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In this paper, we study the well-posedness, ill-posedness and uniqueness of the stationary 3-D radial solution to the bipolar isothermal hydrodynamic model for semiconductors. The density of electron is imposed with sonic boundary and interiorly subsonic case and the density of hole is fully subsonic case. It is difficult to estimate the upper and lower bounds of the holes due to the coupling of electrons and holes and the degeneracy of electrons at the boundary. Thus, we use the topological degree method to prove the well-posedness of solution. We prove the ill-posedness of subsonic solution under some conditions by direct mathematical analysis and contradiction method. The ill-posedness property shows significant difference to the unipolar model. Another highlight of this paper is the application of specific energy method to obtain the uniqueness of solution in two cases. One case is the relaxation time τ = ∞ , namely, the pure Euler-Poisson case; the other case is j τ ≪ 1 , which means that, when the current flow is sufficiently small and the relaxation time is sufficiently large both can satisfy. [ABSTRACT FROM AUTHOR]

In this paper, we consider a diffusive predator-prey system with fear response delay, where a benefit from the ani-predation response in addition to the cost is taken into account. As a step toward understanding the underlying mechanism of the spatiotemporal pattern formations by rigorous mathematical analysis, we are interested in how the joint effects of time delay, the fear effect strength, the carrying capacity, as well as the diffusion rates could affect the complex spatiotemporal pattern formations of the system. In particular, Turing instability of the Hopf bifurcating periodic solutions is investigated in details. To that end, a general formula is derived to determine Turing instability of the Hopf bifurcating periodic solutions for general 2-component delayed-diffusive system. This extends our earlier general results for non-delayed diffusive system to its delayed-diffusive counterparts. Our general formula tends to be new and can be applied to a variety of 2-component delayed-diffusive systems. [ABSTRACT FROM AUTHOR]

According to the Lighthill scheme of approximation procedures, we establish the Blackstock-Cattaneo model in this paper to describe nonlinear acoustics' propagations with second sound phenomena in perfect gases under the irrotational flow, which is an approximated higher-order equation for Navier-Stokes-Cattaneo equations. This paper is devoted to the mathematical analysis of the linear and nonlinear higher-order evolution equations in R n. Concerning the linearized Cauchy problem for the Blackstock-Cattaneo equation, by means of the Fourier analysis and the WKB analysis, we will derive sharp L 2 estimates of solutions with additional L 1 regularity for initial data. For another, we demonstrate global (in time) well-posedness for the nonlinear Blackstock-Cattaneo equation with small initial data and suitable regularity of Sobolev solutions. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of solution of a evolutionary partial differential equation originating from the continuum limit of a coupled process of totally asymmetric simple exclusion process (TASEP) and Langmuir kinetics (LK). In the fields of physics and biology, the TASEP-LK coupled process has been extensively studied by Monte Carlo simulations, numerical computations, and detailed experiments. However, no rigorous mathematical analysis so far has been given for the corresponding differential equations, especially the existence and uniqueness of their solutions. In this paper, we prove the existence of the C ∞ [ 0 , 1 ] steady-state solution by the method of upper and lower solution, and the uniqueness in both W 1 , 2 (0 , 1) and L ∞ (0 , 1) by a generalized maximum principle. We further prove the global existence and uniqueness of the time-dependent solution in C ([ 0 , 1 ] × [ 0 , + ∞)) ∩ C 2 , 1 ([ 0 , 1 ] × (0 , + ∞)) , which, for any continuous initial value, converges to the steady-state solution in C [ 0 , 1 ] (global attractivity). Our results support the numerical calculations and Monte Carlo simulations, and provide theoretical foundations for the TASEP-LK coupled process, especially the most important phase diagram of particle density along the travel track under different model parameters, which is difficult because the boundary layers (at one or both boundaries) and domain wall (separating high and low particle densities) may appear as the length of the travel track tends to infinity. The methods used in this paper may be instructive for studies of the more general cases of the TASEP-LK process, such as the one with multiple travel tracks and/or multiple particle species. [ABSTRACT FROM AUTHOR]

The mathematical analysis on the behavior of the entropy for viscous, compressible, and heat conducting magnetohydrodynamic flows near the vacuum region is a challenging problem as the governing equation for entropy is highly degenerate and singular in the vacuum region. In our previous work [38] , we investigate the global existence of strong solutions to the three-dimensional (3D) compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity. However, it is unknown whether the entropy remains its boundedness or not? Thus, the main novelty of this paper is to give a positive response to this problem. In fact, we show that the uniform boundedness of the entropy and the L 2 regularities of the velocity and temperature can be propagated provided that the initial density decays suitably slow at infinity. Modified De Giorgi type iteration techniques and useful singularly weighted estimates are developed to deduce the lower and upper bounds on the entropy. [ABSTRACT FROM AUTHOR]

Particle tracking in a live cell environment is concerned with reconstructing the trajectories, locations, or velocities of the targeting particles, which holds the promise of revealing important new biological insights. The standard approach of particle tracking consists of two steps: first reconstructing statically the source locations in each time step, and second applying tracking techniques to obtain the trajectories and velocities. In contrast to the standard approach, the dynamic reconstruction seeks to simultaneously recover the source locations and velocities from all frames, which enjoys certain advantages. In this paper, we provide a rigorous mathematical analysis for the resolution limit of reconstructing source number, locations, and velocities by general dynamical reconstruction in particle tracking problems, by which we demonstrate the possibility of achieving super-resolution for the dynamic reconstruction. We show that when the location-velocity pairs of the particles are separated beyond certain distances (the resolution limits), the number of particles and the location-velocity pair can be stably recovered. The resolution limits are related to the cut-off frequency of the imaging system, signal-to-noise ratio, and the sparsity of the source. By these estimates we also derive a stability result for a sparsity-promoting dynamic reconstruction. In addition, we further show that the reconstruction of velocities has a better resolution limit which improves constantly as the particles moving. The result is derived by a crucial observation that the inherent cut-off frequency for the velocity recovery can be viewed as the total observation time multiplies the cut-off frequency of the imaging system, which may lead to a better resolution limit as compared to the one for each diffraction-limited frame. This is discovered for the first time and we hope it can inspire many new reconstruction algorithms that significantly improve the resolution of particle tracking in practice. In addition, we propose super-resolution algorithms for recovering the number and values of the velocities in the tracking problem and demonstrate theoretically or numerically their super-resolution capability. [ABSTRACT FROM AUTHOR]

The paper is devoted to the mathematical analysis of the Cauchy problem for general cross-diffusion systems without any assumption about its entropic structure. A global existence result of nonnegative solutions is obtained by applying a classical Schauder fixed point theorem. The proof is upgraded for enhancing the regularity of the solution, namely its gradient belongs to the space L r ((0 , T) × Ω) for some r > 2. To this aim, the Schauder's strategy is coupled with an extension of Meyers regularity result for linear parabolic equations. We show how this approach allows to prove the well-posedness of the problem using only assumptions prescribing and admissibility range for the ratios between the diffusion and cross-diffusion coefficients. The results are compared with those that are reachable with an additional regularity assumption on the parabolic operator, namely a small BMO assumption for its coefficients. Finally, the question of the maximal principle is also addressed, especially when source terms are incorporated in the equation in order to ensure the confinement of the solution. [ABSTRACT FROM AUTHOR]

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer's confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results. [ABSTRACT FROM AUTHOR]

Neural networks (NN) have generated extensive interest in the field of machine condition monitoring (MCM). Many applications are however adapting structures and approaches from fields where NNs are highly established (e.g., image classification). This is done despite many neural network layers presenting strong similarities to signal processing steps that are well established in MCM. This paper aims at providing a mathematical analysis of typical NN architectures and layers, with specific attention to similarities to common signal processing steps such as filtering, downsampling, and enveloping. The set of strong analogies between NN and traditional MCM tools enables this work to provide MCM-specific guidelines for the design and tuning of NN architectures for MCM-specific applications. Most MCM applications are often characterised by periodic and cyclostationary signals, a situation not frequently encountered in many other NN applications. Hence, the Fourier-based explanation of the typical NN layers will be specific to the solution of typical MCM problems (e.g., diagnostics of gears and bearings). The study is aligned with the current trends of explainable AI and physics informed neural networks (PINN), in the sense that it aims at producing mathematical rules for a rational design of the network, drawing on the expertise accumulated in decades of signal processing for MCM. This is however done without upsetting traditional NN architectures, but rather trying to provide clear reasons for the setting of network parameters. The analytical work will be verified with numerical tests, which will also enable practical considerations, and by application of the suggested guidelines to the popular Case Western Reserve University (CWRU) experimental dataset. [ABSTRACT FROM AUTHOR]

Machine learning methods are widely used in the natural sciences to model and predict physical systems from observation data. Yet, they are often used as poorly understood "black boxes," disregarding existing mathematical structure and invariants of the problem. Recently, the proposal of Hamiltonian Neural Networks (HNNs) took a first step towards a unified "gray box" approach, using physical insight to improve performance for Hamiltonian systems. In this paper, we explore a significantly improved training method for HNNs, exploiting the symplectic structure of Hamiltonian systems with a different loss function. This frees the loss from an artificial lower bound. We mathematically guarantee the existence of an exact Hamiltonian function which the HNN can learn. This allows us to prove and numerically analyze the errors made by HNNs which, in turn, renders them fully explainable. Finally, we present a novel post-training correction to obtain the true Hamiltonian only from discretized observation data, up to an arbitrary order. • The loss of Hamiltonian Neural Networks suffers from an artificial lower bound. • Using a symplectic integrator during training significantly improves performance. • Rigorous mathematical analysis yields an exact handle on the discretization error. • The learnt Hamiltonian can be corrected post-training, to any arbitrary order. [ABSTRACT FROM AUTHOR]

Fast and accurate predictions of uncertainties in the computed dose are crucial for the determination of robust treatment plans in radiation therapy. This requires the solution of particle transport problems with uncertain parameters or initial conditions. Monte Carlo methods are often used to solve transport problems especially for applications which require high accuracy. In these cases, common non-intrusive solution strategies that involve repeated simulations of the problem at different points in the parameter space quickly become infeasible due to their long run-times. Intrusive methods however limit the usability in combination with proprietary simulation engines. In [61] , we demonstrated the application of a new non-intrusive uncertainty quantification approach for Monte Carlo simulations in proton dose calculations with normally distributed errors on realistic patient data. In this paper, we introduce a generalized formulation and focus on a more in-depth theoretical analysis of this method concerning bias, error and convergence of the estimates. The multivariate input model of the proposed approach further supports almost arbitrary error correlation models. We demonstrate how this framework can be used to model and efficiently quantify complex auto-correlated and time-dependent errors. • An efficient importance sampling-based UQ method for MC solvers is proposed. • Analysis of the mathematical properties of expected value, scenario and variance estimators. • Error models are derived for an application to different time-dependent errors in radiation therapy. • Numerical experiments for a proton therapy application are presented. • High agreement with standard UQ methods while reducing the CPU time by more than an order of magnitude. [ABSTRACT FROM AUTHOR]

Weber's law states that the ratio of the smallest perceptual change in an input signal and the background signal is constant. The law is observed across the perception of weight, light intensity, and sound intensity and pitch. To explain Weber's law observed in steady-state responses, two models of perception have been proposed, namely the logarithmic and the linear model. This paper argues in favour of the linear model, which requires the sensory system to generate linear input–output relationship over several orders of magnitude. To this end, a four-node motif (FNM) is constructed from first principles whose series provides almost linear relationship between input signal and the output over arbitrary range of input signal. Mathematical analysis into the origin of this quasi-linear relationship shows that the series of coherent type-1 feed-forward loop (C1-FFL) is able to provide perfectly linear input–output relationship over arbitrary range of input signal. FNM also reproduces the neuronal data of numerosity detection study on the monkey. The series of FNM also provides a mechanism for sensitive detection over arbitrary range of input signal when the output has an upper limit. Further, the series of FNM provides a general basis for a class of bow-tie architecture where the number of receptors is much lower than the range of input signal and the "decoded output". Besides (quasi-)linear input–output relationship, another example of this class of bow-tie architecture that the series of FNM is able to produce is absorption spectra of cone opsins of humans. Further, the series of FNM and C1-FFL, both, can compute logarithm over arbitrary range of input signal. [ABSTRACT FROM AUTHOR]

• Prediction and control of the socio-economic important Olive Quick Decline Syndrome (OQDS). • Mathematical/computational model based on ordinary differential equations. • Qualitative analysis of the mathematical model • Threshold agronomic parameters to tackle OQDS. • Identification of cost-effective and socially sustainable solutions for OQDS. In Southern Italy, since 2013, there has been an ongoing Olive Quick Decline Syndrome (OQDS) outbreak, due to the bacterium Xylella fastidiosa , which has caused a dramatic impact from both socio-economic and environmental points of view. The main players involved in OQDS are represented by the insect vector, Philaenus spumarius , its host plants (olive trees and weeds) and the bacterium, X. fastidiosa. Current agronomic practices are mainly based on uprooting the sick olive trees and their surrounding ones, with later installment of olive cultivars more resistant to the bacterium infection. Unfortunately, both of these practices are having an undesirable impact on the environment (most of these olive trees were monumental ones) and on the economy. Based on a mathematical model expressed in terms of a nontrivial system of ordinary differential equations, our analysis has provided a clear picture of all possible steady states (feasible equilibria) and their stability properties, corresponding to a variety of different parameter scenarios; all of this has been illustrated by a set of computational experiments. A significant original contribution of this paper is the proof of the global asymptotic stability of each of the feasible equilibria under its existence assumptions, a fact that excludes multiple equilibria under the given conditions. It has emerged that the removal of a suitable amount of weed biomass (host plants of the juvenile stages of the insect vector of X. fastidiosa) from olive orchards and surrounding areas leads to the eradication of the epidemic, without requiring neither the removal nor the substitution of the existing olive trees. [ABSTRACT FROM AUTHOR]

• New analysis of the ligand-induced dimerisation process for VEGFR is presented. • A key idea in the mathematical model and analysis is kinetic binding cooperativity. • The nonlinear model is in contrast with the corresponding linear model for GPCRs. • Numerical and asymptotic analysis reveal insights into the underlying LID dynamics. • Our simulations show excellent fit to recent experimental data for VEGFR. The vascular endothelial growth factor (VEGF) receptor (VEGFR) system plays a role in cancer and many other diseases. It is widely accepted that VEGFR receptors dimerise in response to VEGF binding. However, analysis of these mechanisms and their implications for drug development still requires further exploration. In this paper, we present a mathematical model representing the binding of VEGF to VEGFR and the subsequent ligand-induced dimerisation. A key factor in this work is the qualitative and quantitative effect of binding cooperativity, which describes the effect that the binding of a ligand to a receptor has on the binding of that ligand to a second receptor, and the dimerisation of these receptors. We analyse the ordinary differential equation system at equilibrium, giving analytical solutions for the total amount of ligand bound. For time-course dynamics, we use numerical methods to explore possible behaviours under various parameter regimes, while perturbation analysis is used to understand the intricacies of these behaviours. Our simulation results show an excellent fit to experimental data, towards validating the model. [ABSTRACT FROM AUTHOR]