Vertex Method Research Articles

Unified description of collective modes in superconductors and semiconductors with an exciton condensed phase

It is shown that the Bethe-Salpeter approach, the Bardeen, Cooper, and Schrieffer (BCS) based vertex method, and a generalized random-phase approximation (GRPA) to the many-electron problem in the presence of a condensed quantum phase yield the same theoretical excitation spectrum ω(Q) to collective modes. This spectrum reveals a secondary peak in optical absorption in semiconductors that can be understood as signaling the existence of an excitonic Bose-Einstein condensate (BEC). The analysis shows as well that there is an additional, non-trivial linearly-dispersive moving Cooper-pair solution for superconductors in both weak and strong coupling.

A dual-interval vertex analysis method and its application to environmental decision making under uncertainty

In this study, a dual-interval vertex analysis (DIVA) method is developed, through incorporating the vertex method within an interval-parameter programming framework. The developed DIVA method can tackle uncertainties presented as dual intervals that exist in the objective function and the left- and right-hand sides of the modeling constraints. An interactive algorithm and a vertex analysis approach are proposed for solving the DIVA model. Solutions under an associated α-cut level can be generated by solving a series of deterministic submodels. They can help quantify relationships between the objective function value and the membership grade, which is meaningful for supporting in-depth analyses of tradeoffs between environmental and economic objectives as well as those between system optimality and reliability. A management problem in terms of regional air pollution control is studied to illustrate applicability of the proposed approach. The results indicate that useful solutions for planning the air quality management practices have been generated. They can help decision makers to identify desired pollution-abatement strategies with minimized costs and maximized environmental efficiencies.

Instantons and the 5D U(1) gauge theory with extra adjoint

In this paper, we compute the partition function of 5D supersymmetric U(1) gauge theory with extra adjoint matter in general Ω background. It is well known that such partition functions encode very rich topological information. We show in particular that unlike the case with no extra matter, the partition function with extra adjoint at some special values of the parameters directly reproduces the generating function for the Poincare polynomial of the moduli space of instantons. We compare our results with those recently obtained by Iqbal et al (Refined topological vertex, cylindric partitions and the U(1) adjoint theory, arXiv:0803.2260), who used the so-called refined topological vertex method.

Bid-Winning Potential Optimization for Concession Schemes with Imprecise Investment Parameters

This paper portrays the development of a concession-investment optimization model for optimizing the winning potential of a concession-bid from the prospective promoters' viewpoints by taking into account imprecise investment parameters. A financial performance measure has been developed to quantify bid-winning potential. Fuzzy set theory is embedded into a genetic algorithm to optimize bid-winning prospects of concession schemes under uncertainty. The fuzzy vertex method accomplishes propagation of uncertainty in cash flows arising from imprecision in estimation of investment parameters. Genetic algorithm solves the optimization routine. The developed model yields global near-optimal solution of bid-winning potential with contributing values of concessionary items: concession length, base price, and equity level under uncertain investment environment. A numerical example demonstrates model capabilities. The model results would help promoters devise a profitable, yet competitive, financial proposal.

Non-planar ABJM theory and integrability

Using an effective vertex method we explicitly derive the two-loop dilatation generator of ABJM theory in its SU(2) × SU(2) sector, including all non-planar corrections. Subsequently, we apply this generator to a series of finite length operators as well as to two different types of BMN operators. As in = 4 SYM, at the planar level the finite length operators are found to exhibit a degeneracy between certain pairs of operators with opposite parity — a degeneracy which can be attributed to the existence of an extra conserved charge and thus to the integrability of the planar theory. When non-planar corrections are taken into account the degeneracies between parity pairs disappear hinting the absence of higher conserved charges. The analysis of the BMN operators resembles that of = 4 SYM. Additional non-planar terms appear for BMN operators of finite length but once the strict BMN limit is taken these terms disappear.

Analysis of uncertainties in MEMS and their influence on dynamic properties

An application of uncertainty analysis of microelectromechanical resonator is presented. A number of different uncertain parameters have been considered, connected both to geometric characteristics and material property. Sensitivity analysis has been carried out in order to study the influence of each input uncertain parameter on the chosen dynamic characteristics. The propagation of given uncertainties into the variation of studied output parameter has been evaluated by means of the Monte Carlo simulation, the vertex method and genetic algorithms. Finite element model of microresonator has been elaborated. It has taken into account the phenomenon of viscous damping coming from the presence of surrounding air as well as the influence of constant electrostatic field.

REFINED INSTANTON COUNTING AND MACDONALD FUNCTIONS

We argue that the Nekrasov's partition function with two equivariant parameters is reproduced by the method of topological vertex, if we introduce a refined version of the vertex. Our refined topological vertex is expressed in terms the specialization of the Macdonald functions which is related to the equivariant character of the Hilbert scheme of ℂ2. We provide diagrammatic rules for computing the partition function.

A comprehensive model for selecting information system project under fuzzy environment

In general, the development of any information system (IS) project requires large investments of resources, such as human resources, computer software and hardware resources, operational procedures adjustments, and so on. However, IS project selection is difficult because there are lots of factors to be considered, such as business goals, benefits, project risks and limited available resources. In fact, IS project selection takes place in an incomplete, vague and uncertain information environment. The aim of this paper is to present a multiple-criteria decision-making method (MCDM) for selecting an information system project based on the fuzzy measure and the fuzzy integral. In this paper, the subjective opinions of decision makers are described in linguistic terms expressed in trapezoidal fuzzy numbers. After aggregating the fuzzy ratings of all decision makers, the vertex method is applied to transform the aggregated fuzzy rating into a crisp value. And then, a new algorithm is developed to deal with the IS projects selection problems. Finally, a numerical example is given to demonstrate the procedure for the proposed method.

Computing Availability Using Hybrid Approach and Single Value Fuzzy Simulation

This paper discusses availability computation procedures of a mechatronic system under the uncertainties that involve randomness as well as fuzziness. A four state Markov model is considered and 10 cases are modelled that take different nature of failure and repair rates. As in real life, sometimes statistical data is available about failure and repair rates, but in cases where such data is scarce or not available, one have to rely on subjective information or judgment provided by the expert. So this issue where some data is probabilistic and some is of fuzzy nature is stressed and attempted to be resolved in this paper. Various upcoming techniques such as vertex method, hybrid approach (considering probabilistic as well as fuzzy and single value fuzzy simulation are used to arrive at availability value.

Safety assessment of austenitic steel nuclear power plant pipelines against stress corrosion cracking in the presence of hybrid uncertainties

Stress corrosion cracking (SCC) is an important degradation mechanism to be considered for safety assessment of nuclear piping components made of austenitic steels, especially in the heat-affected zones. Damage due to SCC occurs in a susceptible material, in a corrosive environment, in the presence of high temperature and high applied/residual stresses. The operating conditions and the environmental conditions show variations during the lifetime of the power plant. Also, there will be variations in micro-structural properties of the material of piping components. These variations should be taken into account while assessing the safety of the piping component against SCC. This can be accomplished by treating the relevant variables as random or fuzzy depending upon the source and type of uncertainty. In this paper, an attempt has been made to compute the fuzzy failure probabilities of a piping component against SCC with time, using an approach combining the vertex method with the Monte Carlo simulation technique. The initiation and propagation stages of stress corrosion cracks are modelled using a modified PRAISE approach. The degree of sensitisation, material fracture toughness, yield strength, ultimate strength and applied stress are considered as random variables, while operating temperature and oxygen concentration are considered as fuzzy variables. The R6 procedure is used in the computation of the fuzzy failure probabilities. The usefulness of the approach is demonstrated through an example problem.

Characterizing effects of uncertainties in MSW composting process through a coupled fuzzy vertex and factorial-analysis approach

A coupled fuzzy vertex and factorial-analysis approach was developed in this study for systematically characterizing effects of uncertainties in a municipal solid waste composting process. A comprehensive composting process model was also embedded into the system framework and used to address substrate decomposition and biomass growth, as well as the interactions between moisture contents, temperatures, and oxygen concentrations. The applicability of the proposed method was verified through a custom-made pilot-scale composting system. Results from fuzzy simulation indicated that the fuzzy vertex method could effectively communicate implicit knowledge into dynamic simulations and thus provide valuable information for enhancing composting process control under uncertainty. The factorial analysis was effective in quantifying the proportion to which the uncertainty of each single or interactive effect of model parameters contributes to the overall uncertainty of the system outcomes. Thus, sensitive parameters that may lead to errors or unreasonable predictions can be determined. The proposed study system could not only be used in characterizing combined effects of uncertainties for composting processes, but was also applicable to many other environmental modelling systems.

Simulating pharmacokinetic and pharmacodynamic fuzzy-parameterized models: a comparison of numerical methods

Statistical techniques have been traditionally used to deal with parametric variation in pharmacokinetic and pharmacodynamic models, but these require substantial data for estimates of probability distributions. In the presence of limited, inaccurate or imprecise information, simulation with fuzzy numbers represents an alternative tool to handle parametric uncertainty. Existing methods for implementing fuzzy arithmetic may, however, have significant shortcomings in overestimating (e.g., conventional fuzzy arithmetic) and underestimating (e.g., vertex method) the output uncertainty. The purpose of the present study is to apply and compare the applicability of conventional fuzzy arithmetic, vertex method and two recently proposed numerical schemes, namely transformation and optimization methods, for uncertainty modeling in pharmacokinetic and pharmacodynamic fuzzy-parameterized systems. A series of test problems were examined, including empirical pharmacokinetic and pharmacodynamic models, a function non-monotonic in its parameters, and a whole body physiologically based pharmacokinetic model. Our results verified that conventional fuzzy arithmetic can lead to overestimation of response uncertainty and should be avoided. For the monotonic pharmacokinetic and pharmacodynamic models, the vertex method accurately predicted fuzzy-valued output while incurring the least computational cost. It turned out that the choice of a suitable method for fuzzy simulation of the non-monotonic function depended on the required accuracy of the results: the vertex method was capable of eliciting an initial approximate solution with few function evaluations; for more accurate results, the transformation method was the most superior approach in terms of accuracy per unit CPU time.

A Fuzzy Composting Process Model

Composting processes are normally complicated with a variety of uncertainties arising from incomplete or imprecise information obtained in real–world systems. Previously, there has been a lack of studies that focused on developing effective approaches to incorporate such uncertainties within composting process models. To fill this gap, a fuzzy composting process model (FCPM) for simulating composting process under uncertainty was developed. This model was mainly based on integration of a fractional fuzzy vertex method and a comprehensive composting model. Degrees of influence by projected uncertain factors were also examined. Two scenarios were investigated in applying the FCPM method. In the first scenario, model simulation under deterministic conditions was conducted. A pilot–scale experiment was provided for verifications. The result indicated that the proposed composting model could provide an excellent vehicle for demonstrating the complex interactions that occurred in the composting process. In the second scenario, application of the proposed FCPM was conducted under uncertainties. Six input parameters were considered to be of uncertain features that were reflected as fuzzy membership functions. The results indicated that the uncertainties projected in input parameters will result in significant derivations on system predictions; the proposed FCPM can generate satisfactory system outputs, with less computational efforts being required. Analyses on degree of influence of system inputs were also provided to describe the impacts of uncertainties on system responses. Thus, suitable measures can be adopted either to reduce system uncertainty by well–directed reduction of uncertainties of those high–influencing parameters or to reduce the computational requirement by neglecting those negligible factors.

Calculating Functions of Interval Type-2 Fuzzy Numbers for Fault Current Analysis

In this work, functions of type-2 fuzzy numbers are analyzed. For the special case of interval type-2 fuzzy numbers, the type-2 membership function of the output variable is calculated using the lower and upper membership functions of the input variables and the vertex method. This procedure is used in an application where the type-2 fuzzy fault currents of an electric distribution system are calculated. The results are shown and the advantages of the approach are discussed

Groundwater Flow and Contaminant Transport Simulation with Imprecise Parameters

A methodology has been developed in this study wherein a genetic algorithm (GA) is used to find a global optimal solution to a groundwater flow and contaminant problem by incorporating an artificial neural network (ANN) to evaluate the objective function within the genetic algorithm. The study shows that an ANN-GA technique can be used to find the uncertainties in output parameters due to imprecision in input parameters. The ANN-GA methodology is applied to five case studies involving radial flow in a well, one-dimensional solute transport in steady uniform flow, a two-dimensional heterogeneous steady flow, a two-dimensional solute transport, and a two-dimensional unsteady groundwater flow to demonstrate the efficiency and effectiveness of the developed algorithm. The results show that, with this approach, one can successfully measure the uncertainty in groundwater flow and contaminant transport simulations and achieve a considerable reduction in computational effort when compared to the vertex method that has been widely used in the past.

Inverse Possibility Analysis Method for Possibility-Based Design Optimization

DOI: 10.2514/1.16546 Structural analysis and design optimization have recently been extended to the stochastic approach to consider variousuncertainties.However,inareaswhereitisnotpossibletoproduceaccurate statistical information forinput data, the probabilistic method is not appropriate for stochastic structural analysis and design optimization, because improper modeling of uncertainty could cause a greater degree of statistical uncertainty than those of physical uncertainty. For systems with insufficient information for input data, possibility-based (or fuzzy set) methods have recently been introduced in structural analysis and design optimization. Using possibility methods, the extended fuzzy operations are much simpler than of random variables. Possibility-based design optimization will provide more conservative designs than those from probability methods, andwill provide a system level of failure possibility automatically. This paper proposes a new formulation of possibility-based design optimization using the performance measure approach. For the inverse possibility analysis, the maximal possibility search method is proposed to improve numerical efficiency and accuracy comparing with the vertex method and the multilevel-cut method. Two mathematical examples, including a nonmonotonic response and a physical example of vehicle side impact, are used to demonstrate the proposed maximal possibility search method and possibility-based design optimization.

/spl alpha/-cut fuzzy arithmetic: simplifying rules and a fuzzy function optimization with a decision variable

The problems of /spl alpha/-cut fuzzy arithmetic have been shown, like in interval arithmetic, that distinct states of fuzzy parameters (or fuzzy variable values) may be chosen and produce an overestimated fuzziness. Meanwhile, local extrema of a function may exist inside the support of fuzzy parameters and cause an underestimation of fuzziness and an illegal fuzzy number's result. Previous approaches to overcoming these problems have appeared in literature. Yet, the computational burden of these approaches became even heavier. Thus, this paper is based on the vertex method in literature and extensively proposes newly devised rules observed greatly useful for simplifying the vertex method. These rules are devised through a function partitioned into subfunctions, distinguishing the types of fuzzy parameter/variable occurrences, and types of subfunctions or functions with the various observations. The improved efficiency has been found able to significantly reduce the combination (vertex) test of the vertex method for the fuzzy parameters' /spl alpha/-cut endpoints possibly to only a few fuzzy parameters' endpoint combinations. Also as related, a procedure for the fuzzy optimization of fuzzy functions with a fuzzy blurred argument (a single variable) is examined with the vertex method as well. A proper and useful preliminary algorithm is proposed. Numerical examples with results are provided.

An elaborative unit cost structure-based fuzzy economic production quantity model

The purpose of this paper is to investigate and propose a fuzzy extended economic production quantity model based on an elaboratively modeled unit cost structure. This unit cost structure consists of the various lot-size correlative components such as on-line setups, off-line setups, initial production defectives, direct material, labor, and depreciation in addition to lot-size non-correlative items. Thus, the unit cost is correlatively modeled to the production quantity. Therefore, the modeling or the annual total cost function developed consists of not only annual inventory and setup costs but also production cost. Moreover, via the concept of fuzzy blurred optimal argument and the vertex method of the α -cut fuzzy arithmetic (or fuzzy interval analysis), two solution approaches are proposed: (1) a fuzzy EPQ and (2) a compromised crisp EPQ in the fuzzy sense. An optimization procedure, which can simultaneously determine the α -cut-vertex combination of fuzzy parameters and the optimizing decision variable value, is also proposed. The sensitivity model for the fuzzy total cost and thus EPQ to the various cost factors is provided. Finally, a numerical example with the original data collected from a firm demonstrates the usefulness of the new model.

Fuzzy LINMAP method for multiattribute decision making under fuzzy environments

The Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) developed by Srinivasan and Shocker [V. Srinivasan, A.D. Shocker, Linear programming techniques for multidimensional analysis of preference, Psychometrika 38 (1973) 337–342] is one of the existing well-known methods for multiattribute decision making (MADM) problems. However, the LINMAP only can deal with MADM problems in crisp environments. Fuzziness is inherent in decision data and decision making processes, and linguistic variables are well suited to assessing an alternative on qualitative attributes using fuzzy ratings. The aim of this paper is further extending the LINMAP method to develop a new methodology for solving MADM problems under fuzzy environments. In this methodology, linguistic variables are used to capture fuzziness in decision information and decision making processes by means of a fuzzy decision matrix. A new vertex method is proposed to calculate the distance between trapezium fuzzy number scores. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to a fuzzy positive ideal solution (FPIS) which is unknown. The FPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the FPIS can be calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this paper can deal with MADM problems under not only fuzzy environments but also crisp environments.

Large-scale simulation of crack propagation based on continuum damage mechanics and two-step mesh partitioning

In this paper, the damage and crack propagation is numerically simulated based on continuum damage mechanics, and the parallel computing technique is utilized for the accurate analysis through the large-scale structural model. Besides the increase of computing burden resulted from the large-scale model, the complexity of the damage model, which involves highly nonlinear effects, increases the quantity of computation. Therefore, the high performance computing power is obtained from the Departmental Computing Grid (DCG) system composed of PC computers in our laboratory. The methodology of DCG is described and a new two-step mesh-partitioning scheme based on the graph-partitioning method is suggested to enhance the load balancing for parallel finite element analysis in the DCG environment. In the two-step mesh-partitioning scheme, the system performance weights are calculated to reflect the effect of heterogeneous system performances and weighted edge and vertex method (WEVM) is adopted to minimize the increase of communications. Based on these circumstances, damage distribution and crack initiation in an edge-cracked plate under Mode-I, -II, -III loading and fuselage structure in tension are analyzed by more than one million degrees of freedom. The possibility of the utilization of Grid computing system can be also addressed from the numerical results.