Quantum Mechanics over Sets: A pedagogical model with non-commutative finite probability theory as its quantum probability calculus

This paper shows how the classical finite probability theory (with equiprobable outcomes) can be reinterpreted and recast as the quantum probability calculus of a pedagogical or toy model of quantum mechanics over sets (QM/sets). There have been several previous attempts to develop a quantum-like model with the base field of C replaced by Z2. Since there are no inner products on vector spaces over finite fields, the problem is to define the Dirac brackets and the probability calculus. The previous attempts all required the brackets to take values in Z2. But the usual QM brackets ⟨ψ|ϕ⟩ give the ”overlap” between states ψ and ϕ, so for subsets S, T ⊆ U, the natural definition is ⟨S|T ⟩ = |S ∩ T | (taking values in the natural numbers). This allows QM/sets to be developed with a full probability calculus that turns out to be a non-commutative extension of classical Laplace-Boole finite probability theory. The pedagogical model is illustrated by giving simple treatments of the indeterminacy principle, the double-slit experiment, Bell’s Theorem, and identical particles in QM/Sets. A more technical appendix explains the mathematics behind carrying some vector space structures between QM over C and QM/Sets over Z2

Analysis of the Error Propagation Phenomenon in Network Structures

The analysis of error propagation is of fundamental importance to assure safe operation and management of abnormal situations in any distributed information system. In this paper, the quantitative and qualitative methods are proposed to analyze possible error propagation scenarios based on different topologies, error types and probability distributions. The most interesting from our point of view is the course of error propagation in simple structures that are contained in more complex ones. These complex structures, which have attracted the attention of scientists for many decades, are traditionally analyzed with the use of formalisms from graph theory. Certain types of graphs are often used to model naturally occurring complex structures, such as social networks. Graph-theoretic approach proved successful when applied to social networks and other naturally occurring complex networks. The research was verified based on the experiments conducted on simulation model. The results provide some ideas of robustness -- the knowledge how to design the most error resistant architectures in complex environments

How to Express Self-Referential Probability. A Kripkean Proposal

We present a semantics for a language that includes sentences that can talk about their own probabilities. This semantics applies a fixed point construction to possible world style structures. One feature of the construction is that some sentences only have their probability given as a range of values. We develop a corresponding axiomatic theory and show by a canonical model construction that it is complete in the presence of the ω-rule. By considering this semantics we argue that principles such as introspection, which lead to paradoxical contradictions if naively formulated, should be expressed by using a truth predicate to do the job of quotation and disquotation and observe that in the case of introspection the principle is then consistent

An Efficient Quality-Related Fault Diagnosis Method for Real-Time Multimode Industrial Process

Focusing on quality-related complex industrial process performance monitoring, a novel multimode process monitoring method is proposed in this paper. Firstly, principal component space clustering is implemented under the guidance of quality variables. Through extraction of model tags, clustering information of original training data can be acquired. Secondly, according to multimode characteristics of process data, the monitoring model integrated Gaussian mixture model with total projection to latent structures is effective after building the covariance description form. The multimode total projection to latent structures (MTPLS) model is the foundation of problem solving about quality-related monitoring for multimode processes. Then, a comprehensive statistics index is defined which is based on the posterior probability of the monitored samples belonging to each Gaussian component in the Bayesian theory. After that, a combined index is constructed for process monitoring. Finally, motivated by the application of traditional contribution plot in fault diagnosis, a gradient contribution rate is applied for analyzing the variation of variable contribution rate along samples. Our method can ensure the implementation of online fault monitoring and diagnosis for multimode processes. Performances of the whole proposed scheme are verified in a real industrial, hot strip mill process (HSMP) compared with some existing methods

Frequent asymmetric migrations suppress natural selection in spatially structured populations

Natural microbial populations often have complex spatial structures. This can impact their evolution, in particular the ability of mutants to take over. While mutant fixation probabilities are known to be unaffected by sufficiently symmetric structures, evolutionary graph theory has shown that some graphs can amplify or suppress natural selection, in a way that depends on microscopic update rules. We propose a model of spatially structured populations on graphs directly inspired by batch culture experiments, alternating within-deme growth on nodes and migration-dilution steps, and yielding successive bottlenecks. This setting bridges models from evolutionary graph theory with Wright-Fisher models. Using a branching process approach, we show that spatial structure with frequent migrations can only yield suppression of natural selection. More precisely, in this regime, circulation graphs, where the total incoming migration flow equals the total outgoing one in each deme, do not impact fixation probability, while all other graphs strictly suppress selection. Suppression becomes stronger as the asymmetry between incoming and outgoing migrations grows. Amplification of natural selection can nevertheless exist in a restricted regime of rare migrations and very small fitness advantages, where we recover the predictions of evolutionary graph theory for the star graph.Comment: 11 pages of main text, 27 pages of Supplementary materia