Elliptic and K-theoretic stable envelopes and Newton polytopes

In this paper we consider the cotangent bundles of partial flag varieties. We construct the $K$-theoretic stable envelopes for them and also define a version of the elliptic stable envelopes. We expect that our elliptic stable envelopes coincide with the elliptic stable envelopes defined by M. Aganagic and A. Okounkov. We give formulas for the $K$-theoretic stable envelopes and our elliptic stable envelopes. We show that the $K$-theoretic stable envelopes are suitable limits of our elliptic stable envelopes. That phenomenon was predicted by M. Aganagic and A. Okounkov. Our stable envelopes are constructed in terms of the elliptic and trigonometric weight functions which originally appeared in the theory of integral representations of solutions of qKZ equations twenty years ago. (More precisely, the elliptic weight functions had appeared earlier only for the $\frak{gl}_2$ case.) We prove new properties of the trigonometric weight functions. Namely, we consider certain evaluations of the trigonometric weight functions, which are multivariable Laurent polynomials, and show that the Newton polytopes of the evaluations are embedded in the Newton polytopes of the corresponding diagonal evaluations. That property implies the fact that the trigonometric weight functions project to the $K$-theoretic stable envelopes.Comment: Latex, 37 pages; v.2: Appendix and Figure 1 added; v.3: missing shift in Theorem 2.9 added and a proof of Theorem 2.9 adde

Sealed containers in Z

Physical means of securing information, such as sealed envelopes and scratch cards, can be used to achieve cryptographic objectives. Reasoning about this has so far been informal. We give a model of distinguishable sealed envelopes in Z, exploring design decisions and further analysis and development of such models

Optically Thin Core Accretion: How Planets Get Their Gas in Nearly Gas-Free Disks

Models of core accretion assume that in the radiative zones of accreting gas envelopes, radiation diffuses. But super-Earths/sub-Neptunes (1-4$R_\oplus$, 2-20$M_\oplus$) point to formation conditions that are optically thin: their modest gas masses are accreted from short-lived and gas-poor nebulae reminiscent of the transparent cavities of transitional disks. Planetary atmospheres born in such environments can be optically thin to both incident starlight and internally generated thermal radiation. We construct time-dependent models of such atmospheres, showing that super-Earths/sub-Neptunes can accrete their $\sim$1%-by-mass gas envelopes, and super-puffs/sub-Saturns their $\sim$20%-by-mass envelopes, over a wide range of nebular depletion histories requiring no fine tuning. Although nascent atmospheres can exhibit stratospheric temperature inversions effected by atomic Fe and various oxides that absorb strongly at visible wavelengths, the rate of gas accretion remains controlled by the radiative-convective boundary (rcb) at much greater pressures. For dusty envelopes, the temperature at the rcb $T_{\rm rcb} \simeq 2500$ K is still set by ${\rm H}_2$ dissociation; for dust-depleted envelopes, $T_{\rm rcb}$ tracks the temperature of the visible or thermal photosphere, whichever is deeper, out to at least $\sim$5 AU. The rate of envelope growth remains largely unchanged between the old radiative diffusion models and the new optically thin models, reinforcing how robustly super-Earths form as part of the endgame chapter in disk evolution.Comment: accepted to MNRAS, new section 4.2 connects our formation scenario of super-Earths to atmospheric mass los

Imaging the circumstellar envelopes of AGB stars

We report the results of an exploratory program to image the extended circumstellar envelopes of asymptotic giant branch (AGB) stars in dust-scattered galactic light. The goal is to characterize the morphology of the envelopes as a probe of the mass-loss process. The observations consist of short exposures with the VLT and longer exposures with 1-2m telescopes, augmented with archival images from the Hubble Space Telescope. We observed 12 AGB stars and detected the circumstellar envelopes in 7. The detected envelopes have mass loss rates more than about 5 10E-6 solar mass per year, and they can be seen out to distances of about 1 kpc. The observations provide information on the mass loss history on time scales up to about 10,000 years. For the five AGB envelopes in which the circumstellar geometry is well determined by scattered light observations, all except one (OH348.2-19.7) show deviations from spherical symmetry. Two (IRC+10216 and IRC+10011) show roughly spherical envelopes at large radii but asymmetry or bipolarity close to the star; one (AFGL 2514) shows an extended, elliptical envelope, and one (AFGL 3068) shows a spiral pattern. The non-spherical structures are all consistent with the effects of binary interactions. Our observations are in accord with a scenario in which binary companions play a role in shaping planetary nebulae, and show that the circumstellar gas is already partly shaped on the AGB, before evolution to the proto-planetary nebula phase.Comment: Accepted by AA 21 Feb 2006; 18 pages, 14 figs; for high resolution images, contact mauron at graal.univ-montp2.f