Manganite charge and orbitally ordered and disordered states probed by Fe substitution into Mn site in LnBaMn1.96Fe0.04O5, LnBaMn1.96Fe0.04O6 and LnBaMn1.96Fe0.04O5.5 (Ln=Y, Gd, Sm, Nd, Pr, La)

The layered manganese oxides LnBaMn1.96Fe0.04Oy (Ln=Y, Gd, Sm, Nd, Pr, La) have been synthesized for y=5, 5.5 and 6. In the oxygen-saturated state (y=6) they exhibit the charge and orbital order at ambient temperature for Ln=Y, Gd, Sm, but unordered eg-electronic system for Ln=La,Pr,Nd. Fourfold increase of quadrupole splitting was observed owing to the charge and orbital ordering. This is in agreement with the jumplike increase in distortion of the reduced perovskite-like cell for the charge and orbitally ordered manganites compared to the unordered ones. Substitution of 2 percents of Mn by Fe suppresses the temperatures of structural and magnetic transitions by 20 to 50 K. Parameters of the crystal lattices and the room-temperature M\"{o}ssbauer spectra were studied on forty samples whose structures were refined within five symmetry groups: P4/mmm, P4/nmm, Pm-3m, Icma and P2/m. Overwhelming majority of the Fe species are undifferentiated in the M\"{o}ssbauer spectra for most of the samples. Such the single-component spectra in the two-site structures are explained by the preference of Fe towards the site of Mn(III) and by the segmentation of the charge and orbitally ordered domains.Comment: 8 figures; figures 2 and 3 were revise

The conjugacy problem and related problems in lattice-ordered groups

We study, from a constructive computational point of view, the techniques used to solve the conjugacy problem in the "generic" lattice-ordered group Aut(R) of order automorphisms of the real line. We use these techniques in order to show that for each choice of parameters f,g in Aut(R), the equation xfx=g is effectively solvable in Aut(R).Comment: Small update

Local models of Shimura varieties, I. Geometry and combinatorics

We survey the theory of local models of Shimura varieties. In particular, we discuss their definition and illustrate it by examples. We give an overview of the results on their geometry and combinatorics obtained in the last 15 years. We also exhibit their connections to other classes of algebraic varieties such as nilpotent orbit closures, affine Schubert varieties, quiver Grassmannians and wonderful completions of symmetric spaces.Comment: 86 pages, small corrections and improvements, to appear in the "Handbook of Moduli

Topological flatness of local models for ramified unitary groups. I. The odd dimensional case

Local models are certain schemes, defined in terms of linear-algebraic moduli problems, which give \'etale-local neighborhoods of integral models of certain p-adic PEL Shimura varieties defined by Rapoport and Zink. When the group defining the Shimura variety ramifies at p, the local models (and hence the Shimura models) as originally defined can fail to be flat, and it becomes desirable to modify their definition so as to obtain a flat scheme. In the case of unitary similitude groups whose localizations at Q_p are ramified, quasi-split GU_n, Pappas and Rapoport have added new conditions, the so-called wedge and spin conditions, to the moduli problem defining the original local models and conjectured that their new local models are flat. We prove a preliminary form of their conjecture, namely that their new models are topologically flat, in the case n is odd.Comment: Minor revisions, some of which incorporate suggestions of the referee. 27 page

Semigroups, rings, and Markov chains

We analyze random walks on a class of semigroups called ``left-regular bands''. These walks include the hyperplane chamber walks of Bidigare, Hanlon, and Rockmore. Using methods of ring theory, we show that the transition matrices are diagonalizable and we calculate the eigenvalues and multiplicities. The methods lead to explicit formulas for the projections onto the eigenspaces. As examples of these semigroup walks, we construct a random walk on the maximal chains of any distributive lattice, as well as two random walks associated with any matroid. The examples include a q-analogue of the Tsetlin library. The multiplicities of the eigenvalues in the matroid walks are ``generalized derangement numbers'', which may be of independent interest.Comment: To appear in J. Theoret. Proba