Upper bounds for Steklov eigenvalues derived from intersection indices.
problem Finding upper bounds for Steklov eigenvalues of submanifolds in Euclidean space.
method Using intersection indices of submanifolds and their boundaries.
result Explicit upper bounds involving intersection index, volume, and dimensional constants.
The problem on the minimal number (with respect to deformation) of intersection points of two closed curves on a surface is solved. Following the Nielsen approach, we define classes of intersection points and essential classes of intersection points, which "are preserved under deformation" and whose total number is cal…
Generalizes Collins' theorem to products of locally indicable groups.
problem Intersection of conjugates of Magnus subgroups in one-relator groups.
method Generalization to one-relator products of locally indicable groups.
result Result holds for a broader class of groups.
This paper introduces an intersection theory problem for maps into a smooth manifold equipped with a stratification. We investigate the problem in the special case when the target is the unitary group and the domain is a circle. The first main result is an index theorem that equates a global intersection index with a f…
We indicate two short proofs of the Goresky-MacPherson topological invariance of intersection homology. One proof is very short but requires the Goresky-MacPherson support and cosupport axioms; the other is slightly longer but does not require these axioms and so is adaptable to more general perversities.
Gradient-like flows on certain manifolds restrict saddle Morse indices to 1 or n-1.
problem Restricting Morse indices of saddles in gradient-like flows.
method Analyzing invariant manifolds and their intersections for gradient-like flows.
result Morse indices of saddles are either 1 or n-1, no other indices possible.
New parities defined on virtual knots linked to crossing indices.
problem Defining parities on virtual knots.
method Connecting parities to invariant cycles on arcs and quasi-indices on crossings.
result New series of parities on virtual knots defined.
The paper proves conditions for the existence of holomorphic discs in Kähler manifolds.
problem Existence of holomorphic discs for higher A∞ operations. method Showing existence of minimal discs with specific properties implies existence of holomorphic discs.
result Minimal discs in Kähler manifolds with certain boundary conditions are holomorphic.
The paper constructs homotopically non-trivial spheres in complexified spaces.
problem Embedding spheres in complexified spaces defined by hyperplane arrangements.
method Introducing locally consistent systems of half-spaces, embedding a sphere, and computing twisted intersection numbers.
result The constructed sphere is homotopically non-trivial if the half-space system is globally consistent.
CICLAD efficiently mines frequent closed itemsets from data streams with minimal memory usage.
problem Mining frequent closed itemsets from data streams is resource-intensive.
method CICLAD is an intersection-based sliding-window FCI miner that optimizes memory usage while maintaining performance.
result CICLAD achieves significantly lower memory footprint compared to existing methods.
Differential Cohomotopy theory predicts brane interactions via chord diagrams.
problem Quantization of brane charges and moduli spaces.
method Differential refinement of Cohomotopy theory, configuration spaces, chord diagrams.
result Higher observables on brane moduli spaces are given by weight systems on chord diagrams.
This paper tackles efficient cooperative control for large-scale traffic signals using tensor-based deep learning.
problem Efficient training and control for large-scale multi-intersection traffic signals.
method Tensor representation, multi-task learning, imitation learning, proximal policy optimization.
result The proposed model achieves better performance compared to existing methods.
We describe the structure of quasiflats in two-dimensio\-nal Artin groups. We rely on the notion of metric systolicity developed in our previous work. Using this weak form of non-positive curvature and analyzing in details the combinatorics of tilings of the plane we describe precisely the building blocks for quasiflat…
The most general gauge-invariant marginal deformation of four-dimensional abelian BF-type topological field theory is studied. It is shown that the deformed quantum field theory is topological and that its observables compute, in addition to the usual linking numbers, smooth intersection indices of immersed surfaces wh…
The paper improves bounds on geodesic lengths and their simplicity on hyperbolic surfaces.
problem Estimating the distribution of geodesic lengths on hyperbolic surfaces.
method Analyzing the moduli space of hyperbolic surfaces with the Weil-Petersson metric.
result Most geodesics of certain lengths are simple and non-separating, confirming a conjecture.
The paper examines how closed curves on surfaces intersect and how this intersection determines the curves.
problem Determining closed curves on surfaces based on their intersections.
method Constructing and studying k-equivalent curves, analyzing intersections with other curves. result Curves are determined by their intersections with all other curves, but non-simple curves require infinitely many intersections to distinguish.
Abstract perspective on quadratic programming for optimal portfolio allocation.
problem Optimal allocation problems in long portfolio theory.
method Using maximum principles and distinguished boundaries in reproducing kernel Hilbert spaces.
result Support of an optimal distribution lies in a variety intersecting a distinguished boundary.
3-manifold triangulation can be reconstructed from its intersection matrix.
problem Reconstructing the triangulation of 3-manifolds from their intersection matrix.
method Using the intersection matrix of a simplicial complex to determine the triangulation of a 3-manifold up to isomorphism.
result The intersection matrix is sufficient to determine the triangulation of a 3-manifold up to isomorphism.
The paper finds diffeomorphic complex intersections with distinct Hodge numbers.
problem Identifying complex intersections with different Hodge numbers.
method Provided three pairs of 3-dimensional and one pair of 5-dimensional complex complete intersections, all diffeomorphic but with different Hodge numbers.
result Diffeomorphic complex intersections can have different Hodge numbers.
New insights into neural network training efficiency.
problem Understanding the optimal initialization for deep neural networks.
method Exploring the edge of chaos and saturation of tanh activation function.
result The line of uniformity in phase space intersects the edge of chaos, indicating saturation begins to hinder training efficiency.
New polynomials defined for virtual knots, calculated up to crossing 4.
problem Defining and calculating invariants for virtual knots.
method Intersection number of curves on a closed surface.
result Intersection polynomials calculated up to crossing 4.
In this paper we present the algorithms for calculating the differential geometric properties {t,n,b1,b2,b3,k1,k2,k3,k4} along-with geodesic curvature and geodesic torsion of the transversal intersection curve of four hypersurfaces (given by parametric representation) in Euclidean space R^5. In transversal intersection…
We generalize the PL intersection product for chains on PL manifolds and for intersection chains on PL stratified pseudomanifolds to products of locally finite chains on non-compact spaces that are natural with respect to restriction to open sets. This is necessary to sheafify the intersection product, an essential ste…
Conditions for curves on a torus with specific pairwise intersections.
problem Finding curves on a torus with prescribed pairwise intersections.
method Necessary and sufficient conditions for curves on a torus with given pairwise intersections.
result Necessary and sufficient conditions for the existence of curves on a torus with specific pairwise intersections.
Study self-intersections of arcs on a pair of pants, proving natural number spectrum.
problem Understanding self-intersections of arcs on a pair of pants.
method Algorithm to compute self-intersection number, bounds established in terms of word length.
result Spectrum of self-intersection numbers covers all natural numbers.
Virtual knots with same writhe polynomial have equivalent intersection graphs.
problem Equivalence of intersection graphs for virtual knots.
method Proved equivalence through writhe polynomial.
result Intersection graphs of virtual knots with the same writhe polynomial are equivalent.
Study properties of self-similar continua with finite intersection property.
problem Characterize self-similar continua with finite intersection property.
method Prove intersection graph criterion, finite order theorem, and parameter matching theorem.
result All Jordan arcs starting from a intersection point in such continuum on a plane should have the same slope parameter at that point.
Estimates intersection pairing in hyperbolic 4-manifolds.
problem Estimating intersection pairing in hyperbolic 4-manifolds.
method Using Thurston norms of homology classes.
result Proved an estimate on intersection pairing.
By considering a (not necessarily locally-flat) PL knot as the singular locus of a PL stratified pseudomanifold, we can use intersection homology theory to define intersection Alexander polynomials, a generalization of the classical Alexander polynomial invariants for smooth or PL locally-flat knots. We show that the i…
Classifies arcs on a 4-punctured sphere that intersect at most once.
problem Classifying arcs on a 4-punctured sphere with intersection constraints.
method Classification of maximal systems of arcs intersecting at most once.
result Maximal systems of arcs on the 4-punctured sphere identified.
James McClure recently showed that the domain for the intersection pairing of PL chains on a PL manifold M is a subcomplex of C∗(M)⊗C∗(M) that is quasi-isomorphic to C∗(M)⊗C∗(M) and, more generally, that the intersection pairing endows C∗(M) with the structure of a partially-defined commutati…
Novel approach for large genus intersection number asymptotics.
problem Computing intersection numbers in large genus.
method Resurgent analysis of n-point functions with quantum curve.
result Extension of Aggarwal's results and new r-spin and Theta-class intersection numbers. The paper calculates self-intersections on a pair of pants using Bowen and Series' coding.
problem Computing the number of self-intersections of closed geodesics on a pair of pants.
method Used Bowen and Series' coding to compute self-intersections.
result Proved a conjecture and provided bounds for self-intersection numbers.
Generic potential primes have no self-intersections or intersections.
problem Finding non-degenerate periodic orbits without self-intersections.
method Generic convex Hamiltonian approach and Mañé genericity.
result Prime periodic orbits do not intersect or have self-intersections.
Paper proves curves can be smoothed to reduce self-intersection by exactly 1.
problem Prove that the shortest closed geodesic self-intersects exactly k times.
method Carefully smoothing intersection points reduces self-intersection by exactly 1.
result The shortest closed geodesic self-intersects exactly k times for hyperbolic and Riemannian metrics.
The paper extends intersection theory for b-divisors, proving monotonicity and volume inequalities.
problem Intersection theory for b-divisors and monotonicity of intersection products.
method Developed general intersection theory of nef b-divisors, defined restricted volume, proved monotonicity.
result Proved quantitative monotonicity of intersection product and new volume inequalities.
Study intersection polynomials of long virtual knots with supporting genera.
problem Characterize long virtual knots using geometric invariants.
method Define and analyze 1- and 2-supporting genera, and use them to filter long virtual knots. result Provide complete realizability criteria for all twelve intersection polynomials.
Paper computes Atiyah class for DG manifolds of amplitude +1.
problem Computing the Atiyah class for DG manifolds of specific amplitude.
method Computed the Atiyah class by encoding the derived intersection of sections and zero sections of vector bundles.
result Atiyah class vanishes if and only if the intersection is clean.
Minimum algebraic intersection found in hyperbolic surfaces, growing with genus.
problem Finding the minimum algebraic intersection form in hyperbolic surfaces.
method Analyzing algebraic intersection form in moduli space of hyperbolic surfaces.
result Minimum grows in the order of (logg)−2 with genus. We continue here the investigation of the relationship between the intersection of a pair of subgroups of a Kleinian group, and in particular the limit set of that intersection, and the intersection of the limit sets of the subgroups. Of specific interest is the extent to which the intersection of the limit sets being …
In this paper we study relations between intersection numbers on moduli spaces of curves and Hurwitz numbers. First, we prove two formulas expressing Hurwitz numbers of (generalized) polynomials via intersections on moduli spaces of curves. Then we show, how intersection numbers can be expressed via Hurwitz numbers. An…
This paper presents a novel framework for accurate pedestrian intent prediction at intersections. Given some prior knowledge of the curbside geometry, the presented framework can accurately predict pedestrian trajectories, even in new intersections that it has not been trained on. This is achieved by making use of the …
Study the intersection of positive closed currents using tangent currents and King's residue formula.
problem Investigate the intersection of positive closed currents in complex manifolds.
method Employ tangent currents and King's residue formula to establish a natural condition for intersection.
result Derive an integral representation of the intersection of positive closed currents.
The paper tackles intersectional fairness in machine learning, proposing methods to assess and mitigate bias.
problem Achieving optimal predictive performance is not enough; fairness with respect to multiple sensitive attributes is crucial.
method The paper introduces a comprehensive framework for auditing and achieving intersectional fairness in classification problems, including metrics, estimation methods, and post-processing techniques.
result The proposed methods can robustly estimate and mitigate intersectional bias in classification models without compromising predictive performance.
Unified quantum invariants via intersections of embedded Lagrangians.
problem Unified quantum invariants for Uq(sl(2)). method State sum of Lagrangian intersections in configuration spaces.
result Recovery of coloured Jones and Alexander polynomials.
Given a hyperbolic surface §, a classic result of Birman and Series states that for each K, all complete geodesics with at most K self-intersections can only pass through a certain nowhere dense, Hausdorff dimension 1 subset of §. We define a self-intersection function for each complete geodesic, which bounds t…
Positivity of intersections in 4-manifolds leads to taming symplectic structures.
problem Taming symplectic structures in almost complex 4-manifolds.
method Proof of positivity of intersections of pseudoholomorphic curves.
result Positivity of intersections is stable and leads to taming symplectic structures.
The paper explores intersectional fairness in machine learning, proving bounds on it.
problem Intersectional fairness in machine learning, especially when multiple protected attributes are involved.
method Statistical analysis and bounds on intersectional fairness, leveraging marginal fairness.
result Theoretical bounds on intersectional fairness can be computed from marginal fairness and other statistical quantities.