PD_3-groups split as HNN extensions, revealing homology class properties.
problem Understanding PD_3-groups splitting as HNN extensions.
method Analyzing PD_3-groups splitting as HNN extensions and examining homology classes.
result The Poincaré dual of the homology class of a PD_3-group in the kernel of an epimorphism is revealed.
Proves Tits alternative for specific PD(3) groups.
problem Tits alternative for PD(3) groups. method Proving Tits alternative for almost coherent PD(3) groups. result Almost coherent PD(3) groups contain rank 2 free groups. Study pro-p completions of orientable PD_n groups, proving best results in three cases.
problem Understanding pro-p completions of orientable PD_n groups. method Examined four cases of orientable PD_3-groups and some PD_n groups (n≤5), providing examples and proving best results in three cases.
result Best results in three out of four cases of orientable PD_3-groups.
The study of PD3-pairs extends results for aspherical 3-manifolds.
problem Understanding PD3-pairs with aspherical ambient spaces. method Attaching 1-handles to PD3-pairs with aspherical ambient space and π1-injective boundary. result There are only finitely many PD3-pairs with a specific group property. Every PD3-complex bounds a PD4-pair.
problem Bounding PD3-complexes with PD4-pairs. method Constructing PD4-pairs from PD3-complexes. result Every PD3-complex bounds a PD4-pair (Z,P). The paper classifies PD_4-complexes based on their fundamental group properties.
problem Understanding the structure of PD_4-complexes based on their fundamental group properties.
method Analyzing the fundamental group and its modules to classify PD_4-complexes.
result The classification of PD_4-complexes based on their fundamental group properties.
Study extends Elkalla's work on subnormal subgroups to PD3-groups, but L2-Betti numbers need verification.
problem Verifying L2-Betti numbers for PD3-groups and group pairs. method Algebraic arguments extending Elkalla's work, but reliant on unproven L2-Betti number hypothesis. result Need further research on L2-Betti numbers for general PD3-groups. We show that if X is an indecomposable PD3-complex and π1(X)isthefundamentalgroupofareducedfinitegraphoffinitegroupsbutisnotvirtuallycyclicthenXisorientable,theunderlyinggraphisatree,alltheedgegroupsareZ/2Zandallbutatmostoneofthevertexgroupsisdihedraloforder2m…
Study subgroups of pro-p PD^3 groups, finding specific conditions.
problem Characterize subgroups of pro-p PD^3 groups. method Analyzes properties of subnormal and finitely presented subgroups.
result Conditions on subgroups of pro-p PD^3 groups. We show that the orientable double covering space of an indecomposable non-orientable PD3-complex has torsion free fundamental group.
Classifies pro-p PD2 pairs and builds a pro-p curve complex.
problem Classifying pro-p Poincaré duality pairs in dimension two. method Using classification of pro-p PD2 pairs to build a pro-p curve complex. result Established basic properties of the pro-p curve complex. Characterizes groups of branched twist-spun knots.
problem Understanding the groups of branched twist-spun knots.
method Characterization through 3-manifold groups and conjectural algebraic approach.
result Each group is the group of at most finitely many branched twist spins.
New groups prevent certain geometric actions on spaces.
problem Preventing certain geometric actions on spaces.
method Analyzing cyclic orders on boundaries of trees.
result Groups prevent actions on PD(n) spaces.
We define an order relation among oriented PD4-complexes. We show that with respect to this relation, two PD4-complexes over the same complex are homotopy equivalent if and only if there is an isometry between the second homology groups. We also consider minimal objects of this relation.
Proves a theorem about groups and 3-manifolds.
problem Understanding the structure of groups and their relation to 3-manifolds.
method Relatively self-contained proof using algebraic fibration and PD^3-pair properties.
result Groups that fibrate and are part of PD^3-pairs are fundamental groups of fibred compact aspherical 3-manifolds.
Homotopy types of 4-manifolds tied to their fundamental groups.
problem Determining the homotopy type of 4-manifolds based on their fundamental groups.
method Uses the fundamental group, second homotopy group, first Stiefel-Whitney class, and equivariant intersection pairing.
result Homotopy type of 4-manifolds is determined by given group properties.
We consider the homotopy types of PD4-complexes X with fundamental group π such that c.d.π=2 and π has one end. Let β=β2(π;F2) and w=w1(X). Our main result is that (modulo two technical conditions on (π,w)) there are at most 2β orbits of k-invariants determining "strongly minimal" complexes (i.…
We show that there are two homotopy types of PD_3-complexes with fundamental group S_3*_{Z/2Z}S_3, and give explicit constructions for each, which differ only in the attachment of the top cell.
Turaev conjectured that the classification, realization and splitting results for Poincaré duality complexes of dimension 3 (PD3-complexes) generalize to PDn-complexes with (n−2)-connected universal cover for n≥3. Baues and Bleile showed that such complexes are classified, up to oriented homotopy eq…
This paper is a synthesis and extension of three earlier papers on PD4-complexes X with fundamental group π such that c.d.π=2 and π has one end. Our goal is to show that the homotopy types of such complexes are determined by π, the Stiefel-Whitney classes and the equivariant intersection pairing on $π_2(X)…
We study discrete group actions on coarse Poincare duality spaces, e.g. acyclic simplicial complexes which admit free cocompact group actions by Poincare duality groups. When G is an (n-1) dimensional duality group and X is a coarse Poincare duality space of formal dimension n, then a free simplicial action of G on X d…
Abstract: Study of 2-knot groups with restrictions on normal subgroups.
problem Characterizing 2-knot groups based on their normal subgroups.
method Analyzing PD_4-complexes and using properties of π_1(X).
result Characterization of 2-knot groups based on their normal subgroups.
This paper explores the relationship between generalized manifolds and Poincaré duality complexes.
problem Understanding the relationship between generalized manifolds and finite Poincaré duality complexes.
method Introducing Λ-Poincaré duality complexes, constructing 2-patch spaces, and using Gromov-Hausdorff metric.
result Generalized manifolds can be recognized within an enlarged class of Λ-Poincaré duality complexes.
A new kernel for persistence diagrams using Sliced Wasserstein distance.
problem Incorporating persistence diagrams into machine learning pipelines.
method Proposes a new kernel for persistence diagrams based on the Sliced Wasserstein approximation of the Wasserstein distance, demonstrating its stability and discriminative power.
result The proposed kernel is stable and discriminative, outperforming existing kernels on various benchmarks.
We define partial differential (PD in the following), i.e., field theoretic analogues of Hamiltonian systems on abstract symplectic manifolds and study their main properties, namely, PD Hamilton equations, PD Noether theorem, PD Poisson bracket, etc.. Unlike in standard multisymplectic approach to Hamiltonian field the…
The intention with this paper is to provide all the estimation concepts and techniques that are needed to implement a two-phases approach to the parametric estimation of probability of default (PD) curves. In the first phase of this approach, a raw PD curve is estimated based on parameters that reflect discriminatory p…
A new tree-based estimator, FastPD, efficiently estimates PD functions for machine learning models.
problem Efficiently estimating Partial Dependence functions for machine learning models.
method Proposes a new tree-based estimator, FastPD, to estimate PD functions.
result FastPD consistently estimates the desired population quantity and improves complexity from quadratic to linear.
We construct aspherical closed orientable 5-manifolds with perfect fundamental group. This completes part of our study (with D.H.Kochloukova and I.Lima) of PDn-groups with pro-p completion a pro-p Poincaré duality group of dimension ≤n−2. We also consider the question of whether there are any examples wit…
The paper stabilizes PD term structures under forecast uncertainty using a Kalman filter with an anchored observation model.
problem Stable estimation of lifetime PDs under forecast uncertainty.
method Reformulated in state-space framework, introduced an anchored observation model.
result Asymptotic stochastic stability of error dynamics, leading to smoother projections.
In this paper we formalize a combinatorial object for describing link diagrams called a Planar Diagram Code. PD-codes are used by the KnotTheory Mathematica package developed by Bar-Natan, et al. We present the set of PD-codes as a stand alone object and discuss its relationship with link diagrams. We give an explicit …
Extends characterization of PD3-pairs with aspherical boundaries to those with spherical boundaries.
problem Characterizing fundamental triples of PD3-pairs with boundary components of different types. method Extends Turaev and Bleile's work by relaxing the π1-injectivity hypothesis and considering pairs with spherical boundary components. result Characterization of fundamental triples for PD3-pairs with spherical boundary components and c.d.π1(P)≤2. Researchers formalize PD and PFI to relate them to data generating process.
problem Lack of theory linking PD and PFI to data generating process.
method Formalize PD and PFI as estimators of ground truth estimands, account for model variance with learner-PD and learner-PFI.
result PD and PFI estimates deviate from ground truth due to statistical biases, model variance, and Monte Carlo approximation errors.
Paper presents a method for estimating long-term PDs with incomplete data.
problem Estimating long-term PDs with limited and incomplete historical data.
method Single risk factor approach for simultaneous calibration of PDs across sub-portfolios.
result Method yields long-term PDs without requiring complete historical data.
New framework for efficient PD averaging and clustering.
problem Challenges in averaging and clustering persistence diagrams.
method Reformulate PD metrics as optimal transport problems, leveraging recent computational advances.
result Scalable computations of PD barycenters and clustering on thousands of diagrams.
Scaff-PD improves fairness and robustness in federated learning with reduced communication.
problem Improving fairness and robustness in federated learning with limited communication.
method Scaff-PD uses a family of distributionally robust objectives and an accelerated primal dual algorithm with bias-corrected steps.
result Scaff-PD achieves significant gains in communication efficiency and convergence speed while maintaining fairness and robustness.
If the probability of default parameters (PDs) fed as input into a credit portfolio model are estimated as through-the-cycle (TTC) PDs stressed market conditions have little impact on the results of the capital calculations conducted with the model. At first glance, this is totally different if the PDs are estimated as…
PD curve calibration refers to the transformation of a set of rating grade level probabilities of default (PDs) to another average PD level that is determined by a change of the underlying portfolio-wide PD. This paper presents a framework that allows to explore a variety of calibration approaches and the conditions un…
Paper solves open question about non-positive kernels by decomposing them into PD kernels.
problem Can non-positive definite kernels be decomposed into the difference of two positive definite kernels?
method Introduced signed measure to transform positive decomposition into measure decomposition, providing a sufficient and necessary condition.
result First random features algorithm for unbiased estimation of non-positive kernels.
New method converts complex PDs into stable vectors for ML.
problem Complex structure of persistence diagrams makes them hard to use in ML.
method Persistence bag-of-words (BoW) for vectorizing PDs.
result Achieves state-of-the-art performance and speed in ML.
Study on homeomorphism groups of manifolds using set theory.
problem Relationship between set theory and homeomorphism groups of manifolds.
method First-order rigidity, type versus conjugacy, axiom of constructibility, projective determinacy.
result Under V=L, homeomorphism groups of manifolds are first-order rigid and conjugacy class is determined by type.
A new kernel for comparing persistence diagrams without approximation.
problem Lack of suitable kernels for comparing persistence diagrams.
method Persistence Fisher kernel based on Fisher information geometry.
result Proposes a positive definite kernel for persistence diagrams without approximation.
Unreduced PDs can perform similarly to reduced PDs in machine learning tasks.
problem Ignoring much of the information in persistence diagrams in machine learning pipelines.
method Developed methods to generate topological feature vectors from unreduced boundary matrices.
result Unreduced PDs can perform on par with, and sometimes outperform, fully-reduced PDs in machine learning tasks.
Machine learning classifies Parkinson's Disease stages from walker sensors data.
problem Limited cost-effective methods for quantitatively assessing Parkinson's Disease stages.
method Machine learning applied to walker-mounted sensors data, feature selection methods compared.
result Feature selection method using ANOVA provides similar accuracy to full feature set and is clinically interpretable.
Tutorial on interpreting SPECT images for PD recognition using AI.
problem Difficulty in interpreting complex DL models for clinical use.
method Evaluation of six interpretation methods on four DCNN architectures.
result Guided backpropagation and SHAP methods are suitable for PD recognition.
Paper proposes E/PD-Control for better neural network training.
problem Training efficiency and robustness of CNNs in online data flows.
method E/PD-Control combines feedback PD controller with exponential signal.
result Better learning efficiency and robustness demonstrated experimentally.
Machine learning aids in diagnosing Parkinson's disease with higher accuracy.
problem Subjectivity in traditional PD diagnosis methods and missed early symptoms.
method Machine learning applied to various data modalities for PD and control group classification.
result Machine learning methods show high potential for improving PD diagnosis.
AI framework diagnoses Parkinson's disease with 100% accuracy.
problem Expertise-demanding medical imaging procedures for Parkinson's disease diagnosis.
method End-to-end, multi-modality diagnosis framework using T1-MRI and 11C-CFT PET.
result 100% accuracy in PD/NL classification.
Method predicts motor symptoms of Parkinson's disease from daily activities.
problem Objective monitoring of Parkinson's disease symptoms.
method Multi-layer Gaussian process models for three types of movement abnormalities.
result Strong agreement between model predictions and clinical annotations.