Chain-rule

Definitions

  • Chambers's Twentieth Century Dictionary
    • Chain-rule an arithmetical rule, so called from the terms of the problem being stated as equations, and connected, as if by a chain, so as to obtain by one operation the same result as would be obtained by a number of different operations in simple proportion: the rule for solving problems by compound proportion
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Quotations

  • Karl Marx
    Karl%20Marx
    “Let the ruling classes tremble at a Communist revolution. The proletarians have nothing to lose, but their chains. .Workers of the world unite!”

Etymology

Chambers's Twentieth Century Dictionary
Fr. chaine—L. catēna.

Usage

In literature:

To have discovered Europe and to rule as conquerors abroad is well, but it is not enough, if we are led in chains at home.
"Worldly Ways and Byways" by Eliot Gregory
If you chain love down under smug rules you only kill it or distort it.
"Possessed" by Cleveland Moffett
Three chains by which evil propensity is bound: a covenant, a (monastic) rule, law.
"The Triads of Ireland" by Kuno Meyer
Brothers, break the ancient chains of the Popedom, the chains of arbitrary rule!
"The Student-Life of Germany" by William Howitt
Let the ruling classes tremble before a Communist revolution; the workers have nothing to lose by it but their chains.
"The life and teaching of Karl Marx" by M. Beer
One must have courage to live and to rule, and for that one must break all chains.
"Count Brühl" by Joseph Kraszewski
When the Almohades ruled in Seville a great iron chain was drawn across the river, and a tower built on either side to support it.
"Southern Spain" by A.F. Calvert
They were ruled by that terrible trinity, the cowl, the sword and chain.
"The Works of Robert G. Ingersoll, Vol. 4 (of 12) Dresden Edition--Lectures" by Robert G. Ingersoll
Liberty was in chains, honesty in dungeons, while Christian superstition ruled mankind.
"The Works of Robert G. Ingersoll, Vol. 6 (of 12) Dresden Edition--Discussions" by Robert G. Ingersoll
He ruled with whip and chain, with pestilence and fire.
"The Works of Robert G. Ingersoll, Vol. 12 (of 12) Dresden Edition--Miscellany" by Robert G. Ingersoll
As a rule, the fire burned in the centre on a circle of stones, and over it, from the roof, hung chain and hook for the kettle.
"Our Journey to the Hebrides" by Joseph Pennell and Elizabeth Robins Pennell
Four other demon princes rule over the principal mountain chains of the Empire.
"Demonology and Devil-lore" by Moncure Daniel Conway
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In poetry:

Gyved and chained in his father's home,
He toiled 'neath a conqueror's rule;
Bowed to the earth in the land of his birth;
The Slave who was Son of a Fool.
"Son of a Fool" by C J Dennis
Gyved and chained in his father's home,
He toiled 'neath a conqueror's rule;
While they flung in his face the taunt of his race:
A Slave and the Son of a Fool.
"Son of a Fool" by C J Dennis
YES, tyrants, you hate us, and fear while you hate
The self-ruling, chain-breaking, throne-shaking State!
The night-birds dread morning,--your instinct is true,--
The day-star of Freedom brings midnight for you!
""Choose You This Day Whom Ye Will Serve"" by Oliver Wendell Holmes

In news:

Blue catfish rule the food chain at Milford Lake.
(FORTUNE Magazine) – It's one of the immutable rules of the road: Not all chains are created equal.
Cold chain experts agree you have to have people "in country" to help negotiate rules, regulations, and local customs.
The National Association of Chain Drug Stores applauded a federal court ruling that upheld an injunction blocking the implementation of cuts to Medi- Cal, California's Medicaid program.
The paper presents a prototype design tool-chain, called VARDA for the ARD+ conceptual design method for rules.
Rite Aid, reverse stock split , NYSE, New York Stock Exchange, minimum share price listing rule, drug store chain, Russell Redman.
Read this guide to supply chain best practices, and learn the three rules you need to implement in order to deliver the kind of order fulfillment that leads to customer fulfillment.
Three Rules for a Smarter Supply Chain: Align , Adapt, Automate.
Three Rules for a Smarter Supply Chain: Align, Adapt, Automate .
Big chain restaurants must now start posting the number of calories in their dishes, an appeals court ruled yesterday, but their legal battle isn't over yet.
CITY HALL — It was a "win some, lose some" night at the Landmarks Commission for city officials as commissioners made final rulings Monday on two ongoing sagas — the Chez Jay restaurant and " Chain Reaction " statue.
The FDA is proposing new rules that would require chain restaurants to post calorie counts for every food item they sell.
Marriott International Inc won a court ruling preventing the owners of the Eden Roc Renaissance Hotel in Miami Beach, Fla. From removing or replacing the hotel chain as manager of the property.
Blue catfish rule the food chain at Milford Lake.
Barack Obama's signature health reforms, ruling that the owners of a $3 billion arts and crafts chain must provide emergency.
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In science:

Using the chain rule, we have ∂s (cid:19) Q(x, s) |x=s .
Application of Random Matrix Theory to Multivariate Statistics
Figure 4.13: (Top) The starting point: All neurons enabled. (Middle) The optimum configuration by means of the updating rule in (4.16). (Bottom) 2D track segments are found as chains of consecutive enabled links.
Study of accelerator neutrino interactions in a liquid argon TPC
To simplify the understanding of configurations such as in Fig. 11a, we will derive a rule that will replace the Schwinger-boson singlet chains which are completely contained in one side of the partition.
Entanglement entropy of the random spin-1 Heisenberg chain
By chain rule, we may write d(Φ(g )) = (∂y Φ)(g )dg + Pr i=1 ψidfi , where ψi ∈ k [x] for each i.
A generalization of the Shestakov-Umirbaev inequality
By the chain rule of differentiation we have ∂ ¯χ∂χ = Detr (cid:16)X Si ⊗ ¯Si(cid:17) ∂τ ∂σ , where ∂τ ∂σ ≡ Q ∂ 2/∂ τ νµ ∂ σµν .
Superbosonization formula and its application to random matrix theory
The proof of our main result for min-entropy follows the same lines, with a modified chain-rule for min-entropies.
Sampling of min-entropy relative to quantum knowledge
Similarly, a recombination-chain-rule (15) can be proved for min-entropy.
Sampling of min-entropy relative to quantum knowledge
However, the splitting-chain rule (14) is no longer true for min-entropies and has to be replaced by a more subtle statement.
Sampling of min-entropy relative to quantum knowledge
To state this modified splitting-chain-rule, consider a state ρABC with purification |ΨABCD i.
Sampling of min-entropy relative to quantum knowledge
The proof of (ii) is done inductively using subadditivity, the recombination-chain-rule, and the recombination argument outlined above.
Sampling of min-entropy relative to quantum knowledge
In Section 5.4, we establish our central splitting-chain-rule.
Sampling of min-entropy relative to quantum knowledge
An analogous statement cannot be made for the recombination-chain-rule (iv), and we will have to retain the dependence on σC in our arguments.
Sampling of min-entropy relative to quantum knowledge
Having established subadditivity and a recombination-chain-rule, we will address the problem of finding a converse splitting-chain-rule in the next section.
Sampling of min-entropy relative to quantum knowledge
To introduce our splitting-chain-rule, we proceed in two steps: In Section 5.4.1, we show a simplified version which does not restrict the number of states the original state is split into.
Sampling of min-entropy relative to quantum knowledge
The chain-rule we will prove in this section concerns a tripartite state ρABC with purification ρABCD = |ΨABCD ihΨABCD | and an operator σC .
Sampling of min-entropy relative to quantum knowledge
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