We have developed a mathematical model of regulation of expression of the Escherichia coli lac operon, and have investigated bistability in its steady-state induction behavior in the absence of external glucose. Numerical analysis of equations describing regulation by artificial inducers revealed two natural bistability parameters that can be used to control the range of inducer concentrations over which the model exhibits bistability. By tuning these bistability parameters, we found a family of biophysically reasonable systems that are consistent with an experimentally determined bistable region for induction by thio-methylgalactoside (Ozbudak et al. Nature 427:737, 2004). The model predicts that bistability can be abolished when passive transport or permease export becomes sufficiently large; the former case is especially relevant to induction by isopropyl-beta, D-thiogalactopyranoside. To model regulation by lactose, we developed similar equations in which allolactose, a metabolic intermediate in lactose metabolism and a natural inducer of lac, is the inducer. For biophysically reasonable parameter values, these equations yield no bistability in response to induction by lactose; however, systems with an unphysically small permease-dependent export effect can exhibit small amounts of bistability for limited ranges of parameter values. These results cast doubt on the relevance of bistability in the lac operon within the natural context of E. coli, and help shed light on the controversy among existing theoretical studies that address this issue. The results also suggest an experimental approach to address the relevance of bistability in the lac operon within the natural context of E. coli.

This paper presents a discrete-time option pricing model that is rooted in Reinforcement Learning (RL), and more specifically in the famous Q-Learning method of RL. We construct a risk-adjusted Markov Decision Process for a discrete-time version of the classical Black-Scholes-Merton (BSM) model, where the option price is an optimal Q-function, while the optimal hedge is a second argument of this optimal Q-function, so that both the price and hedge are parts of the same formula. Pricing is done by learning to dynamically optimize risk-adjusted returns for an option replicating portfolio, as in the Markowitz portfolio theory. Using Q-Learning and related methods, once created in a parametric setting, the model is able to go model-free and learn to price and hedge an option directly from data, and without an explicit model of the world. This suggests that RL may provide efficient data-driven and model-free methods for optimal pricing and hedging of options, once we depart from the academic continuous-time limit, and vice versa, option pricing methods developed in Mathematical Finance may be viewed as special cases of model-based Reinforcement Learning. Further, due to simplicity and tractability of our model which only needs basic linear algebra (plus Monte Carlo simulation, if we work with synthetic data), and its close relation to the original BSM model, we suggest that our model could be used for benchmarking of different RL algorithms for financial trading applications

The presence of a large number of infected individuals with few or no symptoms is an important epidemiological difficulty and the main mathematical feature of COVID-19. The A-SIR model, i.e. a SIR (Susceptible-Infected-Removed) model with a compartment for infected individuals with no symptoms or few symptoms was proposed by Giuseppe Gaeta, arXiv:2003.08720 [q-bio.PE] (2020). In this paper we investigate a slightly generalized version of the same model and propose a scheme for fitting the parameters of the model to real data using the time series only of the deceased individuals. The scheme is applied to the concrete cases of Lombardy, Italy and São Paulo state, Brazil, showing different aspects of the epidemics. For each case we show that we may have good fits to the data up to the present, but with very large differences in the future behavior. The reasons behind such disparate outcomes are the uncertainty on the value of a key parameter, the probability that an infected individual is fully symptomatic, and on the intensity of the social distancing measures adopted. This conclusion enforces the necessity of trying to determine the real number of infected individuals in a population, symptomatic or asymptomatic.

We prove that for a so-called sticky process $S$ there exists an equivalent probability $Q$ and a $Q$-martingale $\tilde{S}$ that is arbitrarily close to $S$ in $L^p(Q)$ norm. For continuous $S$, $\tilde{S}$ can be chosen arbitrarily close to $S$ in supremum norm. In the case where $S$ is a local martingale we may choose $Q$ arbitrarily close to the original probability in the total variation norm. We provide examples to illustrate the power of our results and present applications in mathematical finance.

"Q Zhang's Problem" is a teaching problem proposed by Qian Zhang, a science teacher at Dongjiao Minxiang Primary School in Dongcheng District, Beijing. In 2022, she proposed that: (1) when explaining the knowledge points of frequency in the "Sound" unit, experiments on the vibration of objects such as rubber bands and steel rulers were used to assist students in learning, but the effect was not obvious, because it was difficult for the naked eye to distinguish the speed of vibration of objects such as rubber bands, and it was difficult to correspond to the high and low frequencies; (2) Students seem to be confused about the difference between frequency and amplitude. When guiding them to make the rubber band vibrate faster, they tend to tug harder at the rubber band, but this actually changes the amplitude rather than the frequency (changing the frequency should be to control its vibrating chord length, similar to the tuning method of a stringed instrument). Therefore, demonstration experiments using objects such as rubber bands as frequencies do not seem suitable and cannot effectively assist students in establishing the concept of frequency. We hope to solicit opinions and solutions (research ideas) on this problem, with a focus on two points: (1) the mathematical/physical explanation of the problem. That is, does simply changing the amplitude really not affect the original vibration frequency of the object (except when the amplitude is 0) (2) explanation from a cognitive perspective: Why do people confuse the two concepts? What is the cognitive mechanism behind it.

Quantifying the neural signatures of consciousness remains a major challenge in neuroscience and AI. Although many theories link consciousness to rich, multiscale, and flexible neural organisation, robust quantitative measures are still lacking. This paper presents a theory-neutral framework that characterises consciousness-related dynamics through three properties: hierarchical integration (H), cross-frequency complexity (D), and metastability (M). Candidate subsystems are identified using predictive information, temporal complexity, and state-space exploration to distinguish structured from unstructured activity. We provide mathematical definitions for all components and implement the framework in a generative model of synthetic EEG, simulating nine brain states ranging from psychedelic and wakeful to dreaming, non-REM sleep, minimally conscious, anaesthetised, and seizure-like regimes. Across single trials and Monte Carlo ensembles, the composite index reliably separates high-consciousness from impaired or non-conscious states. We further validate the framework using real EEG from the Sleep-EDF dataset alongside matched synthetic EEG designed to reproduce state-dependent oscillatory structure. Across Wake, N2, and REM sleep, synthetic data recapitulate the empirical ordering and magnitude of the index, indicating that the index captures stable and biologically meaningful distinctions. This approach provides a principled and empirically grounded tool for quantifying consciousness-related neural organisation with potential applications to both biological and artificial systems.

Membrane potential in a mathematical model of a cardiac myocyte can be formulated in different ways. Assuming a spatially homogeneous myocyte that is strictly charge-conservative and electroneutral as a whole, two methods will be compared: (1) the differential formulation dV/dt=-I/C_m of membrane potential used traditionally; and (2) the capacitor formulation, where membrane potential is defined algebraically by the capacitor equation V=Q/C_m. We examine the relationship between the formulations, assumptions under which each formulation is consistent, and show that the capacitor formulation provides a transparent, physically realistic formulation of membrane potential, whereas use of the differential formulation may introduce unintended and undesirable behavior, such as monotonic drift of concentrations. We prove that the drift of concentrations in the differential formulation arises as a compensation for failure to assign all currents in concentrations. As an example of these considerations, we present an electroneutral, explicitly charge-conservative formulation of Winslow et al. model (1999), and extend it to describe membrane potentials between intracellular compartments.

The nuclear electric quadrupole moment (NQM) of $^{87}$Sr has recently been revisited using high-precision relativistic atomic calculations [B. Lu et al., Phys. Rev. A 100, 012504 (2019)], indicating that the currently accepted value should be revised and that their result may serve as a new reference. In the present work, we determine the NQM of $^{87}$Sr from the molecular method, by combining the experimentally measured nuclear quadrupole coupling constants (NQCCs) of SrO and SrS with highly accurate relativistic calculations of the electric field gradient (EFG) at the Sr nucleus. Electronic correlation is treated at the CCSD(T), CCSD-T and CCSD$\tilde{\text{T}}$ levels. The iterative T contribution of the latter, composite scheme was obtained using a newly implemented parallel scheme where the distributed memory tensor library Cyclops Tensor Framework (CTF) was made available to the DIRAC code for relativistic molecular calculations through TAPP, the new community standard for tensor operations. All correlated calculations are performed using the exact two-component molecular mean-field Hamiltonian (X2C$\mathrm{mmf}$). The Gaunt two-electron interaction is incorporated, an even-tempered optimized quadruple-$ζ$ quality basis set is employed, and vibrational corrections are accounted for. Our best result is $Q($$^{87}$Sr$) = 0.33666 \pm 0.00258$ b, which is about 10% larger than currently accepted standard value, while it is in excellent agreement with recent determinations [Y.-B. Tang, arXiv:2512.07603 [physics.atom-ph] (2025)].

We study the Bogoliubov-Dirac-Fock model introduced by Chaix and Iracane ({\it J. Phys. B.}, 22, 3791--3814, 1989) which is a mean-field theory deduced from no-photon QED. The associated functional is bounded from below. In the presence of an external field, a minimizer, if it exists, is interpreted as the polarized vacuum and it solves a self-consistent equation. In a recent paper math-ph/0403005, we proved the convergence of the iterative fixed-point scheme naturally associated with this equation to a global minimizer of the BDF functional, under some restrictive conditions on the external potential, the ultraviolet cut-off $Λ$ and the bare fine structure constant $α$. In the present work, we improve this result by showing the existence of the minimizer by a variational method, for any cut-off $Λ$ and without any constraint on the external field. We also study the behaviour of the minimizer as $Λ$ goes to infinity and show that the theory is "nullified" in that limit, as predicted first by Landau: the vacuum totally kills the external potential. Therefore the limit case of an infinite cut-off makes no sense both from a physical and mathematical point of view. Finally, we perform a charge and density renormalization scheme applying simultaneously to all orders of the fine structure constant $α$, on a simplified model where the exchange term is neglected.

This report addresses the moments, ${\mathfrak{G}_n}\left( {\mathbf{q}} \right) = \int_{ - \infty }^{ + \infty } {{ω^n}S\left( {\mathbf{q},ω} \right)\mathrm{d}ω},\,n \in \mathbb{N},\,n \geq - 1$, of the quantum mechanical dynamic structure factor $S\left( {\mathbf{q},ω} \right)$ for a one-component Coulomb plasma in thermodynamic equilibrium. The Fluctuation Dissipation Theorem relates these moments to integrals involving the imaginary part of the inverse longitudinal dielectric function, with the odd moments in particular being equivalent to the odd moments of the imaginary part of the inverse dielectric function. Application of the Generalized Plasmon Pole Approximation arXiv:1508.05606 [physics.plasm-ph] to a weakly-coupled non-degenerate plasma, leads to general formulae expressed in terms of polynomial functions. Explicit forms of these functions are given for $n \leq 20$. These formulae are generalized to degenerate and partially degenerate plasmas, in small-$\mathbf{q}$ (long-wavelength) regimes.