SPlit optimizes dataset splitting for better model performance.
problem Improving model performance through optimal dataset splitting.
method Adapting Support Points (SP) algorithm for subsampling and categorical variables in a sequential nearest neighbor approach.
result SPlit significantly improves worst-case testing performance compared to random splitting.
The paper extends keenness concept to bridge splittings and finds conditions for existence.
problem Extending keenness concept to bridge splittings and finding conditions for existence.
method Extending the concept of keenness to bridge splittings and proving existence conditions.
result Existence of strongly keen (g,b)-splitting of a link with distance n for certain integers g, b, and n. Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. Further it is shown that non-split structures appear in the first case as deformations of a split reduction and in the …
Study flippable Heegaard splittings in Seifert fibered spaces.
problem Identifying flippable Heegaard splittings in Seifert fibered spaces.
method Examining isotopies that interchange Heegaard splitting sides in Seifert fibered spaces.
result Characterized which Heegaard splittings are flippable in Seifert fibered spaces.
The paper proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
problem Proving the existence of infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
method Establishing a general sufficient condition for infinitely many topologically non-isotopic splitting 3-spheres in connected sums of 4-manifolds.
result Proves infinitely many non-isotopic splitting 3-spheres for split sphere links in 4D.
Bound on splitting number in terms of four-genus of knots.
problem Understanding the relationship between splitting number and four-genus of knots.
method Provided a bound on the splitting number in terms of the four-genus of related knots.
result Bound on splitting number in terms of four-genus.
We study the self-dual Yang-Mills equations in split signature. We give a special solution, called the basic split instanton, and describe the ADHM construction in the split signature. Moreover a split version of t'Hooft ansatz is described.
Paper proposes a novel SVM method for creating survival trees.
problem Creating non-linear survival trees for right-censored data.
method L2-regularized dipole splitting criteria with kernel methods.
result Non-linear splits using polynomial and Gaussian kernels show similar predictive power but often smaller tree sizes.
New methods improve prediction regions for high-dimensional data.
problem Creating effective prediction regions for high-dimensional data.
method CD-split and HPD-split methods that combine split method and data-driven partition.
result CD-split and HPD-split converge to oracle highest predictive density set and satisfy local and asymptotic conditional validity.
This paper constructs keen weakly reducible Heegaard splittings of arbitrary genus.
problem Creating Heegaard splittings with specific properties.
method Construction of keen weakly reducible splittings.
result Arbitrary genus Heegaard splittings can be constructed.
New spheres can split a 4D link in ways not possible in 3D.
problem Exploring how splitting spheres behave in 4D space.
method Constructing specific 2-component surface-links in S4. result Found non-isotopic splitting spheres in S4∖Lm,n. This paper studies properties of weak reducing pairs in critical Heegaard splittings.
problem Characterize weak reducing pairs in critical Heegaard splittings.
method Analyze the properties of weak reducing pairs in critical Heegaard splittings.
result Provide a necessary condition for a Heegaard surface to be critical.
The paper extends Lorentzian splitting theorems under curvature-dimension bounds.
problem Analyzing Lorentzian spacetimes with curvature-dimension bounds.
method Using Bakry-Émery-Ricci tensor, extending singularity and splitting theorems.
result Theorems hold for all synthetic dimensions, including negative and positive.
We show that if a split link is obtained from a split link L in S3 by 1/n-Dehn surgery along a trivial knot C, then the link L∪C is splittable. That is to say, it is impossible to obtain a split link from a split link via a non-trivial twisting. As its corollary, we completely determine when a trivial li…
New concept of strongly keen Heegaard splittings found.
problem Finding Heegaard splittings with specific properties.
method Introducing and proving existence of strongly keen Heegaard splittings.
result Existence of Heegaard splittings with specified genus and Hempel distance.
Study evaluates when splitting classifiers can improve performance despite disparate treatment.
problem Impact of disparate treatment in classification models.
method Comparison of split classifiers and group-blind classifiers, quantifying performance improvement.
result Proves an equivalent expression for the benefit-of-splitting which can be efficiently computed.
New methods convert complex link presentations to simpler, recognizable forms.
problem Representing split and composite links in a clear, recognizable way.
method Pocket and flip moves to transform link presentations.
result Pocket moves are the only obstruction to representing split links by split plat presentations.
Study shows arc and curve graphs are retracts of free group splitting complexes.
problem Understanding free group splittings through arc and curve graphs.
method Proved arc and curve graphs are coarse Lipschitz retracts of free splitting complexes and other related graphs.
result Arc and curve graphs are retracts of free group splitting complexes.
Powell's conjecture about Goeritz groups for genus 3 splittings is confirmed.
problem Determining the minimum set of elements needed to generate the Goeritz group of Heegaard splittings.
method Proof for splittings of genus 3 and a framework for higher genus splittings.
result Powell's conjecture is correct for splittings of genus 3.
The splitting number of a link is the minimal number of crossing changes between different components required, on any diagram, to convert it to a split link. We introduce new techniques to compute the splitting number, involving covering links and Alexander invariants. As an application, we completely determine the sp…
Lower bounds on splitting number derived from signature and nullity.
problem Determining the minimal number of crossing changes to split a link.
method Using multivariable signature and nullity to derive lower bounds.
result Efficiently computed splitting numbers for many links.
Little is known on the classification of Heegaard splittings for hyperbolic 3-manifolds. Although Kobayashi gave a complete classification of Heegaard splittings for the exteriors of 2-bridge knots, our knowledge of other classes is extremely limited. In particular, there are very few hyperbolic manifolds that are know…
New proof of Lorentzian splitting theorems using elliptic operators.
problem Proving splitting theorems in Lorentzian geometry.
method Using a negative homogeneity elliptic p-d'Alembert operator to prove theorems.
result Lorentzian splitting theorems are proven in a framework similar to Riemannian geometry.
New proof of Giroux Correspondence for tight contact 3-manifolds.
problem Proving the Giroux Correspondence for tight contact 3-manifolds.
method Introducing tight Heegaard splittings, using refinement process, and translating moves between splittings to moves between open books.
result Proves the tight Giroux Correspondence for contact 3-manifolds.
Paper generalizes splitting number to classify plane curve arrangements.
problem Distinguishing embedded topology of plane curves.
method Introduces splitting graph to generalize splitting number.
result Classifies embedded topology of specific plane curve arrangements.
Let M_1 and M_2 be compact, orientable 3-manifolds, and M the manifold obtained by gluing some component F of \bdy M_1 to some component of \bdy M_2 by a homeomorphism φ. We show that when φis "sufficiently complicated" then (1) the amalgamation of low genus, unstabilized, boundary-unstabilized Heegaard splittings of M…
Minimal splitting factors help study scalar curvature constraints.
problem Scalar curvature constraints in geometry.
method Introducing minimal splitting factors with positive scalar curvature.
result Minimal splitting factors have properties similar to area minimizing hypersurfaces.
Non-isotopic Heegaard splittings of non-minimal genus were known previously only for very special 3-manifolds. We show in this paper that they are in fact a wide spread phenomenon in 3-manifold theory: We exhibit a large class of knots and manifolds obtained by Dehn surgery on these knots which admit such splittings. M…
Introduces nonlinear splittings on fibre bundles for generalizing connections.
problem Generalizing connections on fibre bundles.
method Definition and properties of nonlinear splittings, including affine, homogeneous, and principal splittings.
result Curvature map defined for nonlinear splittings, linking to nonholonomic systems and magnetic Lagrangian systems.
Paper strengthens splitting theorem using weak KAM theory.
problem Splitting theorem in warped product spaces.
method Weak KAM theory approach.
result Strengthened splitting theorem.
New method escapes local optima in neural architecture optimization.
problem Escaping local optima in neural architecture optimization.
method Signed neural splitting in steepest descent framework.
result Escapes local optima, leading to better performance.
We define a notion of Hempel distance for one-sided Heegaard splittings and show that the existence of alternate surfaces restricts distance for one-sided splittings in a manner similar to Hartshorn's and Scharlemann-Tomova's results for two-sided splittings. We also show that every geometrically compressible one-sided…
We explicitly compute the lower algebraic K-theory of the split three-dimensional crystallographic groups; i.e., the groups G that act properly and cocompactly on three-dimensional Euclidean space by isometries, such that the natural map from G to O(3) is a split injection onto its image. There are 73 split three-dimen…
Splits a spectrum related to Madsen-Tillmann spectra at prime 2.
problem Homotopy equivalence and splitting of Madsen-Tillmann spectra.
method Uses Steinberg idempotents and Whitehead conjecture.
result Splits MTO(n) off BO(n)+ at prime 2. Complete noncompact solitons split at infinity under certain conditions.
problem Conditions for complete noncompact shrinking gradient Ricci solitons to split at infinity.
method Geometric conditions analysis.
result Conditions under which complete noncompact solitons split at infinity.
We describe an example of a closed orientable 3-manifold with distinct distance three genus two Heegaard splittings. This demonstrates that the constructions of alternate genus two Heegaard splittings of closed orientable 3-manifolds described by Rubinstein and Scharlemann in their 1998 paper Genus Two Heegaard Splitti…
New approach connects stochastic gradient descent to ODE splitting schemes.
problem Improving convergence in stochastic optimization.
method Connection between stochastic gradient descent and ODE splitting schemes.
result Derive a new upper bound on global splitting error.
Casson and Gordon gave the rectangle condition for strong irreducibility of Heegaard splittings [1]. We give a parity condition for irreducibility of Heegaard splittings of irreducible manifolds. As an application, we give examples of non-stabilized Heegaard splittings by doing a single Dehn twist.
Using basic properties of one-sided Heegaard splittings, a direct proof that geometrically compressible one-sided splittings of RP^3 are stabilised is given. The argument is modelled on that used by Waldhausen to show that two-sided splittings of S^3 are standard.
The paper studies Heegaard splittings of knots with high bridge distance and finds stabilization patterns.
problem Stabilization patterns of Heegaard splittings for knots with high bridge distance.
method Analysis of minimal genus Heegaard splittings and their equivalence under stabilizations.
result Two splittings from the same family become equivalent after at most one stabilization, and splittings from different families require up to n−1 stabilizations. Let M be a totally orientable graph manifold with characteristic submanifold T and let M = V cup_S W be a Heegaard splitting. We prove that S is standard. In particular, S is the amalgamation of strongly irreducible Heegaard splittings. The splitting surfaces S_i of these strongly irreducible Heegaard splittings have t…
A new method for kernel tests without data splitting increases power.
problem Lack of power in kernel-based tests due to data splitting.
method Selective inference framework to learn hyperparameters and test on full sample.
result Empirically larger test power without data splitting, regardless of split proportion.
We show that if a Heegaard splitting is obtained by gluing a splitting of Hempel distance at least 4 and the genus-1 splitting of S2×S1, then the Goeritz group of the splitting is finitely generated. To show this, we first provide a sufficient condition for a full subcomplex of the arc complex for a compact …
This paper studies Heegaard splittings of surface bundles via the curve complex of the fibre. The translation distance of the monodromy is the smallest distance it moves any vertex of the curve complex. We prove that the translation distance is bounded above in terms of the genus of any strongly irreducible Heegaard sp…
Extend Kauffman bracket skein module to homology theory using Heegaard splittings
problem Extend Kauffman bracket skein module to homology theory
method Combinatorial approach using Heegaard splittings
result Homology theory depends on Heegaard splittings
Twinning splits data into fast, statistically similar sets.
problem Creating statistically similar data splits for Big Data.
method Twinning is a method based on SPlit for fast, model-independent dataset splitting.
result Twinning is orders of magnitude faster than SPlit.
We construct families of manifolds that have pairs of genus g Heegaard splittings that must be stabilized roughly g times to become equivalent. We also show that when two unstabilized, boundary-unstabilized Heegaard splittings are amalgamated by a "sufficiently complicated" map, the resulting splitting is unstabili…
The study examines stock splits and their effects on companies, managers, and shareholders.
problem Misunderstandings and confounding factors around stock splits and their impacts.
method Selected database analysis of nine recent events, examining market impact, trading volume, and shareholder base.
result Stock splits enhance trading volume, increase shareholder base, and improve market liquidity.