scholarly journals Iterated index and the mean Euler characteristic

2015 ◽  
Vol 07 (03) ◽  
pp. 453-481 ◽  
Author(s):  
Viktor L. Ginzburg ◽  
Yusuf Gören
Keyword(s):  
Euler Characteristic ◽  
Contact Manifold ◽  
Finsler Metric ◽  
Topological Invariants ◽  
Closed Geodesics ◽  
Lefschetz Numbers ◽  
Local Case ◽  
The Mean ◽  
Standard Contact ◽  
Global And Local

The aim of the paper is three-fold. We begin by proving a formula, both global and local versions, relating the number of periodic orbits of an iterated map and the Lefschetz numbers, or indices in the local case, of its iterations. This formula is then used to express the mean Euler characteristic (MEC) of a contact manifold in terms of local, purely topological, invariants of closed Reeb orbits, without any non-degeneracy assumption on the orbits. Finally, turning to applications of the local MEC formula to dynamics, we use it to reprove a theorem asserting the existence of at least two closed Reeb orbits on the standard contact S3 (by Cristofaro–Gardiner and Hutchings in the most general form) and the existence of at least two closed geodesics for a Finsler metric on S2 (Bangert and Long).

scholarly journals On the Mean Euler Characteristic of Gorenstein Toric Contact Manifolds

2018 ◽  
Vol 2020 (14) ◽  
pp. 4465-4495 ◽  
Author(s):  
Miguel Abreu ◽  
Leonardo Macarini
Keyword(s):  
Euler Characteristic ◽  
Chern Class ◽  
Contact Manifold ◽  
Contact Manifolds ◽  
Toric Diagram ◽  
The Mean ◽  
Toric Contact Manifolds ◽  
Symplectic Filling

Abstract We prove that the mean Euler characteristic of a Gorenstein toric contact manifold, that is, a good toric contact manifold with zero 1st Chern class, is equal to half the normalized volume of the corresponding toric diagram and give some applications. A particularly interesting one, obtained using a result of Batyrev and Dais, is the following: twice the mean Euler characteristic of a Gorenstein toric contact manifold is equal to the Euler characteristic of any crepant toric symplectic filling, that is, any toric symplectic filling with zero 1st Chern class.

scholarly journals Identification and isotropy characterization of deformed random fields through excursion sets

2018 ◽  
Vol 50 (3) ◽  
pp. 706-725
Author(s):  
Julie Fournier
Keyword(s):  
Random Field ◽  
Euler Characteristic ◽  
Random Fields ◽  
Weak Form ◽  
Invariance Property ◽  
Basic Domain ◽  
Excursion Sets ◽  
The Mean ◽  
Isotropic Random

Abstract A deterministic application θ:ℝ2→ℝ2 deforms bijectively and regularly the plane and allows the construction of a deformed random field X∘θ:ℝ2→ℝ from a regular, stationary, and isotropic random field X:ℝ2→ℝ. The deformed field X∘θ is, in general, not isotropic (and not even stationary), however, we provide an explicit characterization of the deformations θ that preserve the isotropy. Further assuming that X is Gaussian, we introduce a weak form of isotropy of the field X∘θ, defined by an invariance property of the mean Euler characteristic of some of its excursion sets. We prove that deformed fields satisfying this property are strictly isotropic. In addition, we are able to identify θ, assuming that the mean Euler characteristic of excursion sets of X∘θ over some basic domain is known.

scholarly journals Open books for Boothby-Wang bundles, fibered Dehn twists and the mean Euler characteristic

2014 ◽  
Vol 12 (2) ◽  
pp. 379-426 ◽  
Author(s):  
River Chiang ◽  
Fan Ding ◽  
Otto van Koert
Keyword(s):  
Euler Characteristic ◽  
Open Books ◽  
Dehn Twists ◽  
The Mean

scholarly journals Topological invariants of weighted homogeneous polynomials

1991 ◽  
Vol 33 (3) ◽  
pp. 241-245 ◽  
Author(s):  
Zbigniew Szafraniec
Keyword(s):  
Euler Characteristic ◽  
Homogeneous Polynomial ◽  
Topological Invariants ◽  
Homogeneous Polynomials ◽  
Weighted Homogeneous Polynomial

Let ℝn → ℝ be a weighted homogeneous polynomial such that df(0) = 0, L = {x ∈ Sn−1|f(x) = 0}, and let χ(L) be the Euler characteristic of L. The problem is how to calculate χ(L) in terms of f.

scholarly journals Majorana/Andreev crossover and the fate of the topological phase transition in inhomogeneous nanowires

2021 ◽  
Author(s):  
Pasquale Marra ◽  
Angela Nigro
Keyword(s):  
Bound States ◽  
Transition Probabilities ◽  
Topological Invariants ◽  
Length Scale ◽  
Topological Phase ◽  
Andreev Bound States ◽  
Majorana Bound States ◽  
Topological Phase Transition ◽  
Majorana Modes ◽  
Global And Local

Abstract Majorana bound states (MBS) and Andreev bound states (ABS) in realistic Majorana nanowires setups have similar experimental signatures which make them hard to distinguishing one from the other. Here, we characterize the continuous Majorana/Andreev crossover interpolating between fully-separated, partially-separated, and fully-overlapping Majorana modes, in terms of global and local topological invariants, fermion parity, quasiparticle densities, Majorana pseudospin and spin polarizations, density overlaps and transition probabilities between opposite Majorana components. We found that in inhomogeneous wires, the transition between fully-overlapping trivial ABS and nontrivial MBS does not necessarily mandate the closing of the bulk gap of quasiparticle excitations, but a simple parity crossing of partially-separated Majorana modes (ps-MM) from trivial to nontrivial regimes. We demonstrate that fully-separated and fully-overlapping Majorana modes correspond to the two limiting cases at the opposite sides of a continuous crossover: the only distinction between the two can be obtained by estimating the degree of separations of the Majorana components. This result does not contradict the bulk-edge correspondence: Indeed, the field inhomogeneities driving the Majorana/Andreev crossover have a length scale comparable with the nanowire length, and therefore correspond to a nonlocal perturbation which breaks the topological protection of the MBS.

scholarly journals Global and local interference effects in ensemble encoding are best explained by interactions between summary representations of the mean and the range

2021 ◽  
Author(s):  
Marco A. Sama ◽  
Dilakshan Srikanthan ◽  
Adrian Nestor ◽  
Jonathan S. Cant
Keyword(s):  
Local Level ◽  
Interference Effects ◽  
Single Item ◽  
Ensemble Perception ◽  
Item Level ◽  
The Mean ◽  
Ensemble Encoding ◽  
Average Size ◽  
Local Interference ◽  
Global And Local

AbstractThrough ensemble encoding, the visual system compresses redundant statistical properties from multiple items into a single summary metric (e.g., average size). Numerous studies have shown that global summary information is extracted quickly, does not require access to single-item representations, and often interferes with reports of single items from the set. Yet a thorough understanding of ensemble processing would benefit from a more extensive investigation at the local level. Thus, the purpose of this study was to provide a more critical inspection of global-local processing in ensemble perception. Taking inspiration from Navon (Cognitive Psychology, 9(3), 353-383, 1977), we employed a novel paradigm that independently manipulates the degree of interference at the global (mean) or local (single item) level of the ensemble. Initial results were consistent with reciprocal interference between global and local ensemble processing. However, further testing revealed that local interference effects were better explained by interference from another summary statistic, the range of the set. Furthermore, participants were unable to disambiguate single items from the ensemble display from other items that were within the ensemble range but, critically, were not actually present in the ensemble. Thus, it appears that local item values are likely inferred based on their relationship to higher-order summary statistics such as the range and the mean. These results conflict with claims that local information is captured alongside global information in summary representations. In such studies, successful identification of set members was not compared with misidentification of items within the range, but which were nevertheless not presented within the set.

A Fast-Converging Ensemble Infilling Approach Balancing Global Exploration and Local Exploitation: The Go-Inspired Hybrid Infilling Strategy

2019 ◽  
Vol 142 (2) ◽  
Author(s):  
Liye Lv ◽  
Maolin Shi ◽  
Xueguan Song ◽  
Wei Sun ◽  
Jie Zhang
Keyword(s):  
Adaptive Sampling ◽  
Computational Cost ◽  
Expected Improvement ◽  
Sampling Strategies ◽  
Key Factors ◽  
Engineering Problems ◽  
The Mean ◽  
Global And Local ◽  
Probability Of Improvement ◽  
Better Than

AbstractInfilling strategies have been proposed for decades and are widely used in engineering problems. It is still challenging to achieve an effective trade-off between global exploration and local exploitation. In this paper, a novel decision-making infilling strategy named the Go-inspired hybrid infilling (Go-HI) strategy is proposed. The Go-HI strategy combines multiple individual infilling strategies, such as the mean square error (MSE), expected improvement (EI), and probability of improvement (PoI) strategies. The Go-HI strategy consists of two major parts. In the first part, a tree-like structure consisting of several subtrees is built. In the second part, the decision value for each subtree is calculated using a cross-validation (CV)-based criterion. Key factors that significantly influence the performance of the Go-HI strategy, such as the number of component infilling strategies and the tree depth, are explored. Go-HI strategies with different component strategies and tree depths are investigated and also compared with four baseline adaptive sampling strategies through three numerical functions and one engineering case. Results show that the number of component infilling strategies exerts a larger influence on the global and local performance than the tree depth; the Go-HI strategy with two component strategies performs better than the ones with three; the Go-HI strategy always outperforms the three component infilling strategies and the other four benchmark strategies in global performance and robustness and saves much computational cost.

scholarly journals On the Sign of the Curvature of a Contact Metric Manifold

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 892
Author(s):  
David E. Blair
Keyword(s):  
Positive Curvature ◽  
Contact Manifold ◽  
Contact Metric Manifold ◽  
Reeb Vector ◽  
Reeb Vector Field ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
Expository Article ◽  
Standard Contact ◽  
Negative Sectional Curvature

In this expository article, we discuss the author’s conjecture that an associated metric for a given contact form on a contact manifold of dimension ≥5 must have some positive curvature. In dimension 3, the standard contact structure on the 3-torus admits a flat associated metric; we also discuss a local example, due to Krouglov, where there exists a neighborhood of negative curvature on a particular 3-dimensional contact metric manifold. In the last section, we review some results on contact metric manifolds with negative sectional curvature for sections containing the Reeb vector field.

FINSLER LAPLACIANS AND MINIMAL-ENERGY MAPS

2000 ◽  
Vol 11 (01) ◽  
pp. 1-13 ◽  
Author(s):  
PAUL CENTORE
Keyword(s):  
Riemannian Metric ◽  
Energy Functional ◽  
Mean Value ◽  
Finsler Metric ◽  
Harmonic Mappings ◽  
Laplacian Operator ◽  
Minimal Energy ◽  
Finsler Manifolds ◽  
The Mean ◽  
Energy Maps

For any Finsler manifold, there is a geometrically natural Laplacian operator, called the mean-value Laplacian, which generalizes the Riemannian Laplacian. We show that, like the Riemannian Laplacian (for functions), we can see the vanishing of the mean-value Laplacian at some function f as the minimizing of an energy functional e(f) by f. This energy functional e depends on a Riemannian metric canonically associated to the Finsler metric and on a canonically associated volume form. We relate this construction to a more general construction of Jost, and define a notion of harmonic mappings between Finsler manifolds.

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