Simple Differential Equation Research Articles

Neutrino decoupling beyond the Standard Model: CMB constraints on the Dark Matter mass with a fast and precise Neff evaluation

The number of effective relativistic neutrino species represents a fundamental probe of the thermal history of the early Universe, and as such of the Standard Model of Particle Physics. Traditional approaches to the neutrino decoupling are either very technical and computationally expensive, or assume that neutrinos decouple instantaneously. In this work, we aim to fill the gap between these two approaches by modeling the neutrino decoupling in terms of two simple coupled differential equations for the electromagnetic and neutrino sector temperatures, in which all the relevant interactions are taken into account and which allows for a straightforward implementation of BSM species. Upon including finite temperature QED corrections we reach an accuracy on Neff in the SM of 0.01. We illustrate the usefulness of this approach to the neutrino decoupling by considering, in a model independent manner, the impact of MeV thermal dark matter on Neff. We show that Planck rules out electrophilic and neutrinophilic thermal dark matter particles of m< 3.8 MeV at 95% CL regardless of their spin, and of their annihilation being s-wave or p-wave. We point out that thermal dark matter particles with non-negligible interactions with both electrons and neutrinos are more elusive to CMB observations than purely electrophilic or neutrinophilic ones. In addition, assisted by the accuracy of our approach, we show that CMB Stage-IV experiments will generically test thermal dark matter particles with m ≲ 20 MeV. We make publicly available the codes developed for this study at this https://github.com/MiguelEA/nudec_BSM.

Nonlinear dynamics of a viscous bubbly fluid

A physical model of a three-dimensional flow of a viscous bubbly fluid in an intermediate regime between bubble formation and breakage is presented. The model is based on mechanics and thermodynamics of a single bubble coupled to the dynamics of a viscous fluid as a whole, and takes into account multiple physical effects, including gravity, viscosity, and surface tension. Dimensionless versions of the resulting nonlinear model are obtained, and values of dimensionless parameters are estimated for typical magma flows in horizontal subaerial lava fields and vertical volcanic conduits.Exact solutions of the resulting system of nonlinear equations corresponding to equilibrium flows and traveling waves are analyzed in the one-dimensional setting. Generalized Su–Gardner-type perturbation analysis is employed to study approximate solutions of the model in the long-wave ansatz. Simplified nonlinear partial differential equations (PDE) satisfied by the leading terms of the perturbation solutions are systematically derived. It is shown that for specific classes of perturbations, approximate solutions of the bubbly fluid model arise from solutions of the classical diffusion, Burgers, variable-coefficient Burgers, and Korteweg-de Vries equations.

A mathematical model for chemoimmunotherapy of chronic lymphocytic leukemia

Immunotherapy is currently regarded as the most promising treatment to fight against cancer. This is particularly true in the treatment of chronic lymphocytic leukemia, an indolent neoplastic disease of B-lymphocytes which eventually causes the immune system’s failure. In this and other areas of cancer research, mathematical modeling is pointed out as a prominent tool to analyze theoretical and practical issues. Its lack in studies of chemoimmunotherapy of chronic lymphocytic leukemia is what motivated us to come up with a simple ordinary differential equation model. It is based on ideas of de Pillis and Radunskaya and on standard pharmacokinetics-pharmacodynamics assumptions. In order to check the positivity of the state variables, we first establish an invariant region where these time-dependent variables remain positive. Afterwards, the action of the immune system, as well as the chemoimmunotherapeutic role in promoting cancer cure are investigated by means of numerical simulations and the classical linear stability analysis. The role of adoptive cellular immunotherapy is also addressed. Our overall conclusion is that chemoimmunotherapeutic protocols can be effective in treating chronic lymphocytic leukemia provided that chemotherapy is not a limiting factor to the immunotherapy efficacy.

A Novel Approach for Breast Cancer Investigation and Recognition Using M-Level Set-Based Optimization Functions

Breast cancer identification is the first and foremost step in the journey of proper diagnosis and treatment of this disease. Therefore, many medical examinations and applications are devised and used including different approaches for breast imaging. For medical image analysis, nonparametric algorithm has been used. This work focuses on improving breast cancer recognition in mammogram images using a nonparametric approach based on image pixel intensities (IBCNP). A nonparametric approach is a new expression to deal with uncertainty. In this paper, the nonparametric approach is used to solve the uncertainty in breast cancer detection; uncertainty means no one knows for sure how to detect breast cancer without imaging, which creates an uncertainty for the oncologists. This paper investigates the feasibility of applying the nonparametric approach in mammography (x-ray breast imaging) to detect and hence diagnose and treat breast cancer. Analysing breast images is a very complicated process and requires distinguishing normal and malignant pixel intensities. Maximizing the shared data between image pixel intensities and the desired areas (infected region) is defined as the identification problem, subject to a constraint on the total size of the desired area borders. As part of this research, a comparative study was conducted on the proposed algorithm, Breast Cancer Cells Infestation based on a set of partial differential equations (PDEs), Incoherent Motion in Breast Cancer (IMBC), Global Dynamics of a Breast Cancer Competition Model using five simple differential equations (SDFs), and Estimation of Intensity Using Nonparametric Method (EINP) techniques. These five techniques of medical image segmentation are applied to 156 different mammogram images to obtain the exact measurement of the efficiency of the identification process. This process uses multiple performance metrics such as the overall mammogram image quality (Q), the dice similarity coefficient (DSC), Peak signal-to-noise ratio (PSNR) metric, Haussdorff Distance (HD), and probability of tumor detection (PTD). This paper assumes lack of knowledge of the probability densities in relation to pixel intensities of an image within a given area. It proposes an estimation of non-parametric density that formulates the theoretical data optimization problem by applying curve evolution methods and deriving the correlated gradient flows. It uses level-set techniques to achieve the resulting evolution. The algorithm was applied to 156 sets of different mammogram images from twelve groups with both malignant and/or normal. The experimental findings showed that the proposed algorithm is 94.937% accurately effective in breast cancer detection.

Holographic dark energy through Tsallis entropy

In order to apply holography and entropy relations to the whole universe, which is a gravitational and thus nonextensive system, for consistency one should use the generalized definition for the universe horizon entropy, namely Tsallis nonextensive entropy. We formulate Tsallis holographic dark energy, which is a generalization of standard holographic dark energy quantified by a new dimensionless parameter δ, possessing the latter as a particular sub-case. We provide a simple differential equation for the dark energy density parameter, as well as an analytical expression for its equation-of-state parameter. In this scenario the universe exhibits the usual thermal history, namely the successive sequence of matter and dark-energy epochs, before resulting in a complete dark energy domination in the far future. Additionally, the dark energy equation-of-state parameter presents a rich behavior and, according to the value of δ, it can be quintessence-like, phantom-like, or experience the phantom-divide crossing before or after the present time. Finally, we confront the scenario with Supernovae type Ia and Hubble parameter observational data, and we show that the agreement is very good, with δ preferring a value slightly larger than its standard value 1.

Biomechanical study of magnetohydrodynamic Prandtl nanofluid in a physiological vessel with thermal radiation and chemical reaction

This article is intended to study the peristaltic motion of a Prandtl nanoliquid through an inclined tapered asymmetric channel. The simultaneous effects such as magnetic field, thermal radiation and chemical reactions have been considered. The geometrical model is considered as tapered asymmetric channel because this situation is observed in the flow of uterine fluid in the uterus. The equations governing the flow are simplified under the assumptions of long wavelength and low Reynolds number. The simplified equations are complex in nature, so that the numerical solutions are presented for the simplified nonlinear partial differential equations considering slip and convective boundary conditions using computational software Mathematica via shooting method. The sundry parameters on the flow quantities have been discussed in detail through graphical and tabular forms. The observed results show that rise in the magnetic effects leads to a reduction in velocity. The radiation parameter decreases the temperature and there is an increment in the pressure gradient with an increase in energy Grashof number. This study is encouraged by exploring the nanofluid dynamics in peristaltic transport as symbolized by heat transport in biological flows, novel pharmacodynamic pumps and gastrointestinal motility enhancement.

Tracking intracellular dynamics through extracellular measurements.

Extracellular recordings of neuronal cells are frequently a part of in vitro and in vivo experimental studies as a means of monitoring network-level dynamics. Their connections to intracellular dynamics are not well understood. Single-unit recordings are a more direct way to measure intracellular dynamics, but are typically difficult and expensive. On the other hand, simple differential equations models exist for single neurons. In this article, we apply a recent advance in data assimilation theory, designed to correct bias in general observation functions, toward the reconstruction of model-based intracellular dynamics from extracellular recordings.

RodFIter: Attitude Reconstruction From Inertial Measurement by Functional Iteration

Rigid motion computation or estimation is a cornerstone in numerous fields. Attitude computation can be achieved by integrating the angular velocity measured by gyroscopes, the accuracy of which is crucially important for the dead-reckoning inertial navigation. The state-of-the-art attitude algorithms have unexceptionally relied on the simplified differential equation of the rotation vector to obtain the attitude. This paper proposes a functional iteration technique with the Rodrigues vector (named the RodFIter method) to analytically reconstruct the attitude from gyroscope measurements. The RodFIter method is provably exact in reconstructing the incremental attitude as long as the angular velocity is exact. Notably, the Rodrigues vector is analytically obtained and can be used to update the attitude over the considered time interval. The proposed method gives birth to an ultimate attitude algorithm scheme that can be naturally extended to the general rigid motion computation. It is extensively evaluated under the attitude coning motion and compares favorably in accuracy with the mainstream attitude algorithms. This paper is believed having eliminated the long-standing theoretical barrier in exact motion integration from inertial measurements.

Solving and classifying the solutions of the Yang-Baxter equation through a differential approach. Two-state systems

The formal derivatives of the Yang-Baxter equation with respect to its spectral parameters, evaluated at some fixed point of these parameters, provide us with two systems of differential equations. The derivatives of the R matrix elements, however, can be regarded as independent variables and eliminated from the systems, after which two systems of polynomial equations are obtained in place. In general, these polynomial systems have a non-zero Hilbert dimension, which means that not all elements of the R matrix can be fixed through them. Nonetheless, the remaining unknowns can be found by solving a few number of simple differential equations that arise as consistency conditions of the method. The branches of the solutions can also be easily analyzed by this method, which ensures the uniqueness and generality of the solutions. In this work we considered the Yang-Baxter equation for two-state systems, up to the eight-vertex model. This differential approach allowed us to solve the Yang-Baxter equation in a systematic way and also to completely classify its regular solutions.

Investigation on Laser-Induced Shock Pressure with Condensed Matter Model and Experimental Verification

The laser-induced shock wave generated by the water confined regime in laser shock peening and its propagation in the Al2024 alloys are investigated theoretically and experimentally. Based on the condensed matter model, the pressure of plasma induced by homogenous nano-laser pulse is derived analytically and calculated from a simple partial differential equation. Using the derived plasma pressure as the input condition, the propagation of the shock wave in Al2024 is simulated by finite element method. The shock force at the back surface of Al2024 plate with different thickness were measured with polyvinylidene fluoride (PVDF) transducers and recorded by oscilloscope in the LSP experiment. By using water as a cushion at the Al0204-transducer and BK7-transducer interface, the impedance mismatch effect have been severely reduced. The experimental shock force profiles agree well with the simulated results, which indicates the analytical method based on the condensed matter model is an accurate way to conveniently calculate the laser-induced pressure from the laser power density.

Solving Non-Homogeneous Coupled Linear Matrix Differential Equations in Terms of Matrix Convolution Product and Hadamard Product

We investigate a system of coupled non-homogeneous linear matrix differential equations. By applying the diagonal extraction operator, this system is reduced to a simple vector-matrix differential equation. An explicit formula of the general solution is then obtained in terms of matrix convolution product, Hadamard product, and elementary matrix functions. Moreover, we discuss certain special cases of the main system when initial conditions are imposed.

Revisiting the discharge time of a cylindrical leaking bucket

In [2] we find an exploration of a new mathematical model of the flow in a leaking bucket suitable for beginning students. The model is derived using the non-steady Bernoulli's Principle, and results in a more sophisticated model than the simple ordinary differential equation [EQUATION] derived using the steady Bernoulli's Principle. The simpler model goes sometimes by the name of Torricelli's Law and is very well studied; indeed it is a favourite example in many textbooks. This present paper provides an alternative derivation of the new model that uses an energy balance, and carefully lays out some numerical issues omitted from the treatment in [2]. We also provide an analytic solution in terms of 2 F 1 hypergeometric functions, which, while possibly unfamiliar to the student, are available to them via computer algebra systems. Even before that solution, an intermediate equation [EQUATION] is derived, which already explains the similarity of the solutions to the more sophisticated model to the ones from the simple Torricelli's Law. This paper gives a useful example for use of a CAS in a classroom setting.

Abstract 4264: Mathematical modeling of differential effects of sunitinib on primary tumor and metastatic growth

Abstract Sunitinib is a drug with anti-angiogenic activity used in the treatment of patients with metastases from renal cell carcinoma or gastrointestinal tumors. However, despite clear efficacy in reducing established tumor growth, recent preclinical studies have shown limited, or even opposing, efficacies in preventing metastatic spread. In this work, we evaluated a mathematical metastatic model to describe primary tumor and metastatic dynamics in response to sunitinib in a clinically relevant mouse model of spontaneous metastatic breast cancer that develops after surgical removal of an orthotopically implanted primary tumor. Mice received either vehicle or sunitinib in the neoadjuvant (presurgical) setting according to different schedules. The experimental dataset comprises measurements of primary tumor kinetics, metastatic burden and pre-surgical molecular and cellular biomarkers, including vascular cell Ki67 and CD31 expression, circulating tumor cells (CTCs) and myeloid derived suppressor cells (MDSCs). Estimation of the model's parameters was performed using a mixed-effects population approach. To describe tumor growth under sunitinib treatment, a simple ordinary differential equation model of tumor growth inhibition was used. Population fits obtained modeling the effect of treatment only on primary tumor growth described well the experimental data of all the treated groups considered. On the contrary, simulations of treatment also on metastasis could not reproduce the behavior of the data. By inserting in the model the available biomarkers as covariates, through a Wald-test, measurements of Ki67+/CD31+, CTCs and granulocytic MDSCs were found significantly correlated with the model parameter expressing the metastatic aggressiveness of the tumor. Together, these mathematical models confirm a differential effect of sunitinib on primary (localized) tumors compared to secondary (metastatic) disease. Our results suggest that Ki67+/CD31+, CTCs and MDSCs measurements might help in predicting metastatic potential and thus aid in predicting benefit in overall survival for preoperative antiangiogenic treatments. Citation Format: Chiara Nicolò, Michalis Mastri, Amanda Tracz, John M. Ebos, Sébastien Benzekry. Mathematical modeling of differential effects of sunitinib on primary tumor and metastatic growth [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2018; 2018 Apr 14-18; Chicago, IL. Philadelphia (PA): AACR; Cancer Res 2018;78(13 Suppl):Abstract nr 4264.

Inclusion of Asset Prices: An Argument for Monetary Policy and the Phillips Curve

Abstract This article develops an augmented price index that includes house prices, so that the relationship between inflation and unemployment levels in the traditional Phillips curve can be better represented. This general price index may be considered complementary to the Harmonised Index of Consumer Prices (HICP) and establishes the model-theoretical basis for a new-Keynesian model that derives the conditions for a monetary policy rule in a dynamic stochastic optimization procedure. Based on a simple stochastic differential equation for augmented inflation, we show that the reaction of the central bank depends on the marginal effects on augmented inflation and the output gap of an infinitesimal change in asset prices. This analysis could be interpreted as a way of using asset prices for a general price index, being an adequate method to restore monetary credibility. JEL classifications: E52, E58, G10 Keywords: monetary policy, asset prices, Phillips curve

Models of cytokine dynamics in the inflammatory response of viral zoonotic infectious diseases.

Inflammatory responses to an infection from a zoonotic pathogen, such as avian influenza viruses, hantaviruses and some coronaviruses, are distinctly different in their natural reservoir versus human host. While not as well studied in the natural reservoirs, the pro-inflammatory response and viral replication appear controlled and show no obvious pathology. In contrast, infection in humans results in an initial high viral load marked by an aggressive pro-inflammatory response known as a cytokine storm. The key difference in the course of the infection between the reservoir and human host is the inflammatory response. In this investigation, we apply a simple two-component differential equation model for pro-inflammatory and anti-inflammatory responses and a detailed mathematical analysis to identify specific regions in parameter space for single stable endemic equilibrium, bistability or periodic solutions. The extensions of the deterministic model to two stochastic models account for variability in responses seen at the cell (local) or tissue (global) levels. Numerical solutions of the stochastic models exhibit outcomes that are typical of a chronic infection in the natural reservoir or a cytokine storm in human infection. In the chronic infection, occasional flare-ups between high and low responses occur when model parameters are in a region of bistability or periodic solutions. The cytokine storm with a vigorous pro-inflammatory response and less vigorous anti-inflammatory response occurs in the parameter region for a single stable endemic equilibrium with a strong pro-inflammatory response. The results of the model analyses and the simulations are interpreted in terms of the functional role of the cytokines and the inflammatory responses seen in infection of the natural reservoir or of the human host.

New methodology for parameter estimation of offshore slug models with Hopf bifurcation

Offshore well elevation systems often present limit cycles caused by oil-gas slugging. The latest works in control solutions use models based on simple Ordinary Differential Equations to suppress slug flow and increase overall production. However, obtaining parameters for these models is not straightforward due to the occurrence of Hopf bifurcations. Therefore, this work aims to propose a novel methodology to improve parameter estimation. The limit cycle readings of pressure instruments are condensed into a single characteristic cycle with mean value and variance for each sample. Each iteration then uses a modified weighted sum of least squares. This creates a more convex region around the optimum that improves the performance of optimization algorithms. The results were obtained using previously published data for two models. Using the F-test, it was statistically proven that the parameters obtained from the proposed method provided an equivalent or better model fit by employing strictly numeric methods.

Top-down pulses reduce prey population sizes and persistence

Resource pulses are well documented and have important consequences for population dynamics relative to continuous inputs. However, pulses of top-down factors (e.g. predation) are less explored and appreciated in the ecological literature. Here, we use a simple differential equation population model to show how pulsed removals of individuals from a population alter population size relative to continuous dynamics. Pulsed removals result in lower equilibrium population sizes relative to continuous removals, and the differences are greatest at low population growth rates, high removal rates, and with large, infrequent pulses. Furthermore, the timing of the removal pulses (either stochastic or cyclic) affects population size. For example, cyclic removals are less likely than stochastic removals to result in population eradication, but when eradication occurs, the time until eradication is shorter for cyclic than with stochastic removals.

RESEARCH OF THE CHEMICAL REACTION OPTIMALITY CRITERION FOR TECHNOLOGICAL PROCESS REALIZATION

The optimality criterion of the chemical reaction by the Euler method was investigated in the search for the most suitable reactor design for the realization of the technological process. The optimality criterion corre- sponds to the minimum of the surface of the compared apparatus of a particular chemical reaction with the calculation of a number of reactors, which allows to achieve a minimum investment in the construction in proportion to the material capacity of the apparatus. A system of simple differential equations reflecting the formal model of the chemical reac- tion kinetics was solved by a first-order method. The Euler method scans the time axis with steps, using the recurrence relation to calculate the concentration of the component in the step from the results of calculating the concentration of this component in the previous step. In PascalABC. NET with the ability to determine the necessary duration of the reaction and the output of intermediate values of the concentrations of the reaction components at the reference points, a calculation was made with the ini- tial concentration of the component. The values of the most suitable design of ideal displacement reagents and ideal mixing were determined for this process. The models of ideal mixing of periodic and flowing action, as well as an ideal displacement reactor, are analyzed. Mathematical models were for- mulated and kinetic calculation results were obtained with reaction time values and apparatus volumes at which the yield of the target product was the highest. In the course of the examination, as optimality criterion of the surface of reactor construction was found and, in comparison with these devices, the efficiency of the ideal intermixing model of batch action was noted.

Stochastic Dynamics of Eukaryotic Flagellar Growth.

We study the dynamics of flagellar growth in eukaryotes where intraflagellar transporters (IFT) play a crucial role. First we investigate a stochastic version of the original balance point model where a constant number of IFT particles move up and down the flagellum. The detailed model is a discrete event vector-valued Markov process occurring in continuous time. First the detailed stochastic model is compared and contrasted with a simple scalar ordinary differential equation (ODE) model of flagellar growth. Numerical simulations reveal that the steady-state mean value of the stochastic model is well approximated by the ODE model. Then we derive a scalar stochastic differential equation (SDE) as a first approximation and obtain a "small noise" approximation showing flagellar length to be Gaussian with mean and variance governed by simple ODEs. The accuracy of the small noise model is compared favorably with the numerical simulation results of the detailed model. Secondly, we derive a revised SDE for flagellar length following the revised balance point model proposed in 2009 in which IFT particles move in trains instead of in isolation. Small noise approximation of the revised SDE yields the same approximate Gaussian distribution for the flagellar length as the SDE corresponding to the original balance point model.

Spinning solutions in general relativity with infinite central density

This paper presents general relativistic numerical simulations of uniformly rotating polytropes. Equations are developed using MSQI coordinates, but taking a logarithm of the radial coordinate. The result is relatively simple elliptical differential equations. Due to the logarithmic scale, we can resolve solutions with near-singular mass distributions near their center, while the solution domain extends many orders of magnitude larger than the radius of the distribution (to connect with flat space-time). Rotating solutions are found with very high central energy densities for a range of adiabatic exponents. Analytically, assuming the pressure is proportional to the energy density (which is true for polytropes in the limit of large energy density), we determine the small radius behavior of the metric potentials and energy density. This small radius behavior agrees well with the small radius behavior of large central density numerical results, lending confidence to our numerical approach. We compare results with rotating solutions available in the literature, which show good agreement. We study the stability of spherical solutions: instability sets in at the first maximum in mass versus central energy density; this is also consistent with results in the literature, and further lends confidence to the numerical approach.