We introduce a framework that can be used to model both mathematics and human reasoning about mathematics. This framework involves stochastic mathematical systems (SMSs), which are stochastic processes that generate pairs of questions and associated answers (with no explicit referents). We use the SMS framework to define normative conditions for mathematical reasoning, by defining a “calibration” relation between a pair of SMSs. The first SMS is the human reasoner, and the second is an “oracle” SMS that can (...) be interpreted as deciding whether the question–answer pairs of the reasoner SMS are valid. To ground thinking, we understand the answers to questions given by this oracle to be the answers that would be given by an SMS representing the entire mathematical community in the infinite long run of the process of asking and answering questions. We then introduce a slight extension of SMSs to allow us to model both the physical universe and human reasoning about the physical universe. We then define a slightly different calibration relation appropriate for the case of scientific reasoning. In this case the first SMS represents a human scientist predicting the outcome of future experiments, while the second SMS represents the physical universe in which the scientist is embedded, with the question–answer pairs of that SMS being specifications of the experiments that will occur and the outcome of those experiments, respectively. Next we derive conditions justifying two important patterns of inference in both mathematical and scientific reasoning: (i) the practice of increasing one’s degree of belief in a claim as one observes increasingly many lines of evidence for that claim, and (ii) abduction, the practice of inferring a claim’s probability of being correct from its explanatory power with respect to some other claim that is already taken to hold for independent reasons. (shrink)

Quantum stochastic models are developed within the framework of a measure entity. An entity is a structure that describes the tests and states of a physical system. A measure entity endows each test with a measure and equips certain sets of states as measurable spaces. A stochastic model consists of measurable realvalued function on the set of states, called a generalized action, together with measures on the measurable state spaces. This structure is then employed to compute quantum (...) probabilities of test outcomes. We characterize those measure entities that are isomorphic to a quantum probability space. We also show that stochastic models provide a phase space description of quantum mechanics and a realistic model of spin. (shrink)

In this work we study the behavior of a time discrete multiregional stochastic model for a population structured in age classes and spread out in different spatial patches between which individuals can migrate. The dynamics of the population is controlled both by reproduction-survival and by migration. These processes take place at different time scales in the sense of the latter being much faster than the former. We incorporate the effect of demographic stochasticity into the population, which results in (...) both dynamics being modelled by multitype Bienaymé–Galton–Watson branching processes. We present a multitype global model that incorporates the effect of both processes and, making use of the existence of different time scales for demography and migration, build a reduced model in which the variables correspond to the total population in each age class. We extend previous results that relate the behavior of the original and the reduced model showing that, given a large enough separation of time scales between demography and migration, we can obtain information about the behavior of the multitype global model through the study of the simpler reduced model. We concentrate on the case where the two systems are supercritical and therefore the expected number of individuals grows to infinity, and show that we can approximate the asymptotic structure of the population vector and the asymptotic population size of the original system through the study of the reduced model. (shrink)

We prove almost sure exponential stability for the disease-free equilibrium of a stochastic differential equations model of an SIR epidemic with vaccination. The model allows for vertical transmission. The stochastic perturbation is associated with the force of infection and is such that the total population size remains constant in time. We prove almost sure positivity of solutions. The main result concerns especially the smaller values of the diffusion parameter, and describes the stability in terms of an (...) analogue $$\mathcal{R}_\sigma$$ of the basic reproduction number $$\mathcal{R}_0$$ of the underlying deterministic model, with $$\mathcal{R}_\sigma \le \mathcal{R}_0$$. We prove that the disease-free equilibrium is almost sure exponentially stable if $$\mathcal{R}_\sigma <1$$. (shrink)

As the world population ages, the patients per physician ratio keeps on increasing. This is even more important in the domain of chronic pathologies where people are usually monitored for years and need regular consultations.To address this problem, we propose an automated system to monitor a patient population, detecting anomalies in instantaneous data and in their temporal evolution, so that it could alert physicians. By handling the population of healthy patients autonomously and by drawing the physicians' attention to the patients–at-risk, (...) the system allows physicians to spend comparatively more time with patients who need their services. In such a system, the interaction between the patients, the diagnosis module, and the physicians is very important. We have based this system on a combination of stochastic models, fuzzy filters, and strong medical semantics. (shrink)

Recently developed large language models (LLMs) perform surprisingly well in many language-related tasks, ranging from text correction or authentic chat experiences to the production of entirely new texts or even essays. It is natural to get the impression that LLMs know the meaning of natural language expressions and can use them productively. Recent scholarship, however, has questioned the validity of this impression, arguing that LLMs are ultimately incapable of understanding and producing meaningful texts. This paper develops a more optimistic view. (...) Drawing on classic externalist accounts of reference, it argues that LLMs are very likely capable of reference. Not only that: The combination of a popular externalist account of reference and recent experimental data in machine psychology even suggests that LLMs might play a role in shifting what our words refer to. (shrink)

In this paper, we study a stochastic epidemic model with double epidemics which includes white noise and telegraph noise modeled by Markovian switching. Sufficient conditions for the extinction and persistence of the diseases are established. In the end, some numerical simulations are presented to demonstrate our analytical results.

In a remarkable confirmation of OT in an empirical domain for which it was not originally intended, phonological and morphological variation has been successfully modeled by partially ranked categorical constraints (Anttila 1997, 2002). Poetic meter is a good place to extend and test this approach to variation, because there is abundant and diverse quantitative data available for it, and because it is typically governed by a relatively small number of well-understood constraints. I report the results of four such studies here. (...) They confirm that choices among metrical options are governed by the interaction of partially ranked constraints, in each case constraints that are grounded, and motivated independently of variation data by related systems in which they have a fixed rank. The partially ranked constraint systems turned out to predict not only the relative preferences among metrical options, but also their actual frequencies in the corpora, with surprising accuracy. These findings support the partial ranking model of variation, and provide an explanatory benchmark beyond the reach of intrinsically weaker stochastic approaches that posit a statistical component for metrical competence (Hayes & MacEachern 1998). (shrink)

The electron is conceived here as a complex structure composed of a subparticle that is bound to a nearly circular motion. Although in quantum mechanics the spin is not representable, in classical stochastic physics this corresponds to the angular momentum of the subparticle. In fact, assuming Schrödinger-type hydrodynamic equations of motion for the subparticle, the spin-1/2 representation in configuration space and the corresponding Pauli matrices for the electron are obtained. The Hamiltonian of Pauli's theory as the nonrelativistic limit of (...) Dirac's equation is also derived. (shrink)

We investigate the dynamics of a nonautonomous stochastic SIS epidemic model with nonlinear incidence rate and double epidemic hypothesis. By constructing suitable stochastic Lyapunov functions and using Has’minskii theory, we prove that there exists at least one nontrivial positive periodic solution of the system. Moreover, the sufficient conditions for extinction of the disease are obtained by using the theory of nonautonomous stochastic differential equations. Finally, numerical simulations are utilized to illustrate our theoretical analysis.

The locomotor pattern of a goldfish may be described by 17 locomotor variables whose time series are known to exhibit distinct patterns of autocorrelation. The present model reduces the set of 17 variables to a set of seven predictor variables which optimize the prediction of future locomotor behavior. These seven variables are then grouped into two clusters by a dendogram analysis, and it is shown that the clusters are also characterized by their strength of prediction. A locomotor control (...) class='Hi'>model is then formulated where one cluster is interpreted to be a level of activity factor and the other is hypothesized to be a turning behavior factor. (shrink)

Deterministic models in population dynamics often are really approximations to stochastic models, justified by an appeal to the law of large numbers. It is proposed to call such models pseudodeterministic. Four questions are discussed in this article: (1) What errors may be made by equating deterministically predicted values to expectations? (2) When, and in what sense, may numbers be assumed to be large? (3) How large are the variances, coefficients of variations, etc., as assigned to the variables in the (...) stochastic versions of the models? (4) What role may pseudodeterministic models play in empirical research, where problems of statistical reliability arise?As an example, a modified Nicholson-Bailey model of the interaction between insect parasitoids and their hosts is discussed; the modification consists of assigning a random (density-independent) mortality to the parasitoid population. A stochastic version of this model is discussed. The expectation of the final host density is compared with the value computed from the deterministic model. The latter value is systematically lower than the former. The magnitude of the difference depends on parameter values. The variability to be expected with the stochastic model is characterized by the coefficient of variation of the final host density; its dependence on parameter values and initial conditions is discussed. It is concluded that it is worthwhile in practical applications to estimate parasitoid mortality, and that the coefficient of variation in real situations may be far from negligible. (shrink)

Atmospheric pollution is deteriorating, which has affected the evolution of respiratory disease for the exposed human worldwide. Thus, exploring the influence of air pollution on the evolution of disease transmission dynamics is a significant issue. In this article, a stochastic susceptible-infective epidemic model in a polluted atmospheric environment is investigated. The existence and uniqueness of the global positive solution are established. In virtue of the aggregation methods and Lyapunov function, the sufficient conditions of disease extinction, persistence, and existence (...) of the stationary distribution are established, respectively. TakingPM2.5concentration as the air pollutant index, numerical simulations are carried out to support these results. Our results indicated that the disease transmission dynamics are significantly associated with the environmental atmospheric pollution and fluctuation. (shrink)

This paper investigates a stochastic two-patch predator-prey model with ratio-dependent functional responses. First, the existence of a unique global positive solution is proved via the stochastic comparison theorem. Then, two different methods are used to discuss the long-time properties of the solutions pathwise. Next, sufficient conditions for extinction and persistence in mean are obtained. Moreover, stochastic persistence of the model is discussed. Furthermore, sufficient conditions for the existence of an ergodic stationary distribution are derived by (...) a suitable Lyapunov function. Next, we apply the main results in some special models. Finally, some numerical simulations are introduced to support the main results obtained. The results in this paper generalize and improve the previous related results. (shrink)

This paper provides an experimental test of stochastic choice models of decisions. Models that admit Fechnerian structure are tested through the repeated pairwise choice problems. Results refute the Fechner hypothesis that characterizing the probability of selecting a given prospect increases in how strongly it is preferred to alternative choices. However, the experimental data lend support to characterizing an individual’s binary choice probability as some scalable functions of the von Neumann–Morgenstern utilities in the risky context.

The paper develops models of statistical experiments that combine propensities with frequencies, the underlying theory being the branching space-times (BST) of Belnap (1992). The models are then applied to analyze Bell's theorem. We prove the so-called Bell-CH inequality via the assumptions of a BST version of Outcome Independence and of (non-probabilistic) No Conspiracy. Notably, neither the condition of probabilistic No Conspiracy nor the condition of Parameter Independence is needed in the proof. As the Bell-CH inequality is most likely experimentally falsified, (...) the choice is this: contrary to the appearances, experimenters cannot choose some measurement settings, or some transitions, with spacelike related initial events, are correlated; or both. (shrink)

Solvency assessment is the core content of insurance supervision. In this paper, from the perspective of capital flow, the insurance company’s capital flow is regarded as a dynamic system, the stochastic differential equations model is established to describe its flow characteristics, and the existence of positive equilibrium point of the system is proved, as well as the conditions of stability at equilibrium point, that is, the requirements of the insurance company’s solvency. Furthermore, by using the numerical simulation method, (...) we get the strategy of insurance companies to deal with the solvency shortage when facing the change of external environment, and the strategy of insurance company to deal with solvency shortage is obtained. (shrink)

In this paper, we formulate and investigate a two-microorganism and single nutrient chemostat model with Monod-Haldane response function and random perturbation. First, for the corresponding deterministic system, we introduce the conditions of the stability of the equilibrium points. Then, using Lyapunov function and Itô’s formula, we investigate the existence and uniqueness of the global positive solution of the stochastic chemostat model. Furthermore, we explore and obtain the criterions of the extinction and the permanence for the stochastic (...) model. Finally, numerical simulations are carried out to illustrate our main results. (shrink)

Philosophers of biology have worked extensively on how we ought best to interpret the probabilities which arise throughout evolutionary theory. In spite of this substantial work, however, much of the debate has remained persistently intractable. I offer the example of Bayesian models of divergence time estimation as a case study in how we might bring further resources from the biological literature to bear on these debates. These models offer us an example in which a number of different sources of uncertainty (...) are combined to produce an estimate for a complex, unobservable quantity. These models have been carefully analyzed in recent biological work, which has determined the relationship between these sources of uncertainty, both quantitatively and qualitatively. I suggest here that this case shows us the limitations of univocal analyses of probability in evolution, as well as the simple dichotomy between “subjective” and “objective” probabilities, and I conclude by gesturing toward ways in which we might introduce more sophisticated interpretive taxonomies of probability as a path toward advancing debates on probability in the life sciences. (shrink)