Random World and Quantum Mechanics

Quantum mechanics (QM) predicts probabilities on the fundamentallevel which are, via Born probability law, connected to the formal randomnessof infinite sequences of QM outcomes. Recently it has been shown thatQM is algorithmic 1-random in the sense of Martin-L¨of. We extend this resultand demonstrate that QM is algorithmic ω-random and generic, precisely asdescribed by the ’miniaturisation’ of the Solovay forcing to arithmetic. Thisis extended further to the result that QM becomes Zermelo–Fraenkel Solovayrandom on infinite-dimensional Hilbert spaces. Moreover, it is more likely thatthere exists a standard transitive ZFC model M, where QM is expressed in reality,than in the universe V of sets. Then every generic quantum measurementadds to M the infinite sequence, i.e. random real r ∈ 2ω, and the model undergoesrandom forcing extensions M[r]. The entire process of forcing becomesthe structural ingredient of QM and parallels similar constructions applied tospacetime in the quantum limit, therefore showing the structural resemblanceof both in this limit. We discuss several questions regarding measurability andpossible practical applications of the extended Solovay randomness of QM.The method applied is the formalization based on models of ZFC; however,this is particularly well-suited technique to recognising randomness questionsof QM. When one works in a constant model of ZFC or in axiomatic ZFCitself, the issues considered here remain hidden to a great extent.

Random World and Quantum Mechanics

Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω , to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

Random World and Quantum Mechanics

Quantum mechanics (QM) predicts probabilities on the fundamental level which are, via Born probability law, connected to the formal randomness of infinite sequences of QM outcomes. Recently it has been shown that QM is algorithmic 1-random in the sense of Martin-Löf. We extend this result and demonstrate that QM is algorithmic ω-random and generic precisely as described by the ’miniaturisation’ of the Solovay forcing to arithmetic. This is extended further to the result that QM becomes Zermelo-Fraenkel Solovay random on infinite dimensional Hilbert spaces. Moreover it is more likely that there exists a standard transitive model of ZFC M where QM is expressed in reality than in the universe V of sets. Then every generic quantum measurement adds the infinite sequence, i.e. random real r ∈ 2ω, to M and the model undergoes random forcing extensions, M[r]. The entire process of forcing becomes the structural ingredient of QM and parallels similar constructions applied to spacetime in the quantum limit. This shows the structural resemblance of both in the limit. We discuss several questions regarding measurability and eventual practical applications of the extended Solovay randomness of QM. The method applied is the formalization based on models of ZFC, however, this is particularly well-suited technique to recognising randomness questions of QM. When one works in a constant model of ZFC or in axiomatic ZFC itself the issues considered here become mostly hidden.

On Adaptive Haar Approximations of Random Flows

The adaptive approximations for some characteristic of random functions defined on arbitrary irregular grids are discussed in this paper. The mentioned functions can be examined as flows of random real values associated with an irregular grid. This paper considers the question of choosing an adaptive enlargement of the initial grid. The mentioned enlargement essentially depends on the formulation of the criterion in relation to which adaptability is considered. Several criteria are considered here, among which there are several criteria applicable to the processing of random flows. In particular, the criteria corresponding to the mathematical expectation, dispersion, as well as autocorrelation and cross-correlation of two random flows are considered. It is possible to consider criteria corresponding to various combinations of the mentioned characteristics. The number of knots of the initial (generally speaking, irregular) grid can be arbitrary, and the main grid can be any subset of the initial one. Decomposition algorithms are proposed, taking into account the nature of the changes in the initial flow. The number of arithmetic operations in the proposed algorithms is proportional to the length of the initial flow. Sequential processing of the initial flow is possible in real time.

RANDOM REAL BRANCHED COVERINGS OF THE PROJECTIVE LINE

In this paper, we construct a natural probability measure on the space of real branched coverings from a real projective algebraic curve $(X,c_X)$ to the projective line $(\mathbb{C} \mathbb {P}^1,\textit{conj} )$ . We prove that the space of degree d real branched coverings having “many” real branched points (for example, more than $\sqrt {d}^{1+\alpha }$ , for any $\alpha>0$ ) has exponentially small measure. In particular, maximal real branched coverings – that is, real branched coverings such that all the branched points are real – are exponentially rare.

Real-Time Safety Audits of Neonatal Delivery Room Resuscitation Areas: Are We Sufficiently Prepared?

Objective This study aimed to use real-time safety audits to establish whether preparation of the equipment required for the stabilization and resuscitation of newborns in the delivery room areas is adequate. Study Design This was a descriptive, multicenter study performed at five-level III-A neonatal units in Madrid, Spain. For 1 year, one researcher from each center performed random real-time safety audits (RRTSAs), on different days and during different shifts, of at least three neonatal stabilization areas, either in the delivery room or in the operating room used for caesarean sections. Three factors in each area were reviewed: the set-up of the radiant warmer, the materials, and medication available. The global audit was considered without defect when no errors were detected in any of the audited factors. Possible differences in the results were analyzed as a function of the study month, day of the week, or shift during which the audit had been performed. Results A total of 852 audits were performed. No defects were detected in any of the three factors analyzed in the 534 (62.7%, 95% confidence interval [CI]: 59.3–65.9) cases. Slight defects were detected in 98 (11.5%, 95% CI: 9.4–13.8) cases and serious defects capable of producing adverse events in the newborn during resuscitation were found in 220 (25.8%, 95% CI: 22.9–28.9) cases. No statistically significant differences in the results were found according to the day of the week or time during which the audits were performed. However, the percentage of RRTSAs without defect increased as the study period progressed (first quarter 38.1% vs. the last quarter 84.2%; p < 0.001). Conclusion The percentage of adequately prepared resuscitation areas was low. RRTSAs made it possible to detect errors in the correct availability of the neonatal stabilization areas and improved their preparation by preventing errors from being perpetuated over time. Key Points