846,700 research outputs found
Convex Relaxation of Optimal Power Flow, Part II: Exactness
This tutorial summarizes recent advances in the convex relaxation of the
optimal power flow (OPF) problem, focusing on structural properties rather than
algorithms. Part I presents two power flow models, formulates OPF and their
relaxations in each model, and proves equivalence relations among them. Part II
presents sufficient conditions under which the convex relaxations are exact.Comment: Citation: IEEE Transactions on Control of Network Systems, June 2014.
This is an extended version with Appendex VI that proves the main results in
this tutoria
Convex Relaxation of Optimal Power Flow, Part I: Formulations and Equivalence
This tutorial summarizes recent advances in the convex relaxation of the
optimal power flow (OPF) problem, focusing on structural properties rather than
algorithms. Part I presents two power flow models, formulates OPF and their
relaxations in each model, and proves equivalence relations among them. Part II
presents sufficient conditions under which the convex relaxations are exact.Comment: Citation: IEEE Transactions on Control of Network Systems,
15(1):15-27, March 2014. This is an extended version with Appendices VIII and
IX that provide some mathematical preliminaries and proofs of the main
result
The Structure of Differential Invariants and Differential Cut Elimination
The biggest challenge in hybrid systems verification is the handling of
differential equations. Because computable closed-form solutions only exist for
very simple differential equations, proof certificates have been proposed for
more scalable verification. Search procedures for these proof certificates are
still rather ad-hoc, though, because the problem structure is only understood
poorly. We investigate differential invariants, which define an induction
principle for differential equations and which can be checked for invariance
along a differential equation just by using their differential structure,
without having to solve them. We study the structural properties of
differential invariants. To analyze trade-offs for proof search complexity, we
identify more than a dozen relations between several classes of differential
invariants and compare their deductive power. As our main results, we analyze
the deductive power of differential cuts and the deductive power of
differential invariants with auxiliary differential variables. We refute the
differential cut elimination hypothesis and show that, unlike standard cuts,
differential cuts are fundamental proof principles that strictly increase the
deductive power. We also prove that the deductive power increases further when
adding auxiliary differential variables to the dynamics
The Case Against Powers
Powers ontologies are currently enjoying a resurgence. This would be dispiriting news for the moderns; in their eyes, to imbue bodies with powers is to slide back into the scholastic slime from which they helped philosophy crawl. I focus on Descartesâs âlittle soulsâ argument, which points to a genuine and, I think persisting, defect in powers theories. The problem is that an Aristotelian power is intrinsic to whatever has it. Once this move is accepted, it becomes very hard to see how humble matter could have such a thing. It is as if each empowered object were possessed of a little soul that directs it and governs its behavior. Instead of attempting to resurrect the Aristotelian power theory, contemporary philosophers would be best served by taking their inspiration from its early modern replacement, devised by John Locke and Robert Boyle. On this view, powers are internal relations, not monadic properties intrinsic to their bearers. This move at once drains away the mysterious directedness of Aristotelian powers and solves the contemporary version of the little souls argument, Neil Williamsâs âproblem of fit.
The Steinmann Cluster Bootstrap for N=4 Super Yang-Mills Amplitudes
We review the bootstrap method for constructing six- and seven-particle
amplitudes in planar super Yang-Mills theory, by exploiting
their analytic structure. We focus on two recently discovered properties which
greatly simplify this construction at symbol and function level, respectively:
the extended Steinmann relations, or equivalently cluster adjacency, and the
coaction principle. We then demonstrate their power in determining the
six-particle amplitude through six and seven loops in the NMHV and MHV sectors
respectively, as well as the symbol of the NMHV seven-particle amplitude to
four loops.Comment: 36 pages, 4 figures, 5 tables, 1 ancillary file. Contribution to the
proceedings of the Corfu Summer Institute 2019 "School and Workshops on
Elementary Particle Physics and Gravity" (CORFU2019), 31 August - 25
September 2019, Corfu, Greec
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