Spark complexes defined on good effective orbifold atlases.
problem Constructing a structured representation for effective orbifolds.
method Defining good atlases and constructing spark complexes categorically.
result Spark character 2-functor factors through the constructed 2-functor.
Fine-grained atlases improve fMRI analysis of brain activity.
problem Large fMRI datasets require scalable brain network summaries.
method Trained on millions of fMRI volumes, DiFuMo dictionaries of 64-1024 networks.
result Fine-grained atlases enhance classic fMRI analysis pipelines.
New method creates personalized brain atlases from large datasets.
problem Limited generalizability and spatial specificity of traditional probabilistic atlases.
method Data-driven clustering of regions using point distribution models.
result Personalized probabilistic atlases adapt quickly to new subjects.
We give a definition of atlases for ineffective orbifolds, and prove that this definition leads to the same notion of orbifold as that defined via topological groupoids.
Starting with some motivating examples (classical atlases for a manifold, space of leaves of a foliation, group orbits), we propose to view a Lie groupoid as a generalized atlas for the "virtual structure" of its orbit space, the equivalence between atlases being here the smooth Morita equivalence. This "structure" kee…
Analyzes how suborbifolds relate to groupoid embeddings.
problem Understanding the relationship between suborbifolds and groupoid embeddings.
method Examines the correspondence between suborbifolds and groupoid embeddings via atlases and effective orbifold groupoids.
result Identifies classes of suborbifolds that naturally lead to groupoid embeddings.
New method combines prior knowledge and brain atlases for fMRI analysis.
problem Matrix factorization formulation of task-related fMRI problem.
method Incorporates prior knowledge from experimental design and brain atlases, uses novel sparsity promoting constraint.
result Efficiently copes with uncertainties and selection of sparsity parameters.
We study the number of Darboux charts needed to cover a closed connected symplectic manifold (M,ω), and effectively estimate this number from below and from above in terms of the Lusternik--Schnirelmann category of M and the Gromov width of (M,ω).
We construct an infinite sequence of projectively flat manifolds by using castling transformations of prehomogeneous vector spaces. We also give a classification of manifolds equipped with a flat projective structure obtained by a finite number of castling transformations, and describe these flat projective structures …
The Jacobian conjecture is simplified using polynomial mappings.
problem Simplifying the Jacobian conjecture over the real field.
method Using polynomial mappings to restrict transitions on manifolds.
result An equivalent statement of the Jacobian conjecture.
New brain atlas method improves classification accuracy.
problem Creating accurate brain atlases from connectomes.
method Connectivity-based hierarchical clustering and consensus aggregation.
result Consensus parcellation outperforms existing atlases in classification tasks.
Wider networks learn more interpretable features and improve performance during fine-tuning.
problem Transferability of learned features between tasks and the effect of network width on feature learning.
method Activation atlases to visualize and analyze features learned by wide and narrow networks.
result The hidden state of a wide network contains more information about the inputs than a narrow network, leading to improved performance during fine-tuning.
New method uses entropy to generate multiple plausible causal maps.
problem Learning causal relationships from noisy data can lead to artifacts in DAGs.
method Entropy-based inference to generate an ensemble of plausible causal graphs.
result Multiple causal maps consistent with underlying data variability.
Toric quasifolds extend toric geometry to non-rational polytopes.
problem Extending toric geometry to non-rational polytopes.
method Introduced toric quasifolds to solve symplectic extension problem.
result Illustrated toric quasifolds and their atlases.
We describe a bicategory (RedOrb) of reduced orbifolds in the framework of classical differential geometry (i.e. without any explicit reference to notions of Lie groupoids or differentiable stacks, but only using orbifold atlases, local lifts and changes of charts). In order to construct such a …
New bridge between diffeology and noncommutative geometry.
problem Connecting diffeology and noncommutative geometry.
method Embedding quasifolds into diffeology and associating C*-algebras.
result Morita classes of C*-algebras associated with diffeomorphic quasifolds.
Accumulation of standardized data collections is opening up novel opportunities for holistic characterization of genome function. The limited scalability of current preprocessing techniques has, however, formed a bottleneck for full utilization of contemporary microarray collections. While short oligonucleotide arrays …
In this paper we compute the minimal number of Darboux chart needed to cover a Hermitian symmetric space of compact type in terms of the degree of their embeddings in CPN. The proof is based on the recent work of Y. B. Rudyak and F. Schlenk [18] and on the symplectic geometry tool developed by the first au…
DET unifies geometric and functional alignment for high-dimensional scientific data.
problem Challenges in nonrigid registration for high-dimensional, irregular data.
method Domain Elastic Transform (DET) treats data as functions on irregular domains, using a Bayesian framework for elastic motion registration.
result DET achieves 92% topological preservation on MERFISH data and successfully registers whole-embryo Stereo-seq atlases.
We study the minimal number C(M,ξ) of contact charts that one needs to cover a closed connected contact manifold (M,ξ). Our basic result is C(M,ξ) \le \dim M + 1. We compute C(M,ξ) for all closed connected contact 3-manifolds: C (M,ξ) = 2 if M = S^3 and ξis tight, 3 if M = S^3 and ξis overtwisted or if M = #_k (S^2 \ti…
Resting-state functional Magnetic Resonance Imaging (R-fMRI) holds the promise to reveal functional biomarkers of neuropsychiatric disorders. However, extracting such biomarkers is challenging for complex multi-faceted neuropatholo-gies, such as autism spectrum disorders. Large multi-site datasets increase sample sizes…
We analyze the possibility of defining infinite-dimensional manifolds as ringed spaces. More precisely, we consider three definitions of manifolds modeled on locally convex spaces: in terms of charts and atlases, in terms of ringed spaces, and in terms of functored spaces, as introduced by Douady in his thesis. It is s…
'Kuranishi spaces' were introduced in the work of Fukaya, Oh, Ohta and Ono in symplectic geometry (see e.g. arXiv:1106.4882), as the geometric structure on moduli spaces of J-holomorphic curves. An alternative to Kuranishi spaces is the 'polyfolds' of Hofer, Wysocki and Zehnder (see e.g. arXiv:1407.3185). Finding a s…
The first, second and fourth Painlevé equations are studied by means of dynamical systems theory and three dimensional weighted projective spaces $\C P^3(p,q,r,s)$ with suitable weights (p,q,r,s) determined by the Newton diagrams of the equations or the versal deformations of vector fields. Singular normal forms of t…
The paper develops a theory of Ehresmann structures in positive characteristic.
problem Developing a theory for Ehresmann structures in positive characteristic.
method Comparing Frobenius-Ehresmann structures with Cartan geometries and studying their equivalence.
result Formulating and proving the Ehresmann-Weil-Thurston principle for Frobenius-Ehresmann structures.
Novel framework for medical image segmentation using deep learning.
problem Class imbalance and domain adaptation in medical image segmentation.
method Biophysics-based domain adaptation and automatic segmentation of white, gray, and cerebrospinal fluid.
result Improved segmentation performance, especially with the biophysics-based domain adaptation.
Ensemble learning improves rs-fMRI predictions using 3D CNNs.
problem Improving specificity and sensitivity of rs-fMRI measurements through better parcellation schemes.
method Ensemble learning with 3D CNNs to combine predictions from different parcellations.
result Ensemble learning with 3D CNNs outperforms traditional methods in rs-fMRI classification and regression tasks.
The study classifies rational 1-forms on the Riemann sphere with simple poles.
problem Classifying rational 1-forms on the Riemann sphere with specified pole conditions.
method Recognized three equivalent atlases, proved submanifold properties, and used PSL(2,C) action.
result Quotients of isochronous 1-forms admit stratified orbit types.
Proves sufficient condition for 2D orbifolds to be good.
problem Characterizing 2D orbifolds as good.
method Analyzes orbifold fundamental groups for goodness.
result Connected 2D orbifolds with infinite orbifold fundamental group are good.
The paper defines conditions for good involutions in generalized Alexander quandles.
problem Determining conditions for good involutions in generalized Alexander quandles.
method Analyzing the structure of generalized Alexander quandles and their involutions.
result Classification of all good involutions in connected generalized Alexander quandles.
Graph Neural Network identifies ASD biomarkers from fMRI data.
problem Finding biomarkers for Autism Spectrum Disorder (ASD).
method Graph Neural Network (GNN) for analyzing task-fMRI brain networks, 2-stage pipeline to interpret feature importance.
result GNN achieves high accuracy in identifying ASD biomarkers and reveals their association with social behaviors.
3D good continuation model explains stereo vision using neurogeometry.
problem Understanding how the brain processes 3D visual correspondence.
method Developed a neurogeometric model involving spatial and orientation disparities.
result Provides insight into neural organization and correspondence problem.
We study convex risk measures describing the upper and lower bounds of a good deal bound, which is a subinterval of a no-arbitrage pricing bound. We call such a convex risk measure a good deal valuation and give a set of equivalent conditions for its existence in terms of market. A good deal valuation is characterized …
Proves correspondence between harmonic and Higgs bundles.
problem Connecting harmonic and Higgs bundles for study.
method Kobayashi-Hitchin correspondence for polystable bundles.
result Establishes correspondence between good wild harmonic bundles and polystable good filtered λ-flat bundles. The study describes good involutions in quandles and Alexander quandles.
problem Characterizing and enumerating good involutions in quandles and Alexander quandles.
method Completely describing good involutions of free and subquandles of twisted conjugation quandles of groups, including Alexander quandles.
result Explicit mappings for good involutions of linear quandles up to order 23.
The study proves symplectic quandles cannot have good involutions.
problem Existence of good involutions in symplectic quandles.
method Investigation of necessary and sufficient conditions for good involutions.
result Nonexistence of good involutions in symplectic quandles.
This paper studies an environment of simultaneous, separate, first-price auctions for complementary goods. Agents observe private values of each good before making bids, and the complementarity between goods is explicitly incorporated in their utility. For simplicity, a model is presented with two first-price auctions …
FF algorithm uses goodness as a likelihood-ratio test for scalar normalization.
problem Training each layer locally with scalar goodness.
method FF algorithm uses a likelihood-ratio test with squared goodness as the sufficient statistic.
result The FF algorithm generalizes to anisotropic and heavy-tailed populations.
We study a notion of good-deal hedging, that corresponds to good-deal valuation for generalized good-deal constraints. Under model uncertainty about the market prices of risk of hedging assets, a robust approach leads to a reduction or even elimination of a speculative component in good-deal hedging, which is shown to …
Paper tackles good arm identification in stochastic bandits.
problem Identifying good arms with minimal samples.
method Proposes DGAI, a differentiable algorithm to improve sample complexity.
result DGAI outperforms baseline algorithms in synthetic and real-world datasets.
FF algorithm uses goodness as a measure of input quality, derived from likelihood-ratio tests.
problem Training each layer locally with a goodness measure.
method FF algorithm uses a likelihood-ratio test to define goodness, which is the sum of squared activations normalized between layers.
result The goodness measure is a sufficient statistic for a likelihood-ratio test, explaining the FF algorithm's performance.
Classifies good involutions in conjugation subquandles and racks.
problem Classifying quandles with good involutions for applications in surface-knot theory.
method Study of subquandles of conjugation quandles, including core quandles; analysis of good involutions of faithful racks.
result Sharp bounds on the number of good involutions of racks in these families.
New method freely slices good boundary links with specific conditions.
problem Slicing good boundary links with multiple components.
method Using a Seifert surface and homotopically trivial plus assumption.
result Provides new freely slice links and subsumes previous methods.
We shall provide in this paper good deal pricing bounds for contingent claims induced by the shortfall risk with some loss function. Assumptions we impose on loss functions and contingent claims are very mild. We prove that the upper and lower bounds of good deal pricing bounds are expressed by convex risk measures on …
A good cover in R^d is a collection of open contractible sets in R^d such that the intersection of any subcollection is either contractible or empty. Motivated by an analogy with convex sets, intersection patterns of good covers were studied intensively. Our main result is that intersection patterns of good covers are …
Study lenient regret and good-action identification in Gaussian process bandits.
problem Optimizing function values above a certain threshold in Gaussian process bandits.
method Study lenient regret notions and introduce algorithms for finding good actions.
result Upper and lower bounds on lenient regret for GP-UCB and elimination algorithms.
Kinetic models predict speculators' strategy can affect market prices.
problem Understanding how speculators' behavior affects market prices in a multi-agent exchange system.
method Developed kinetic equations to model interactions between dealers and speculators, using utility functions and mean quantities.
result Speculators' strategy can drive the price of goods towards a zone with marked utility for their group.
New algorithms find all ε-good arms in stochastic bandits.
problem Finding all arms with means above a specified threshold in stochastic bandits.
method Two algorithms introduced to identify all ε-good arms.
result Demonstrated great empirical performance on large datasets.