This work introduces Lie 2-algebra structures for multiplicative sections of LA-groupoids.
problem Understanding algebraic structures of sections in LA-groupoids.
method Introducing and proving natural strict Lie 2-algebra structures on the category of multiplicative sections.
result The Lie algebra of geometric vector fields is described in various cases.
Defines basic sections of LA-groupoids for simpler modeling.
problem Modeling sections of stacky Lie algebroids.
method Introduces basic sections with injective core-anchor map.
result Basic sections are Morita invariant and equivalent to multiplicative sections.
Develops multivalued Jacobi fields for stable submanifolds.
problem Stability of minimally immersed submanifolds in Riemannian manifolds.
method Defines and studies multiple valued Jacobi fields as minimizers of a second variation functional.
result Any Q-valued Jacobi field can be decomposed into classical Jacobi fields except on a singular set of codimension at least two.
The study extends cobordism theory to complex sections, defining and calculating cobordism groups.
problem Understanding when almost complex manifolds can have complex sections.
method Defined complex section cobordism, determined groups, and introduced an obstruction.
result The obstruction vanishes for certain multiplicative generators in the complex cobordism ring.
The work deals with the risk assessment theory. An unitary risk algorithm is elaborated. The algorithm is based on parallel curves. The basic curve of risk is a hyperbolic curve, obtained as a multiplication between the probability of occurrence of certain event and its impact. Section 1 contains the problem formulatio…
Some optimization problems coming from the Differential Geometry, as for example, the minimal submanifolds problem and the harmonic maps problem are solved here via interior solutions of appropriate multitime optimal control problems. Section 1 underlines some science domains where appear multitime optimal control prob…
The paper studies the metric and algebraic structures on section rings of projective manifolds.
problem Understanding the relationship between metric and algebraic structures on section rings.
method Analyzes the section ring of projective manifolds and ample line bundles, proving approximate isometry properties under various norms.
result Characterizes L2-norms associated with continuous plurisubharmonic metrics and refines the theorem of Phong-Sturm. Estimates higher order derivatives using Lie derivatives and combinatorics.
problem Estimating higher order derivatives of Lie derivatives.
method Combines Lie derivatives, combinatorics of forests, and Dyck polynomials.
result Provides an estimate for higher order covariant derivatives of multiple Lie derivatives.
Researchers find multiple supporting minimal Lefschetz pencils for a complex fibration.
problem Finding multiple supporting minimal Lefschetz pencils for a specific Lefschetz fibration.
method Analyzing maximal disjoint (−1)-sections of the Matsumoto-Cadavid-Korkmaz Lefschetz fibration. result The Matsumoto-Cadavid-Korkmaz Lefschetz fibration has more than one supporting minimal Lefschetz pencils.
Revisits and proves a reparametrization theorem for multi-valued graphs in higher codimension.
problem Analyzing multi-valued sections of vector bundles and proving a reparametrization theorem.
method Develops properties of Q-multisections and provides a geometric proof. result Elementary and purely geometric proof of a reparametrization theorem for multi-valued graphs.
This thesis predicts the distribution of smoothed zeros of random sections on line bundles.
problem Predicting the distribution of smoothed zeros of random sections on line bundles.
method Developing smoothing operators on discrete surfaces and computing the expected sum of indices on each face.
result Predictions on the distribution of smoothed section's signed zeros with multiplicity.
This paper studies torsion obstructions to complex sections on manifolds.
problem Torsion obstructions to finding complex sections on almost complex manifolds.
method Calculations using the Adams-Novikov spectral sequence for Thom spectra.
result Torsion obstructions for finding r complex sections of order p vanish for r<p2−p. We prove that a sequence of solutions of the Seiberg-Witten equation with multiple spinors in dimension three can degenerate only by converging (after rescaling) to a Fueter section of a bundle of moduli spaces of ASD instantons.
Compactifies moduli space of Seiberg-Witten monopoles on a surface times a circle.
problem Compactify moduli space of Seiberg-Witten monopoles with multiple spinors.
method Define signed count of points in moduli space, construct analytic and algebro-geometric compactifications, and show invariance under perturbations.
result Signed count of isolated points in moduli space is independent of initial parameters, computed for low-genus surfaces.
The paper addresses how to add points to existing configurations on surfaces without disrupting continuity.
problem Adding points to existing configurations on orientable surfaces without disrupting continuity.
method Algebraic approach for g≥1 and m≥2, geometric approach for g≥1 and m=1. result A necessary condition for the existence of a section is that n must be a multiple of m+(2g−2) for g≥1 and m≥2. A new imputation model for clinical data captures both cross-sectional and temporal correlations.
problem Missing values in multivariable time series clinical data.
method Integrates Gaussian processes with mixture models and individualized mixing weights.
result The proposed model provides more accurate imputation than benchmarks on real-world and synthetic datasets.
New strategies help CNNs recognize tissue features across different stains.
problem Training deep learning models for images with multiple stains is challenging and expensive.
method Presented unsupervised training strategies that leverage one staining modality to improve performance on images with multiple stains.
result CNNs trained with these strategies outperform standard training methods on images with multiple stains.
Unified framework for complex financial networks using lattice theory.
problem Complex financial networks with multiple currencies and dependencies.
method Recast classical financial clearing model into lattice liability networks.
result Lattice-valued clearing sections form a complete lattice, enabling tractable analysis.
Study infinitesimal automorphisms of VB-groupoids and algebroids.
problem Understand transformations preserving the structure of VB-groupoids and algebroids.
method Examine vector fields generating flows that preserve both linear and groupoid/algebroid structures.
result Infinitesimal automorphisms of a special class are multiplicative sections of a derivation groupoid/algebroid.
We give a complete proof of a propagation theorem of multiplicity-free property from fibers to spaces of global sections for holomorphic vector bundles. The propagation theorem is formalised in three ways, aiming for producing various multiplicity-free theorems in representation theory for both finite and infinite dime…
The paper solves a problem related to exterior multiplication with singularities.
problem Finding sections of a p-exterior product given sections of a vector bundle.
method Analyzes the sufficiency of a condition involving regular singularities and depth of ideals.
result Shows that the condition is sufficient for existence of a continuous right inverse in smooth, real analytic, and holomorphic categories.
Variant of Seiberg-Witten equations for multiple-spinors connects to stability of holomorphic bundles.
problem Detecting stability of holomorphic vector bundles using Seiberg-Witten equations.
method Abelian gauge-theoretic variant of Seiberg-Witten equations for multiple-spinors.
result Constructs a numerical invariant related to φ−stability of SU(n)−holomorphic vector bundles. Kähler-Ricci solitons can be immersed into complex space forms if and only if the manifold is Einstein.
problem Characterizing Kähler-Ricci solitons that can be immersed into complex space forms.
method Proving that a Kähler-Ricci soliton's metric is Einstein if it can be immersed into a complex space form.
result The Kähler-Ricci soliton's metric is an Einstein metric if it can be immersed into a complex space form.
Method counts zeros of Betti map for elliptic surface sections.
problem Counting zeros of Betti map for elliptic surface sections.
method Differential-geometric approach using Kähler metrics.
result Explicit linear estimates of Betti map multiplicities.
The paper generalizes virtual knot theory using multiple types of virtual crossings.
problem Generalizing virtual knot theory to include multiple types of virtual crossings.
method Starting with graph theory, the paper reviews previous work and then constructs multi-virtual knots and links.
result The multiplicity of virtual crossings allows for a broader application of the Penrose evaluation to all trivalent graphs.
Defines new invariants for Riemann-Finsler manifolds, generalizing Preissman's theorem.
problem Finding metrics with negative sectional curvature on compact products.
method Defining a Q-valued deformation invariant and using it to generalize Preissman's theorem. result First and mostly sharp generalizations of Preissman's theorem on non-existence of negative sectional curvature metrics.
We prove that any rational linear combination of Pontryagin numbers that is not a multiple of the signature is unbounded on connected closed oriented manifolds of nonnegative sectional curvature. Combining our result with Gromov's finiteness result for the signature yields a new characterization of the L-genus.
We show that the skew-symmetrized product on every Leibniz algebra E can be realized on a reductive complement to a subalgebra in a Lie algebra. As a consequence, we construct a nonassociative multiplication on E which, when E is a Lie algebra, is derived from the integrated adjoint representation. We apply this constr…
Paper introduces a new risk measure for multivariate residual estimation.
problem Quantifying residual estimation risk in complex financial models.
method Developed a multivariate framework for residual estimation risk, defined using various risk measures, and proposed a back-testing criterion.
result Demonstrated the effectiveness of the new measure through back-testing on retail credit portfolios.
In this paper we study geometric coincidence problems in the spirit of the following problems by B. Grünbaum: How many affine diameters of a convex body in Rn must have a common point? How many centers (in some sense) of hyperplane sections of a convex body in Rn must coincide? One possible approa…
Motivated by the literature on investment flows and optimal trading, we examine intraday predictability in the cross-section of stock returns. We find a striking pattern of return continuation at half-hour intervals that are exact multiples of a trading day, and this effect lasts for at least 40 trading days. Volume, o…
Pipeline integrates cross-sectional and longitudinal multi-omics data for IBD research.
problem Integrating diverse data types from the same individuals for disease understanding.
method Statistical and deep learning methods for variable selection, feature extraction, and joint integration.
result Identified microbial pathways, metabolites, and genes discriminating IBD status.
We propose a new Bayesian tracking and parameter learning algorithm for non-linear non-Gaussian multiple target tracking (MTT) models. We design a Markov chain Monte Carlo (MCMC) algorithm to sample from the posterior distribution of the target states, birth and death times, and association of observations to targets, …
Curvature interpretation for WDVV equation in Frobenius manifolds.
problem Understanding the WDVV equation in statistical manifolds.
method Analyzing the curvature of statistical manifolds and their tangent spaces.
result WDVV equation is equivalent to zero sectional K-curvature. Quantizes Toda systems using geometric methods.
problem Quantizing Toda systems with geometric quantization.
method Geometric quantization of Toda systems as a coadjoint orbit of a group of matrices.
result Found unitary and non-unitary finite dimensional quantum Hilbert spaces.
Unified R packages for forecast reconciliation of constrained series.
problem Improving accuracy and coherence of forecasts for linearly constrained multiple time series.
method Classical and machine learning-based linear reconciliation approaches for cross-sectional, temporal, and cross-temporal frameworks.
result Unified toolbox for forecast reconciliation in R.
Study of choosing distinct points on cubic curves, proving impossibility.
problem Choosing distinct points on cubic plane curves continuously.
method Topological fiber bundle analysis, proving non-existence of continuous sections.
result Proves impossibility for non-multiple of 9 points.
Classifies orientation-reversing homeomorphisms of even periods on surfaces.
problem Classifying orientation-reversing homeomorphisms of even periods on surfaces.
method Following the approach of [1] and correcting errors in [1] for the case of periods multiple of 4.
result Classification for orientation-reversing homeomorphisms of periods multiple of 4.
Divide and conquer quantizes neural networks, improving accuracy.
problem Quantizing neural networks to reduce memory and compute.
method Divide a pretrained network into sections, train each section independently, then stitch them.
result Improves quantized training accuracy by 21.6% on average.
Solutions found for specific curvature problems in geometric settings.
problem Finding hypersurfaces with prescribed mean curvature in geometric settings.
method Proving existence of solutions to the asymptotic Plateau problem in Cartan-Hadamard manifolds.
result Existence of solutions to the asymptotic Plateau problem for hypersurfaces of prescribed mean curvature.
Representative investors whose behaviour is modelled by a deterministic finite automaton generate complexity both in the time series of each asset and in the cross-sectional correlation when the rule governing their behaviour is schizophrenic, meaning the investor must hold multiple seemingly contradictory beliefs simu…
Study covariant derivatives of eigenfunctions on curved spaces, proving they are scalar multiples of the functions.
problem Understanding covariant derivatives of eigenfunctions on curved spaces.
method Analyzing the Laplace-Beltrami operator on Riemannian manifolds with constant curvature.
result Covariant derivatives of eigenfunctions are scalar multiples of the functions, and these scalars are polynomials in the eigenvalue.
New method controls false discoveries in financial asset pricing.
problem Controlling false discoveries in time series with unknown correlations.
method Double bootstrapping method to control false discovery rate.
result Superior statistical power and controlled false discovery rate.
This work proposes a geometric approach to identify slow invariant manifolds in dynamical systems.
problem Identifying slow invariant manifolds in multiple time-scale dynamical systems.
method Differential geometric concepts for submanifolds, sectional curvature, flow invariance.
result Necessary condition for slow invariant manifold invariance stated in terms of differential geometry.
Given an integral symplectic manifold, we construct a family of "coherent state" maps into complex projective space. The maps are built from sections of the tensor powers of a hermitian line bundle whose curvature is a multiple of the symplectic form. We show that this family is an almost-complex version of the Kodiara…
Deep learning models can generalize well even when they fit training data perfectly.
problem Generalization in over-parameterized deep learning models.
method Combining empirical risk minimization with capacity control, exploring inductive biases and smooth empirical risk minimizers.
result Double descent phenomenon: test error can decrease after interpolation point.
Lp-cohomology of rank one symmetric spaces of noncompact type is shown to be Hausdorff for values of p where this does not follow from curvature pinching. Using the multiplicative structure on Lp-cohomology, it is shown that no simply connected Riemannian manifold with strictly -1/4-pinched sectional curvature …
Study uses CSIE to estimate portfolio volatility relative to market.
problem Estimating relative volatility risk of stock portfolios.
method Cross-sectional intrinsic entropy (CSIE) model to estimate cross-sectional volatility.
result Discover sets of symbols that outperform market indices in terms of return with similar or lower risk.