Orbital Integrals Research Articles

An equivariant Atiyah–Patodi–Singer index theorem for proper actions II: the K-theoretic index

AbstractConsider a proper, isometric action by a unimodular locally compact group G on a Riemannian manifold M with boundary, such that M/G is compact. Then an equivariant Dirac-type operator D on M under a suitable boundary condition has an equivariant index $${{\,\mathrm{index}\,}}_G(D)$$ index G ( D ) in the K-theory of the reduced group $$C^*$$ C ∗ -algebra $$C^*_rG$$ C r ∗ G of G. This is a common generalisation of the Baum–Connes analytic assembly map and the (equivariant) Atiyah–Patodi–Singer index. In part I of this series, a numerical index $${{\,\mathrm{index}\,}}_g(D)$$ index g ( D ) was defined for an element $$g \in G$$ g ∈ G , in terms of a parametrix of D and a trace associated to g. An Atiyah–Patodi–Singer type index formula was obtained for this index. In this paper, we show that, under certain conditions, $$\begin{aligned} \tau _g({{\,\mathrm{index}\,}}_G(D)) = {{\,\mathrm{index}\,}}_g(D), \end{aligned}$$ τ g ( index G ( D ) ) = index g ( D ) , for a trace $$\tau _g$$ τ g defined by the orbital integral over the conjugacy class of g. This implies that the index theorem from part I yields information about the K-theoretic index $${{\,\mathrm{index}\,}}_G(D)$$ index G ( D ) . It also shows that $${{\,\mathrm{index}\,}}_g(D)$$ index g ( D ) is a homotopy-invariant quantity.

Exp: N-body integration using basis function expansions

ABSTRACT We present the N-body simulation techniques implemented in the exp code. exp uses empirically chosen basis functions to expand the potential field of an ensemble of particles. Unlike other basis function expansions, the derived basis functions are adapted to an input mass distribution, enabling accurate expansion of highly non-spherical objects, such as Galactic discs. We measure the force accuracy in three models, one based on a spherical or aspherical halo, one based on an exponential disc, and one based on a bar-based disc model. We find that exp is as accurate as a direct-summation or tree-based calculation, and in some ways is better, while being considerably less computationally intensive. We discuss optimizing the computation of the basis function representation. We also detail numerical improvements for performing orbit integrations, including time-steps.

Gaia early DR3 systemic motions of Local Group dwarf galaxies and orbital properties with a massive Large Magellanic Cloud

Aims. We perform a comprehensive determination of the systemic proper motions of 74 dwarf galaxies and dwarf galaxy candidates in the Local Group based on Gaia early data release 3. The outputs of the analysis for each galaxy, including probabilities of membership, will be made publicly available. The analysis is augmented by a determination of the orbital properties of galaxies within 500 kpc. Methods. We adopt a flexible Bayesian methodology presented in the literature, which takes into account the location of the stars on the sky, on the colour-magnitude diagram, and on the proper motion plane. We applied some modifications, in particular to the way the colour-magnitude diagram and spectroscopic information are factored in, for example, by including stars in several evolution phases. The bulk motions were integrated in three gravitational potentials: two where the Milky Way was treated in isolation and has a mass 0.9 & 1.6 × 1012 M⊙, and a time-varying potential, which includes the infall of a massive Large Magellanic Cloud (LMC). Results. We were able to determine bulk proper motions for 73 systems, and we consider 66 to be reliable measurements. For the first time, systemic motions are presented for galaxies out to a distance of 1.4 Mpc in the NGC 3109 association. The inclusion of the infall of a massive LMC significantly modifies the orbital trajectories of the objects, with respect to orbit integration in static Milky-Way-only potentials, and this leads to six galaxies likely being associated with the LMC, three possibly being associated with it, and one recently captured object. We discuss the results of the orbit integration in the context of the relation of the galaxies to the system of Milky Way satellites, implications for the too-big-to-fail problem, the impact on star formation histories, and tidal disruption.

Подходы к определению коэффициента геометрии сетевых организационных структур

The article analyzes the methods of determining the geometry coefficient in network organizational structures and offers its own ways of calculating it. The geometry coefficient is presented as an integral indicator which consolidates the coefficients of filling orbits, connectivity and criticality of connections. The orbit filling coefficient characterizes the organizational network through the reserve production capacities realized by the enterprises of the organizational network in the free market. The connectivity coefficient characterizes excess mileage, transport load, fuel consumption and the remoteness of organizational network elements from the integrator. The coefficient of criticality of connections characterizes the organizational network through availability of subsuppliers in the elements belonging to the first from the integrator orbit of the network structure. The developed methodology for determining the organizational network geometry coefficient makes it possible to evaluate the work of the network from the side of geometric indicators. The existing methods of determining the coefficient in network organizational structures are also analyzed. The disadvantages and features of calculating the geometric characteristics of organizational networks are revealed. The author’s methods for determining the indicators included in the coefficient of the geometry of network structures are proposed. The geometry coefficient is presented as an integral indicator that consolidates the filling coefficients of orbits, connectivity and criticality of connections. The orbit filling coefficient characterizes the organizational network through the reserve production capacities realized by the enterprises of the organizational network in the free market. The greater the volume of reserve production capacities possessed by enterprises of the first orbit integrator, the greater the filling of the organizational network as a whole. The connectivity coefficient characterizes excess mileage, transport load, fuel consumption and the remoteness of organizational network elements from the integrator. The coefficient of criticality of connections characterizes the organizational network through availability of subsuppliers in the elements belonging to the first from the integrator orbit of the network structure. The developed methodology for determining the organizational network geometry coefficient allows us to evaluate the work of the network from the geometric haracteristics side. The methodology can be applied when forming a network organizational structure and quantifying its elements.

Pitch angle scattering of fast particles by low frequency magnetic fluctuations

The adiabatic invariance of the magnetic moment during particle motion is of fundamental importance to the dynamics of magnetized plasma. The related rate of pitch angle scattering is investigated here for fast particles that thermally stream through static magnetic perturbations. For a uniform magnetic field with a localized perturbation, it is found that the curvature parameter κ2=min(Rc/ρL) does not predict the level of pitch angle scattering. Instead, based on numerical integration of particle orbits in prescribed magnetic fields, we derive predictions for the particle scattering rates, which can be characterized by the relative perturbation amplitude A≃δB/B, and the particle Larmor radius normalized by the field-aligned wavelength of the perturbations, ρL/λ||. Particles with ρL/λ||≃1 are subject to strong pitch angle scattering, while the level of scattering vanishes for both the limits of ρL/λ||≪1 and ρL/λ||≫1. The results are summarized in terms of a scattering operator, suitable for including the described scattering in basic kinetic models.

An infinitesimal variant of the Guo–Jacquet trace formula, II

We establish an infinitesimal variant of Guo-Jacquet trace formula for the case of a central simple algebra over a number field $F$ containing a quadratic field extension $E/F$. It is an equality between a sum of geometric distributions on the tangent space of some symmetric space and its Fourier transform. To prove this, we need to define an analogue of Arthur's truncation and then use the Poisson summation formula. We describe the terms attached to regular semi-simple orbits as explicit weighted orbital integrals. To compare them to those for another case studied in our previous work, we state and prove the weighted fundamental lemma at the infinitesimal level by using Labesse's work on the base change for $GL_n$.

An Equivariant Atiyah–Patodi–Singer Index Theorem for Proper Actions I: The Index Formula

AbstractConsider a proper, isometric action by a unimodular locally compact group $G$ on a Riemannian manifold $M$ with boundary, such that $M/G$ is compact. For an equivariant, elliptic operator $D$ on $M$, and an element $g \in G$, we define a numerical index ${\operatorname {index}}_g(D)$, in terms of a parametrix for $D$ and a trace associated to $g$. We prove an equivariant Atiyah–Patodi–Singer index theorem for this index. We first state general analytic conditions under which this theorem holds, and then show that these conditions are satisfied if $g=e$ is the identity element; if $G$ is a finitely generated, discrete group, and the conjugacy class of $g$ has polynomial growth; and if $G$ is a connected, linear, real semisimple Lie group, and $g$ is a semisimple element. In the classical case, where $M$ is compact and $G$ is trivial, our arguments reduce to a relatively short and simple proof of the original Atiyah–Patodi–Singer index theorem. In part II of this series, we prove that, under certain conditions, ${\operatorname {index}}_g(D)$ can be recovered from a $K$-theoretic index of $D$ via a trace defined by the orbital integral over the conjugacy class of $g$.

Accurate Analysis of Anisotropic Carrier Mobility and Structure-property Relationships in Organic BOXD Crystalline Materials.

Charge mobility is an essential factor of organic crystalline materials. Although many investigators have made important progress, the exact relationship between the crystal structure and carrier mobility remains to be clarified. Fortunately, a series of bis-1,3,4-oxadiazole derivatives have been successfully prepared and reported. They have similar main molecular fragments but different crystal packing modes, which provide an ideal research objective for studying the effect of molecular packing on charge mobility in organic photoelectric conversion systems. In this work, the charge mobilities of these molecules are systematically evaluated from the perspective of first-principles calculation, and the effect of a molecular overlap on orbital overlap integral and final charge carrier mobility is fully discussed. It can be seen that the small intermolecular distance (less than 6 Å) is the decisive factor to achieve high electron mobility in π stacking, and better mobility can be obtained by increasing the hole migration distance appropriately. A larger dihedral angle of anisotropy is an important point limiting the charge mobility in the herringbone arrangement. It is hoped that the correlation results between the crystal structure and mobility can assist the experimental study and provide an effective way to improve the photoelectric conversion efficiency of the organic semiconductor devices and multiple basis for multiscale material system characterization and material information.

TRUNCATED AFFINE SPRINGER FIBERS AND ARTHUR’S WEIGHTED ORBITAL INTEGRALS

AbstractWe explain an algorithm to calculate Arthur’s weighted orbital integral in terms of the number of rational points on the fundamental domain of the associated affine Springer fiber. The strategy is to count the number of rational points of the truncated affine Springer fibers in two ways: by the Arthur–Kottwitz reduction and by the Harder–Narasimhan reduction. A comparison of results obtained from these two approaches gives recurrence relations between the number of rational points on the fundamental domains of the affine Springer fibers and Arthur’s weighted orbital integrals. As an example, we calculate Arthur’s weighted orbital integrals for the groups ${\textrm {GL}}_{2}$ and ${\textrm {GL}}_{3}$ .

Relativistic effects on triple black holes: Burrau’s problem revisited

ABSTRACT We explore, using numerical simulations, the influence of mass and distance on the evolution of triple black hole systems. Following in the direction of Burrau’s famous 3,4,5 problem, black holes are initially placed at the vertices of Pythagorean triangles. Numerical integration of orbits was conducted using relativistic corrections (post-Newtonian) up to the 2.5th order with ARCcode. As a descriptor of the evolution of the systems, the lifetimes, the number of two-body encounters and the number of mergers were all analysed. We found that as the mass unit of the black holes increased, there was strong positive correlation with the fraction of mergers (0.9868), strong negative correlation with the average number of two-body encounters (−0.9016), and the average lifetimes of the triple systems decayed exponentially (determination coefficient of 0.9986). Around the mass unit range of 105.5–105.6M⊙, there was a transition from escape dominated dynamics to merger dominated dynamics. However, in the mass unit range of 106–109M⊙ with 1 pc distance unit, we find that 25 per cent of cases resulted in the escape of a supermassive black hole (SMBH) which may be a cause for wandering SMBH’s found in some galaxies/galactic merger remnants.

Numerical integration of particle orbits in discontinuous fields using VENUS-LEVIS and SPEC

The orbit code VENUS-LEVIS is adapted to follow particles in plasma equilibria with discontinuous fields generated by the Stepped Pressure Equilibrium Code (SPEC). The latter is an implementation of the Multi-Region relaxed MHD model, which efficiently computes Taylor states in a series of nested toroidal volumes and supports the formation of magnetic islands and chaotic field regions. To adapt VENUS-LEVIS, an event location procedure is implemented in the existing numerical integrator, which ensures the particle sees the correct field along its trajectory, regardless of the discontinuities present in the Stepped Pressure Equilibrium model. The algorithm is tested in the case where the magnetic field is uniform in the upper and lower half-spaces but has a discontinuity in its direction (shear) on the plane z=0. Particle drifts due to the discontinuity are studied. The convergence properties of the numerical scheme are highlighted by the numerical accuracy, and conservation of the system's invariants, such as energy and momentum. Simulations and convergence studies using the SPEC-LEVIS interface in axisymmetric geometry are then presented. Finally, illustrative particle drifts due to the discontinuity are studied and explained: we examine drifts associated with the change in Larmor radius of passing particles with small excursion from flux surfaces.

On the stabilization of relative trace formulae: Descent and the fundamental lemma

Motivated by the study of periods of automorphic forms and relative trace formulae, we develop the theory of descent necessary to study orbital integrals arising in the fundamental lemma for a general class of symmetric spaces over a p-adic field F. More precisely, we prove that a connected symmetric space over F enjoys a notion of topological Jordan decomposition, which may be of independent interest, and establish a relative version of a lemma of Kazhdan that played a crucial role in the proof of the Langlands–Shelstad fundamental lemma.As our main application, we use these results to prove the endoscopic fundamental lemma for the unit element of the Hecke algebra for the symmetric space associated to unitary Friedberg–Jacquet periods.

The two phases of core formation – orbital evolution in the centres of ellipticals with supermassive black hole binaries

ABSTRACT The flat stellar density cores of massive elliptical galaxies form rapidly due to sinking supermassive black holes (SMBHs) in gas-poor galaxy mergers. After the SMBHs form a bound binary, gravitational slingshot interactions with nearby stars drive the core regions towards a tangentially biased stellar velocity distribution. We use collisionless galaxy merger simulations with accurate collisional orbit integration around the central SMBHs to demonstrate that the removal of stars from the centre by slingshot kicks accounts for the entire change in velocity anisotropy. The rate of strong (unbinding) kicks is constant over several hundred Myr at $\sim 3 \ \mathrm{ M}_\odot\, \rm yr^{-1}$ for our most massive SMBH binary (MBH = 1.7 × 1010 M⊙). Using a frequency-based orbit classification scheme (box, x-tube, z-tube, rosette), we demonstrate that slingshot kicks mostly affect box orbits with small pericentre distances, leading to a velocity anisotropy of β ≲ −0.6 within several hundred Myr as observed in massive ellipticals with large cores. We show how different SMBH masses affect the orbital structure of the merger remnants and present a kinematic tomography connecting orbit families to integral field kinematic features. Our direct orbit classification agrees remarkably well with a modern triaxial Schwarzschild analysis applied to simulated mock kinematic maps.

Linking nearby stellar streams to more distant halo overdensities

Context. It has recently been shown that the halo near the Sun contains several kinematic substructures associated with past accretion events. For the more distant halo, there is evidence of large-scale density variations – in the form of stellar clouds or overdensities. Aims. We study the link between the local halo kinematic groups and three of these stellar clouds: the Hercules-Aquila cloud, the Virgo Overdensity, and the Eridanus-Phoenix overdensity. Methods. We perform orbital integrations in a standard Milky Way potential of a local halo sample extracted from Gaia EDR3 with the goal of predicting the location of the merger debris elsewhere in the Galaxy. We specifically focus on the regions occupied by the three stellar clouds and compare their kinematic and distance distributions with those predicted from the orbits of the nearby debris. Results. We find that the local halo substructures have families of orbits that tend to pile up in the regions where the stellar clouds have been found. The distances and velocities of the cloud’s member stars are in good agreement with those predicted from the orbit integrations, particularly for Gaia-Enceladus stars. This is the dominant contributor of all three overdensities, with a minor part stemming from the Helmi streams and to an even smaller extent from Sequoia. The orbital integrations predict no asymmetries in the sky distribution of halo stars, and they pinpoint where additional debris associated with the local halo substructures may be located.

The young Hobson family: Possible binary parent body and low-velocity dispersal

Context.Asteroid families with ages younger than 1 Myr offer an interesting possibility of studying the outcomes of asteroid disruptions that are little modified by subsequent evolutionary processes.Aims.We analyze a very young asteroid family associated with (18777) Hobson in the central part of the main belt. We aim at (i) understanding its peculiar size distribution, and (ii) setting an upper limit on the characteristic dispersal velocity at subkilometer sizes corresponding to the smallest visible Hobson members.Methods.We identified the Hobson family using an up-to-date asteroid catalog. A significant increase in the number of its known members allowed us to study their size distribution and compare it with computer simulations of catastrophic disruptions. Backward orbital integrations of the heliocentric orbits allowed us to confirm the previously suggested age of Hobson and helped to estimate limits of the ejection speed.Results.The Hobson family has an unusual size distribution: two nearly equal-size bodies, followed by a population of smaller asteroids, whose distribution takes a characteristic power law. There are two possibilities to explain these data. Either a canonical impact onto a single parent body, requiring fine-tuned impact conditions that have not been studied so far, or an unconventional model for the parent object of the Hobson family, namely a binary with ≃7−9 km primary and a ≃2.5 km secondary. In the latter case, the primary was disrupted, leaving behind the largest remnant (18777) Hobson and a suite of subkilometer asteroids. The second largest asteroid, (57738) 2001 UZ160, is the nearly intact satellite of the parent binary. The excellent convergence of nominal orbits of Hobson members sets an upper limit of ≃(10−20) m s−1for the initial dispersal velocity of the known members, which is consistent with both formation models. The Hobson family provides a so far rare opportunity of studying disruptions of small asteroids in a situation in which both the material strength and reaccumulation efficiency play an important role.

Transfer factors for Jacquet-Mao's fundamental lemma

In an earlier paper we proved Jacquet-Mao’s metaplectic fundamental lemma which is the identity between two orbital integrals (one is defined on the space of symmetric matrices and another one is defined on the 2-fold cover of the general linear group) corrected by a transfer factor. But we restricted to the case where the relevant representative is a diagonal matrix. Now, we show that we can extend this result for the more general relevant representative. Our proof is based on the concept of Shalika germs for certain Kloosterman integrals.

Orbits of 152 globular clusters of the MilkyWay galaxy constructed from Gaia DR2

We present orbits and their properties for 152 globular clusters of the Milky Way galaxy obtained using average Gaia DR2 proper motions and other astrometric data from the list of Vasiliev. For orbital integration we have applied the axisymmetric model of the Galactic potential based on the Navarro-Frenk- White dark halo, and modified by Bajkova & Bobylev utilizing circular velocities of Galactic objects in a wide region of Galactocentric distances (up to 200 kpc) from the Bhattacharjee et al. catalog. Based on the analysis of the obtained orbits, we have modified the composition of the subsystems of globular clusters presented in Massari et al.

Isolation of the cuspidal spectrum: The function field case

Isolating cuspidal automorphic representations from the whole automorphic spectrum is a basic problem in the trace formula approach. For example, matrix coefficients of supercuspidal representations can be used as test functions for this. However, they kill a large class of interesting cuspidal automorphic representations. For the case of number fields, multipliers of the Schwartz algebra are used in the recent work (see Beuzart-Plessis et al. (2019)) to isolate all the cuspidal spectrum. In particular, they are suitable for the comparison of orbital integrals. These multipliers are then applied to the proof of the Gan-Gross-Prasad conjecture for unitary groups (see Beuzart-Plessis et al. (2019, 2020)). In this article, we prove the similar result on isolating the cuspidal spectrum in Beuzart-Plessis et al. (2019) for the function field case.

An index theorem for higher orbital integrals

An index theorem for higher orbital integrals

Minimum perihelion distances and associated dwell times for near-Earth asteroids

ABSTRACT The observed near-Earth asteroid (NEA) population contains very few objects with small perihelion distances, say, $q \lesssim 0.2\, \mathrm{au}$. NEAs that currently have orbits with larger q might be hiding a past evolution during which they have approached closer to the Sun. We present a probabilistic assessment of the minimum q that an asteroid has reached during its orbital history. At the same time, we offer an estimate of the dwell time, that is, the time q has been in a specific range. We have re-analysed orbital integrations of test asteroids from the moment they enter the near-Earth region until they either collide with a major body or are thrown out from the inner Solar system. We considered a total disruption of asteroids at certain q as a function of absolute magnitude (H). We calculated the probability that an asteroid with given orbital elements and H has reached a q smaller than a given threshold value and its respective dwell time in that range. We have constructed a look-up table that can be used to study the past orbital and thermal evolution of asteroids as well as meteorite falls and their possible parent bodies. An application to 25 meteorite falls shows that carbonaceous chondrites typically have short dwell times at small q, whereas for ordinary chondrites it ranges from $10\, 000$ to $500\, 000$ yr. A dearth of meteorite falls with long dwell times and small minimum q supports a supercatastrophic disruption of asteroids at small q.