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Note on sums involving the Euler function
In this note, we provide refined estimates of the following sums involving
the Euler totient function: where denotes the integral part of real
. The above summations were recently considered by Bordell\`es et al. and
Wu.Comment: Bull. Aust. Math. Soc., accepte
On the 'Reality' of Observable Properties
This note contains some initial work on attempting to bring recent
developments in the foundations of quantum mechanics concerning the nature of
the wavefunction within the scope of more logical and structural methods. A
first step involves generalising and reformulating a criterion for the reality
of the wavefunction proposed by Harrigan & Spekkens, which was central to the
PBR theorem. The resulting criterion has several advantages, including the
avoidance of certain technical difficulties relating to sets of measure zero.
By considering the 'reality' not of the wavefunction but of the observable
properties of any ontological physical theory a novel characterisation of
non-locality and contextuality is found.
Secondly, a careful analysis of preparation independence, one of the key
assumptions of the PBR theorem, leads to an analogy with Bell locality, and
thence to a proposal to weaken it to an assumption of
`no-preparation-signalling' in analogy with no-signalling. This amounts to
introducing non-local correlations in the joint ontic state, which is, at
least, consistent with the Bell and Kochen-Specker theorems. The question of
whether the PBR result can be strengthened to hold under this relaxed
assumption is therefore posed.Comment: 8 pages, re-written with new section
Unlimited parity alternating partitions
We introduce a new type of partitions that consists of partitions whose
different parts alternate in parity (e.g., ). Various properties of
this partition function are studied. In particular, we obtain its asymptotic
behavior by employing Ingham's Tauberian theorem
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