scholarly journals DESCENT AND C0-RIGIDITY OF SPECTRAL INVARIANTS ON MONOTONE SYMPLECTIC MANIFOLDS

2012 ◽  
Vol 04 (04) ◽  
pp. 481-498 ◽  
Author(s):  
SOBHAN SEYFADDINI
Keyword(s):  
Symplectic Manifolds ◽  
Spectral Invariants ◽  
Open Set ◽  
Continuity Properties ◽  
Hofer Metric

We obtain estimates showing that on monotone symplectic manifolds (asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty open set, U, descend from [Formula: see text] to Hamc(M\U). Furthermore, we show that these invariants are continuous with respect to the C0-topology on Hamc(M\U).We apply the above results to Hofer geometry and establish unboundedness of the Hofer diameter of Hamc(M\U) for stably displaceable U. We also answer a question of F. Le Roux about C0-continuity properties of the Hofer metric.

scholarly journals DUALITY IN FILTERED FLOER–NOVIKOV COMPLEXES

2010 ◽  
Vol 02 (02) ◽  
pp. 233-258 ◽  
Author(s):  
MICHAEL USHER
Keyword(s):  
Chain Complex ◽  
Bilinear Pairing ◽  
Symplectic Manifolds ◽  
Duality Relation ◽  
Spectral Invariants ◽  
Floer Theory ◽  
Intersection Pairing ◽  
Period Map ◽  
New Formula ◽  
Special Case

We prove that a certain bilinear pairing (analogous to the Poincaré–Lefschetz intersection pairing) between filtered sub- and quotient complexes of a Floer-type chain complex and of its "opposite complex" is always nondegenerate on homology. This implies a duality relation for the Oh–Schwarz-type spectral invariants of these complexes which (in Hamiltonian Floer theory) was established in the special case that the period map has discrete image by Entov and Polterovich. The duality relation served as a key lemma in Entov and Polterovich's construction of a Calabi quasimorphism on certain rational symplectic manifolds, and the result that we prove here implies that their construction remains valid when the rationality hypothesis is dropped. Apart from this, we also use the nondegeneracy of the pairing to establish a new formula for what we have previously called the boundary depth of a Floer chain complex; this formula shows that the boundary depth is unchanged under passing to the opposite complex.

scholarly journals Spectral invariants of distance functions

2016 ◽  
Vol 08 (04) ◽  
pp. 655-676 ◽  
Author(s):  
Suguru Ishikawa
Keyword(s):  
Symplectic Manifold ◽  
Symplectic Form ◽  
Floer Homology ◽  
Symplectic Manifolds ◽  
Distance Functions ◽  
Spectral Invariants ◽  
Euclidian Space ◽  
Closed Riemann Surface ◽  
Genus Formula ◽  
Spectral Invariant

Calculating the spectral invariant of Floer homology of the distance function, we can find new superheavy subsets in symplectic manifolds. We show if convex open subsets in Euclidian space with the standard symplectic form are disjointly embedded in a spherically negative monotone closed symplectic manifold, their complement is superheavy. In particular, the [Formula: see text] bouquet in a closed Riemann surface with genus [Formula: see text] is superheavy. We also prove some analogous properties of a monotone closed symplectic manifold. These can be used to extend Seyfaddni’s result about lower bounds of Poisson bracket invariant.

The smooth case

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Homotopy Type ◽  
Unit Vector ◽  
Smooth Case ◽  
Open Set ◽  
Birational Invariant

This chapter examines the simplifications occurring in the proof of the main theorem in the smooth case. It begins by stating the theorem about the existence of an F-definable homotopy h : I × unit vector X → unit vector X and the properties for h. It then presents the proof, which depends on two lemmas. The first recaps the proof of Theorem 11.1.1, but on a Zariski dense open set V₀ only. The second uses smoothness to enable a stronger form of inflation, serving to move into V₀. The chapter also considers the birational character of the definable homotopy type in Remark 12.2.4 concerning a birational invariant.

ON A NEW CLASS OF SEMI GENERALIZED CLOSED SETS IN STRONG GENERALIZED TOPOLOGICAL SPACES

2020 ◽  
Vol 9 (11) ◽  
pp. 9353-9360
Author(s):  
G. Selvi ◽  
I. Rajasekaran
Keyword(s):  
Topological Spaces ◽  
Closed Set ◽  
Basic Properties ◽  
Closed Sets ◽  
New Class ◽  
Open Set ◽  
Continuous Maps

This paper deals with the concepts of semi generalized closed sets in strong generalized topological spaces such as $sg^{\star \star}_\mu$-closed set, $sg^{\star \star}_\mu$-open set, $g^{\star \star}_\mu$-closed set, $g^{\star \star}_\mu$-open set and studied some of its basic properties included with $sg^{\star \star}_\mu$-continuous maps, $sg^{\star \star}_\mu$-irresolute maps and $T_\frac{1}{2}$-space in strong generalized topological spaces.

Symplectic capacities

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Euclidean Space ◽  
Generating Functions ◽  
Weinstein Conjecture ◽  
Finite Dimensional ◽  
Hofer Metric

This chapter returns to the problems which were formulated in Chapter 1, namely the Weinstein conjecture, the nonsqueezing theorem, and symplectic rigidity. These questions are all related to the existence and properties of symplectic capacities. The chapter begins by discussing some of the consequences which follow from the existence of capacities. In particular, it establishes symplectic rigidity and discusses the relation between capacities and the Hofer metric on the group of Hamiltonian symplectomorphisms. The chapter then introduces the Hofer–Zehnder capacity, and shows that its existence gives rise to a proof of the Weinstein conjecture for hypersurfaces of Euclidean space. The last section contains a proof that the Hofer–Zehnder capacity satisfies the required axioms. This proof translates the Hofer–Zehnder variational argument into the setting of (finite-dimensional) generating functions.

Constructing symplectic manifolds

2017 ◽  
Author(s):  
Dusa McDuff ◽  
Dietmar Salamon
Keyword(s):  
Final Section ◽  
Symplectic Manifolds ◽  
Homotopy Equivalence ◽  
Codimension Two ◽  
Open Manifolds ◽  
Connected Sums ◽  
Symplectic Forms ◽  
Ambient Manifold ◽  
Blowing Up ◽  
Four Manifolds

This chapter examines various ways to construct symplectic manifolds and submanifolds. It begins by studying blowing up and down in both the complex and the symplectic contexts. The next section is devoted to a discussion of fibre connected sums and describes Gompf’s construction of symplectic four-manifolds with arbitrary fundamental group. The chapter also contains an exposition of Gromov’s telescope construction, which shows that for open manifolds the h-principle rules and the inclusion of the space of symplectic forms into the space of nondegenerate 2-forms is a homotopy equivalence. The final section outlines Donaldson’s construction of codimension two symplectic submanifolds and explains the associated decompositions of the ambient manifold.

scholarly journals R-continuous functions

1992 ◽  
Vol 15 (1) ◽  
pp. 57-64 ◽  
Author(s):  
Ch. Konstadilaki-Savvapoulou ◽  
D. Janković
Keyword(s):  
Continuous Functions ◽  
Strong Form ◽  
Topological Spaces ◽  
Strong Continuity ◽  
Closed Graph ◽  
Continuity Properties

A strong form of continuity of functions between topological spaces is introduced and studied. It is shown that in many known results, especially closed graph theorems, functions under consideration areR-continuous. Several results in the literature concerning strong continuity properties are generalized and/or improved.

Data-driven sector coupling in 5G-based smart networks

2021 ◽  
Vol 22 (1) ◽  
pp. 53-68
Author(s):  
Guenter Knieps
Keyword(s):  
Data Privacy ◽  
Service Providers ◽  
Policy Implications ◽  
Network Industries ◽  
5G Networks ◽  
Privacy And Security ◽  
Open Set ◽  
Application Service ◽  
Downstream Application

5G attains the role of a GPT for an open set of downstream IoT applications in various network industries and within the app economy more generally. Traditionally, sector coupling has been a rather narrow concept focusing on the horizontal synergies of urban system integration in terms of transport, energy, and waste systems, or else the creation of new intermodal markets. The transition toward 5G has fundamentally changed the framing of sector coupling in network industries by underscoring the relevance of differentiating between horizontal and vertical sector coupling. Due to the fixed mobile convergence and the large open set of complementary use cases, 5G has taken on the characteristics of a generalized purpose technology (GPT) in its role as the enabler of a large variety of smart network applications. Due to this vertical relationship, characterized by pervasiveness and innovational complementarities between upstream 5G networks and downstream application sectors, vertical sector coupling between the provider of an upstream GPT and different downstream application industries has acquired particular relevance. In contrast to horizontal sector coupling among different application sectors, the driver of vertical sector coupling is that each of the heterogeneous application sectors requires a critical input from the upstream 5G network provider and combines this with its own downstream technology. Of particular relevance for vertical sector coupling are the innovational complementarities between upstream GPT and downstream application sectors. The focus on vertical sector coupling also has important policy implications. Although the evolution of 5G networks strongly depends on the entrepreneurial, market-driven activities of broadband network operators and application service providers, the future of 5G as a GPT is heavily contingent on the role of frequency management authorities and European regulatory policy with regard to data privacy and security regulations.

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

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