A single instance of the Pythagorean theorem implies the parallel postulate

2002 ◽  
Vol 33 (4) ◽  
pp. 596-600
Author(s):  
David E. Dobbs
Keyword(s):  
Pythagorean Theorem ◽  
Single Instance

scholarly journals Sequential decoding of a general classical-quantum channel

2013 ◽  
Vol 469 (2157) ◽  
pp. 20130259 ◽  
Author(s):  
Mark M. Wilde
Keyword(s):  
Quantum Measurement ◽  
Failure Probability ◽  
High Probability ◽  
Polar Codes ◽  
Sequential Decoding ◽  
Single Instance ◽  
Small Failure Probability ◽  
Classical Quantum ◽  
Union Bound ◽  
Sequential Measurements

Because a quantum measurement generally disturbs the state of a quantum system, one might think that it should not be possible for a sender and receiver to communicate reliably when the receiver performs a large number of sequential measurements to determine the message of the sender. We show here that this intuition is not true, by demonstrating that a sequential decoding strategy works well even in the most general ‘one-shot’ regime, where we are given a single instance of a channel and wish to determine the maximal number of bits that can be communicated up to a small failure probability. This result follows by generalizing a non-commutative union bound to apply for a sequence of general measurements. We also demonstrate two ways in which a receiver can recover a state close to the original state after it has been decoded by a sequence of measurements that each succeed with high probability. The second of these methods will be useful in realizing an efficient decoder for fully quantum polar codes, should a method ever be found to realize an efficient decoder for classical-quantum polar codes.

Behold! The Pythagorean Theorem via Mean Proportions

1986 ◽  
Vol 17 (5) ◽  
pp. 422
Author(s):  
Michael Hardy
Keyword(s):  
Pythagorean Theorem

Rhetorical Word-Play in Chaucer

PMLA ◽  
1954 ◽  
Vol 69 (4-Part1) ◽  
pp. 937-952
Author(s):  
Helge Kökeritz
Keyword(s):  
Word Play ◽  
Troilus And Cressida ◽  
Single Instance

In the opinion of the late Thomas R. Lounsbury, Chaucer was virtually “free from these verbal quibbles which characterize to so marked a degree the language of the Elizabethan dramatists.” “The single instance,” he went on to say, “in which he furnishes any noticeable example of this sort is the play upon the word ‘style’ in the Squire's tale; though there is possibly one of the same character in a line in ‘Troilus and Cressida,‘ where it is said that‘This Calkas knew by calkulynge,‘ i., 71that Troy was to be taken. Still, from conceits of all kinds and of all grades Chaucer's language, at every period of his literary career, was in general wholly free.“

91.22 A new generalisation of the Pythagorean theorem

2007 ◽  
Vol 91 (520) ◽  
pp. 128-131
Author(s):  
Gyula Darvasi
Keyword(s):  
Pythagorean Theorem
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