Mathematical Treatment Research Articles

Analytical Analysis of Power-Constrained Repeaters’ Insertion in Large-Scale CMOS Chips

As the die area of CMOS integrated circuits continues to increase, interconnects will become dominant in determining the performance of the circuits from the standpoint of speed and power consumption. Uniform repeater insertion is an effective method used to reduce the propagation delay of a signal in long resistive-capacitive lines. However, non-optimal repeaters’ insertion yields non-optimal circuit performance. In this work, we provide a mathematical treatment for optimal repeater insertion with power consumption constraints. In particular, a closed-form expression for the optimum number and size of repeaters is given for a two-stage buffer used as a repeater. The validation of the analytical solution is assessed by means of circuit simulations, by comparing the theoretical optimal number and size of the repeaters to be placed in the long resistive-capacitive line with the simulated values.

NIR spectroscopy prediction model for capsaicin content estimation in chilli: A rapid mining tool for trait-specific germplasm screening

Chilli is a widely produced crop, highly valued for its capsaicin content, a key economic trait. Traditional wet chemistry methods for estimating capsaicin are time-taking and laborious, while non-destructive methods like NIRS coupled with chemometrics, offer efficient alternatives, simplifying and accelerating biochemical assessments. This study is the first to develop and validate and tested for applicability of NIRS-based prediction model for capsaicin content in Indian chilli germplasm using MPLS regression. Various mathematical treatments were performed, and the most suited model was selected based on high RSQexternal, RPD and lower SEP values in the external validation set, indicating strong prediction accuracy and minimal error. The model achieved high RSQexternal value of 0.808, RPD value of 2.088 and low SEP value of 3.415 for capsaicin content, demonstrating excellent prediction performance. A paired sample t-test p-value of 0.757 (p > 0.05) showed non-significant difference between wet lab and predicted values, confirming the model’s accuracy. The applicability of the model was validated on fresh harvest germplasm the following year, showing a higher reliability score of 0.949, further confirming model’s reliability. This model would aid in high-throughput, accurate screening of chilli germplasm for capsaicin, accelerating chilli crop improvement programs and the development of new high-capsaicin varieties.

An exactly solvable model for RNA polymerase during the elongation stage

We consider a Markovian model for the kinetics of RNA Polymerase (RNAP) which provides a physical explanation for the phenomenon of cooperative pushing during transcription elongation observed in biochemical experiments onEscherichia coliand yeast RNAP. To study how backtracking of RNAP affects cooperative pushing we incorporate into this model backward (upstream) RNAP moves. With a rigorous mathematical treatment of the model we derive conditions on the mutual static and kinetic interactions between RNAP under which backtracking preserves cooperative pushing. This is achieved by exact computation of several key properties in the steady state of this model, including the distribution of headway between two RNAP along the DNA template and the average RNAP velocity and flux.

A Universal Target Plate Design Scheme for Stellarators: theoretical basis and its application to heat load control

Abstract This study introduces a novel divertor target design scheme for stellarators, grounded in mathematical treatments and tailored to control toroidal heat load distributions. Initially, a differential equation characterizing toroidally uniform heat load distribution has been established in a two-dimensional (2D) slab configuration, and its analytic solution has been obtained. Subsequently, a numerical scheme has been developed to adapt the analytic solution into the 3D surface shape of stellarator target. The effectiveness of this design scheme has been validated through simulations of the Chinese First Quasi-axisymmetric Stellarator (CFQS) using a suite of codes including HINT, FLARE and EMC3-EIRENE, where a toroidally uniform heat load distribution has been achieved with an island configuration. Further, the effects of input parameters on the target shape and heat load distribution have been studied. The robustness of the designed target has been investigated by simulation results with varying magnetic island configurations, confirming that the toroidal uniformity of heat load distribution is insensitive to changes in island configurations. Moreover, the designed target has been assessed with gas puffing of neon, which shows that neon injections effectively reduce the heat loads without altering the toroidal uniformity of heat load distributions. The proposed scheme highlights the importance of theoretical and mathematical foundations of target design, offering an advantageous alternative/complement to the traditional numerical schemes.

Natural convection effects in molten LiF-NaF-KF

Understanding mass transfer is of great importance to molten salt electrochemistry. However, natural convection has long been neglected in electrochemical measurements in stagnant molten salt systems due to the complex mathematical treatments. Here we reported the notable natural convection effects in molten LiF-NaF-KF during cyclic voltammetry and chronoamperometry tests. The established model showed that the convection-diffusion layer (CDL) convection dominated mass transfer under long time-scale tests in molten LiF-NaF-KF-EuF3. However, in the case of molten LiF-NaF-KF-K2TiF6, CDL convection predominated at dilute redox concentrations while density-driven convection came into dominance at high redox concentrations, because of the severe coordinate structure change of the redox couple. The derived thickness of the convection-diffusion layer, δconv., ranged from 135 to 183 μm, much higher than that in aqueous solutions (220–250 μm). Moreover, the increase in system temperature greatly enhanced natural convection effects in molten fluorides. Lastly, the use of microelectrode (re<39.7 μm for δconv.=160 μm) was demonstrated to be an effective way to suppress the intense natural convection effects.

Atmospheric footnotes: Ada Lovelace on climate

This article brings to light correspondence between Augusta Ada King, Countess of Lovelace (1815–1852) and Sir John Frederick William Herschel (1792–1871) from the Royal Society archives. The 1848 letters reveal the extent of her contribution to an article authored by her husband, William King, Lord Lovelace (1805–1893), on the role of climate on crop growth. This involvement uncovers little-known facets of her scientific pursuits: her interest in the application of photography to meteorological instrumentation, her budding study of astronomy, and her development of mathematical treatments to model the environmental conditions required for plants to grow. The exchange situates Ada Lovelace in the larger context of Victorian science and reflects more widely on the complicated access of women to institutional science.

Simulation of a temporally dynamic atmospheric wavefront using the modified von-Kármán spectrum and Zernike-based method

In the simulation of the atmospheric wavefront, the Kolmogorov spectrum facilitates the mathematical treatments, but this model gives inaccurate predictions in cases where the effects of outer- and inner-scale can not be neglected. Therefore, we derive the covariance formula of Zernike coefficients based on the modified von-Kármán turbulence spectrum to improve the Zernike-based method. Further, using the frozen flow hypothesis, the temporal power spectra of Zernike coefficients are derived, and the effects of wind velocity and direction, as well as the outer and inner scales, are also taken into account. Then, we introduce a method for generating the temporally dynamic atmospheric wavefront using the inverse Fourier transform to draw random time-evolving coefficients from the temporal spectra. We evaluate the performance on simulated atmospheric wavefront, and the results confirm that accurate spatial and temporal statistics are obtained.

A general method to synthesize planar overconstrained mechanism

Traditionally, planar overconstrained mechanisms has been explained through special geometry, which often lacks clarity and specificity. This paper addresses this challenge by introducing two explicit criteria for analyzing and synthesizing planar overconstrained mechanisms. With the criteria, one may synthesize new planar overconstrained systematically and confidently. To create new planar overconstrained mechanisms, three approaches are employed: the parallelogram-aided approach, the stretch-rotation approach, and an optimization-based method. Given that the mobility of mechanisms is independent of the choice of the ground link and that the mathematical treatment of function generation is straightforward, this paper proposes a systematic approach for synthesizing novel function cognates. By combining these approaches, new types of overconstrained mechanisms are discovered. Leveraging advanced computational tools, a special geometric condition for function cognates of inverted slider cranks is found for the first time. Through inversion, the coupler cognates for slider cranks are also identified. The success of these results indicates that the proposed criteria are explicit and crucial for discovering new mechanisms, and the concept may be extended to spherical or spatial cases.

New insights on the origin of the Richardson-Richards equation

ABSTRACT The Richardson-Richards equation (RRE), despite known shortcomings especially in regard to preferential flow, provides the basis for the vast majority of unsaturated flow models in use today. L.F. Richardson published this equation in 1922, nine years before L.A. Richards. Whereas Richards approached this problem directly from the groundbreaking developments of Edgar Buckingham, Richardson, surprisingly, cited as his starting point only the earlier work of L.J. Briggs. Collectively, these four scientists’ published and unpublished work reveals that: (1) Briggs’ work, though qualitative, captured the essential physical principles needed for quantifying unsaturated flow; (2) Buckingham came very close to deriving the RRE and explained why he stopped short of doing so; (3) derivation of the RRE from the work of either Briggs or Buckingham required only modest developmental work; and (4) besides deriving the RRE, Richards carried through much of the experimental agenda that Buckingham considered a necessary precursor to mathematical treatment.

A mathematical treatment of the impact of p53 signalling on the expression of the transcription factor E2F in the CycE/Cdk2 subsystem in cancerous cells

Complex networks of proteins and genes control cell growth until mitosis. These components play crucial roles in determining the characteristics and timing of each reaction in the cell. A family of cyclins and cyclin-dependent kinases regulates the progression of the cell from one stage to the next, while certain inhibitors regulate cell proliferation, significantly impacting this process. We examine the impact of p53 protein production in the cyclin E/cyclin-dependent kinase 2 (CycE/CDK2) subsystem. Although the precise functioning of the p53 protein is not fully understood, it is known to exhibit pulsatile behaviour, triggered in response to DNA damage. Consequently, the expression of the p53 protein within the cell system increases with the severity of DNA damage. Similarly, levels of CycE/CDK2 are substantially elevated in instances of DNA damage.To investigate this subsystem, we have developed a mathematical model of the CycE/CDK2 subsystem, presented as non-linear ordinary differential equations. Our primary aim is to determine the conditions that could theoretically result in tumorigenesis. We conduct a dynamical systems analysis to investigate the presence of stable limit cycles, which indicate the standard operational states of cells. Our investigation reveals the levels of p53 protein at which the cell can correct defects, undergo apoptosis, or enter quiescence, and where the system exhibits a Hopf-bifurcation (limit cycles). This is the novelty of our study, as previous research has primarily focused on numerical simulations.Using MATLAB simulations, we produced various results in response to different expressions and concentrations of the p53 protein, allowing us to assess its influence on malignant cells. By altering the concentration of protein p53, we ascertain the specific cellular circumstances that facilitate its natural growth, self-correction in the presence of defective activity, and entry into a state of quiescence.This investigation is motivated by the significant role of oscillations in driving, controlling, and managing the cellular oversight machinery. These aspects are incorporated into developing a mathematical model that emulates cellular operations. From this research, we establish that E2F is a critical target to mitigate oncogenic activity, achievable by managing the pulsatile behaviour of the inhibitor p53 or inhibiting its production process within the cell machinery.

Atmospheric pressure-induced three-dimensional surface wave propagation in the compressible ocean: Effect of static compression

Under the assumptions of linearized water wave theory, we build a three-dimensional mathematical model that couples atmospheric pressure waves and surface ocean waves, including water compressibility and its static part, to simulate Meteotsunami propagation in the ocean. The solution uses the Laplace–Fourier double transformation technique, emphasizing axisymmetry of the mathematical problem and rigorous treatment of a fairly complicated dispersion relation while using inverse transformations. A novel derivation of the axisymmetric atmospheric pressure front is shown. The impact of water compressibility is shown through a comparative graphical representation against the incompressible case. Faster travel of free-surface waves is observed in the incompressible ocean, followed by the cases with and without static compression of the compressible ocean, respectively. The static compression shifts the phase of the acoustic-gravity modes. The locked wave is hardly influenced by the water compressibility and is entangled with the moving pressure front. The model is validated with the observational pressure data and agrees well with our computed pressure profile. Then, the locked wave profile generated from our model agrees well with the deep-ocean assessment and reporting of tsunami data.

GGR Handbook of Rock and Mineral Analysis Chapter 6 (Part 1) Principles and Practice of X‐Ray Fluorescence Spectrometry –1: Fundamentals of XRF and Matrix Corrections

This chapter (Principles and Practice of X‐Ray Fluorescence Spectrometry – 1: Fundamentals of XRF and Matrix Corrections) is a contribution to the Geostandards and Geoanalytical Research Handbook of Rock and Mineral Analysis – an online textbook that is a fully revised and updated edition of the Handbook of Silicate Rock Analysis (P. J. Potts, 1987, Blackie, Glasgow).Chapter 6, Part 1 (from Section 2 of the handbook dealing with techniques for the determination of major and trace elements) considers the fundamentals of XRF spectrometry and matrix corrections in detail. Part 2 deals with wavelength dispersive and energy dispersive instrumentation. Following an introduction dealing with the analytical characteristics of XRF spectrometry, Part 1 continues with a detailed consideration of the origin and excitation of X‐ray spectra. Sub‐chapters that tackle the interaction of X‐rays with matter and the matrix effect in geological materials follow this. Part 1 is concluded with a mathematical treatment of the correction of absorption‐enhancement effects.

Improper Background Treatment Underestimates Thermometric Performance of Rare Earth Vanadate and Phosphovanadate Nanocrystals.

Luminescence thermometry is the state-of-the-art technique for remote nanoscale temperature sensing, offering numerous promising cutting-edge applications. Advancing nanothermometry further requires rational design of phosphors and well-defined, comprehensive mathematical treatment of spectral information. However, important questions regarding improper signal processing in ratiometric luminescence thermometry are continuously overlooked in the literature. Here, we demonstrate that systematic errors arising from background/signal superposition impact the calculated thermometric quality parameters of ratiometric thermometers. We designed ultraviolet-excitable (Y,Eu)VO4 and (Y,Eu)(P,V)O4 nanocrystals showing overlapped VO4 3- and Eu3+ emissions to discuss systematically how uncorrected background emissions cause magnified (∼10×) temperature uncertainties and undervalued (∼60%) relative thermal sensitivities. Adequate separation of spectral contributions from the VO4 3- background and the Eu3+ signals via baseline correction is necessary to prevent underestimation of the thermometric performances. The described approach can be potentially extended to other luminescent thermometers to account for signal superposition, thus enabling to circumvent computation of apparent, miscalculated thermometric parameters.

Bound-state evolution of Dirichlet waveguide with Neumann window(s) in transverse electric fields

The influence of electric field E on a straight Dirichlet waveguide with Neumann window(s) of arbitrary length L is studied theoretically for 2D and 3D geometries. The energy spectrum for the corresponding bound states was obtained and their dependence on E was analysed. At low E , the boundary of the window(s) influenced the properties of the particle contrary to the high E case, where the energies and localization of particle were seen similar in all the geometries. The critical lengths L cr , at which higher excited state emerges into the continuum, at different E were found and their behavior was explained qualitatively. It was found that at high E , the critical lengths of Dirichlet-Neumann boundary window approach the Neumann-Neumann boundary window. This is due to the increase in the fundamental propagation threshold of transverse modes in the wall and window region. On the other hand, the Neumann-Dirichlet boundary window showed an opposite behavior. Furthermore, the polarization of ground state was obtained for different geometries. It was proved that at small L, the polarization reduces to its 1D case. A mathematical treatment and semi classical explanation was provided. Comparative analysis of the 2D and 3D geometries reveals their qualitative similarity and quantitative differences.

DLS homemade setup: reviewing first and second-order coherence and autocorrelation concepts of a light source in the context of nanoparticle sizes synthesized by PLAL

The size of the nanoparticles (NP) is one of the most important and essential characteristics to know the properties of the synthesized nanostructures. The most common characterization procedures are related to Scanning Electronic Microscopy (SEM), Transmission Electronic Microscopy (TEM), and Atomic Force Microscopy (AFM). Unfortunately, from a practical point of view, they represent a time-consuming procedure and require expensive equipment, which limits its application to specialized research groups. Significant attention has been paid to Dynamic Light Scattering (DLS) as a simple, fast, and reproducible method for sizing nanoparticles. However, inadequate representation of the fundamental principles of DLS and data interpretation represents two of the most important challenges related to this technique. In this work we try to provide the fundamental principles of the DLS technique, the fundamental mathematical treatment of data obtained during the optical scattering studies and provide the MATLAB code to configure non-commercial DLS equipment. Additionally, analyzes of nanoparticles obtained by pulsed laser ablation of Ag, Au, Si and W and commercial Au nanoparticles were carried out. The particle size results are compared with SEM images to calculate the percentage error of the DLS measurements. The results show an error of 5%, 3.8%, 2.1% for the Ag, Au and Si nanoparticles respectively, which proves to be an excellent approximation to the real values of nanoparticle diameter. Meanwhile, the error in size for W nanoparticles by the same technique and commercial Au nanoparticle is 29% and 12%, which shows the effect of the hydrodynamic diameter of the nanoparticles. This work ends with the analysis of the concentration of nanoparticles and its importance in reliable results of DLS measurements.

Analytic Modelling of 2-D Transient Heat Conduction with Heat Source Under Mixed Boundary Constraints by Symplectic Superposition

Abstract Many studies have been conducted on 2-D transient heat conduction, but analytic modelling is still uncommon for the cases with complex boundary constraints due to the mathematical challenge. With an unusual symplectic superposition method, this paper reports new analytic solutions to 2-D isotropic transient heat conduction problems with heat source over a rectangular region under mixed boundary constraints at an edge. With the Laplace transform, the Hamiltonian governing equation is derived. The applicable mathematical treatments, e.g., the variable separation and the symplectic eigenvector expansion in the symplectic space, are implemented for the fundamental solutions whose superposition yields the ultimate solutions. Benchmark results obtained by the present method are tabulated, with verification by the finite element solutions. Instead of the conventional Euclidean space, the present symplectic-space solution framework has the superiority on rigorous derivations without pre-determining solution forms, which may be extended to more issues with the complexity caused by mixed boundary constraints.

Optical fibers with a frequency-dependent Kerr nonlinearity: Theory and applications

This review provides a detailed discussion of both the mathematical treatment and the impact of a frequency-dependent Kerr nonlinearity on the propagation of short pulses in optical fibers. We revisit the theoretical framework required to deal with the frequency dependence of the nonlinear response without incurring any physical inconsistencies, such as the non-conservation of the photon number. Then, we point out the role of the zero-nonlinearity wavelength, its interplay with the zero-dispersion wavelength, and their influence on evolution of optical pulses in optical fibers, specifically by looking at soliton propagation and the ensuing generation of Cherenkov radiation. Finally, by means of a space–time analogy involving the collision of a weak control pulse and an intense soliton, we describe an all-optical switching scheme in the presence of a zero-nonlinearity wavelength within a photon-conserving framework.

Calibrating ITC instruments: Problems with weak base neutralization

Modern isothermal titration calorimetry instruments give great precision, but for comparable accuracy they require chemical calibration. For the heat factor, one recommended process is HCl into the weak base TRIS. In studying this reaction with a VP-ITC and two Nano-ITCs, we have encountered some problems, most importantly a titrant volume shortfall Δv ≈ 0.3 μL, which we attribute to diffusive loss of HCl in the syringe tip. This interpretation is supported by a mathematical treatment of the diffusion problem. The effect was discovered through a variable-v protocol, which thus should be used to properly allow for it in any reaction that similarly approaches completion. We also find that the effects from carbonate contamination and from OH− from weak base hydrolysis can be more significant that previously thought. To facilitate proper weighting in the least-squares fitting of data, we have estimated data variance functions from replicate data. All three instruments have low-signal precision of σ ≈ 1 μJ; titrant volume uncertainty is a factor of ∼2 larger for the Nano-ITCs than for the VP-ITC. The final heat factors remain uncertain by more than the ∼1 % precision of the instruments and are unduly sensitive to the HCl concentration.

Addressing variability in cell electrorotation through holographic imaging and correction factors

This study addresses the variations observed in electrorotation measurements due to cell positioning and movement. Electrorotation provides a non-disruptive method for inferring the electrical properties of individual cells. However, its widespread adoption is hindered by significant variation in the observed speed. By mitigating the impact of positional dependencies and other influencing factors, our methodology opens avenues for broader applications of electrorotation in single-cell analysis without the need for complex setups to trap and retain the cell in place. Our novel approach combines multi-plane imaging with mathematical treatment of rotation data. This method uses a conventional quadrupole chip and lens-free imaging to track cell movement, resulting in a simpler design and set-up. Through numerical simulations incorporating cell coordinates, chip design, and experimental parameters, we calculate the variation in torque for each position. These values serve as the basis for the correction factors. Validation experiments with T-lymphocytes and fibroblasts show that the correction factors reduce electrorotation speed variation due to cell movement, with an average reduction to 21% and 18%, respectively. These corrections also revealed previously concealed changes in cell properties, in response to external stimuli, thereby enhancing the reliability of measurements and enabling broader applications in single-cell analysis.

A new competing risks model with applications to blood cancer data

Competing risks models are invaluable probabilistic models in survival analysis, a particular part of medical research. Such models deal with research problems that involve multiple potential risk factors that compete with each other to cause death (i.e., failure from a statistical perspective). Competing risks models' flexibility must be thoroughly explored to guarantee suitability for multifaceted risk scenarios (e.g., modeling data of malicious diseases where factors such as treatment response and progression are intertwined). This article considers a new competing risks model called the additive generalized linear-exponential (AGLE) competing risks model. The proposed model is expected to be more robust and superior to other well-known models when modeling real-life data. A mathematical treatment for the properties of the new model is first considered. Afterward, model parameters estimation via various estimation methods is discussed. The critical role of estimating shape parameters in understanding blood cancer's survival and failure mechanisms is highlighted. Furthermore, the bias and root mean square error of different estimation methods are examined numerically through Monte Carlo simulations. The simulation study indicated that the maximum product of spacings estimation method for the model parameters has the optimal balance of bias and variance as the sample sizes increase. Two real-life blood cancer data sets are analyzed to illustrate the application of the proposed model. The analysis outcomes support the AGLE model's robustness, with low Kolmogorov-Smirnov statistics and high p-value, affirming its superior fit for blood cancer data compared to other well-known models.