Rigorous Mathematical Expressions Research Articles

Modeling of the electrical interaction between desert dust particles and the Earth’s atmosphere

In the current work, the two distinct charging mechanisms of spherical dust particles settling in the Earth’s atmosphere, i.e. ion attachment and contact electrification, are studied in parallel. A novel 1D numerical model has been developed that parametrizes the charging processes in the presence of a large scale electric field, under stagnant atmospheric conditions where wind contribution is neglected. The model is able to self-consistently calculate the modification of atmospheric ion densities and the subsequent alteration of the large scale electric field, when dust particles are present and atmospheric ions attach to them, and is further updated to account for the particle charging due to the efficient collisions between particles of sizes ranging from fine (less than 1μm in radius) to giant mode (larger than 50μm in radius). Due to a lack of a rigorous mathematical expression for the quantification of the particle collision process, the mechanical and electrostatic effects are modeled independently, based on works in the past literature, and then are combined by a simple superposition principle. Additionally, the electrostatic effects are studied in the presence of weak external electric fields (less than 1–2 kV/m). Moreover, for the same reasons a simple superposition principle has been applied for the quantification of the ion attachment process. All these assumptions lead to an approximation of the actual mechanisms quantification, but this method can lead to physically reasonable results, highlighting the impact of several processes to the eventual charging of the dust particles. By binning integration of a realistic particle size distribution in the model, the acquired electrical charge on the dust particles is calculated in the range of 0.5 to 2000 elementary charges. The particles become on average negatively charged, but under specific conditions giant mode particles can be locally and temporarily positive, congruently to previously reported laboratory experiments. Moreover, the electrical force that is applied on the particles by each mechanism individually and through the superposition of both, is extracted and compared to the gravitational force acting on each particle. However, the 1D results indicate that the electrical force is not enough to significantly influence the gravitational settling of the charged particles, as it is more than one order of magnitude less than gravity. This designates that although the electrical processes alone are accounted for, significant meteorological processes such as updrafts, wind shears, or horizontal winds need to be included in the model to accurately represent the modulation of particle electrification and so as to represent the impact of electricity on particle dynamics in a more realistic way.

Multiple-pole solutions and degeneration of breather solutions to the focusing nonlinear Schrödinger equation

Based on the Hirota’s method, the multiple-pole solutions of the focusing Schrödinger equation are derived directly by introducing some new ingenious limit methods. We have carefully investigated these multi-pole solutions from three perspectives: rigorous mathematical expressions, vivid images, and asymptotic behavior. Moreover, there are two kinds of interactions between multiple-pole solutions: when two multiple-pole solutions have different velocities, they will collide for a short time; when two multiple-pole solutions have very close velocities, a long time coupling will occur. The last important point is that this method of obtaining multiple-pole solutions can also be used to derive the degeneration of N-breather solutions. The method mentioned in this paper can be extended to the derivative Schrödinger equation, Sine-Gorden equation, mKdV equation and so on.

Limit analysis for the seismic stability of three-dimensional rock slopes using the generalized Hoek-Brown criterion

The parameters that influence slope stability and their criteria of failure are fairly understood but over-conservative design approaches are often preferred, which can result in excessive overburden removal that may jeopardize profitability in the context of open pit mining. Numerical methods such as finite element and discrete element modelling are instrumental to identify specific zones of stability, but they remain approximate and do not pinpoint the critical factors that influence stability without extensive parametric studies. A large number of degrees of freedom and input parameters may make the outcome of numerical modelling insufficient compared to analytical solutions. Existing analytical approaches have not tackled the stability of slopes using non-linear plasticity criteria and three-dimensional failure mechanisms. This paper bridges this gap by using the yield design theory and the Hoek-Brown criterion. Moreover, the proposed model includes the effect of seismic forces, which are not always taken into account in slope stability analyses. The results are presented in the form of rigorous mathematical expressions and stability charts involving the loading conditions and the rock mass properties emanating from the plasticity criterion.

Metering Method and Measurement Uncertainty Evaluation of Underwater Positioning System in Six Degrees of Freedom Space

Meterage is an important action to ensure the performance of underwater acoustic positioning system, but the commonly used measurement methods are mostly based on the low degrees of freedom (DOF) measurement and control platforms which could not realize the all-round and high efficiency evaluation of this equipment. In this paper, we develop a new metering method base on the conventional standard 4-DOF measurement and control platform to realize the whole exploration space high efficiency and high accuracy measurement of underwater acoustic positioning systems in 6-DOF space. First, a 4-DOF space to 6-DOF space quantity value propagation model is proposed, and a rigorous mathematical expression is established. Next, a measurement procedure is formed based on the building of a standard base-line field, so the channel of transforming measurable position coordinate of the measured field point in 4-DOF space to the 6-DOF space is established. Finally, the evaluation of the measurement uncertainty is done using both the Guide to the expression of uncertainty in measurement method (GUM method) and Adaptive Monte Carlo method (AMCM method), and the evaluation results of GUM method are verified by AMCM method correspondingly, the effectiveness of this metering theory and the corresponding uncertainty evaluation methods are also illustrated by analyzing the length of coverage interval and numerical tolerances of measurement uncertainty. The research results show that the relative combined uncertainty of the metering system is not higher than 0.18% with the parameters given in this paper.

Stabilization of Mode-Dependent Impulsive Hybrid Systems Driven by DFA With Mixed-Mode Effects.

This paper is concerned with mode-dependent impulsive hybrid systems driven by deterministic finite automaton (DFA) with mixed-mode effects. In the hybrid systems, a complex phenomenon called mixed mode, caused in time-varying delay switching systems, is considered explicitly. Furthermore, mode-dependent impulses, which can exist not only at the instants coinciding with mode switching but also at the instants when there is no system switching, are also taken into consideration. First, we establish a rigorous mathematical equation expression of this class of hybrid systems. Then, several criteria of stabilization of this class of hybrid systems are presented based on semi-tensor product (STP) techniques, multiple Lyapunov-Krasovskii functionals, as well as the average dwell time approach. Finally, an example is simulated to illustrate the effectiveness of the obtained results.

Glucose solution determination based on liquid photoacoustic resonance.

Noninvasive blood glucose determination has received considerable attention in the past from both patients and scientists all over the world, and it is becoming increasingly important as a research focus. The two most difficult problems leading to no breakthrough in this area are sensitivity and specificity in determination. In order to obtain reliable measurement results of blood glucose levels, we propose a new liquid photoacoustic resonance theory that can significantly enhance the intensity of the signal and improve the sensitivity. This paper demonstrates the theory of liquid photoacoustic resonance, gives a rigorous mathematical expression, and analyzes the variation of the transducer output in the case of liquid photoacoustic resonance. A signal processing method is demonstrated at the same time under the liquid photoacoustic resonance condition. Meanwhile, the feasibility and validity are verified by experiments with different concentrations of glucose solution. The result shows that liquid photoacoustic resonance can strengthen the signal, and the resolution achieves 20 mg/dL. This method overcomes the issue of low sensitivity and the inaccurate detection in the nonresonant case, and gets accurate results. This result could provide a theoretical basis for realization of noninvasive measurement of blood glucose.

Theoretical estimates for flat voids coalescence by internal necking

Coalescence of voids by internal necking is in most cases the last microscopic event related to ductile fracture and corresponds to a localized plastic flow between adjacent voids. Macroscopic load associated to the onset of coalescence is classically estimated based on limit analysis. However, a rigorous upper-bound mathematical expression for the limitload required for flat voids coalescence that remains finite for penny-shaped voids/cracks is still unavailable. Therefore, based on limit analysis, theoretical upper-bound estimates - both integral expression and closed-form formula - are obtained for the limit-load of cylindrical flat voids in cylindrical unit-cell subjected to boundary conditions allowing the assessment of coalescence, for axisymmetric stress state. These estimates, leading to finite limit-loads for pennyshaped cracks, are shown to be in very good agreement with numerical limit analysis, for both cylindrical and spheroidal voids. Approximate formula is also given for coalescence under combined tension and shear loading. These coalescence criteria can thus be used to predict onset of coalescence of voids by internal necking in ductile fracture modelling.

The One-persistent Multichannel CSMA Protocol with Monitoring Functions Based on Conflict Resolution Algorithm in WSN

In the Wireless Sensor Network, a new MAC protocol: one-persistent CSMA protocol with monitoring functions and multichannel mechanism for Wireless Sensor Network based on conflict resolution algorithm is proposed. Protocol from the system throughput, collision rate, better service quality for higher priority and other aspects improve the traditional CSMA system to perfect the controllability of the system, making the system more stable for the big data, better adapted to different business. The average cycle method is used to derive rigorous mathematical expressions of specific relevant parameters. Using MATLAB simulation tool for the accuracy of the agreement is demonstrated.

Organizing principles as tools for bridging the gap between system theory and biological experimentation.

Twentieth-century theoretical efforts towards the articulation of general system properties came short of having the significant impact on biological practice that their proponents envisioned. Although the latter did arrive at preliminary mathematical formulations of such properties, they had little success in showing how these could be productively incorporated into the research agenda of biologists. Consequently, the gap that kept system-theoretic principles cut-off from biological experimentation persisted. More recently, however, simple theoretical tools have proved readily applicable within the context of systems biology. In particular, examples reviewed in this paper suggest that rigorous mathematical expressions of design principles, imported primarily from engineering, could produce experimentally confirmable predictions of the regulatory properties of small biological networks. But this is not enough for contemporary systems biologists who adopt the holistic aspirations of early systemologists, seeking high-level organizing principles that could provide insights into problems of biological complexity at the whole-system level. While the presented evidence is not conclusive about whether this strategy could lead to the realization of the lofty goal of a comprehensive explanatory integration, it suggests that the ongoing quest for organizing principles is pragmatically advantageous for systems biologists. The formalisms postulated in the course of this process can serve as bridges between system-theoretic concepts and the results of molecular experimentation: they constitute theoretical tools for generalizing molecular data, thus producing increasingly accurate explanations of system-wide phenomena.

Practical Valuation of Options on Durable Goods

Manufacturers and retailers in markets with a secondary market continually struggle to differentiate their products and increase the perceived value of their offerings to consumers. Prior literature presented the marketing rationale for manufacturers and dealers in the car industry to offer new car buyers options to buy a new car at a predetermined price or a discount, or to sell the car they buy now at a guaranteed fixed price in the future, on the next purchase event. Previous publications often avoided a systematic valuation of such options. An exception is Afik, Lowengart, and Yosef (2012), (ALY), presenting rigorous mathematical expressions for mathematically tractable price processes. When the valuation of more complex options is required, developing mathematical tools such as those of ALY becomes a Herculean effort and often mathematically impossible. In this paper, on the other hand, we focus our attention on the development and application of Monte Carlo valuation schemes for such options, presenting a very flexible and easy methodology to apply numerical solution to this mathematical challenge.

The stoichiometry and energetics of oxygenic phototrophic growth

The values of gross metabolic flows in cells are essentially interconnected due to conservation laws of chemical elements and interrelations of biochemical coupling. Therefore, the overall stoichiometry of cellular metabolism, such as the biomass quantum yield, the ratio between linear and circular flows via the electron transport chain, etc., can be calculated using balances of metabolic flows in the network branching points and coupling ratios related to ATP formation and expenditures. This work has studied the energetic stoichiometry of photosynthetic cells by considering the transfer of reductivity in the course of biochemical reactions. This approach yielded rigorous mathematical expressions for biomass quantum yield and other integral bioenergetic indices of cellular growth as functions of ATP balance parameters. The effect of cellular substance turnover has been taken into account. The obtained theoretical estimation of biomass quantum yield is rather close to experimental data which confirms the predictive capacity of this approach.

A Movement based Algorithm for Determining Handoff in Heterogeneous Wireless Networks

The Next Generation wireless networks generally called 4G Networks will most likely be actualized by the Integration of present day available heterogeneous wireless networks. Integration of these networks will largely depend on the process of vertical handoff. Vertical handoff occurs when an active communication session is transferred between different radio access technologies. An efficient assessment system is essential for appropriate network selection.Unlike horizontal handoff, vertical handoff only takes place in cell edges. There are many other factors that should be considered for a proper vertical handoff i.e., triggering factor, critical factor and influencing factors. For that an efficient assessment system is essential for appropriate network selections. The vertical handover problem has been becoming increasingly interesting due to its practical value. The convergence of heterogeneous wireless access technologies characterizes the 4G wireless networks. In such systems the seamless and efficient handoff between different access technologies is essential and remains a challenging problem. This paper presents a Movement based algorithm (MBA) for determining handoff in heterogeneous wireless networks. Our algorithm addresses to cellular as well as broadband mix systems for randomly generated users. Our algorithm determines pathloss, throughput and signal-tonoise- ratio (SNR) between randomly generated users, Access points (APs) and Base stations (BS). The proposed method is well suited for networks systems, it eliminates the rigorous mathematical expressions and its design is simple and straightforward.The main goal of our algorithm is to enhance or improve the system performance in terms of higher throughput for the required networks (BWN and WiMax). Simulation work has been carried out in the MATLAB environment. Simulation results significantly outperform the existing algorithms. We haven’t used any MATLAB toolbox. In fact we developed our own simulation platform in MATLAB. Simulation results confirm that our algorithm improves higher throughput by 14.8 % compared to other existing approaches.

Metal flux through consuming interfaces in ligand mixtures: boundary conditions do not influence the lability and relative contributions of metal species

In a mixture of metal ions and complexes, it is difficult to predict ecological risk without understanding the contribution of each metal species to biouptake. For microorganisms, the rate of uptake (internalization flux) has not only a major influence on the total metal flux but also on the bioavailability of the various metal species and their relative contributions to the total flux. In this paper, the microorganism is considered as a consuming interface, which interacts with the metal ion, M, via the Michaelis-Menten boundary conditions. The contribution of each metal complex to the overall metal flux, in relation to its lability, is examined for a number of important boundary parameters (the equilibrium constant K(a) of metal with transport sites, internalization rate constant k(int) and total transport sites concentration {R}(t)). Computations were performed for Cu(II) complexes, in a multicomponent culture medium for microoganisms. For a one-ligand system, results were acquired using rigorous mathematical expressions, whereas approximate expressions, based on the reaction layer approximation (RLA) and rigorous numerical computations (computer codes MHEDYN and FLUXY), were employed for ligand mixtures. Under the condition of ligand excess, as often found in the natural environment, the relative contribution of each metal species to the total flux is shown to be independent of the boundary conditions. This finding has important implications, including an improved basis for relating the analytical signals of dynamic metal speciation sensors to metal bioavailability.

Theoretical results for chemotactic response and drift of E. coli in a weak attractant gradient

The bacterium Escherichia coli ( E. coli) moves in its natural environment in a series of straight runs, interrupted by tumbles which cause change of direction. It performs chemotaxis towards chemo-attractants by extending the duration of runs in the direction of the source. When there is a spatial gradient in the attractant concentration, this bias produces a drift velocity directed towards its source, whereas in a uniform concentration, E. coli adapts, almost perfectly in case of methyl aspartate. Recently, microfluidic experiments have measured the drift velocity of E. coli in precisely controlled attractant gradients, but no general theoretical expression for the same exists. With this motivation, we study an analytically soluble model here, based on the Barkai–Leibler model, originally introduced to explain the perfect adaptation. Rigorous mathematical expressions are obtained for the chemotactic response function and the drift velocity in the limit of weak gradients and under the assumption of completely random tumbles. The theoretical predictions compare favorably with experimental results, especially at high concentrations. We further show that the signal transduction network weakens the dependence of the drift on concentration, thus enhancing the range of sensitivity.

Complex counterpart of Chern-Simons-Witten theory and holomorphic linking

In this paper we are begining to explore the complex counterpart of the Chern-Simon-Witten theory. We define the complex analogue of the Gauss linking number for complex curves embedded in a Calabi-Yau 3-fold using the formal path integral that leads to a rigorous mathematical expression. We give an analytic and geometric interpretation of our holomorphic linking following the parallel with the real case. We show in particular that the Green kernel that appears in the explicit integral for the Gauss linking number is replaced by the Bochner-Martinelli kernel. We also find canonical expressions of the holomorphic linking using the Grothendieck-Serre duality in local cohomology, the latter admits a generalization for an arbitrary field.

Theoretical model of interactions between ligand-binding sites in a dimeric protein and its application for the analysis of thiamine diphosphate binding to yeast transketolase

The binding of thiamin diphosphate (ThDP) to yeast dimeric apotransketolase (apoTK) is accompanied by the appearance of a band in the absorption spectrum with maximum at 320 nm. The saturation function has been analyzed using a scheme that involves binding of ThDP to each subunit followed by the conformational transition of this subunit. It is assumed that the binding of ThDP to one subunit may affect the conformational transition of the other subunit. Rigorous mathematical expressions describing the dependence of the optical absorption on the total concentration of ThDP are first developed. Equilibrium constants and corresponding rate constants for the binding of ThDP to apoTK have been estimated. The negative cooperativity in the ThDP binding has been characterized by the function reflecting the dependence of the conformational change on the saturation of apoTK by ThDP.

Towards a paradigm for fuzzy logic control

The purpose of this paper is to lay a foundation for fuzzy logic control (FLC) design, constructing a class of fuzzy logic controllers that is suitable for a broad range of controls applications. Some scalar definitions in FLC are extended to the n-dimensional case, including vector fuzzy number and membership vector. A rigorous mathematical expression is given for the function g( x) (called the ‘reasoning surface’) manufactured by a fuzzy associative memory (FAM). For the existence and uniqueness of solutions in any closed-loop system with a fuzzy logic controller, it is shown that the FAM function g( x) must be Lipschitz: this places restrictions on the allowed rules in the FLC. It is argued that, for effective control design, the membership functions should not be uniformly spaced, the triangular membership functions generally suffice, and product inferencing should be used. Under these conditions, the FLC manufactures a piecewise-multilinear Lipschitz FAM function g( x) that is a generalized proportional-plus-derivative-plus-offset controller. The scalar and two-dimensional cases are studied in detail.

Deficit Irrigation. I: Analytical Framework

The economics of deficit irrigation are examined. The concepts developed in the heuristic discussion are developed into a set of rigorous mathematical expressions for determination of optimum water use under deficit irrigation. These expressions also can be used to estimate the range of water use within which deficit irrigation would be more profitable than full irrigation. The expressions are completely general in the sense that they can be used with any crop production function and cost function that the analyst chooses. The general expressions are used to derive a set of specific expressions for a particular case study involving a quadratic production function and linear cost function.

The use of differential equilibrium functions for interpretation of metal binding in complex ligand systems: Its relation to site occupation and site affinity distributions

This paper compares the capabilities of site affinity distribution functions (SADF) (also called affinity spectra) and differential equilibrium functions (DEF) for interpreting metal complexation by natural heterogeneous complexants using simulated data. The various steps and factors that influence the transformation of an experimental titration curve into SADF or DEF functions are discussed in detail and both practical and theoretical limitations are emphasized. Although an SADF enables one, in principle, to extract concentration and equilibrium constants of actual sites, the concentration and K values given by experimentally determined SADF always correspond to average values resulting from the influence of several sites at a given point in a titration. The averaging process depends on experimental errors and on the characteristics of the analytical method used (in particular the “width” of the analytical window). Consequently the concentrations and intensity parameters obtained by a SADF curve cannot be related, by rigorous mathematical expressions, to the individual concentration and equilibrium constants of actual sites. The Differential Equilibrium Function, on the other hand, can be defined by rigorous mathematical expressions without reference to experimental errors. Its exact meaning is discussed in detail with regard to the concentrations and complexation constants of the actual sites present. The effects of the relative concentrations of the various site types, the difference in their log K values, and the distribution shape of continuously distributed sites are studied in detail. The particular case of a linear logarithmic distribution (leading to the Freundlich isotherm followed by many natural complexing agents) and Sips distribution are also considered. The DEF for such continuous distributions is shown to be a promising means of approximating the underlying cumulative site distribution.

Predicted steady-state cell size distributions for various growth models

The question of how an individual bacterial cell grows during its life cycle remains controversial. In 1962 Collins and Richmond derived a very general expression relating the size distributions of newborn, dividing and extant cells in steady-state growth and their growth rate; it represents the most powerful framework currently available for the analysis of bacterial growth kinetics. The Collins-Richmond equation is in effect a statement of the conservation of cell numbers for populations in steady-state exponential growth. It has usually been used to calculate the growth rate from a measured cell size distribution under various assumptions regarding the dividing and newborn cell distributions, but can also be applied in reverse--to compute the theoretical cell size distribution from a specified growth law. This has the advantage that it is not limited to models in which growth rate is a deterministic function of cell size, such as in simple exponential or linear growth, but permits evaluation of far more sophisticated hypotheses. Here we employed this reverse approach to obtain theoretical cell size distributions for two exponential and six linear growth models. The former differ as to whether there exists in each cell a minimal size that does not contribute to growth, the latter as to when the presumptive doubling of the growth rate takes place: in the linear age models, it is taken to occur at a particular cell age, at a fixed time prior to division, or at division itself; in the linear size models, the growth rate is considered to double with a constant probability from cell birth, with a constant probability but only after the cell has reached a minimal size, or after the minimal size has been attained but with a probability that increases linearly with cell size. Each model contains a small number of adjustable parameters but no assumptions other than that all cells obey the same growth law. In the present article, the various growth laws are described and rigorous mathematical expressions developed to predict the size distribution of extant cells in steady-state exponential growth; in the following paper, these predictions are tested against high-quality experimental data.