Schubert classes in the equivariant cohomology of the Lagrangian Grassmannian

Let $LG_n$ denote the Lagrangian Grassmannian parametrizing maximal isotropic (Lagrangian) subspaces of a fixed symplectic vector space of dimension $2n.$ For each strict partition $\lambda=(\lambda_1,...,\lambda_k)$ with $\lambda_1\leq n$ there is a Schubert variety $X(\lambda).$ Let $T$ denote a maximal torus of the symplectic group acting on $LG_n.$ Consider the $T$-equivariant cohomology of $LG_n$ and the $T$-equivariant fundamental class $\sigma(\lambda)$ of $X(\lambda).$ The main result of the present paper is an explicit formula for the restriction of the class $\sigma(\lambda)$ to any torus fixed point. The formula is written in terms of factorial analogue of the Schur $Q$-function, introduced by Ivanov. As a corollary to the restriction formula, we obtain an equivariant version of the Giambelli-type formula for $LG_n.$ As another consequence of the main result, we obtained a presentation of the ring $H_T^*(LG_n).

Excited Young diagrams and equivariant Schubert calculus

We describe the torus-equivariant cohomology ring of isotropic Grassmannians by using a localization map to the torus fixed points. We present two types of formulas for equivariant Schubert classes of these homogeneous spaces. The first formula involves combinatorial objects which we call ``excited Young diagrams'' and the second one is written in terms of factorial Schur $Q$- or $P$-functions. As an application, we give a Giambelli-type formula for the equivariant Schubert classes. We also give combinatorial and Pfaffian formulas for the multiplicity of a singular point in a Schubert variety.Comment: 29 page

The economic theory of quasi-exclusive territory

This paper introduces the economic theory of gquasi-exclusive territory.h We consider vertical dealings with two upstream firms and four downstream firms that compete in two separate markets. Under quasi- exclusive territory, downstream firms are bound to pay additional charges when selling goods beyond their territorial areas. We find that with respect to the two-part tariffs comprising a marginal price and a fixed fee, quasi-exclusive territory is more beneficial for upstream firms and more harmful for consumers than conventional exclusive territory. Moreover, we note that quasi-exclusive territory is in practice in various vertical dealings and that its regulation is a difficult task.Exclusive territory, Quasi-exclusive territory, Vertical dealings, Two-part tariff

Thinning or thickening? Multiple rheological regimes in dense suspensions of soft particles

The shear rheology of dense colloidal and granular suspensions is strongly nonlinear, as these materials exhibit shear-thinning and shear-thickening, depending on multiple physical parameters. We numerically study the rheology of a simple model of soft repulsive particles at large densities, and show that nonlinear flow curves reminiscent of experiments on real suspensions can be obtained. By using dimensional analysis and basic elements of kinetic theory, we rationalize these multiple rheological regimes and disentangle the relative impact of thermal fluctuations, glass and jamming transitions, inertia and particle softness on the flow curves. We characterize more specifically the shear-thickening regime and show that both particle softness and the emergence of a yield stress at the jamming transition compete with the inertial effects responsible for the observed thickening behaviour. This allows us to construct a dynamic state diagram, which can be used to analyze experiments.Comment: 6 pages, 3 figure

Garnet-biotite geothermometry of a pelitic gneiss from the Lutzow-Holm Complex in Skallen, East Antarctica: Constraints on retrograde metamorphism

A garnet-rich pelitic gneiss was examined to estimate retrograde conditions of the Lutzow-Holm Complex in Skallen, East Antarctica. Porphyroblastic grains of garnet show a chemical zoning of an outward decrease in Mg/Fe, and include biotite, plagioclase and quartz. Local depletion in Mg/Fe of garnet is recognized where garnet is in contact with biotite occurring as inclusions and in the matrix. A pair of garnet-biotite geothermometry and garnet-plagioclase-sillimanite-quartz (GASP) geobarometry were applied for garnet rim and matrix biotite that are not in contact with each other, which demonstrates a retrograde condition of 5.6 kbar and 760℃. Pairs of garnet and adjacent biotite yield temperatures of ca. 660℃ at a nominal pressure of 6 kbar irrespective of whether biotiteoCcurs in the matrix or as inclusions, which represents the closure temperature for the exchange reaction between garnet and biotite