Building from work by Cedzich et al. and Suzuki et al., we consider topological and index-theoretic properties of chiral unitaries, which are an abstraction of the time evolution of a ...chiral-symmetric self-adjoint operator. Split-step quantum walks provide a rich class of examples. We use the index of a pair of projections and the Cayley transform to define topological indices for chiral unitaries on both Hilbert spaces and Hilbert $C^*$-modules. In the case of the discrete time evolution of a Hamiltonian-like operator, we relate the index for chiral unitaries to the index of the Hamiltonian. We also prove a double-sided winding number formula for anisotropic split-step quantum walks on Hilbert $C^*$-modules, extending a result by Matsuzawa.
Coherent boundaries of Lagrangian vortices in fluid flows have recently been identified as closed orbits of line fields associated with the Cauchy–Green strain tensor. Here, we develop a fully ...automated procedure for the detection of such closed orbits in large-scale velocity datasets. We illustrate the power of our method on ocean surface velocities derived from satellite altimetry.
We give an abstract formulation of the Dixmier trace formula for the density of states. This recovers prior versions and allows us to provide a Dixmier trace formula for the density of states of ...second order elliptic differential operators on manifolds of bounded geometry satisfying a certain geometric condition. This formula gives a new perspective on Roe's index on open manifolds.
We develop index and degree theories for extensive form games allowing the identification of equilibria that are robust to payoff perturbations directly from the extensive form. Our approach is based ...on index and degree theories for games where the strategy sets are polytopes (and not necessarily simplices) and payoff functions are multiaffine. Polytope strategy sets arise naturally from topologically identifying equivalent mixed strategies of a normal form game.
We prove an index theorem of Atiyah–Singer type for Dirac operators on manifolds with a Lie structure at infinity (Lie manifolds for short). With the help of a renormalized supertrace, defined on a ...suitable class of regularizing operators, the proof of the index theorem relies on a rescaling technique similar in spirit to Getzler's rescaling. With a given Lie manifold we associate an appropriate integrating Lie groupoid. We then describe the heat kernel of a geometric Dirac operator via a functional calculus with values in the convolution algebra of sections of a rescaled bundle over the adiabatic groupoid. Finally, we calculate the right coefficient in the heat kernel expansion by deforming the Dirac operator into a polynomial coefficient operator over the rescaled bundle and applying the Lichnerowicz theorem to the fibers of the groupoid and the Lie manifold.
Nous prouvons un théorème de l'indice de type Atiyah–Singer pour des opérateurs de Dirac sur des variétés munies d'une structure de Lie à l'infini (en abrégé : variétés de Lie). À l'aide d'une supertrace renormalisée, définie sur une classe appropriée d'opérateurs régularisants, la preuve du théorème de l'indice repose sur une technique de rééchelonnement similaire à celle de Getzler. À une variété de Lie donnée, nous associons un groupoïde de Lie intégrant approprié. Nous décrivons ensuite le noyau de chaleur d'un opérateur de Dirac géométrique par un calcul fonctionnel à valeurs dans l'algèbre de convolution des sections d'un fibré rééchelonné sur le groupoïde adiabatique. Enfin, nous calculons le coefficient approprié du développement du noyau de chaleur en déformant l'opérateur de Dirac dans un opérateur de coefficients polynomiaux sur le fibré rééchelonné et en appliquant le théorème de Lichnerowicz aux fibres du groupoïde et à la variété de Lie.
Given a closed smooth manifold M which carries a positive scalar curvature metric, one can associate an abelian group P(M) to the space of positive scalar curvature metrics on this manifold. The ...group of all diffeomorphisms of the manifold naturally acts on P(M). The moduli group P˜(M) of positive scalar curvature metrics is defined to be the quotient abelian group of this action, i.e. the coinvariant of the action. P˜(M) measures the size of the moduli space of positive scalar curvature metrics on M. In this paper, we use the higher rho invariant and the finite part of the K-theory of the group C⁎-algebra of π1(M) to give a lower bound of the rank of the moduli group P˜(M). The key idea of our proof is the use of higher rho invariant; this is a secondary invariant associated to Dirac operators and in particular depends on the choice of Riemannian metric. We show that the higher rho invariant remains unchanged in a certain K-theory group under the action of the diffeomorphism group, allowing us to distinguish elements in P˜(M).
In this paper, we consider the linear stability of the elliptic relative equilibria of the restricted 4-body problem where the three primaries form a Lagrangian triangle. By reduction, the linearized ...Poincaré map is decomposed to the essential part, the Keplerian part and the elliptic Lagrangian part where the last two parts have been studied in literature. The linear stability of the essential part depends on the masses parameters α, β and the eccentricity e∈0,1). Via ω-Maslov index theory and linear differential operator theory, we obtain the full bifurcation diagram of linearly stable and unstable regions with respect to α, β and e. Especially, two linearly stable sub-regions are found.
The aim of this paper is investigating the existence of weak solutions of the quasilinear elliptic model problem{−div(A(x,u)|∇u|p−2∇u)+1pAt(x,u)|∇u|p=f(x,u)in Ω,u=0on ∂Ω, where Ω⊂RN is a bounded ...domain, N≥2, p>1, A is a given function which admits partial derivative At(x,t)=∂A∂t(x,t) and f is asymptotically p-linear at infinity.
Under suitable hypotheses both at the origin and at infinity, and if A(x,⋅) is even while f(x,⋅) is odd, by using variational tools, a cohomological index theory and a related pseudo-index argument, we prove a multiplicity result if p>N in the non-resonant case.
In this paper, we use localization algebras to study higher rho invariants of closed spin manifolds with positive scalar curvature metrics. The higher rho invariant is a secondary invariant and is ...closely related to positive scalar curvature problems. The main result of the paper connects the higher index of the Dirac operator on a spin manifold with boundary to the higher rho invariant of the Dirac operator on the boundary, where the boundary is endowed with a positive scalar curvature metric. Our result extends a theorem of Piazza and Schick 27, Theorem 1.17.
We develop an index theory for linear self-adjoint operator equation with the linear self-adjoint operator having no compact resolvent. As applications, we consider the existence and multiplicity of ...periodic solutions of wave equations and beam equations.
PDF
