A new distance measure for circular Heegaard splittings helps understand knot exteriors.
problem Understanding the structure of knot exteriors using circular Heegaard splittings.
method Defining and analyzing circular distance for circular Heegaard splittings.
result Circular distance bounds properties of knot exteriors, like the uniqueness of minimal-genus Seifert surfaces.
Random survival forests (RSF) are a powerful method for risk prediction of right-censored outcomes in biomedical research. RSF use the log-rank split criterion to form an ensemble of survival trees. The most common approach to evaluate the prediction accuracy of a RSF model is Harrell's concordance index for survival d…
Paper establishes a new formula for Atiyah-Patodi-Singer index using eta invariants.
problem Calculating the Atiyah-Patodi-Singer index without invertibility of boundary operator.
method Using an asymptotic gluing formula for eta invariants and a splitting principle.
result Formula expressing index in terms of eta invariants of domain-wall massive Dirac operators.
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…
The study proves how groups can be split with limited complexity.
problem Understanding the complexity of group splittings.
method Analyzing trees with finite stabilizers and their quotient structures.
result Deformation spaces of trees have maximal complexity.
Minimal Heegaard surfaces are shown to have index 1 for generic metrics.
problem Existence and properties of minimal Heegaard surfaces in 3-manifolds.
method Generic metrics and min-max procedure for 3-spheres, strongly irreducible Heegaard sweepouts for 3-manifolds.
result Strongly irreducible Heegaard splittings can be isotoped to minimal surfaces of index at most 1 or one-sided minimal Heegaard surfaces.
Efficient algorithm achieves tightest bounds for active learning.
problem Active learning of general concept classes.
method Efficient algorithm that realizes the tightest upper and lower bounds.
result Empirically demonstrates good performance of the algorithm.
The Birman exact sequence cannot be split over any finite-index subgroup.
problem The inability to split the Birman exact sequence over finite-index subgroups.
method Proof by contradiction and analysis of the Torelli group.
result The Birman exact sequence does not virtually split.
The study shows that surface groups are the only non-free infinite index subgroups of certain hyperbolic groups.
problem Identifying the only non-free infinite index subgroups of specific hyperbolic and one-relator groups.
method Careful analysis of free and cyclic splittings of cubulated groups.
result Proves that surface groups are the only non-free infinite index subgroups of certain hyperbolic and one-relator groups.
We consider a curve of Fredholm pairs of Lagrangian subspaces in a fixed Banach space with continuously varying (weak) symplectic structures. Assuming vanishing index, we obtain intrinsically a continuously varying splitting of the total Banach space into pairs of symplectic subspaces. Using such decompositions we defi…
We describe a relation between Atiyah-Patodi-Singer boundary condition and a global elliptic boundary condition which naturally appears in formulating a splitting formula for a spectral flow, when we decompose the manifold into two components. Then we give a variant of the splitting formula with the Hoermander index as…
The construction of efficient and effective decision trees remains a key topic in machine learning because of their simplicity and flexibility. A lot of heuristic algorithms have been proposed to construct near-optimal decision trees. ID3, C4.5 and CART are classical decision tree algorithms and the split criteria they…
Study price impact in OTC credit index market without order book.
problem Estimate price impact in OTC credit index market with no order book.
method Applied propagator technique to classify trades and correct for errors.
result Price impact is mainly permanent in OTC credit index market.
The paper proves conditions for non-uniform expansion in partially hyperbolic systems.
problem Conditions for non-uniform expansion in partially hyperbolic systems.
method Analysis of Lyapunov exponents and dominated splittings.
result Existence of physical SRB measure under specific conditions.
New proof of splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
problem Proving splitting theorem and finite ends of minimal hypersurfaces in nonnegative curvature manifolds.
method New proof of splitting theorem and construction of weighted minimizing geodesics at infinity.
result Minimal hypersurfaces with finite index in manifolds with nonnegative biRic curvature must have finite ends.
We derive a decomposition formula for the spectral flow of a 1-parameter family of self-adjoint Dirac operators on an odd-dimensional manifold M split along a hypersurface Σ (M=X∪ΣY). No transversality or stretching hypotheses are assumed and the boundary conditions can be chosen arbitrarily. The formula tak…
Let (V,W;F) be a weakly reducible, unstabilized, genus three Heegaard splitting in an orientable, irreducible 3-manifold M. In this article, we prove that either the disk complex D(F) is contractible or F is critical. Hence, the topological index of F is two if F is topologically minimal.
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.
The paper is devoted to finding conditions to the existence of a self-indexing energy function for Morse-Smale diffeomorphisms on a 3-manifold. These conditions involve how the stable and unstable manifolds of saddle points are embedded in the ambient manifold. We also show that the existence of a self-indexing energy …
Paper proves bounds on knot indices using bisected vertex leveling of plane graphs.
problem Upper bounds on knot indices using plane graph embeddings.
method Bisected vertex leveling of plane graphs to prove upper bounds on braid index and arc index.
result Quadratic upper bound on minimal crossing number of delta diagrams.
Extends a theorem for first-order elliptic operators on manifolds.
problem Proving the relative index theorem for general first-order elliptic operators.
method Using boundary value problems and graphical decomposition of elliptically regular boundary conditions.
result Proves the relative index theorem for general first-order elliptic operators.
First, we prove a local spectral flow formula (Theorem 3.7) for a differentiable curve of selfadjoint Fredholm operators. This formula enables us to prove in a simple way a general spectral flow formula. Secondly, we prove a splitting formula (Theorem 4.12) for the spectral flow of a curve of selfadjoint elliptic opera…
Formula for spectrum linking braid and bridge indices.
problem Understanding the relationship between braid and bridge indices.
method Foliation theory applied to booklink embeddings.
result Formula for link spectra of braid and bridge indices.
A closed, orientable, splitting surface in an oriented 3-manifold is a topologically minimal surface of index n if its associated disk complex is (n−2)-connected but not (n−1)-connected. A critical surface is a topologically minimal surface of index 2. In this paper, we use an equivalent combinatorial definit…
Formula for Z_2-valued index of symmetric operators on manifolds.
problem Index of symmetric operators on manifolds with boundary.
method Cohomological formula for Z_2-valued index.
result A formula for the Z_2-valued index of operators on manifolds.
Affirmative answer to splitting question for open manifolds with nonnegative Ricci curvature and linear volume growth.
problem Understanding the structure of universal covers of open manifolds with nonnegative Ricci curvature and linear volume growth.
method Proving the universal cover splits off an isometric R-factor. result If an open manifold with nonnegative Ricci curvature has linear volume growth, its universal cover is isometric to a metric product RkimesN. Authors prove a conjecture about the Goeritz group of 3-sphere Heegaard splittings.
problem The Goeritz group of a genus g Heegaard splitting of the 3-sphere.
method Proof relies on the topological minimality of Heegaard surfaces and their disk complexes.
result The Goeritz group is generated by four specific elements for g ≥ 3.
The study improves inequalities for link diagrams and introduces weak rectangular diagrams.
problem Improving inequalities for link diagrams and understanding their properties.
method Introducing weak rectangular diagrams and proving new inequalities.
result Generalizes and subsumes many known inequalities related to multi-crossing numbers.
The paper challenges the validity of cluster validity measures in unsupervised learning.
problem The validity of cluster validity measures in selecting optimal clusterings.
method The authors investigate the use of cluster validity measures as objective functions in unsupervised learning and introduce a new variant of the Dunn index.
result Many cluster validity measures promote clusterings that do not match expert knowledge well.
The paper proves properties of minimal surfaces and Heegaard splittings in 3-manifolds.
problem Characterizing minimal surfaces and Heegaard splittings in 3-manifolds.
method Analyzes strongly irreducible Heegaard splittings and proves properties of minimal surfaces.
result Strongly irreducible Heegaard splittings are either isotopic to minimal surfaces of index at most one or to non-orientable minimal surfaces with a vertical handle.
Study submanifolds with relative nullity in space forms using splitting tensor.
problem Characterize submanifolds with relative nullity in space forms.
method Use splitting tensor and Codazzi equation to express second fundamental form.
result Derive new strong consequences in hyperbolic and Euclidean spaces.
Knots from a specific band sum have similar homologies but are distinct.
problem Cosmetic crossing conjecture for knots from a specific band sum.
method Analyzing knot homologies (Heegaard, instanton, Khovanov) to verify distinctness.
result Knots from the band sum are distinct despite similar homologies.
We prove a semi-Riemannian version of the celebrated Morse Index Theorem for geodesics in semi-Riemannian manifolds; we consider the general case of both endpoints variable on two submanifolds. The key role of the theory is played by the notion of the {\em Maslov index} of a semi-Riemannian geodesic, which is a homolog…
Decision tree classifiers are a widely used tool in data stream mining. The use of confidence intervals to estimate the gain associated with each split leads to very effective methods, like the popular Hoeffding tree algorithm. From a statistical viewpoint, the analysis of decision tree classifiers in a streaming setti…
Unified representation for tree ensembles indexed by nodes
problem Unifying geometric object for tree ensembles indexed by nodes
method KPP indexes feature map by nodes, weighted by path metric
result Unified non-diagonal Gram for prediction, additive attribution, robust radius, and risk bounds
Improved linear upper bound for ribbonlength of knots.
problem Estimating the ribbonlength of knots and links.
method Using four-page open book decompositions and spanning trees of checkerboard graphs, constructing a four-page presentation with at most 2c(K) arcs.
result Proved that ribbonlength is bounded above by the four-page index, leading to the linear bound Rib(K) ≤ 2c(K).
Let M be an orientable, irreducible 3-manifold and (V,W;F) a weakly reducible, unstabilized Heegaard splitting of M of genus at least three. In this article, we define an equivalent relation ∼ on the set of the generalized Heegaard splittings obtained by weak reductions and find special…
The paper studies a splitting theorem for a specific invariant of 4-manifolds.
problem Obstructing metrics of positive scalar curvature and finding integral homology 3-spheres.
method Proves a splitting formula for the Seiberg-Witten invariant in terms of the Frøyshov invariant and Lefschetz number.
result New classes of integral homology 3-spheres with infinite order in the homology cobordism group.
We introduce a monoid corresponding to knotted surfaces in four space, from its hyperbolic splitting represented by marked diagram in braid like form. It has four types of generators: two standard braid generators and two of singular type. Then we state relations on words that follows from topological Yoshikawa moves. …
New tree and forest methods use oblique splits for better risk bounds.
problem Improving risk bounds for regression algorithms.
method Randomized decision trees and forests with oblique splits.
result Oblique splits lead to better risk bounds for multi-index models.
Critical surfaces are defined by Bachman as topological index 2 surfaces, generalizing incompressible surfaces and strongly irreducible surfaces. In this paper we give a condition to obtain critical Heegaard surfaces by amalgamation. As a special case, we obtain critical Heegaard surfaces by boundary stabilization. It …
Proposes an efficient method for sparse index tracking with ℓ0-norm constraints.
problem Constructing a sparse portfolio to track a financial index.
method Formulates a new problem using ℓ0-norm constraints, develops an efficient algorithm based on primal-dual splitting. result Demonstrates effectiveness through experiments on S&P500 and Russell3000 datasets.
We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the spacetime. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to an integral line of the standar…
We examine three key conjectures in 3-manifold theory: the virtually Haken conjecture, the positive virtual b_1 conjecture and the virtually fibred conjecture. We explore the interaction of these conjectures with the following seemingly unrelated areas: eigenvalues of the Laplacian, and Heegaard splittings. We first gi…
The paper refines the three-page index for links, proving a new bound and characterizing specific links.
problem Investigating the three-page index invariant for links and proving bounds.
method Constructing three-page presentations from reduced link diagrams via binding circles and contractible subcomplexes.
result Proves a new bound for the three-page index and characterizes links achieving equality.
Paper uses neural networks to analyze oil price impact on Iranian stock and industry indices.
problem Impact of oil price volatility on Tehran stock and industry indices.
method Feed-forward neural networks analysis of two periods: sanctions and post-sanctions.
result Neural networks predict stock and industry indices well, showing significant oil price volatility impact.
Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
New minimal surface theory disproves a conjecture in symmetric spaces.
problem Proving the existence of unstable minimal maps in symmetric spaces.
method Using Hitchin representations and equivariant maps, with a new index bound.
result Disproves the Labourie conjecture for PSL(n,R) with n≥4.