Morse theory of the moment map for representations of quivers

The results of this paper concern the Morse theory of the norm-square of the moment map on the space of representations of a quiver. We show that the gradient flow of this function converges, and that the Morse stratification induced by the gradient flow co-incides with the Harder-Narasimhan stratification from algebraic geometry. Moreover, the limit of the gradient flow is isomorphic to the graded object of the Harder-Narasimhan-Jordan-H\"older filtration associated to the initial conditions for the flow. With a view towards applications to Nakajima quiver varieties we construct explicit local co-ordinates around the Morse strata and (under a technical hypothesis on the stability parameter) describe the negative normal space to the critical sets. Finally, we observe that the usual Kirwan surjectivity theorems in rational cohomology and integral K-theory carry over to this non-compact setting, and that these theorems generalize to certain equivariant contexts.Comment: 48 pages, small revisions from previous version based on referee's comments. To appear in Geometriae Dedicat

Briquets from wood waste

Revised September 1960

Wood fuel combustion practice

Information reviewed and reaffirmed May 1961

Volume loss from inaccurate sawing

This report originally published in Southern Lumberman, Sept. 15, 1954

Incorporation of Pendant Bases into Rh(diphosphine)_2 Complexes: Synthesis, Thermodynamic Studies, And Catalytic CO_2 Hydrogenation Activity of [Rh(P_2N_2)_2]^+ Complexes

A series of five [Rh(P_2N_2)_2]^+ complexes (P_2N_2 = 1,5-diaza-3,7-diphosphacyclooctane) have been synthesized and characterized: [Rh(P^(Ph)_2N^(Ph)_2)_2]^+ (1), [Rh(P^(Ph)_2N^(Bn)_2)_2]^+ (2), [Rh(P^(Ph)_2N^(PhOMe)_2)_2]^+ (3), [Rh(P^(Cy)_2N^(Ph)_2)_2]^+ (4), and [Rh(P^(Cy)_2N^(PhOMe)_2)_2]^+ (5). Complexes 1–5 have been structurally characterized as square planar rhodium bis-diphosphine complexes with slight tetrahedral distortions. The corresponding hydride complexes 6–10 have also been synthesized and characterized, and X-ray diffraction studies of HRh(P^(Ph)_2N^(Bn)_2)_2 (7), HRh(P^(Ph)_2N^(PhOMe)_2)_2 (8) and HRh(P^(Cy)_2N^(Ph)_2)_2 (9) show that the hydrides have distorted trigonal bipyramidal geometries. Equilibration of complexes 2–5 with H_2 in the presence of 2,8,9-triisopropyl-2,5,8,9-tetraaza-1-phosphabicyclo[3,3,3]undecane (Verkade’s base) enabled the determination of the hydricities and estimated pK_a’s of the Rh(I) hydride complexes using the appropriate thermodynamic cycles. Complexes 1–5 were active for CO_2 hydrogenation under mild conditions, and their relative rates were compared to that of [Rh(depe)_2]^+, a nonpendant-amine-containing complex with a similar hydricity to the [Rh(P_2N_2)_2]^+ complexes. It was determined that the added steric bulk of the amine groups on the P_2N_2 ligands hinders catalysis and that [Rh(depe)_2]^+ was the most active catalyst for hydrogenation of CO_2 to formate

Quiver Structure of Heterotic Moduli

We analyse the vector bundle moduli arising from generic heterotic compactifications from the point of view of quiver representations. Phenomena such as stability walls, crossing between chambers of supersymmetry, splitting of non-Abelian bundles and dynamic generation of D-terms are succinctly encoded into finite quivers. By studying the Poincar\'e polynomial of the quiver moduli space using the Reineke formula, we can learn about such useful concepts as Donaldson-Thomas invariants, instanton transitions and supersymmetry breaking.Comment: 38 pages, 5 figures, 1 tabl

How to Compute Worst-Case Execution Time by Optimization Modulo Theory and a Clever Encoding of Program Semantics

International audienceIn systems with hard real-time constraints, it is necessary to compute upper bounds on the worst-case execution time (WCET) of programs; the closer the bound to the real WCET, the better. This is especially the case of synchronous reactive control loops with a fixed clock; the WCET of the loop body must not exceed the clock period. We compute the WCET (or at least a close upper bound thereof) as the solution of an optimization modulo theory problem that takes into account the semantics of the program, in contrast to other methods that compute the longest path whether or not it is feasible according to these semantics. Optimization modulo theory extends satisfiability modulo theory (SMT) to maximization problems. Immediate encodings of WCET problems into SMT yield formulas intractable for all current production-grade solvers; this is inherent to the DPLL(T) approach to SMT implemented in these solvers. By conjoining some appropriate "cuts" to these formulas, we considerably reduce the computation time of the SMT-solver. We experimented our approach on a variety of control programs, using the OTAWA analyzer both as baseline and as underlying microarchitectural analysis for our analysis, and show notable improvement on the WCET bound on a variety of benchmarks and control programs