Elliptic Pairing Research Articles

Characterization and low-order representation of the vortex meandering motion in upswept afterbody wakes

Vortex meandering is often characterized by seemingly stochastic motions of the vortex core. An elucidation of the underlying mechanisms of core movement aids not only in characterizing the meandering phenomenon but also forms a starting point to guide flow control techniques. This paper investigates the complex motion of longitudinal vortices behind an abstracted cargo aircraft fuselage. In contrast to wingtip vortices that arise in relative isolation, in this case the large aft region upsweep generates a meandering counter-rotating vortex pair that retains its coherence for long distances downstream. High-resolution spatio-temporal datasets obtained from large-eddy simulation of a cylindrical fuselage with longitudinally aligned axis and a sharp-edged base are used for this purpose. The circulation of each vortex in the pair initially increases immediately after formation and asymptotes to a near constant value downstream. Although the time-mean form of the vortices becomes nearly axisymmetric, the transient disorganized motions include time-local regions of opposite vorticity. The complicated meandering motion is decomposed using proper-orthogonal decomposition (POD) modes as primary building blocks. The first two modes constitute a mutually orthogonal |m|=1 elliptic pair, accounting for about 35% of the total energy; individually these modes account for the displacement of vortex cores along straight lines whose slopes coincide with their respective dipole axes. The wake outside of the immediate vicinity of the base displays low-rank behavior, in that the number of modes required to reproduce the flow to a given degree of accuracy diminishes rapidly. Beyond two fuselage (cylinder) diameters downstream, the two leading POD modes can reconstruct the dominant meandering motion and spatial structure in the LES data with less than 15% error using a suitable loss measure.

Blown-up toric surfaces with non-polyhedral effective cone

Abstract We construct projective toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone. As a consequence, we prove that the pseudo-effective cone of the Grothendieck–Knudsen moduli space M ¯ 0 , n \overline{M}_{0,n} of stable rational curves is not polyhedral for n ≥ 10 n\geq 10 . These results hold both in characteristic 0 and in characteristic 𝑝, for all primes 𝑝. Many of these toric surfaces are related to an interesting class of arithmetic threefolds that we call arithmetic elliptic pairs of infinite order. Our analysis relies on tools of arithmetic geometry and Galois representations in the spirit of the Lang–Trotter conjecture, producing toric surfaces whose blow-up at a general point has a non-polyhedral pseudo-effective cone in characteristic 0 and in characteristic 𝑝, for an infinite set of primes 𝑝 of positive density.

On the geometry of elliptic pairs

An elliptic pair(X,C) is a projective rational surface X with log terminal singularities, and an irreducible curve C contained in the smooth locus of X, with arithmetic genus 1 and self-intersection 0. They are a useful tool for determining whether the pseudo-effective cone of X is polyhedral [1], and interesting algebraic and geometric objects in their own right. Especially of interest are toric elliptic pairs, where X is the blow-up of a projective toric surface at the identity element of the torus. In this paper, we classify all toric elliptic pairs of Picard number two. Strikingly, it turns out that there are only three of these. Furthermore, we study a class of non-toric elliptic pairs coming from the blow-up of P2 at nine points on a nodal cubic, in characteristic p. This construction gives us examples of surfaces where the pseudo-effective cone is non-polyhedral for a set of primes p of positive density, and, assuming the generalized Riemann hypothesis, polyhedral for a set of primes p of positive density.

Close Major-merger Pairs at z = 0: Bulge-to-total Ratio and Star Formation Enhancement

We present a study of the bulge-to-total ratio (B/T) of a Ks-band-selected sample of 88 close major-merger pairs of galaxies (H-KPAIR), based on two-dimensional decompositions of SDSS r-band images with galfit. We investigate the dependence of the interaction-induced specific star formation rate enhancement (sSFRenh) on the B/T ratio, and the effects of this dependence on the differences between star-forming galaxies (SFGs) in spiral+spiral (S+S) and spiral+elliptical (S+E) pairs. Of all the 132 spiral galaxies in H-KPAIR, the 44 in S+E pairs show higher B/T than those in the 44 S+S pairs, with means of B/T = 0.35 ± 0.05 and B/T = 0.26 ± 0.03, respectively. There is a strong negative dependence of sSFRenh on the B/T ratio, and only paired SFGs with B/T < 0.3 show significant (>5σ) enhancement. Paired SFGs in S+S pairs show a similar trend, and many disky SFGs (B/T < 0.1) in S+S pairs have strong sSFR enhancements (sSFRenh > 0.7 dex). For SFGs in S+E pairs, the sSFR has no clear B/T dependence, nor any significant enhancement in any B/T bin. Disky SFGs in S+S pairs show significant (>4σ) enhancement in molecular gas content (), while SFGs in S+E pairs have no such enhancement in any B/T bin. No significant enhancement of the total gas content (M gas/M star) is found in any B/T bin for paired galaxies. The star formation efficiency of either the total gas (SFEgas = SFR/M gas) or molecular gas () does not depend on the B/T ratio. The only significant (> 4σ) SFE enhancement found for paired SFGs is the SFEgas for disky SFGs in S+S pairs.

How Coordinate Bethe Ansatz Works for Inozemtsev Model

Three decades ago, Inozemtsev found an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane-Shastry (HS) spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero-Sutherland model. Though Inozemtsev's spin chain is widely believed to be quantum integrable, the underlying algebraic reason for its exact solvability is not yet well understood. As a step in this direction we refine Inozemtsev's `extended coordinate Bethe ansatz' and clarify various aspects of the model's exact spectrum and its limits. We identify quasimomenta in terms of which the $M$-particle energy is close to being (functionally) additive, as one would expect from the limiting models; our expression is additive iff the energy of the elliptic Calogero-Sutherland system is so. This enables us to rewrite the energy and Bethe-ansatz equations on the elliptic curve, turning the spectral problem into a rational problem as might be expected for an isotropic spin chain. We treat the $M=2$ particle sector and its limits in detail. We identify an $S$-matrix that is independent of positions. We show that the Bethe-ansatz equations reduce to those of Heisenberg in one limit and give rise to the `motifs' of HS in the other limit. We show that, as the interpolation parameter changes, the `scattering states' from Heisenberg become Yangian highest-weight states for HS, while bound states become ($\mathfrak{sl}_2$-highest weight versions of) affine descendants of the magnons from $M=1$. For bound states we find a generalisation of the known equation for the `critical length' for the Heisenberg spin chain. We discuss completeness for $M=2$ by passing to the elliptic curve. Our review of the two-particle sectors of the Heisenberg and HS spin chains may be of independent interest.

NOEMA Observations of CO Emission in Arp 142 and Arp 238

Abstract Previous studies have shown significant differences in the enhancement of the star formation rate (SFR) and star formation efficiency (SFE = SFR/M mol) between spiral–spiral and spiral–elliptical mergers. In order to shed light on the physical mechanism of these differences, we present NOEMA observations of the molecular gas distribution and kinematics (linear resolutions of ∼2 kpc) in two representative close major-merger star-forming pairs: the spiral–elliptical pair Arp 142 and the spiral–spiral pair Arp 238. The CO in Arp 142 is widely distributed over a highly distorted disk without any nuclear concentration, and an off-center ringlike structure is discovered in channel maps. The SFE varies significantly within Arp 142, with a starburst region (region 1) near the eastern tip of the distorted disk showing an SFE ∼ 0.3 dex above the mean of the control sample of isolated galaxies and the SFE of the main disk (region 4) 0.43 dex lower than the mean of the control sample. In contrast, the CO emission in Arp 238 is detected only in two compact sources at the galactic centers. Compared to the control sample, Arp 238-E shows an SFE enhancement of more than 1 dex, whereas Arp 238-W has an enhancement of ∼0.7 dex. We suggest that the extended CO distribution and large SFE variation in Arp 142 are due to an expanding large-scale ring triggered by a recent high-speed head-on collision between the spiral galaxy and the elliptical galaxy, and the compact CO sources with high SFEs in Arp 238 are associated with nuclear starbursts induced by gravitational tidal torques in a low-speed coplanar interaction.

A new design of the Lobe pump is based on the meshing principle of elliptical gear pairs

The paper presents a new rotor design of an external coupling Lobe pump driven by pair of elliptical gears. The new rotor is a four-tooth elliptical gear with tooth profile is a improve cylocid curve. The improve cylocid curve is the locus of the fixed point on the generation circle, when the circle a pure rolling without slipping on the elliptical centrode of the rotor. The conditions of the rotor addendum and dedendum profiles are also considered. The limited supply angle addendum and dedendum rotor profiles are determined through an iterative algorithm when the generation circle makes a pure rolling without slipping on the ellipse base of the rotor. From there, we proceed to determine the pump design parameters according to the characteristic design parameters forming the rotor profile. The flow rate of the pump is determined by the area of the pockets on a cross-section perpendicular to the pump shaft. On that basis, a Matlab program is written from the mathematical model established by the paper to calculate the rotor design. In addition, the paper also investigates the effect of the coefficient l (semi-major axis divided semi-minor axis of the elliptical centrode ) on the average flow and axis distance. Survey results show that the design at l = 0.5 flow is 52.17% larger and the axis distance is reduced by 21.43% when compared to the traditional design at l = 1. This is the advantage of the new design proposed by this study.

CO observations of major merger pairs at z = 0: molecular gas mass and star formation

We present CO observations of 78 spiral galaxies in local merger pairs. These galaxies represent a subsample of a Ks-band-selected sample consisting of 88 close major-merger pairs (HKPAIRs), 44 spiral–spiral (S+S) pairs, and 44 spiral–elliptical (S+E) pairs, with separation &lt;20 h−1 kpc and mass ratio &lt;2.5. For all objects, the star formation rate (SFR) and dust mass were derived from Herschel PACS and SPIRE data, and the atomic gas mass, MHI, from the Green Bank Telescope HI observations. The complete data set allows us to study the relation between gas (atomic and molecular) mass, dust mass, and SFR in merger galaxies. We derive the molecular gas fraction (MH2/M*), molecular-to-atomic gas mass ratio (MH2/MHI), gas-to-dust mass ratio and SFE (= SFR/MH2) and study their dependences on pair type (S+S compared to S+E), stellar mass, and the presence of morphological interaction signs. We find an overall moderate enhancement (∼2×) in both molecular gas fraction (MH2/M*) and molecular-to-atomic gas ratio (MH2/MHI) for star-forming galaxies in major-merger pairs compared to non-interacting comparison samples, whereas no enhancement was found for the SFE nor for the total gas mass fraction ((MHI + MH2)/M*). When divided into S+S and S+E, low mass and high mass, and with and without interaction signs, there is a small difference in SFE, a moderate difference in MH2/M*, and a strong difference in MH2/MHI between subsamples. For the molecular-to-atomic gas ratio MH2/MHI, the difference between S+S and S+E subsamples is 0.55 ± 0.18 dex and between pairs with and without interaction sign 0.65 ± 0.16 dex. Together, our results suggest that (1) star formation enhancement in close major-merger pairs occurs mainly in S+S pairs after the first close encounter (indicated by interaction signs) because the HI gas is compressed into star-forming molecular gas by the tidal torque; and (2) this effect is much weakened in the S+E pairs.

A General Elliptical Formulation of Hybrid-POLSAR System Ambiguities

In this letter, we present a new matrix formulation of range and azimuth ambiguities in polarimetric synthetic aperture radar. We consider the general hybrid case, where the scattering matrix is measured by transmitting a general orthogonal elliptical pair as interlaced pulse trains with simultaneous linear H and V receive. Earlier papers have shown some potential advantages for ambiguity suppression in this mode, and here, we confirm and expand these and show two key new ideas: first that successful suppression of range ambiguities is strongly phase dependent and second that suppression can be achieved for a set of elliptical polarizations along a great circle of the Poincare sphere

Dirac cohomology, the projective supermodules of the symmetric group and the Vogan morphism

Abstract We derive an explicit description of the genuine projective representations of the symmetric group Sn using Dirac cohomology and the branching graph for the irreducible genuine projective representations of Sn. Ciubotaru and He [D. Ciubotaru and X. He, Green polynomials of Weyl groups, elliptic pairings, and the extended index. Adv. Math., 283:1–50, 2015], using the extended Dirac index, showed that the characters of the projective representations of Sn are related to the characters of elliptic-graded modules. We derive the branching graph using Dirac theory and combinatorics relating to the cohomology of Borel varieties ℬe of g and are able to use Dirac cohomology to construct an explicit model for the projective representations. We also describe Vogan’s morphism for Hecke algebras in type A using spectrum data of the Jucys–Murphy elements.

Simultaneous Angular Rate Estimates Extracted From a Single Axisymmetric Resonator

An axisymmetric planar microelectromechanical (MEM) resonator resonator is configured, such that the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n = 2$ </tex-math></inline-formula> elliptical pair of modes near 13.5kHz, and the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n = 3$ </tex-math></inline-formula> pair of modes near 23.8kHz, are both degenerate, i.e., the frequency difference within a given pair of modes is close to the resonance bandwidth. This configuration enables not only the exploitation of the standard elliptical pair of modes for angular rate sensing but also permits the operation of the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n = 3$ </tex-math></inline-formula> pair of modes as a high-sensitivity Coriolis vibratory gyro. The performance for each pair of modes separately acting as a vibratory gyro is quantified, however, the control architecture also facilitates the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">simultaneous</i> operation of both pairs of modes. In this scenario, two measurements of angular rate are extracted from a single resonator and although the short-term rate noise associated with the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n = 3$ </tex-math></inline-formula> pair is an order of magnitude larger than the rate extracted from the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n = 2$ </tex-math></inline-formula> pair, the long-term drift in the rate offsets are correlated. Thus, a filter architecture for fusing the rate measurements is proposed and it is shown how the derived rate estimate possesses superior offset stability but also retains the low short-term noise associated with the rate measurement from the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$n = 2$ </tex-math></inline-formula> pair.

Attacks on “a provably secure and efficient authentication scheme for access control in mobile pay-TV systems”

Mobile pay-TV systems represent an application of important electronic commerce, providing mobile users with the multimedia services. In 2016, Farash and Attari employed the elliptic curves and bilinear pairing technologies to design a one-to-many authentication scheme for mobile pay-TV systems. They claimed that their scheme provides a high level of efficiency and can resist most attacks on mobile pay-TV systems. In this paper, we point out their scheme cannot achieve the fundamental requirement of authentication that is, the mutual authentication. In addition, their scheme still suffers from three security weaknesses.

Elliptic minuscule pairs and splitting abelian varieties

We partially answer, in terms of monodromy, Murty and Patankar's question: Given an absolutely simple abelian variety over a number field, does it have simple specializations at a set of places of positive Dirichlet density? The answer is based on the classification of pairs (G,V) consisting of a semi-simple algebraic group G over a non-archimedean local field and an absolutely irreducible representation V of G such that G admits a maximal torus acting irreducibly on V.

HERSCHEL OBSERVATIONS OF MAJOR MERGER PAIRS AT z = 0: DUST MASS AND STAR FORMATION

ABSTRACT We present Herschel PACS and SPIRE far-infrared (FIR) and submillimeter imaging observations for a large K-band selected sample of 88 close major-merger pairs of galaxies (H-KPAIRs) in 6 photometric bands (70, 100, 160, 250, 350, and 500 μm). Among 132 spiral galaxies in the 44 spiral–spiral (S+S) pairs and 44 spiral–elliptical (S+E) pairs, 113 are detected in at least 1 Herschel band. The star formation rate (SFR) and dust mass (M dust) are derived from the IR SED fitting. The mass of total gas (M gas) is estimated by assuming a constant dust-to-gas mass ratio of 0.01. Star-forming spiral galaxies (SFGs) in S+S pairs show significant enhancements in both specific star formation rate (sSFR) and star formation efficiency (SFE), while having nearly the same gas mass compared to control galaxies. On the other hand, for SFGs in S+E pairs, there is no significant sSFR enhancement and the mean SFE enhancement is significantly lower than that of SFGs in S+S pairs. This suggests an important role for the disk–disk collision in the interaction-induced star formation. The M gas of SFGs in S+E pairs is marginally lower than that of their counterparts in both S+S pairs and the control sample. Paired galaxies with and without interaction signs do not differ significantly in their mean sSFR and SFE. As found in previous works, this much larger sample confirms that the primary and secondary spirals in S+S pairs follow a Holmberg effect correlation on sSFR.

Euler-Poincaré pairing, Dirac index and elliptic pairing for Harish-Chandra modules

Let G be a connected real reductive group with maximal compact subgroup K of equal rank, and let M be the category of Harish-Chandra modules for G. We relate three differentely defined pairings between two finite length modules X and Y in M : the Euler-Poincare pairing, the natural pairing between the Dirac indices of X and Y , and the elliptic pairing of [2]. (The Dirac index IDir(X) is a virtual finite dimensional representation of K, the spin double cover of K.) Analogy with the case of Hecke algebras studied in [7] and [6] and a formal (but not rigorous) computation lead us to conjecture that the first two pairings coincide. In the second part of the paper, we show that they are both computed as the indices of Fredholm pairs (defined here in an algebraic sense) of operators acting on the same spaces. We construct Euler-Poincare functions fX for any finite length Harish-Chandra module X. These functions are very cuspidal in the sense of Labesse, and their orbital integrals on elliptic elements coincide with the character of X. From this we deduce that the Dirac index pairing coincide with the elliptic pairing. These results are the archimedean analog of results of Schneider-Stuhler [21] for p-adic groups.

Dirac cohomology for symplectic reflection algebras

We define uniformly the notions of Dirac operators and Dirac cohomology in the framework of the Hecke algebras introduced by Drinfeld (Anal i Prilozhen 20(1):69–70, 1986). We generalize in this way, the Dirac cohomology theory for Lusztig’s graded affine Hecke algebras defined in Barbasch et al. (Acta Math 209(2):197–227, 2012) and further developed in Barbasch et al. (Acta Math 209(2):197–227, 2012), Ciubotaru et al. (J Inst Math Jussieu 13(3):447–486, 2014), Ciubotaru (J Reine Angew Math 671:199–222, 2012), Ciubotaru and He (Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index, Adv. Math. 283:1–50, 2015), Chan (On a twisted Euler-Poincare pairing for graded affine Hecke algebras, preprint 2014, arXiv:1407.0956). We apply these constructions to the case of the symplectic reflection algebras defined by Etingof and Ginzburg (Invent Math 147:243–348, 2002), particularly to rational Cherednik algebras for real or complex reflection groups. As applications, we give criteria for unitarity of modules in category \({\mathcal {O}}\) and we show that the 0-fiber of the Calogero–Moser space admits a description in terms of a certain “Dirac morphism” originally defined by Vogan for representations of real reductive groups.

Green polynomials of Weyl groups, elliptic pairings, and the extended Dirac index

In this paper, we give a uniform construction of irreducible genuine characters of the Pin cover W˜ of a Weyl group W, and put them into the context of theory of Springer representations. In the process, we provide a direct connection between Springer theory, via Green polynomials, the irreducible representations of W˜, and an extended Dirac operator for graded Hecke algebras. We also introduce a q-elliptic pairing for W with respect to the reflection representation V. These constructions are of independent interest. The q-elliptic pairing is a generalization of the elliptic pairing of W introduced by Reeder, and it is also related to S. Kato's notion of (graded) Kostka systems for the semidirect product AW=C[W]⋉S(V).

Geometric Anisotropic Spatial Point Pattern Analysis and Cox Processes

ABSTRACTWe consider spatial point processes with a pair correlation function, which depends only on the lag vector between a pair of points. Our interest is in statistical models with a special kind of ‘structured’ anisotropy: the pair correlation function is geometric anisotropic if it is elliptical but not spherical. In particular, we study Cox process models with an elliptical pair correlation function, including shot noise Cox processes and log Gaussian Cox processes, and we develop estimation procedures using summary statistics and Bayesian methods. Our methodology is illustrated on real and synthetic datasets of spatial point patterns.

Regularity for elliptic pairs over $\mathbb C [[\hbar ]

We extend the results of Schapira and Schneiders [12] on relative regularity, finiteness and duality (in the smooth case) of elliptic pairs to the framework of \mathcal D [[\hbar ]] -modules and constructible sheaves of \mathbb C [[\hbar ]] -modules.

Microlocal Euler classes and Hochschild homology

AbstractWe define the notion of a trace kernel on a manifold $M$. Roughly speaking, it is a sheaf on $M\times M$ for which the formalism of Hochschild homology applies. We associate a microlocal Euler class with such a kernel, a cohomology class with values in the relative dualizing complex of the cotangent bundle ${T}^{\ast } M$ over $M$, and we prove that this class is functorial with respect to the composition of kernels.This generalizes, unifies and simplifies various results from (relative) index theorems for constructible sheaves, $\mathscr{D}$-modules and elliptic pairs.