutions. At the heart of the system lies the stock exchange mechanism, which establishes a time-evolving network of trades among companies and individu…
A BV algebra is a formal framework within which the BV quantization algorithm is implemented. In addition to the gauge symmetry, encoded in the BV master equation, the master action often exhibits further global symmetries, which may be in turn gauged. We show how to carry this out in a BV algebraic set up. Depending o…
Survey on Allen-Cahn equations and systems, focusing on multiplicity results and geometric interpretation.
problem Multiplicity results for Allen-Cahn equations and systems in singular perturbation regime.
method Photography method, variational-topological approach based on localized approximate solutions and barycenter maps.
result Encoding of topology into multiplicity results through variational-topological approach.
Survey on manifold complexities and motion planning in robotics.
problem Understanding topological complexities of manifolds in robotic motion planning.
method Overview of topological complexities, geodesic motion planning, and connections to critical point theory.
result Estimation of motion planning complexity using Riemannian geometry and critical point theory.
Study shows infinite kernels in topological monodromy for curve families.
problem Understanding kernels of topological monodromy representations.
method Extending Kuno's arguments and using Carlson-Toledo techniques.
result Kernels are infinite for certain linear systems on surfaces.
We solve Euler equations on graph manifolds, classifying steady flows with Morse-Bott Bernoulli functions.
problem Classifying steady Euler flows with Morse-Bott Bernoulli functions.
method Constructing non-vanishing steady solutions using integrable systems and topology.
result Steady Euler flows with Morse-Bott Bernoulli functions exist only on graph three-manifolds.
The paper introduces invariants to describe period-doubling routes to chaos in dynamical systems.
problem Understanding the dynamics of period-doubling routes to chaos in complex systems.
method Introducing three topological invariants to describe the topology of period-doubling routes to chaos.
result Ascribed symbolic dynamics to perturbations of the Shilnikov homoclinic scenario and dynamics of the Henon map.
Category theory generalizes finite type invariants using diagrams systems.
problem Generalizing finite type invariants using category theory.
method Relating generating sets for generalized finite type theories with diagrams systems.
result Demonstrates the correspondence between finite type theories and diagrams systems.
New approach uses MST and copula-DCC-GARCH for systemic risk analysis in European insurance sector.
problem Analyzing systemic risk in European insurance sector through indirect connections.
method Combining copula-DCC-GARCH model and Minimum Spanning Trees (MST) for interlinkage dynamics analysis.
result Proposed approach useful for systemic risk analysis in insurance sector, with MST topological indicators as predictors.