Small Extended Formulation for Knapsack Cover Inequalities from Monotone Circuits

Abstract Results from full scale fatigue tests of offshore mooring chains performed in recent years have revealed considerable influence of both mean load and corrosion condition on the fatigue capacity. It has been shown that a reduction of the mean load gives an increase in fatigue life, whereas the corrosion experienced by used chains have a significant negative impact. Neither of these effects are properly addressed by current S-N design curves or design practice. This paper suggests an extended S-N curve formulation, that includes the effects of mean load and corrosion condition. The parameters of the extended formulation are estimated empirically from mooring chain test data that includes new and used chains, with various mean loads and with different degrees of corrosion. The fitted capacity model is then used for fatigue calculation for the mooring system of a semi-submersible, showing the importance of using realistic mean loads and mooring chain corrosion in fatigue assessments.

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Randomized feasible interpolation and monotone circuits with a local oracle

Journal of Mathematical Logic ◽

10.1142/s0219061318500125 ◽

2018 ◽

Vol 18 (02) ◽

pp. 1850012 ◽

Cited By ~ 1

Author(s):

Jan Krajíček

Keyword(s):

Lower Bounds ◽

Cutting Planes ◽

Communication Complexity ◽

Interpolation Theorem ◽

Proof System ◽

Symbolic Logic ◽

Approximate Counting ◽

Small Width ◽

Randomized Protocols ◽

Monotone Circuits

The feasible interpolation theorem for semantic derivations from [J. Krajíček, Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic, J. Symbolic Logic 62(2) (1997) 457–486] allows to derive from some short semantic derivations (e.g. in resolution) of the disjointness of two [Formula: see text] sets [Formula: see text] and [Formula: see text] a small communication protocol (a general dag-like protocol in the sense of Krajíček (1997) computing the Karchmer–Wigderson multi-function [Formula: see text] associated with the sets, and such a protocol further yields a small circuit separating [Formula: see text] from [Formula: see text]. When [Formula: see text] is closed upwards, the protocol computes the monotone Karchmer–Wigderson multi-function [Formula: see text] and the resulting circuit is monotone. Krajíček [Interpolation by a game, Math. Logic Quart. 44(4) (1998) 450–458] extended the feasible interpolation theorem to a larger class of semantic derivations using the notion of a real communication complexity (e.g. to the cutting planes proof system CP). In this paper, we generalize the method to a still larger class of semantic derivations by allowing randomized protocols. We also introduce an extension of the monotone circuit model, monotone circuits with a local oracle (CLOs), that does correspond to communication protocols for [Formula: see text] making errors. The new randomized feasible interpolation thus shows that a short semantic derivation (from a certain class of derivations larger than in the original method) of the disjointness of [Formula: see text], [Formula: see text] closed upwards, yields a small randomized protocol for [Formula: see text] and hence a small monotone CLO separating the two sets. This research is motivated by the open problem to establish a lower bound for proof system [Formula: see text] operating with clauses formed by linear Boolean functions over [Formula: see text]. The new randomized feasible interpolation applies to this proof system and also to (the semantic versions of) cutting planes CP, to small width resolution over CP of Krajíček [Discretely ordered modules as a first-order extension of the cutting planes proof system, J. Symbolic Logic 63(4) (1998) 1582–1596] (system R(CP)) and to random resolution RR of Buss, Kolodziejczyk and Thapen [Fragments of approximate counting, J. Symbolic Logic 79(2) (2014) 496–525]. The method does not yield yet lengths-of-proofs lower bounds; for this it is necessary to establish lower bounds for randomized protocols or for monotone CLOs.

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Lifted r-cover inequalities

Design of Survivable Networks - Lecture Notes in Mathematics ◽

10.1007/bfb0088972 ◽

1992 ◽

pp. 101-112

Author(s):

Mechthild Stoer

Keyword(s):

Cover Inequalities

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Empathic Imagery in Keats and Shelley

PMLA ◽

10.1632/459226 ◽

1946 ◽

Vol 61 (1) ◽

pp. 163-191

Author(s):

Richard H. Fogle

Keyword(s):

External World ◽

Extended Formulation ◽

Living Body ◽

The World ◽

Bodily Sensations ◽

The Dead

Empathy, the involuntary projection of oneself into an object, received its first extended formulation in the Mikrokosmos of Hermann Lotze (1858). To Lotze Einfühlung, or empathy as it has been termed in English, was a phenomenon which accounts for our knowledge of the external world. “The world,” he said, “becomes alive to us through this power to see in forms the joy and sorrow of existence that they hide: there is no shape so coy that our fancy cannot sympathetically enter into it.” In this knowledge our consciousness of our own bodily sensations is a factor: “Unquestionably the vividness of these perceptions is added to by our abiding remembrance of the activity of our own body … every movement which we execute, every attitude in which we repose, has its meaning rendered plain to us by the feeling of exertion or of enjoyment.” Entering thus into our own sensations, by means of them we are also enabled to know the feelings of creatures and objects beyond their immediate range:… we, thus aided by our sentience, assuredly can comprehend also the alien silent form. Nor is it only into the peculiar vital feelings of that which in nature is near to us that we enter into the joyous flight of the singing bird or the graceful fleeting of the gazelle; we not only countract our mental feelers to the most minute creatures, to enter in reverie into the narrow round of existence of a mussel-fish and the monotonous bliss of its openings and shuttings, we not only expand into the slender proportions of the tree whose twigs are animated by the pleasure of graceful bending and waving; nay, even to the inanimate do we transfer these interpretative feelings, transforming through them the dead weights and supports of buildings into so many limbs of a living body whose inner tensions pass over into ourselves.

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