New constructions from non-separating planar graphs improve understanding of graph linkability and knotability.
problem Understanding linkability and knotability of graph complements.
method Using maximal non-separating planar graphs to construct examples of maximal linkless and knotless graphs, and analyzing their Colin de Verdière invariant.
result The Colin de Verdière invariant of the complement of a maximal non-separating planar graph satisfies μ(cG) ≤ n-4, and equality holds.
The complement of a non-separating planar graph contains a K_n minor.
problem Characterizing the structure of complements of planar graphs.
method Analyzing the structure of complements of non-separating planar graphs and using examples to illustrate hypotheses.
result The order 2n-3 is the lowest possible for a non-separating planar graph whose complement contains a K_n minor.
Minimal simplicial complexes in high dimensions always contain complex links.
problem Existence of complex links in high-dimensional embeddings.
method Demonstrated through minimal simplicial complexes in R2n. result Minimal simplicial n-complexes inevitably contain a nonsplittable two-component link. Determining the space of free discrete two generator groups of Möbius transformations is an old and difficult problem. In this paper we show how to construct large balls of full dimension in this space. To do this, we begin with a marked discrete group of non-separating disjoint circle type. Such a group determines thr…
Surgery on knots can produce non-separating spheres, using Heegaard Floer homology.
problem Conditions for surgery on knots to produce non-separating spheres.
method Heegaard Floer homology
result Sufficient conditions for a knot to be unknotted.
Finite rigid sets found in complex of curves for surfaces.
problem Finding finite rigid sets in curve complexes of surfaces.
method Exhaustion by finite rigid sets proved for surfaces of finite type and genus ≥3.
result Finite rigid sets exist in the non-separating curve complex of surfaces.
Neural networks can separate non-separable data using feature maps.
problem Non-separable data in neural networks.
method Characterization of feedforward neural networks and use of feature maps.
result ReLU neural networks can separate concentric data.
Paper examines Dehn twists on non-orientable surfaces and their limitations.
problem Limitations of generating Dehn twists on non-orientable surfaces.
method Analyzes the level 2 mapping class group of non-orientable surfaces and their subgroups.
result Dehn twist subgroup of M2(Ng) cannot be generated by squares of Dehn twists about non-separating curves. We construct Bing houses in all dimensions n≥3, obtaining non-separating PL immersions of Sn→Rn+1.
A non-separating multicurve of a surface S of genus g with m punctures is a multicurve c so that S-c is connected. For k>0 define the graph of non-separting k-multicurves to be the graph whose vertices are non-separating multicurves with k components and where two such multicurves are connected by an edge if they can b…
New non-separable covariance kernels for spatiotemporal data derived from harmonic oscillator physics.
problem Capturing complex spatiotemporal dependencies in Gaussian processes.
method Hybrid spectral method based on the harmonic oscillator, deriving explicit covariance kernels.
result Explicit non-separable covariance kernels with space-time interactions.
If a 3--manifold Y contains a non-separating sphere, then some twisted Heegaard Floer homology of Y is zero. This simple fact allows us to prove several results about Dehn surgery on knots in such manifolds. Similar results have been proved for knots in L--spaces.
We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to…
Study shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.
problem Computing shortest non-separating simple closed curves on non-orientable surfaces.
method Developed tools for computing shortest curves, proving NP-hardness and tractability.
result Proved NP-hardness and fixed-parameter tractability for computing shortest orienting curves, and polynomial-time algorithm for non-orienting curves.
Gradient descent converges to perfect classification in neural nets for non-separable data.
problem Classifying linearly non-separable data using neural networks.
method Analysis of gradient descent dynamics in neural networks with sufficient but not large number of neurons.
result Gradient descent converges to global minima with perfect classification in the landscape of minimization problems.
We obtain a finite generating set for the level 2 twist subgroup of the mapping class group of a closed non-orientable surface. The generating set consists of crosscap pushing maps along non-separating two-sided simple loops and squares of Dehn twists along non-separating two-sided simple closed curves. We also prove t…
This paper restricts efficient geodesics to non-separating curves.
problem Finding efficient geodesics in the complex of curves.
method Analysis of the dot graph and surgeries.
result Efficient geodesics can be restricted to the non-separating curve complex.
For n >2, we shall show that the group Aut(NS(M)) of simplicial automorphisms of the complex NS(M) of non-separating embedded spheres in the manifold M,connected sum of n copies of S^2 X S^1, isomorphic to the group Out(F_n) of outer automorphisms of the free group F_n, where Fn is identified with the fundamental gr…
New algorithms reduce slate bandit regret for large slates, outperforming existing methods.
problem Non-separable reward functions in slate bandits with many slates.
method Design of algorithms with sub-linear regret.
result Sub-linear regret with respect to the time horizon for large number of slates.
Logistic regression is one of the most popular methods in binary classification, wherein estimation of model parameters is carried out by solving the maximum likelihood (ML) optimization problem, and the ML estimator is defined to be the optimal solution of this problem. It is well known that the ML estimator exists wh…
The paper proves local laws for non-separable sample covariance matrices.
problem Analyzing non-separable sample covariance matrices with dependent or nonlinearly transformed data.
method Tensor network framework for analyzing fluctuation averaging in the presence of higher-order cumulant structure.
result Optimal averaged local law and full anisotropic local law for non-separable sample covariance matrices.
We prove integral rigidity for Seiberg-Witten invariants of 4-manifolds with specific hypersurfaces.
problem Integral rigidity of Seiberg-Witten invariants in 4-manifolds with non-separating hypersurfaces.
method Floer theoretic conditions and interplay between irreducible and reducible solutions to Seiberg-Witten equations.
result Sum of Seiberg-Witten invariants is determined cohomologically for specific 4-manifolds.
Safe screening rule improves Group SLOPE efficiency.
problem Efficiently selecting groups of predictors in high-dimensional sparse learning.
method Safe screening rule for Group SLOPE, addressing block non-separable group effects.
result Significant computational efficiency gains without sacrificing accuracy.
In the curve complex for a surface, a handlebody set is the set of loops that bound properly embedded disks in a given handlebody bounded by the surface. A boundary set is the set of non-separating loops in the curve complex that bound two-sided, properly embedded surfaces. For a Heegaard splitting, the distance betwee…
Reduces connectivity problem for genus-4 Heegaard surface in 3-sphere.
problem Connectivity problem in reducing sphere complex for genus-4 Heegaard surface.
method Presented a sufficient condition for a non-separating weak reducing pair to be separated by a reducing sphere.
result Reduced connectivity problem to showing disjointness of representative reducing spheres from a fixed disk.
For an infinite cardinal κ let ℓ2(κ) be the linear hull of the standard othonormal base of the Hilbert space ℓ2(κ) of density κ. We prove that a non-separable convex subset X of density κ in a locally convex linear metric space if homeomorphic to the space (i) ℓ2f(κ) if and only if X can be…
We discuss the ζ−regularized determinant of elliptic boundary value problems on a line segment. Our framework is applicable for separated and non-separated boundary conditions.
The unknot U in S^4 has non-unique smooth spanning 3-balls up to isotopy fixing U. Equivalently there are properly embedded non-separating 3-balls in S^1xB^3 not properly isotopic to 1xB^3. More generally there exist non-separating 3-spheres in S^1xS^3 not isotopic to 1xS^3 and non trivial elements of π_0 Diff_0(S^1xS^…
New graph kernels capture spatio-temporal interactions.
problem Lack of justified spatio-temporal graph kernels for graph problems.
method Derive graph kernels via SPDEs for spatio-temporal modelling.
result Non-separable spatio-temporal graph kernels outperform existing ones.
Characterizes minor-minimal separating projective planar graphs and their generalizations.
problem Understanding projective planar graphs and their properties.
method Analyzing minors, embeddings, and specific link types.
result Partial characterization of minor-minimal separating projective planar graphs and their generalizations.
We introduce an iterative optimization scheme for convex objectives consisting of a linear loss and a non-separable penalty, based on the expectation-consistent approximation and the vector approximate message-passing (VAMP) algorithm. Specifically, the penalties we approach are convex on a linear transformation of the…
To any compact Riemann surface of genus g one may assign a principally polarized abelian variety of dimension g, the Jacobian of the Riemann surface. The Jacobian is a complex torus, and a Gram matrix of the lattice of a Jacobian is called a period Gram matrix. This paper provides upper and lower bounds for all the ent…
We construct an isotopy of a planar compactum that is not the restriction of an isotopy of any planar continuum.
We generalize the classical Szpiro inequality to the case of a semistable family of hyperelliptic curves. We show that for a semistable symplectic Lefschetz fibration of hyperelliptic curves of genus g, the number N of non-separating vanishing cycles and the number D of singular fibers satisfy the inequality $N \…
We present a necessary and sufficient condition for existence of a contractible, non-separating and noncontractible separating Hamiltonian cycle in the edge graph of polyhedral maps on surfaces. In particular, we show the existence of contractible Hamiltonian cycle in equivelar triangulated maps. We also present an alg…
Study examines how changing regions affects planar graphs.
problem Effect of region crossing change on planar trivalent graphs.
method Investigation of region crossing changes on planar trivalent graphs.
result Effect of region crossing change on planar trivalent graphs.
In her seminal 2008 paper, Maryam Mirzakhani showed that the ratio that two topological types of curves occur in is a rational number. In this paper we describe the process by which we obtained experimental evidence that separating and non-separating curves on the surface of genus two occur in the ratio 1 : 48.
In this paper, we introduce the notions of an iterated planar Lefschetz fibration and an iterated planar open book decomposition and prove the Weinstein conjecture for contact manifolds supporting an open book that has iterated planar pages. For n≥1, we show that a (2n+1)-dimensional contact manifold M suppor…
New inequalities for planar convex domains' Laplacian eigenvalues.
problem Neumann eigenvalues of the Laplacian on planar convex domains.
method Established two new universal inequalities.
result New inequalities for Laplacian eigenvalues on convex domains.
Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
problem Quasi-transitive graphs quasi-isometric to planar graphs need to be upgraded to Cayley graphs.
method Upgrading a planar graph to a Cayley graph.
result Quasi-transitive graphs quasi-isometric to planar graphs can be upgraded to Cayley graphs.
Identifies Heegaard Floer homology solid tori via Dehn fillings.
problem Characterizing Heegaard Floer homology solid tori.
method Using Dehn fillings to identify solid tori.
result Characterized Seifert fibered Heegaard Floer solid tori.
Modifying the method of [21], we compute the perturbed HF+ for some special classes of fibered three manifolds in the second highest spinc-structures Sg−2. The special classes considered in this paper include the mapping tori of Dehn twists along a single non-separating curve and along a transverse pair of c…
Study of higher-dimensional contact manifolds and their properties.
problem Understanding contact manifolds in higher dimensions.
method Topological and symplectic methods, including open books.
result Many contact manifolds can be realized as iterated planar contact manifolds.
Proves planar graphs' configuration spaces have highest topological complexity.
problem Proving Farber's conjecture for planar graphs.
method Generic maximality argument for topological complexities.
result Generic maximality of topological complexities for planar graphs.
We find an invariant characterization of planar webs of maximum rank. For 4-webs, we prove that a planar 4-web is of maximum rank three if and only if it is linearizable and its curvature vanishes. This result leads to the direct web-theoretical proof of the Poincaré's theorem: a planar 4-web of maximum rank is lineari…
We characterize those planar Peano continua that are homotopy equivalent to 1-dimensional sets. While many planar Peano continua are not homotopically 1-dimensional, we prove that each has fundamental group that embeds in the fundamental group of a 1-dimensional planar Peano continuum. We leave open the following quest…
Planar multilinks prove rational singularities in surface geometry.
problem Characterizing surface singularities using planar multilinks.
method Combining topological and combinatorial approaches, including Min--Roy--Wang's work.
result Planar multilinks imply rational singularities and sandwiched singularities.
Paper introduces a new invariant for planar knotoids.
problem Defining an invariant for planar knotoids.
method Using Gauss diagrams and transcendental functions.
result The invariant is a Vassiliev invariant of order one.