We define 2-crossed module bundle 2-gerbes related to general Lie 2-crossed modules and discuss their properties. A 2-crossed module bundle 2-gerbe over a manifold is defined in terms of a so called 2-crossed module bundle gerbe, which is a crossed module bundle gerbe equipped with an extra sructure. It is shown that s…
This paper studies the structure of a specific neighborhood in minimal 2-crossing charts.
problem Understanding the structure of a minimal 2-crossing chart with specific neighborhoods.
method Analyzes the neighborhoods of Γα∪Γβ and proposes a normal form for 2-crossing minimal n-charts. result Proposes a normal form for 2-crossing minimal n-charts. Paper classifies surface-links using charts with specific properties.
problem Classifying surface-links with specific charts.
method Using quandle colorings to differentiate charts representing different surface-links.
result Charts in the second class represent different surface-links.
We define the thin fundamental Gray 3-groupoid S3(M) of a smooth manifold M and define (by using differential geometric data) 3-dimensional holonomies, to be smooth strict Gray 3-groupoid maps S3(M)→C(H), where H is a 2-crossed module of Lie groups and C(H) is the Gray 3-groupoid naturally constructed f…
We show that the maximal number of singular moves required to pass between any two regularly homotopic planar or spherical curves with at most n crossings, grows quadratically with respect to n. Furthermore, this can be done with all curves along the way having at most n+2 crossings.
New bounds on odd multicrossing numbers of knots and links are established.
problem Determining bounds on the number of crossings in knots and links.
method Proved inequalities involving the (2k+1)-crossing number, 3-genus, and number of components of a link. result Established new bounds on the odd crossing numbers of torus knots and links.
We describe principal 3-bundles with adjusted connections using Lie algebras and groupoids.
problem Describing principal 3-bundles with adjusted connections.
method Derived explicit forms of adjustment data for 3-term L∞-algebras, integrated action Lie 3-algebroids to Lie 3-groupoids, and used differential cohomology. result Explicit description of principal 3-bundles with adjusted connections in terms of differential cohomology.
We prove that any diagram of the unknot with c crossings may be reduced to the trivial diagram using at most (236 c)^{11} Reidemeister moves. Moreover, every diagram in this sequence has at most (7 c)^2 crossings. We also prove a similar theorem for split links, which provides a polynomial upper bound on the number of …
We define an invariant of tangles and framed tangles given a finite crossed module and a pair of functions, called a Reidemeister pair, satisfying natural properties. We give several examples of Reidemeister pairs derived from racks, quandles, rack and quandle cocycles, 2-crossed modules and braided crossed modules. We…
In the 3-gauge theory, a 3-connection is given by a 1-form A valued in the Lie algebra g, a 2-form B valued in the Lie algebra h and a 3-form C valued in the Lie algebra l, where (g,h,l) constitutes a differential 2-crossed modu…
Study minimizes crossing points of up to 12 curves on a genus 2 surface.
problem Minimizing intersection points of curves on a surface.
method Analyzes systems of up to 12 simple closed curves on a genus 2 surface to find the minimum crossing number.
result Determines the minimal crossing number of up to 12 curves on a genus 2 surface and proves the minimization systems are unique.
New index principle shows indistinguishability of certain knot crossings.
problem Identifying indistinguishable crossings in knot diagrams.
method Developed a universal index function on knot diagrams that respects Reidemeister moves and sign of crossings.
result Crossings of the same sign in a classical knot diagram cannot be distinguished by any inherent property.
Polynomial bound on Reidemeister moves for each link type.
problem Recognizing whether a given link diagram represents a specific link type.
method Showed existence of a polynomial pK such that any two diagrams of a link type differ by at most pK(c1)+pK(c2) Reidemeister moves. result The problem of recognising a link type is in NP and can be completed in exponential time.
Polynomial algorithm found for alternating link equivalence.
problem Link equivalence of alternating links in 3-space.
method Tait flyping conjectures, observations from graph theory, and topological graph theory.
result Alternating link equivalence has a polynomial algorithm.
Machine learning portfolios perform well with simple imputation of missing data.
problem Handling missing values in machine learning portfolios constructed from cross-sectional return predictors.
method Simple imputation with cross-sectional means compared to rigorous expectation-maximization methods.
result Simple imputation performs well due to the structure of missing data.
This paper compares deep transfer learning with classical ML in low-shot text classification.
problem Low-shot text classification with limited labeled data.
method Comparison of BERT and top classical ML approaches on a sentiment classification task.
result BERT outperforms classical ML by 9.7% on average with 100 labeled examples per class.
CopulaSMOTE addresses class imbalance in diabetes prediction models.
problem Class imbalance in diabetes prediction models, especially with fewer confirmed cases.
method Copula-based oversampling approach that models joint dependence structure.
result CopulaSMOTE improves minority-class recovery in larger diabetes datasets.
Masanao Aoki developed a new methodology for a basic problem of economics: deducing rigorously the macroeconomic dynamics as emerging from the interactions of many individual agents. This includes deduction of the fractal / intermittent fluctuations of macroeconomic quantities from the granularity of the mezo-economic …
Proves links can be simplified to trivial form in few changes, limiting Milnor's invariants.
problem Limits of Milnor's invariants in understanding unlinking number.
method Sequence of links, crossing changes, and geometric filtration.
result Crossing changes to Ck-trivial link grows quadratically with number of components.