Integration Representation Method Research Articles

Solution of the Interpretation Tomography Problem in Geophysics under the Linear Integral Representation Method and Theories of Discrete Gravity and the Magnetic Potential

Solution of the Interpretation Tomography Problem in Geophysics under the Linear Integral Representation Method and Theories of Discrete Gravity and the Magnetic Potential

Integral representations of conformal cylindrical microstrip radiators radiation fields

The article is devoted to the development of a method of integral representations of electric and magnetic field strengths of conformal cylindrical microstrip emitters. The proposed integral representations, when the corresponding boundary conditions on the surface of the emitters are substituted in them, turn into a singular (hypersingular) integral equation, the numerical solution of which is a mathematically correct problem. The proposed method makes it possible to conduct electrodynamic analysis of conformal cylindrical microstrip emitting structures very effectively, to determine such characteristics as the distribution of the radiation field in space, the radiation pattern, the directional coefficient, the current distribution across the emitters, the input impedance, the polarization characteristic, etc.

A model for green-resilient supplier selection: fuzzy best-worst multi-criteria decision-making method and its applications.

Supplier selection is regarded as the primary goal of supply chain management (SCM) because it affects its performance, productivity, pleasure, flexibility, and system speed in lockdown. A new method is proposed based on a multi-stage fuzzy sustainable supplier index (FSSI). Experts can use the triple bottom line (TBL) criteria to select the best supplier. In addition, the worst method is proposed based on trapezoidal membership and fuzzy membership functions, which can cover uncertainties and ambiguous environments. Because it collects the related criteria and sub-criteria and uses a direct fuzzy methodology, this research has impacted the SCM literature because it helps solve previous expert methods' computational difficulties. In addition, an ordered mean integration representation method has been implemented to prioritize the selection of the best supplier (SS) based on the sustainability performance of the best supplier, which improves the selection accuracy compared to the previous ranking method. This study can be used as a benchmark to determine which supplier is the best in sustainability. To provide the superiority and broad applicability of the proposed model, a practical case study was completed. On the other hand, the COVID-19 pandemic harms productivity, company performance, and selecting the best suppliers based on sustainability performance. The lockdown situation caused by the COVID-19 pandemic hurts company performance and management.

MODEL LOGISTIK FUZZY DENGAN ADANYA PEMANENAN PROPORSIONAL

The logistic growth model with proportional harvesting is a population growth model that takes into account harvesting factors. In real life, not all conditions can be known with certainty, such as different growth rates in each population and harvest rates depending on the needs of the harvester. To overcome these conditions, there is a concept that accommodates the problem of uncertainty, namely the fuzzy concept. This concept can be induced into a logistic model with proportional harvesting which assumes the intrinsic growth rate and the harvest rate is expressed by fuzzy numbers. The purpose of this research is to form a logistic model with fuzzy proportional harvesting, analyze the stability of the model, and form a numerical simulation. This study uses the alpha-cut method to generalize the intrinsic growth rate and harvest rate from crisp numbers to fuzzy numbers, then the Graded Mean Integration Representation (GMIR) method to defuzzify the model, and the linearization method to analyze the stability of the model. The results of this study obtained a logistic model with proportional harvesting. Then the model was developed into a logistic model with fuzzy proportional harvesting by assuming the intrinsic growth rate and the harvest rate expressed by fuzzy numbers. From the model obtained 2 equilibrium points, namely the first equilibrium point is unstable and the second equilibrium point is asymptotically stable under certain conditions. Model simulation is given to show illustration of stability analysis. From the simulation, it can also be shown that the higher the graded mean value, the lower the population.

Integral representation method based efficient rule optimizing framework for anti-money laundering

PurposeThis paper aims to introduce a framework for optimizing rule-based anti-money laundering systems with a clear economic interpretation, and the authors introduce the integral representation method.Design/methodology/approachBy using a microeconomic model, the authors reformulate the threshold optimization problem as a decision problem to gain insights from economics regarding the main properties of the optimum. The authors used algorithmic considerations to find an efficient implementation by using a kind of weak mode estimate of the distribution and the authors extend this approach to classes of alerts or cases.FindingsThe method provides a new and efficient alternative for the sampling method or the multidimensional optimization technique described in the literature to decrease the bias emanating from multiple alerts by smoothing the number of alerts across classes in the optimum and decrease the overlapping between scenarios at the case level. Using the method for real bank data, the authors were able to decrease the number of false positives cases by about 18% while retaining almost 98% of the true-positive cases.Research limitations/implicationsThe model assumes that alerts from different scenarios are indifferent to the bank. To include scenario-specific preferences or constraints demands further research.Originality/valueThe new framework presented in the paper is a flexible extension of the usual above-the-line method, which makes it possible to include bank preferences and use the parallelization capabilities of modern processors.

Various Series Related to the Polylogarithmic Function

We derive some series related to the polylogarithmic function, and we also give a new proof to the existing series. Our approach is based on using the summation and integral representation methods. We obtain various interesting series as a consequence.

On the solution of a nonstationary problem of heat and mass transfer in a multilayer medium by the method of integral representations

On the solution of a nonstationary problem of heat and mass transfer in a multilayer medium by the method of integral representations

Fuzzy Inventory Model with Variable Production and Selling Price Dependent Demand under Inflation for Deteriorating Items

The main purpose of this paper is to develop an inventory model under fuzzy approach by considering the effect of inflation and time value of money, to determine the optimal time period for inventory cycle and minimum total average costs. The model is integrated production inventory model developed where; the Demand has a direct linear impact on production rate. The model can be divided into four stages. In the first two stages with original production rate and subsequent change in production rate, inventory level rises. Third stage is time after the accumulation of inventory and before the deterioration starts, where demand which selling price dependent is depreciating the inventory level, while in the fourth stage deterioration occurs, which is considered to follow two parameter Weibull distribution. The back-order is not considered. Hexagonal fuzzy numbers are used to derive optimum solution and defuzzification by graded mean integration representation method. A numerical example is given to demonstrate the applicability of the purposed model and sensitivity analysis is carried out to reveal the impact of change in parameter values.

A New Approach to Analytical Modeling of Mars’s Magnetic Field

ABSTRACT A new three-staged technique based on various versions of the linear integral representation method (S-, F- and R-approximations) is applied to solve the problem of analytical modelling of the magnetic field on the surface of Mars. At the first step, we build an analytical approximation from the given data set (in local or regional cases). In the second step, we analyze the previous stage results and solve an ordinary differential equation system to find the integral curve of the gravity or magnetic field. The points of this curve are included in an extended data set, and we repeat the procedure described above for the first stage. The final approximation of the anomalous field elements or the topographic data allows us to find various linear transformations of the field under investigation, e.g. higher derivatives of the potential or Fourier-spectra of the field elements. Analytical downward continuations of the magnetic field of Mars at various distances, including the planet's surface is presented.

A Branch-and-Bound-based solution method for solving vehicle routing problem with fuzzy stochastic demands

In this paper, a capacitated vehicle routing problem (CVRP) with fuzzy stochastic demands has been presented. Discrete fuzzy random variables have been used to represent the demands of the customers. The objective of CVRP with fuzzy stochastic demands is to obtain a set of routes that originates as well as terminates at the source node and while traversing the route, the demands of all the customers present in the network are satisfied. The task here is to carry out all these operations with minimum cost. CVRP in imprecise and random environment has been considered here, and an a priori route construction technique has been adopted for which Branch and Bound algorithm has been used. The recourse policy used in this work is reactive, i.e. recourse to depot is done only upon the occurrence of the failure. The delivery policy considered here is unsplit delivery. Demands of the customers are the only source of impreciseness and randomness in the problem under construction. Parametric graded mean integration representation (PGMIR) method has been used for the comparison purposes, whenever required. A numerical example with four customers has been solved to present the proposed methodology.

Analyzing The Concept Of Graded K-Preference Integration Representation Method

Generally, the fuzzy set concept could be used to deal with the problems with the qualities of ambiguity as well as vagueness. In the decision making process, the reference comparisons for criteria & options tend to be more appropriate to make use of the linguistic variables rather than crisp values in some instances. Meanwhile, the GMIR technique is utilized for the constrained trouble construction to derive the weights of options & criteria, which accomplishes the extension of fuzzy environment. Here in this paper we will study about some basic terms related to K-preference Graded Integration method. We will discuss the fuzzy inventory models under decision maker’s preference (k-preference), and find the optimal solutions of these models, the optimal crisp order quantity or the optimal fuzzy order quantity.

On functions with the boundary Morera property in domains with piecewise-smooth boundary

The problem of holomorphic extension of functions defined on the boundary of a domain into this domain is actual in multidimensional complex analysis. It has a long history, starting with the proceedings of Poincaré and Hartogs. This paper considers continuous functions defined on the boundary of a bounded domain $ D $ in $ \mathbb C ^ n $, $ n> 1 $, with piecewise-smooth boundary, and having the generalized boundary Morera property along the family of complex lines that intersect the boundary of a domain. Morera property is that the integral of a given function is equal to zero over the intersection of the boundary of the domain with the complex line. It is shown that such functions extend holomorphically to the domain $ D $. For functions of one complex variable, the Morera property obviously does not imply a holomorphic extension. Therefore, this problem should be considered only in the multidimensional case $ (n> 1) $. The main method for studying such functions is the method of multidimensional integral representations, in particular, the Bochner-Martinelli integral representation.

Fermion and photon gap-equations in Minkowski space within the Nakanishi integral representation method

The approach based on the Nakanishi integral representation of n-leg transition amplitudes is extended to the treatment of the self-energies of a fermion and an (IR-regulated) vector boson, in order to pave the way for constructing a comprehensive application of the technique to both gap- and Bethe-Salpeter equations, in Minkowski space. The achieved result, namely a 6-channel coupled system of integral equations, eventually allows one to determine the three Källén–Lehman weights for fully dressing the propagators of fermion and photon. A first consistency check is also provided. The presented formal elaboration points to embed the characteristics of the non-perturbative regime at a more fundamental level. It yields a viable tool in Minkowski space for the phenomenological investigation of strongly interacting theories, within a QFT framework where the dynamical ingredients are made transparent and under control.

Modified method S-, and R-approximations in solving the problems of Mars’s Morphology

The connection of different modifications of the method of linear integral representation is studied. Solutions of the related inverse problems based upon a ‘hybrid version of two approximations’ of the topography and geopotential fields enable more refined tuning of the method in solving the inverse problems of geophysics and geomorphology and more complete allowance for the a priori information about the surface elevation data and elements of anomalous fields. The technique for finding a stable approximate solution for the inverse problem of determining the mass distributions equivalent by the external gravitational (or any other potential) field is presented. The method is applied to study anomalous densities beneath one of the highest elevation Elysium Mons and deep Hellas basin on Mars. The results of the mathematical experiment are discussed.

A multi-stage multi-criteria hierarchical decision-making approach for sustainable supplier selection

Sustainable supplier selection is known as a crucial objective in supply chains due to its impact on profitability, adorability, flexibility, and agility of the system. This study proposes a new multi-stage hierarchical fuzzy index-based approach with which decision-makers are empowered to select the most sustainable supplier based on sustainability triple bottom line criteria. Besides, a new fuzzy extension for the best-worst method is proposed considering trapezoidal fuzzy membership functions that can cover uncertainty under imprecise environments. This study makes a contribution to the literature of sustainable supply chains in the sense that it facilitates the computational complexity of previous decision-making approaches by collecting the most relevant criteria and a straightforward fuzzy process. Besides, the graded mean integration representation method has been adopted for prioritizing the supplier based on their performance of sustainable development, which enhances the accuracy of selection compared with previous ranking methods. The proposed study can be utilized as a benchmark for sustainability evaluations between suppliers. A real-world case study is resolved to illustrate the superiority and broad application of the proposed model.

Mathematical Model of a Two-Element Microstrip Radiating Structure with a Chiral Metamaterial Substrate

Abstract—A mathematical model of a two-element array is presented, the substrate of which is made of a chiral metamaterial. The method of singular integral representation of a field is used to obtain a system of integral equations for the unknown distribution functions of the surface current densities of the array emitters. The results of calculating the current functions are given, as well as the dependence of isolation between the emitters on the chirality parameter.

QED calculation of the dipole polarizability of helium atom

The QED contribution to the dipole polarizability of the $^{4}\mathrm{He}$ atom was computed, including the effect of finite nuclear mass. The computationally most challenging contribution of the second electric-field derivative of the Bethe logarithm was obtained using two different methods: the integral representation method of Schwartz and the sum-over-states approach of Goldman and Drake. The results of both calculations are consistent, although the former method turned out to be much more accurate. The obtained value of the electric-field derivative of the Bethe logarithm, equal to 0.048 557 2(14) in atomic units, confirms the small magnitude of this quantity found in the only previous calculation [G. \L{}ach, B. Jeziorski, and K. Szalewicz, Phys. Rev. Lett. 92, 233001 (2004)], but differs from it by about 5%. The origin of this difference is explained. The total QED correction of the order of ${\ensuremath{\alpha}}^{3}$ in the fine-structure constant $\ensuremath{\alpha}$ amounts to $30.6671(1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, including the $0.1822\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ contribution from the electric-field derivative of the Bethe logarithm and the $0.011\phantom{\rule{0.16em}{0ex}}12(1)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ correction for the finite nuclear mass, with all values in atomic units. The resulting theoretical value of the molar polarizability of helium-4 is $0.517\phantom{\rule{0.16em}{0ex}}254\phantom{\rule{0.16em}{0ex}}08(5)\phantom{\rule{4pt}{0ex}}{\mathrm{cm}}^{3}/\text{mol}$ with the error estimate dominated by the uncertainty of the QED corrections of order ${\ensuremath{\alpha}}^{4}$ and higher. Our value is in agreement with but an order of magnitude more accurate than the result $0.517\phantom{\rule{0.16em}{0ex}}254\phantom{\rule{0.16em}{0ex}}4(10)\phantom{\rule{0.28em}{0ex}}{\mathrm{cm}}^{3}/\text{mol}$ of the most recent experimental determination [C. Gaiser and B. Fellmuth, Phys. Rev. Lett. 120, 123203 (2018)].

The Valuation of Executive Stock Option Using the Integral Representation Method

This study derives a closed-form approximate solution for Bermudan reset executive stock options with cash dividends. The complicated payoff characteristic of this exotic option is caused by its particular stylistic features, including the reset mechanism, cash dividends, early exercises, and increasing volatilities. Using the integral representation method, a closed-form approximate solution was successfully derived. Moreover, this study discusses how specific critical factors affect executive stock option values and arrives at some interesting results by way of a numerical calibration.

ИНТЕГРАЛЬНОЕ ПРЕДСТАВЛЕНИЕ РЕШЕНИЙ ОДНОГО ОБЫКНОВЕННОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ И УРАВНЕНИЯ ЛЕВНЕРА – КУФАРЕВА

The article presents a method of integral representation of solutions of ordinary differential equations and partial differential equations with a polynomial right-hand side part, which is an alternative to the construction of solutions of differential equations in the form of different series. The method is based on the introduction of additional analytical functions establishing the equation of connection between the introduced functions and the constituent components of the original differential equation. The implementation of the coupling equations contributes to the representation of solutions of the differential equation in the integral form, which allows solving some problems of mathematics and mathematical physics. The first part of the article describes the coupling equation for an ordinary differential equation of the first order with a special polynomial part of a higher order. Here, the integral representation of the solution of a differential equation with a second-order polynomial part is indicated in detail. In the second part of the paper, we consider the integral representation of the solution of a partial differential equation with the polynomial second-order part of the Loewner–Kufarev equation, which is an equation for univalent functions.

Inventory model for deteriorating items with negative exponential demand, probabilistic deterioration and fuzzy lead time under partial back logging

The effect of lead time plays an important role in inventory management. It is also important to study the optimal strategies when the lead time is not precisely known to the decision makers. The aim of this paper is to examine the inventory model for deteriorating items with fuzzy lead time, negative exponential demand, and partially backlogged shortages. This model is unique in its nature due to probabilistic deterioration along with fuzzy lead time. The fuzzy lead time is assumed to be triangular, parabolic, trapezoidal numbers and the graded mean integration representation method is used for the defuzzification purpose. Moreover, three different types of probability distributions, namely uniform, triangular and Beta are used for rate of deterioration to find optimal time and associated total inventory cost. The developed model is validated numerically and values of optimal time and total inventory cost are given in tabular form, corresponding to different probability distribution and fuzzy lead-time. The sensitivity analysis is performed on variation of key parameters to observe its effect on the developed model. Graphical representations are also given in support of derived optimal inventory cost vs. time.