Euler Obstruction and Defects of Functions on Singular Varieties

Several authors have proved Lefschetz type formulae for the local Euler obstruction. In particular, a result of this type is proved in [BLS].The formula proved in that paper turns out to be equivalent to saying that the local Euler obstruction, as a constructible function, satisfies the local Euler condition (in bivariant theory) with respect to general linear forms. The purpose of this work is to understand what prevents the local Euler obstruction of satisfying the local Euler condition with respect to functions which are singular at the considered point. This is measured by an invariant (or ``defect'') of such functions that we define below. We give an interpretation of this defect in terms of vanishing cycles, which allows us to calculate it algebraically. When the function has an isolated singularity, our invariant can be defined geometrically, via obstruction theory. We notice that this invariant unifies the usual concepts of {\it the Milnor number} of a function and of the {\it local Euler obstruction} of an analytic set.Comment: 18 page

Small resolutions of Schubert varieties and Kazhdan-Lusztig polynomials

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Small resolutions of Schubert varieties in symplectie and orthogonal Grassmannians

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Accidental SUSY: Enhanced Bulk Supersymmetry from Brane Back-reaction

We compute how bulk loops renormalize both bulk and brane effective interactions for codimension-two branes in 6D gauged chiral supergravity, as functions of the brane tension and brane-localized flux. We do so by explicitly integrating out hyper- and gauge-multiplets in 6D gauged chiral supergravity compactified to 4D on a flux-stabilized 2D rugby-ball geometry, specializing the results of a companion paper, arXiv:1210.3753, to the supersymmetric case. While the brane back-reaction generically breaks supersymmetry, we show that the bulk supersymmetry can be preserved if the amount of brane-localized flux is related in a specific BPS-like way to the brane tension, and verify that the loop corrections to the brane curvature vanish in this special case. In these systems it is the brane-bulk couplings that fix the size of the extra dimensions, and we show that in some circumstances the bulk geometry dynamically adjusts to ensure the supersymmetric BPS-like condition is automatically satisfied. We investigate the robustness of this residual supersymmetry to loops of non-supersymmetric matter on the branes, and show that supersymmetry-breaking effects can enter only through effective brane-bulk interactions involving at least two derivatives. We comment on the relevance of this calculation to proposed applications of codimension-two 6D models to solutions of the hierarchy and cosmological constant problems.Comment: 49 pages + appendices. This is the final version to appear in JHE

Order and Disorder in AKLT Antiferromagnets in Three Dimensions

The models constructed by Affleck, Kennedy, Lieb, and Tasaki describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin S is an integer multiple M of half the lattice coordination number. The equal time quantum correlations in their ground states may be computed as finite temperature correlations of a classical O(3) model on the same lattice, where the temperature is given by T=1/M. In dimensions d=1 and d=2 this mapping implies that all AKLT states are quantum disordered. We consider AKLT states in d=3 where the nature of the AKLT states is now a question of detail depending upon the choice of lattice and spin; for sufficiently large S some form of Neel order is almost inevitable. On the unfrustrated cubic lattice, we find that all AKLT states are ordered while for the unfrustrated diamond lattice the minimal S=2 state is disordered while all other states are ordered. On the frustrated pyrochlore lattice, we find (conservatively) that several states starting with the minimal S=3 state are disordered. The disordered AKLT models we report here are a significant addition to the catalog of magnetic Hamiltonians in d=3 with ground states known to lack order on account of strong quantum fluctuations.Comment: 7 pages, 2 figure

Comprehensive review of various corrosion behaviours on 316 stainless steel

Corrosion is a destructive process that converts the pure metal into a chemically stabled form by hydroxide or sulphide and it is a slow process of destruction on the material by the chemical or electrochemical reaction in the environmental space. This kind of destruction has been typically produced from oxides or salt content on the material and it results in distinctive orange coloration. The classifications of corrosion act on atmospheric air and liquids as well as on contact of two solids. To resist the corrosion rate, stainless steel 316 has been chosen because of the presence of 2-3% molybdenum content and the presence of molybdenum plays a vital role in corrosion resistance. In this study, literature related to various works has been reviewed to explain the corrosion behaviour on cavitation, crevice, electrochemical, erosion, fatigue, galvanic, uniform, pitting, and stress corrosion which act on 316 stainless steel. In the present work, several coating processes and the additives, that have been added to SS 316 to enhance the outcomes according to various corrosion causes, are discussed