We give here a self contained and elementary introduction to the Conley-Zehnder index for a path of symplectic matrices. We start from the definition of the index as the degree of a map into the circle for a path starting at the identity and ending at a matrix for which 1 is not an eigenvalue. We prove some properties …
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48 results for “Conley-Zehnder index”
The aim of this paper is to give an explicit formula in order to compute the Maslov index of the fundamental solution of a linear autonomous Hamiltonian system, in terms of the Conley-Zehnder index and the time one flow.
Let be a non-degenerate Ustilovsky geodesic in generated by . We give a simple proof of a generalization of the conjecture stated in \cite{virtmorse}, relating the Morse index of , as a critical point of the Hofer length functional, with the Conley Zehnder index of the extremizers of , consid…
Paper proves ellipticity of certain Reeb orbits and estimates ECH spectrum on lens spaces.
problem Proving ellipticity of Reeb orbits in lens spaces and estimating ECH spectrum.
method Using rational self-linking number, Conley-Zehnder index, and ECH computations.
result First ECH spectrum on dynamically convex L(3,1) is estimated and shown to be equal to contact area infimum.
The first two authors showed in~\cite{AM1} how the Conley-Zehnder index of any contractible periodic Reeb orbit of a non-degenerate toric contact form on a good toric contact manifold with zero first Chern class, i.e. a Gorenstein toric contact manifold, can be explicitly computed using moment map data. In this paper w…
We study the classical action functional $\SMC_V$ on the free loop space of a closed, finite dimensional Riemannian manifold and the symplectic action $\AMC_V$ on the free loop space of its cotangent bundle. The critical points of both functionals can be identified with the set of perturbed closed geodesics in .…
We exhibit sufficient conditions for a finite collection of periodic orbits of a Reeb flow on a closed -manifold to bound a positive global surface of section with genus zero. These conditions turn out to be -generically necessary. Moreover, they involve linking assumptions on periodic orbits with Conley-Z…
Let M be a weakly monotone symplectic manifold, and H be a time-dependent Hamiltonian; we assume that the periodic orbits of the corresponding time-dependent Hamiltonian vector field are non-degenerate. We construct a refined version of the Floer chain complex associated to these data and any regular covering of M, and…
New techniques reveal tight contact manifolds with vanishing contact homology.
problem Understanding closed tight contact manifolds with vanishing contact homology.
method Developed algebraic tools and techniques to study holomorphic curves in surgery cobordisms.
result First known examples of closed tight contact manifolds with vanishing contact homology.
A new clustering evaluation index based on density estimation.
problem Improving internal clustering evaluation indices.
method The index is a mixture of Ambiguous and Similarity sub-indices, calculated using density estimation.
result The new index significantly outperforms other internal clustering evaluation indices.
Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.
problem Investigating elliptic operators with a specific symmetry and their index modulo 2.
method Analysis of Callias-type operators on non-compact manifolds, establishing mod 2 versions of index theorems.
result Established mod 2 versions of the Gromov-Lawson relative index theorem, Callias index theorem, and Boutet de Monvel's index theorem for Toeplitz operators.
An equivariant Atiyah-Patodi-Singer index theorem for proper actions II: the -theoretic indexmath.KT
New index formula connects numerical and -theoretic indices.
problem Equivariant index for proper group actions on manifolds.
method Developed a trace on group conjugacy classes to relate numerical and -theoretic indices.
result Shows that numerical index equals -theoretic index under certain conditions.
The paper explores global index formulas for one-dimensional holomorphic foliations.
problem Global index formulas for one-dimensional holomorphic foliations.
method Microlocal point of view and short proofs for existing index formulas.
result Generalizations of existing index formulas.
Explain Arnold's proof of the Morse index theorem using Maslov index.
problem Proving the Morse index theorem in Riemannian geometry.
method Using symplectic arguments and the Maslov index.
result Self-contained exposition of Arnold's proof.
Paper introduces danceability index as a new bridge index definition.
problem Defining the bridge index in various mathematical contexts.
method Proves danceability index as equivalent to bridge index, extends to virtual knots.
result Danceability index is a new equivalent definition of the bridge index.
The p-index improves investment performance for NYSE stocks but not for SSE stocks.
problem Improving investment performance for stocks using the p-index.
method Comparing different p-ratio strategies and empirical efficient frontiers for SSE and NYSE stocks.
result The p-index enhances investment performance for NYSE stocks but not for SSE stocks.
Study Whittle index learning algorithms for restless bandits with constant stepsizes.
problem Optimizing decisions in restless multi-armed bandits with constant stepsizes.
method Developed Q-learning algorithms with constant stepsizes for index learning in restless bandits, extending to DQN and function approximations.
result The algorithms learn the Whittle index effectively.
Diversifying an Indexq-fin.PM
A new index rebalancing strategy reduces large constituent weights without undesirable effects.
problem Undesirable effects of current Nasdaq-100 index rebalancing.
method A simple rebalancing strategy that avoids undesirable effects.
result Preserves the order of index weights and prevents maximum weight increase.
We study bounded pseudoconvex domains in complex Euclidean space. We define an index associated to the boundary and show this new index is equivalent to the Diederich-Fornæss index defined in 1977. This connects the Diederich-Fornæss index to boundary conditions and refines the Levi pseudoconvexity. We also prove the $…
Study on symmetric braid index of ribbon knots, deriving bounds and characterizations.
problem Understanding the symmetric braid index of ribbon knots.
method Defining symmetric braid index, using Khovanov homology, and calculating bounds.
result Existence of knots with symmetric braid index greater than braid index.
Study proves bridge and braid indices match for twist positive knots.
problem Determining when bridge and braid indices are equal for knots.
method Used knot Floer torsion order to prove for all twist positive knots.
result Bridge and braid indices coincide for all twist positive knots.
Minimal grid diagrams for 15,735 knots with 14 crossings and arc index 14.
problem Representing prime knots with 14 crossings and specific arc indices using grid diagrams.
method Enumerated all prime knots with 14 crossings, categorized by arc index, and found minimal grid diagrams for those with arc index 14.
result 8,027 knots with arc index 13 and 15,735 knots with arc index 14 were represented by minimal grid diagrams.
Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples th…
Proves the index of a Möbius band in 4D ball equals 5.
problem Determining the Morse index of critical non-orientable surfaces.
method Comparison theorem between Steklov spectral index and energy index.
result Proves the index of critical Möbius band in equals 5.
Finite index subgroups of relatively hyperbolic groups have equal index.
problem Finite index subgroups of relatively hyperbolic groups have equal index.
method Demonstrating that the number of simplices in a simplicial classifying space grows linearly with index.
result Finite index subgroups of relatively hyperbolic groups have equal index.
Enhanced indexation uses equity and index options for better performance.
problem Improving portfolio performance through enhanced indexation.
method Integrating index options into an enhanced indexation strategy based on second-order stochastic dominance.
result Introducing option strategies in enhanced indexation leads to improved out-of-sample performance.
The abstract discusses connecting quantum mechanics and algebraic index theories.
problem Exploring the connection between quantum mechanics and algebraic index theories.
method Explains how the classical algebraic index theorem can be proved in terms of BV quantization of topological quantum mechanics and 2d chiral CFT.
result Shows how the generating function of all genus Gromov-Witten invariants on elliptic curves is mirror equivalent to an elliptic chiral index.
Researchers construct an index map for contact manifolds using K-theory.
problem Constructing an index for maximally hypoelliptic operators on contact manifolds.
method Using Higson's construction for symbol class in K-theory, they derive a series of maps whose induced map in K-theory is the Heisenberg Atiyah-Singer index map.
result Explicit construction of a series of maps leading to the Heisenberg Atiyah-Singer index map.
New index formula for hypoelliptic operators on manifolds.
problem Index computation for hypoelliptic differential operators.
method Generalized index formula for *-maximally hypoelliptic operators.
result Explicit index computations for Hormander's sum of squares operators.
Deep learning predicts market sensitivities for cost-effective index tracking.
problem Costly and impractical replication of index funds.
method Learning to predict market sensitivities using deep learning models.
result Significant reduction in prediction errors compared to historical methods.
Study on minimal surfaces with free boundary in a half-space, improving index estimates.
problem Non-existence of index two embedded minimal surfaces with free boundary in a half-space.
method Improved estimates of Neumann and Dirichlet indices, simplified proof of lower bounds.
result Answered Ambrozio et al.'s question and provided new lower bounds.
Listed 19,513 prime knots with arc index 12-16.
problem Tabulating prime knots with specific arc indices.
method Provided list of prime knots with minimal grid diagrams.
result 19,513 prime knots with arc index 12-16.
An upper bound of the superbridge index of the connected sum of two knots is given in terms of the braid index of the summands. Using this upper bound and minimal polygonal presentations, we give an upper bound in terms of the superbridge index and the bridge index of the summands when they are torus knots. In contrast…
A new stock index model simplifies high-dimensional stock data.
problem Reflecting the overall stock market activity in high-dimensional data.
method Manifold learning and feature detection on discrete Laplace-Beltrami operator.
result The MF index series approximates the stock market better and has lower risk.
New method accurately reconstructs Russell 3000 index, revealing crowded portfolios.
problem Crowding in index portfolios during reconstitution events.
method Developed a Python package for accurate index reconstruction using CRSP US Stock data.
result Annual Russell 3000 portfolios are more crowded than quarterly ones, suggesting lower transaction costs.
In this paper we study the chord index of virtual knots, which can be thought of as an extension of the chord parity. We show how to use the chord index to define finite type invariants of virtual knots. The notions of indexed Jones polynomial and indexed quandle are introduced, which generalize the classical Jones pol…
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.
We introduce a notion of cobordism of Callias-type operators over complete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a s…
We study differential operators on complete Riemannian manifolds which act on sections of a bundle of finite type modules over a von Neumann algebra with a trace. We prove a relative index and a Callias-type index theorems for von Neumann indexes of such operators. We apply these results to obtain a version of Atiyah's…
Proves a lattice version of the Atiyah-Singer index theorem.
problem Index problems of Wilson-Dirac operators on lattice approximations of manifolds.
method Formulates and proves a -theoretic formula for an index-type invariant.
result Main theorem gives a formula for an index-type invariant of operators on lattice approximations of closed integral affine manifolds.
Improved upper bound on superbridge index from 5b-3 to 3b-1.
problem Improving bounds on superbridge index for knots.
method Improved upper bound formula for superbridge index.
result Upper bound on superbridge index reduced from 5b-3 to 3b-1.
We establish a mod 2 index theorem for real vector bundles over 8k+2 dimensional compact pin manifolds. The analytic index is the reduced invariant of (twisted) Dirac operators and the topological index is defined through -theory. Our main result extends the mod 2 index theorem of Atiyan and Singer to non-o…
We introduce \textcolor{red}{general} new techniques for computing the geometric index of a link in the interior of a solid torus . These techniques simplify and unify previous ad hoc methods used to compute the geometric index in specific examples \textcolor{red}{ and allow the simple computation of geometric i…
Using equivariant Toeplitz operator calculus, we give a new proof of the Atiyah-Weinstein conjecture on the index of Fourier integral operators and the relative index of CR structures.
Gini index needs auto-calibration for consistent decision-making.
problem Gini index's inconsistency in decision-making.
method Restrict Gini index to auto-calibrated regression models.
result Gini index becomes strictly consistent with auto-calibration.
The notion of pseudo-differential operators with coefficients in a continuous trace algebra over a manifold are introduced and their index theory is studied. The algebra of principal symbols in this calculus provides an abstract Poincaré dual to the continuous trace algebra. Index formulas for pseudo-differential opera…
For an immersed minimal surface in , we show that there exists a lower bound on its Morse index that depends on the genus and number of ends, counting multiplicity. This improves, in several ways, an estimate we previously obtained bounding the genus and number of ends by the index. Our new estimate resol…
The Rosenberg index vanishes if a manifold admits a wide Riemannian band or cube-like domain.
problem Proving the Rosenberg index does not vanish for certain manifolds.
method Analyzing isometric immersions of wide Riemannian bands and cube-like domains on spin manifolds.
result Closed spin manifolds with infinite -width have non-vanishing Rosenberg index.