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. 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.
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 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 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). 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. 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. 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.
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)…
Study of Tannakian categories for integrable connections on Kaehler manifolds.
problem Understanding Tannakian categories for integrable connections on Kaehler manifolds.
method Analyzing pairs (E, D) where E is a trivial holomorphic vector bundle and D is an integrable holomorphic connection.
result The pro-algebraic affine group scheme uniquely determines the isomorphism class of compact Riemann surfaces.
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.…
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.
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…
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.
CFR-Pro enhances treatment effect estimation by incorporating local proximity.
problem Treatment selection bias in HTE estimation from observational data.
method Proximity-enhanced CounterFactual Regression (CFR-Pro) with pair-wise proximity regularizer and subspace projector.
result Significantly outperforms competitors in HTE estimation accuracy.
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.
Given a pair of number fields with isomorphic rings of adeles, we construct bijections between objects associated to the pair. For instance we construct an isomorphism of Brauer groups that commutes with restriction. We additionally construct bijections between central simple algebras, maximal orders, various Galois co…
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. 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.
Study of p-subgroups in 3-manifold groups' completions.
problem Characterizing p-subgroups in 3-manifold group completions. method Complete description of finitely generated pro-p subgroups. result Completely described pro-p subgroups of 3-manifold group completions. Paper compares topological and pro-étale fundamental groups.
problem No specific problem stated; comparing two fundamental groups.
method Constructs a comparison map between topological and pro-étale fundamental groups.
result Establishes a map between fundamental groups.
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.
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.
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…
New method quantifies financial risk pro-cyclicality, identifying key factors.
problem Pro-cyclicality in financial risk measurement.
method Introduced a new indicator based on SQP, evaluated using 11 stock indices.
result Identified clustering and return-to-the-mean of volatility as key factors.
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.
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 …
Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the c…
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.
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.
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.
Introduces pro-rata rationing for groundwater markets to resolve supply-demand imbalances.
problem Resolving supply-demand imbalances in groundwater markets.
method Introduces a pro-rata rationing mechanism and analyzes its properties in markets with exogenous restrictions and a leader-follower setting.
result Pro-rata approach provides a unique and fair rationing device.
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 representation for braid groups and surface braid groups, extending Lawrence-Krammer-Bigelow.
problem Constructing representations for braid groups and surface braid groups.
method Pro-nilpotent tower of representations, starting with the original LKB representation.
result 3-variable enrichment of the Lawrence-Krammer-Bigelow representation.
The Burde--de Rham theorem is extended to finitely presented pro-p groups with specific conditions.
problem Extending the Burde--de Rham theorem to pro-p groups with certain constraints. method Assumption of total degrees of relators being 0, concrete examples, and cohomological interpretations.
result The theorem is extended to finitely presented pro-p groups under specified conditions. We show that the orientable double covering space of an indecomposable non-orientable PD3-complex has torsion free fundamental group.
We show that if X is an indecomposable PD3-complex and π1(X)isthefundamentalgroupofareducedfinitegraphoffinitegroupsbutisnotvirtuallycyclicthenXisorientable,theunderlyinggraphisatree,alltheedgegroupsareZ/2Zandallbutatmostoneofthevertexgroupsisdihedraloforder2m…
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.
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.
Khovanov homology for pro-tangles and spectral sequences
problem Developing a framework for Khovanov homology for pro-tangles and spectral sequences
method Using pro-tangles, simplicial presheaves, and spectral sequences
result Establishing a fully faithful embedding and an algebraic spectral sequence for pro-tangles