Explicit Analytical Functions Research Articles

A Mathematical Model for the Treatment of Melanoma with the BRAF/MEK Inhibitor and Anti-PD-1

Skin cancer treatment is a combination of BRAF and MEK kinase inhibitors administered as tablets, along with immunotherapy treatment (treatment into the vein) with a group of drugs that inhibit the activity of the immune barrier proteins PD-1 and PDL1. Here, we propose a new approach to the therapy for melanoma with the BRAF/MEK inhibitor and anti-PD-1. With the help of explicit analytical functions, we were able to model this combined treatment and present the treatment in a mathematical model described by a system of differential equations including variables, such as Treg, IL12, Il10, TGF-β, and cytokine, which are significant variables that are all critical factors which determine the effectiveness of therapies. The most significant advantage of a treatment described by a mathematical model with explicit analytical functions is the control of parameters, such as time and dose, which are variable critical parameters in the treatment, that is, these parameters can be adapted to the patient’s personalized treatment. In the current study, we showed that by simultaneously changing and combining these two parameters, we could decrease the tumor volume. To validate the numerical results, we computed the relative error between the results obtained from the mathematical model and clinical data.

A mathematical model for cancer treatment based on combination of anti-angiogenic and immune cell therapies

In this study, we investigated a mathematical model that integrated mechanisms of tumor angiogenesis and tumor-targeted cytotoxicity in immune cells. The model considered the interaction of cancer cells with the immune system and the treatment that combines unlicensed dendritic cells and anti-vascular endothelial growth factor antibodies. We proposed an unconventional treatment protocol; for now, treatment is purely theoretical and has not been practically attempted. The treatment has been described by analytical functions that depend on time and dose. This was a function of t and q, where q is the amount (dose) of the drug. A customized treatment is a significant advantage of explicit analytical functions. We compared the results obtained from the mathematical model with the proposed treatment to the clinical results and observed an almost complete match between the treatments.In addition, we applied the global quasi-linearization GQL method to the mathematical model, which enabled us to expose the hierarchy of the system and, hence, split the system into fast and slow subsystems. This decomposition enabled us to analytically compute the equilibrium points of the model and their stability, which is crucial for the application.

Explicit analytical shape function calculation for macro-element in glue connection

This manuscript presents a simplified approach to adhesive joints calculation. Aviation and space rocket engineering constructions joints deals with coaxial cylindrical pipes that can be connected by adhesive with advantages in terms of tightness, aerodynamic efficiency, manufacturability, low weight. Glue joints are, therefore, considered and simplified in this article by the analytical calculation of the shape functions for a macro-element applied to the Volkersen’s theory. The mathematical solution makes possible to calculate the explicit form of the stiffness matrix of the macro-element. The analysis proposed shows the perfect matching of the solution for a single element under classical mechanical loads.

Modeling the Chance of Commerciality of Petroleum Assets for Economic Development

The chance to discover hydrocarbon volumes of economic quantity diminishes with progressive discovery in explored basins. Given the preponderance of smaller deposits in extensively explored basins and the cost implications of discovering deposits less than the required Minimum Economic Reserves (MER), explorationists and investors in exploration activities need a framework to evaluate the chance of a successful petroleum resources discovery to minimize the risk of unsuccessful exploration. This study develops a new framework to evaluate the chance of discovery of at least a minimum economic reserves volume in an extensively explored basin. It leverages on the postulation for the determination of probability of hydrocarbon economic success as a building block for the new framework. The model combines the concepts of Minimum Economic Reserves, Discovery Efficiency and Probability to derive an explicit analytical function for discovery efficiency and hydrocarbon probability for a commercial discovery. It digitalizes existing Risk Table to ease the complexity to obtain geological chance of success and hydrocarbon asset evaluation for commerciality. Nine Case studies from the prolific Niger Delta basin of Nigeria are used to validate the model. The result of the semi-digital solution of the model shows that three of the studied cases are commercial whereas the remaining six cases are sub-commercial. The study recommends the application of the new framework for hydrocarbon asset evaluation for chance of commerciality to complement models like the cream off curve to predict chance of commercial discovery of hydrocarbon assets.

New exact analytical solutions for the transient surface temperature of solids subjected to a non-uniform axisymmetric circular heat source

This paper presents a set of new explicit analytical solutions for determining the transient surface temperature of semi-infinite solids subjected to a non-uniform axisymmetric circular heat source. The uniform heat flux, semi-elliptic one and flux with singularity (equivalent to a uniform temperature in steady-stae) are considered. These solutions are simple to use and can therefore be applied in many industrial areas such as electronic components, laser treatments and any problem of diffusion other than heat transfer. They can also be used to study the constriction phenomenon. Validations of these analytical solutions by comparison with numerical modeling are performed and show an excellent agreement. We show that for the same case of heat flux distribution, surface temperatures at two different radii can be connected by an explicit analytical transfer function which does not require the knowledge of the flux imposed by the heat source.

English

Consider a second order differential linear periodic equation. The friction coefficient is real positive constant. Some transformation of the solution and its first derivative allow writing two-order differential equations with void friction coefficients. The solutions of these equations are periodic functions or sum of periodic function and an oscillating function with monotone linear increasing amplitude. The second order equation with linear friction is recast as a first order system. The coefficients of the principal fundamental matrix solution of the system are explicit analytical functions.

Magnetizing Curve Identification for Induction Motors at Standstill Without Assumption of Analytical Curve Functions

It is necessary to obtain accurate knowledge about the magnetizing curve of an induction machine so that the magnetizing inductance can be updated according to different flux references for either efficiency optimization or field weakening control. This paper proposes a computational saving method to obtain a one-to-one correspondence magnetizing curve without the need for conventional curve-fitting strategies, which require the assumption of explicit analytical curve functions. A detailed comparison between the proposed method and the conventional curve-fitting method has been done. It shows great accuracy improvements of this proposed method. The identification is easy to implement with high precision while maintaining the motor at standstill. No additional sensor devices are needed except the sensors carried by the drive itself. This proposed method is verified by simulation. Further experimental results of this method are utilized in a real control strategy with a 2.2-kW 50-Hz 4-pole induction motor to confirm its validity and feasibility.

Perturbative many-body expansion for electrostatic energy and field for system of polarizable charged spherical ions in a dielectric medium.

An analytical solution for the electrostatic energy and potential for a system of charged, polarizable spheres in a dielectric medium is developed from a multiple scattering expansion that is equivalent to a formal solution to Poisson's equation for the system. The leading contributions emerge in the form of effective two-, three-, and four-body interactions that are explicit analytical functions of the sphere positions, charges, and internal dielectric constants and the external dielectric constant, thereby also enabling analytical computation of the electrostatic forces on the ions. Tests of successive terms demonstrate their rapid convergence. Similar methods can be used to evaluate higher order contributions and the expansion for the electrostatic field. The results will prove far more efficient for MD and MC simulations with spherical particles than current approximate methods that require the computation of surface polarization charge distributions but that apply also for systems with complex geometries.

Motion of a spherical body in a viscous suspension

The problem of drag of an arbitrary-size solid sphere with its motion in a uniform monodispersed viscous suspension is considered in the Stokes approximation. The expression for the effective suspension viscosity is derived in the first approximation over the volume concentration of the viscous suspension. The coefficient before the concentration is found in the form of an explicit analytical function, which depends on the ratio of sizes of the dispersed particles and the body. The found coefficient coincides with Einstein’s result at the limit of “point” dispersed particles, the size of which is negligibly small compared with the size of the moving sphere, but can substantially differ from it in the case of finite-size particles.

Characterization of Thin-Film SOI PIN Diodes from Cryogenic to Above Room Temperatures Using an Explicit I-V Multi-Branch Model

A multi-branch model is proposed to describe multiple conduction mechanisms in thin-film SOI PIN diodes with parasitic series resistance over a wide operating temperature range, from 90 to 390 K. The model is composed of the parallel combination of three branches, each of them incorporating a diode and a series resistance. The model's unique advantage is its explicit nature which allows the terminal current to be continuously expressed as an explicit analytical function of the applied terminal voltage. The resulting explicit equation is convenient for repetitive simulation applications as well as for analytic differentiation and integration, in contrast to conventional models which only allow numerical solutions. The model's suitability has been assessed by parameter extraction and subsequent playback on real SOI PIN diode forward I-V characteristics.

An explicit multi-exponential model for semiconductor junctions with series and shunt resistances

An alternative explicit multi-exponential model is proposed to describe multiple, arbitrary ideality factor, conduction mechanisms in semiconductor junctions with parasitic series and shunt resistances. This Lambert function based model allows the terminal current to be expressed as an explicit analytical function of the applied terminal voltage, in contrast to the implicit-type conventional multi-exponential model. As a result this model inherently offers a higher computational efficiency than conventional models, making it better suited for repetitive simulation and parameter extraction applications. Its explicit nature also allows direct analytic differentiation and integration. The model’s applicability has been assessed by parameter extraction and subsequent playback using synthetic and experimental diode forward I–V characteristics.

MLP bilinear separation

In this paper, we present a thorough mathematical analysis of the use of neural networks to solve a specific classification problem consisting of a bilinear boundary. The network under consideration is a three-layered perceptron with two hidden neurons having the sigmoid serving as the activation function. The analysis of the hidden space created by the outputs of the hidden neurons will provide results on the network’s capacity to isolate two classes of data in a bilinear fashion, and the importance of the value of the sigmoid parameter is highlighted. We will obtain an explicit analytical function describing the boundary generated by the network, thus providing information on the effect each parameter has on the network’s behavior. Generalizations of the results are obtained with additional neurons, and a theorem concerned with analytical reproducibility of the boundary function is established.

Anisotropic semblance analysis and NMO corrections for long offset data

Sedimentary rocks can be regarded as transversely isotropic (TI) media. One difficulty in seismic processing is how to flatten the events for long offset data. P-wave reflections from horizontal reflectors in transversely isotropic media have non-hyperbolic moveout. For a multi-layered model, the reflection moveout formula is usually expressed as a Taylor series with higher order terms ignored. Alkhalifah and Tsvankin (1995) developed a three term Taylor series formula to calculate reflection travel times from a horizontal reflector in TI media with vertical symmetry axis (VTI). Using this formula, NMO correction works well for short spread lengths, but not so well for long spreads. Zhang and Uren (2001a, b) developed an approximate explicit analytical P-wave ray velocity function for transversely isotropic (TI) media. From this ray velocity function, a reflection travel time formula from a horizontal reflector in TI media was derived. This formula can be used for anisotropic NMO correction. It works well for both large offsets and small offsets. In order to obtain the unknown parameters required for seismic processing, a 3D-semblance analysis technique has been developed. We tested this method with numerical data from TI models with a single horizontal reflector and from multi-horizontal reflector models. The method was also tested with an isotropic model with multi-horizontal reflectors. The results show that the events can be completely flattened even for very large offsets and that both multi-layered TI and isotropic models may appear to be anisotropic.

Anisotropic semblance analysis and NMO corrections for long-offset data

Sedimentary rocks can be regarded as transversely isotropic (TI) media. P-wave reflections from horizontal reflectors in transversely isotropic media have non-hyperbolic moveout. One difficulty in seismic processing is how to flatten such reflection events for long-offset data. For a multi-layered model, the reflection moveout formula is usually expressed as a Taylor series with higher order terms ignored. Alkhalifah and Tsvankin (1995) developed a three-term Taylor-series formula to calculate reflection travel times from a horizontal reflector in TI media with a vertical symmetry axis (VTI). Using this formula, NMO correction works well for short spread lengths, but not so well for long spreads.Zhang and Uren (2001a, b) developed an approximate explicit analytical P-wave ray velocity function for transversely isotropic (TI) media. From this ray velocity function, a reflection travel-time formula for a horizontal reflector in TI media was derived. This formula can be used for anisotropic NMO correction. It works well for both large offsets and small offsets. In order to obtain the unknown parameters required for seismic processing, a 3D-semblance analysis technique has been developed. We tested this method on numerical data from TI models with a single horizontal reflector and from multiple horizontal reflector models. The method was also tested on an isotropic model with multiple horizontal reflectors. The results show that the events can be completely flattened even for very large offsets, and that both multi-layered TI and isotropic models may appear to be anisotropic.

Explicit Anisotropic P-wave Ray Velocity Functions

Sedimentary rocks encountered in seismic exploration frequently exhibit directional seismic wave velocity dependence. These anisotropic rocks often can be regarded as transversely isotropic (TI) media in which the ray velocity surface of SH-waves is elliptical, while those of P- and SV-waves are non-elliptical. Transversely isotropic phase velocity expressions exist for P-, SH- and SV-waves. Unfortunately, until now no explicit exact analytical ray velocity functions exist for P- or SV-waves propagating in TI media. In exploration seismology, explicit ray velocity functions are needed for efficient migration, velocity analysis and other seismic processing procedures.Approximate explicit analytical P-wave ray velocity functions in transversely isotropic media have now been developed. These ray velocity functions include a new dimensionless constant, A, which depends on elastic parameters C11, C33, C44, and C13 or Thomsen's parameters ε, δ, α0 and (β0. The accuracy of the functions has been tested for all the materials published by Thomsen (1986). From these ray velocity functions, the travel time between any two points in homogeneous transversely isotropic media may be calculated. Reflection travel times from a single horizontal reflector overlain by transversely isotropic media with vertical symmetrical axis (VTI) are also calculated and compared with those obtained by Tsvankin (1996). The results show that the proposed explicit ray velocity functions have high accuracy.

Modeling of the collector epilayer of a bipolar transistor in the MEXTRAM model

A new model description for the behaviour of epitaxial collectors in bipolar transistors is given. This is part of MEXTRAM, a compact model for circuit simulation, and it gives the voltage drop and stored minority carrier charge in the collector epilayer as a function of the bias conditions. It covers (total) depletion and quasi-saturation for both ohmic and space charge conditions in the end region of the epilayer. New features are that the collector current and stored charge are given as explicit analytical functions, thus guaranteeing continuity also for their derivatives, and that collector current spreading is taken into account. >

Mass Spectrometer Sample Line Diffusion Processes

We observed transients during the isotopic analysis of carbon dioxide, which could not be attributed to mixing processes because of their asymmetry, and to explain these we were led to consider the effects of diffusion on gas flow in the sample line. In particular, we investigated the propagation by diffusion of a concentration anomaly against a steady gas flow. If fractionation of isotopic(or chemical)species at the leak is to cause a change with time of the composition of the gas in the sample reservoir, then the effects of the fractionation must diffuse backwards as far as the gas reservoir.Using a one-dimensional Green's function, we obtained explicit analytical functions for the steady state concentration variation along a connecting tube between the leak and the reservoir, and for the composition variations with time at the leak. It is shown that the isotopic composition in the sample reservoir need not vary with time, even with a fractionating leak, provided that bulk gas velocity in the connecting tube is large enough. It is also shown that the diffusion process in combination with a fractionating leak can lead to the type of transients we observed.