Mathematical Considerations Research Articles

Arithmetization and Rigor as Beliefs in the Development of Mathematics

With the arrival of the nineteenth century, a process of change guided the treatment of three basic elements in the development of mathematics: rigour, the arithmetization and the clarification of the concept of function, categorised as the most important tool in the development of the mathematical analysis. In this paper we will show how several prominent mathematicians contributed greatly to the development of these basic elements that allowed the solid underpinning of mathematics and the consideration of mathematics as an axiomatic way of thinking in which anyone can deduce valid conclusions from certain types of premises. This nineteenth century stage shares, possibly with the Heroic Age of Ancient Greece, the most revolutionary period in all history of mathematics.

Some Mineral Exploration Problems that can be Solved by Mathematics and Geostatistics

Databases usually consist of information drawn from drill hole and similar data. These data very often present some problems which can be solved by using mathematical and geostatistical methods. The present paper deals with the problems of estimating the proportions of lithological types within selective mining units, the problem ofan addition of new drill-holes in an explored deposit and the results in the estimation of its quality or quantity and the problem of safe sampling of gold deposits (sentence is too long). For the first issue, the paper gives an overview of how indicator kriging can be used to make estimates of the proportions of the various lithology within blocks of ground, given the lithology's occurring in the samples. The paper deals with and answers the questions that arise when evaluating an ore deposit after the completion of a drilling or sampling campaign. With the aid of geostatistics one can justify or reject a supplementary sampling program at the same deposit, with all the consequences of such a decision. One of the problems of the sampling procedures during a mineral exploration campaign is the size and the weight of the samples to be taken. The sample proportions matter especially in case ofa gold deposit, due to the rarity and the high value of the mineral. Using the modifiedGy's sampling formula, this paper presents the mathematical considerations and transformations to eventually produce a mathematical formula that enables the exploration personnel to safely sample a deposit of valuable ore.

Mathematical consideration of the pipe model theory in woody plant species

Key message Using the pipe model theory, a quantitative relationship between leaves and non-green plant organs has been defined that depends on tree size.

Inverse shape design via a new physical-based iterative solution strategy

Inverse shape design in the context of fluid flow problems is commonly referred to the determination of the boundary shape corresponding to a given target surface pressure. Designers naturally turn to this class of problems whenever there are concerns regarding pressure-related phenomena such as cavitation, separation, shock waves, surface loading, etc. Numerical solution is often unavoidable and, therefore, three computational tools, i.e. a grid generator, a flow field solver and a shape updater, are required in an iterative solution procedure. In all existing iterative inverse design methods, the shape updater comes from separate mathematical or physical considerations that are not derived solely from the equations governing the flow field. In this paper, the currently used strategies to solve inverse shape design problems are categorized and reviewed, and then a truly physical-based iterative inverse shape design method is introduced in which the governing equations are not only used in the flow solver, but are also employed to update the shape. To explore the features of the proposed method, it is used to solve a number of shape design problems. The previously developed direct shape design method and a typical iterative solution approach are also explained and used for the solution of the same problems for the purpose of comparison. Computational results reveal that the proposed algorithm has a two to three times faster convergence rate as compared to the iterative algorithm. The convergence rate of the proposed algorithm usually stalls after three or two orders of magnitude reduction of the residual. For the test cases in this study, the direct design method is the fastest and most accurate method as compared to other algorithms. Both in-house developed computer codes and commercial software are used as a field solver to show the general applicability of the proposed method in its current state of development.

A hybrid method for inversion of 3D DC resistivity logging measurements.

This paper focuses on the application of hp hierarchic genetic strategy (hp–HGS) for solution of a challenging problem, the inversion of 3D direct current (DC) resistivity logging measurements. The problem under consideration has been formulated as the global optimization one, for which the objective function (misfit between computed and reference data) exhibits multiple minima. In this paper, we consider the extension of the hp–HGS strategy, namely we couple the hp–HGS algorithm with a gradient based optimization method for a local search. Forward simulations are performed with a self-adaptive hp finite element method, hp–FEM. The computational cost of misfit evaluation by hp–FEM depends strongly on the assumed accuracy. This accuracy is adapted to the tree of populations generated by the hp–HGS algorithm, which makes the global phase significantly cheaper. Moreover, tree structure of demes as well as branch reduction and conditional sprouting mechanism reduces the number of expensive local searches up to the number of minima to be recognized. The common (direct and inverse) accuracy control, crucial for the hp–HGS efficiency, has been motivated by precise mathematical considerations. Numerical results demonstrate the suitability of the proposed method for the inversion of 3D DC resistivity logging measurements.

Physics on the boundary between classical and quantum mechanics

Nature's laws in the domain where relativistic effects, gravitational effects and quantum effects are all comparatively strong are far from understood. This domain is called the Planck scale. Conceivably, a theory can be constructed where the quantum nature of phenomena at such scales can be attributed to something fundamentally simpler. However, arguments that quantum mechanics cannot be explained in terms of any classical theory using only classical logic seem to be based on sound mathematical considerations: there can't be physical laws that require "conspiracy". It may therefore be surprising that there are several explicit quantum systems where these considerations apparently do not apply. In the lecture we will show several such counterexamples. These are quantum models that do have a classical origin. The most curious of these models is superstring theory. This theory is often portrayed as to underly the quantum field theory of the subatomic particles, including the "Standard Model". So now the question is asked: how can this model feature "conspiracy", and how bad is that? Is there conspiracy in the vacuum fluctuations?

The statistics of local motion signals in naturalistic movies.

Extraction of motion from visual input plays an important role in many visual tasks, such as separation of figure from ground and navigation through space. Several kinds of local motion signals have been distinguished based on mathematical and computational considerations (e.g., motion based on spatiotemporal correlation of luminance, and motion based on spatiotemporal correlation of flicker), but little is known about the prevalence of these different kinds of signals in the real world. To address this question, we first note that different kinds of local motion signals (e.g., Fourier, non-Fourier, and glider) are characterized by second- and higher-order correlations in slanted spatiotemporal regions. The prevalence of local motion signals in natural scenes can thus be estimated by measuring the extent to which each of these correlations are present in space-time patches and whether they are coherent across spatiotemporal scales. We apply this technique to several popular movies. The results show that all three kinds of local motion signals are present in natural movies. While the balance of the different kinds of motion signals varies from segment to segment during the course of each movie, the overall pattern of prevalence of the different kinds of motion and their subtypes, and the correlations between them, is strikingly similar across movies (but is absent from white noise movies). In sum, naturalistic movies contain a diversity of local motion signals that occur with a consistent prevalence and pattern of covariation, indicating a substantial regularity of their high-order spatiotemporal image statistics.

High performing extraction procedure for the one-diode model of a photovoltaic panel from experimental I–V curves by using reduced forms

In the last years, several numerical techniques were proposed in literature for the identification of the one diode model of photovoltaic (PV) panels from manufacturers’ data sheets or experimental data. In this paper we present a fast and accurate method for obtaining the five parameters of the one diode model by starting from the experimental I–V curve of the PV panel. In particular we exploit the adoption of the reduced forms of the original five-parameter model. These reduced forms take advantage of important mathematical considerations through which the five parameters of the model are splitted in independent and dependent unknowns, in order to reduce the dimensions of the search space. Thus, the fitting of experimental data can be performed by using a restricted number of unknowns (namely two) and this provides strong benefits in terms of convergence, computational costs and execution times. In this way, the approach allows to characterize a PV panel from its measured I–V curve with an accuracy never obtained before in literature, employing few steps and execution times of few milliseconds on a simple notebook computer. Suitable validations on two important case studies are presented for comparing our procedure with the most recent and effective techniques proposed in literature.

A Novel Toxicokinetic Modeling of Cypermethrin and Permethrin and Their Metabolites in Humans for Dose Reconstruction from Biomarker Data

To assess exposure to pyrethroids in the general population, one of most widely used method nowadays consists of measuring urinary metabolites. Unfortunately, interpretation of data is limited by the unspecified relation between dose and levels in biological tissues and excreta. The objective of this study was to develop a common multi-compartment toxicokinetic model to predict the time courses of two mainly used pyrethroid pesticides, permethrin and cypermethrin, and their metabolites (cis-DCCA, trans-DCCA and 3-PBA) in the human body and in accessible biological matrices following different exposure scenarios. Toxicokinetics was described mathematically by systems of differential equations to yield the time courses of these pyrethroids and their metabolites in the different compartments. Unknown transfer rate values between compartments were determined from best fits to available human data on the urinary excretion time courses of metabolites following an oral and dermal exposure to cypermethrin in volunteers. Since values for these coefficients have not yet been determined, a mathematical routine was programmed in MathCad to establish the possible range of values on the basis of physiological and mathematical considerations. The best combination of parameter values was then selected using a statistic measure (reliability factor) along with a statistically acceptable range of values for each parameter. With this approach, simulations provided a close approximation to published time course data. This model allows to predict urinary time courses of trans-DCCA, cis-DCCA and 3-PBA, whatever the exposure route. It can also serve to reconstruct absorbed doses of permethrin or cypermethrin in the population using measured biomarker data.

PV cell modeling의 수학적 고찰

PV cell modeling의 수학적 고찰

Elastic fields due to dislocation arrays in anisotropic bimaterials

Based on the single-dislocation Green’s function, analytical solutions of the elastic fields due to dislocation arrays in an anisotropic bimaterial system are derived by virtue of the Cottrell summation formula. The singularity in the Peach–Koehler (P–K) force is removed by both rigorous mathematical approach and physical energy consideration. Numerical results for dislocation arrays in the Cu/Nb bimaterial with Kurdjumov–Sachs (K–S) orientation show that: (1) the traction continuity and periodic condition are both satisfied; (2) the maximum magnitude of the traction at the interface due to a mixed dislocation array is smaller than that due to a single mixed dislocation. In other words, the traction at the interface could be suppressed by the corresponding array with a relatively high density (L<10nm); however, the shear stress on the glide plane increases with increasing dislocation density; (3) the Cu/Nb interface attracts the mixed dislocation array in copper and repels the screw one there. This implies that the P–K force depends not only on the material properties, but also on the crystal orientation and the type of Burgers vector, among others.

Rational Design of a Face‐Centred Square‐Cuboid Coordination Cage

The simple synthetic conversion of a 90°-angled bis-pyridyl ligand into a tripodal tris-pyridyl ligand leads to the formal transformation of a cubic (a=b=c) into a square-cuboid (a=b≠c) coordination cage. Mathematical considerations associated with the ligand design, together with X-ray structure results, NMR spectroscopic and mass spectrometric characterization and molecular modeling of both coordination cages are presented and discussed.

Physical properties and analytical models of band-to-band tunneling in low-bandgap semiconductors

Low-bandgap semiconductors, such as InAs and InSb, are widely considered to be ideal for use in tunnel field-effect transistors to ensure sufficient on-current boosting at low voltages. This work elucidates the physical and mathematical considerations of applying conventional band-to-band tunneling models in low-bandgap semiconductors, and presents a new analytical alternative for practical use. The high-bandgap tunneling generates most at maximum field region with shortest tunnel path, whereas the low-bandgap generations occur dispersedly because of narrow tunnel barrier. The local electrical field associated with tunneling-electron numbers dominates in low-bandgap materials. This work proposes decoupled electric-field terms in the pre-exponential factor and exponential function of generation-rate expressions. Without fitting, the analytical results and approximated forms exhibit great agreements with the sophisticated forms both in high- and low-bandgap semiconductors. Neither nonlocal nor local field is appropriate to be used in numerical simulations for predicting the tunneling generations in a variety of low- and high-bandgap semiconductors.

Very Fast and Accurate Procedure for the Characterization of Photovoltaic Panels from Datasheet Information

In recent years several numerical methods have been proposed to identify the five-parameter model of photovoltaic panels from manufacturer datasheets also by introducing simplification or approximation techniques. In this paper we present a fast and accurate procedure for obtaining the parameters of the five-parameter model by starting from its reduced form. The procedure allows characterizing, in few seconds, thousands of photovoltaic panels present on the standard databases. It introduces and takes advantage of further important mathematical considerations without any model simplifications or data approximations. In particular the five parameters are divided in two groups, independent and dependent parameters, in order to reduce the dimensions of the search space. The partitioning of the parameters provides a strong advantage in terms of convergence, computational costs, and execution time of the present approach. Validations on thousands of photovoltaic panels are presented that show how it is possible to make easy and efficient the extraction process of the five parameters, without taking care of choosing a specific solver algorithm but simply by using any deterministic optimization/minimization technique.

Mathematical consideration of depopulated area of gas station

縮小時代にある現代日本において，インフラをどう維持して行くのかが喫緊の課題となってきている．給油所もまた，その供給財の性質から代替不可な公共インフラの側面を持つといえる．本論文ではまず，既存研究を用いることで自治体における適正な給油所供給量を明らかにし，自治体ごとの給油所の多寡についての検証を行った．またモデル式を変形することで，公共インフラに対する投資効果指標の作成も行った．次に，「給油所過疎地域」検出のための指標を住民の最寄り給油所までの距離と定義し，茨城県北地域を対象として，自治体単位では把握できない過疎地域の存在を具体的に示した．最後に，そうした給油所過疎地域抑制のための具体的施策を示し，移動距離の変化量という指標によってその効果の評価を行った．

Some Mathematical Considerations on the Natural Protection of the Danube Delta Coastal Zone in Romania

Today, more and more scientists agree with the hypothesis concerning the raising of the planetary ocean level due to global climate warming and consequently due to glaciers melting. Therefore, as a consequence of climate change phenomenon, the Black Sea level is raising as well, which led to the loss of significant areas of land due to marine erosion, but also the production of some cracks and penetration of salt water into the Razelm - Sinoe freshwater ecosystem in Romania.This paper uses a mathematical tool, which addresses the maintenance of the ecological balance of freshwater ecosystem, the transit of freshwater through the Razelm - Sinoe hydrotechnical complex, and from here into the Black Sea, required to ensure the hydraulic gradient from freshwater to saltwater. This would help maintaining vegetation as natural means for protection against the sea erosion, as well as to attenuate the coastal energy of waves, so as to ensure the shielding role together with the other coastal engineering construction. Besides, the strong reed vegetation represents a natural element of protection of levees already made or those in process of development, and also a retention element of fine particles brought by waves at times when sea level is high. The mathematical algorithm used in this paper take into account the principle of fluid flows in porous media. The vegetation of littoral from Razelm - Sinoe ecosystem was considered a porous media. The research results show the capacity of the vegetation to attenuate the waves’ energy in the study area. The mathematical model was especially conceived for a unique littoral area, which separates the Black Sea from Razelm - Sinoe ecosystem. The solution of protection exploits the natural environment; it is not expensive and falls into the natural-ecological aspect of the coastal zone.

Improved Numerical Determination of the Effective Stress within the Framework of Local Approach Cleavage Models

For the determination of the Weibull stress or analogous effective stress measures in the assessment of the cleavage failure probability based on finite element results, robust and numerically stable procedures are necessary. For the quantitative assessment of the numerical errors in the approximate integration procedures for the determination of the Weibull stress or corresponding stress measures, an appropriate error indicator in analogy to adaptive finite element meshing methods was derived. The mathematical considerations reveal that an evaluation of the cleavage probability by means on the classical Beremin (1983) model using the standard one-point Gauss integration scheme may result in distinct numerical errors. Therefore, an enhanced numerical integration formula was derived and implemented. This formula facilitates a numerical evaluation of the integrals with improved accuracy.

On rising Sun von mises and rising Sun wrapped cauchy circular models

Abstract By applying wrapping (Dattatreya Rao et al (2007)), offsetting (Girija et al (2013 and Radhika et al (2013)) and stereographic projection (Phani et al (2012)) have constructed circular models. Further the circular models based on the Rising Sun function are motivated by purely mathematical considerations as a smoothing function and possible application. This work takes a further step in this direction using several mathematical tools such as Real Analysis along with MATLAB and are applied to enlarge the horizon of Mathematical Statistics. Here an attempt is made to construct new circular models using the Rising Sun function and also the circular rising sun lemma is proposed

Revision of the Dissolution Kinetics of Aggregated Settling Particles

The dissolution kinetics theory of settling particles, which was developed to express the elemental concentration variation as a function of diatom productivity, is revised to allow further consideration on the presence of voids. In this revision, the number of diatom frustules which form an aggregate is introduced as a basic variable to secure the rigidity of mathematical consideration, (unlike the previous paper, where productivity is employed as one of the basic variations - Akagi et al., 2011). No exchange of elements is assumed between seawater surrounding the aggregates and seawater in the voids of the aggregates. It is shown that the presence of voids reduces the settling velocity, but does not affect the dissolution/alteration behavior of settling particles as long as the particles are collected at a certain depth.

Some Mathematical Considerations on Solid State Physics in the Framework of the Phase Space Formulation of Quantum Mechanics

We propose, for a finite crystal, that the entire system be expressed in terms of the phase space, including momentum, configuration position and spin. This is to be done both classically and quantum mechanically. We illustrate this with a silicon crystal. Then, quantum mechanically measuring with a wire containing electrons, we obtain a theoretically good approximation for an electron in one semiconductor. As well, we outline what has to be done for any crystal.