In this paper we execute the **solution** procedure for second order linear fuzzy **difference** **equation** by Lagrange’s multiplier method. In crisp sense the **difference** **equation** are easy to solve, but when we take in fuzzy sense it forms a system **of** **difference** **equation** which is not so easy to solve. By the help **of** Lagrange’s multiplier we can solved it easily. The results are illustrated by two different numerical examples and followed by two applications.

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Based on the analysis **of** the solutions **of** the hybrid new Keynesian IS curve I arrive at the conclusion that this curve is backward looking regardless **of** the size **of** the forward coefficient. The backward **solution** **of** the new Keynesian IS curve expectational **difference** **equation** implies that the current interest rate as well as the expected **future** real interest rates do not matter for the determination **of** current output gap. In the canonical new Keynesian Taylor- rule model, the central bank promises to make nominal interest higher when inflation is above its target. When the elasticity **of** substitution is less than or equal to one (σ ≤ 1) the higher interest rate does not bring about a reduced level **of** aggregate demand in the current period. Instead, it yields a higher level **of** aggregate demand in the current period and in the next period. Current inflation would increase due to a positive current output gap and a higher expected inflation, unless inflation jumps to a locally bounded equilibrium. 12

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promoting actions is a good tool for product life cycle management for increasing the patent life **of** existing products. Fast Disintegrating Tablets is the best delivery system when needs a rapid onset **of** action. Today, for the treatment **of** allergies, cold, and flu symptoms, ODTs are more widely available as OTC products. The target population has expanded to those who want convenient dosing anywhere, anytime, without water. The potential for such dosage forms is promising because **of** the availability **of** new technologies combined with strong market acceptance and patient demand. By paying close attention to advances in technologies, pharmaceutical companies can take advantage **of** ODTs for product line extensions or for first-to-market products. With continued development **of** new pharmaceutical excipients, one can expect the emergence **of** more novel technologies for ODTs in the days to come.

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The examined model is a computational solver from the NuscaS system [10], which is used for simulating thermomechanical phenomena in solidifying castings. It is based upon the heat conduction **equation** with a source term, solved using enthalpy formulation **of** solidification [11]. The finite element method is applied for spatial discretization, which allows to obtain a system **of** matrix equations with the derivative with respect to time. Integration is performed using the two-step Dupont II scheme [12].

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The Schrödinger **equation** being a partial differential **equation** describes the quantum state **of** a physical system and the changes in that system with respect to time. It was first time formulated by Schrödinger in 1925(Schrödinger, 1926) . In the sense **of** classical mechanics, the governing **equation** predicts the behavior **of** a system mathematically at any time after the initial state **of** the system is set and this corresponds to the Newton’s law F ma . In the language **of** quantum mechanics, the Schrödinger **equation** is analogous to Newton’s law for quantum mechanical system (which usually involves molecules, atoms, sub-atomic particles, whether moving freely, bounded or localized). It is not as simple algebraic operation but in general form it is a linear partial differential **equation** that describes the evaluation **of** time **of** the wave function system (Griffiths, 2004).

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In this section, we test the Belov-Chaltikian lattice **equation** to verify the efficiency **of** the GDTM-Padé technique. We compare the performance **of** the GDTM-Padé technique with the original GDTM algorithm. All the numerical computations are performed by Ma- thematica 7.0.

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In this paper, a new method is introduced to solve a two-dimensional Fredholm integral **equation**. The method is based on the approximation by Gaussian radial basis functions and triangular nodes and weights. Also, a new quadrature is introduced to approximate the two dimensional integrals which is called the triangular method. The results **of** the example illustrate the accuracy **of** the proposed method increases.

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Consumers are willing to pay a premium for products made in socially and environmentally responsible ways. Consumers claim that they are willing to pay a higher price for products **of** socially responsible firms, but are not willing to spend time on figuring out which firm is socially responsible. At the same time, a segment **of** consumers continue to buy products from firms with well acknowledged bad ethical conducts. Consumers do not act as what they claim. A small number **of** highly ethical consumers do exist. They care much about producers‘ CSR in their buying and consumption choices (Carrigan & Attalla, 2001). Carrigan and Attalla (2001) categorized consumers into four types based on their ethical awareness and ethical purchase intention. Consumers with low ethical awareness and low ethical purchase intention are called oblivious consumers. Confused and uncertain consumers are those with low ethical awareness but show high ethical purchase intention. Consumers with high ethical awareness who do not make purchase intention accordingly are cynical and disinterested consumers. The last type is caring and ethical consumers who obtain sufficient CSR information and are highly ethical in their purchase activities. Carrigan and Attalla (2001) also pointed out that consumers are not the only stakeholders **of** a business. Other stakeholders such as employees, environmental organizations may care much about a firm‘s CSR. Each stakeholder group has a focus on firms‘ CSR. The focus is usually what influences their interest most. When addressing CSR, firms need always engage the target stakeholders and their focus **of** concerns. Philanthropy seen everywhere without any consumer engagement has little contribution to consumer choice. Consumers‘ awareness **of** firms‘ CSR is limited in general, so opportunities exist in the communication **of** corporate social performance and social responsibility initiatives with consumers. Even though consumers‘ choices are not significantly influenced by a firm‘s CSR, it is still important to be socially responsible in business operations. Consumers may have more CSR information and become more ethical in their purchasing behavior in the **future**. In addition, the new generation may behave differently to socially responsible companies‘ products.

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QAP is a NP-hard problem that has been studied by a number **of** researchers in combinatorial optimization through the second half **of** the 20 th century and which continues to excite the curiosity **of** many others, both in the study **of** exact algorithms and through the formulation **of** information mechanisms for heuristics. The exact approach is severely limited by the high complexity **of** the problem, a difficult 30-order instance (Nug30) having been solved exactly only through a huge work **of** metacomputing. More recent years have seen an increased interest in the use **of** metaheuristic schemes: some examples are [AQB99], [BT94], [AOT00], [CMMT97], [Ma00], [TRFR01] and [TS95].

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tive α, β, as |ω| → ∞), real parts **of** both functions are bounded by positive **value**, and their the Fourier trans- forms satisfy the H¨older condition on R. The Wiener- Hopf integral **equation** is an integral **equation**, which in 75 years **of** its history has been impressed all who use it to almost all branches **of** engineering, mathemat- ical physics, and applied mathematics. The technique to solve a Wiener-Hopf integral **equation**, named the Riemann-Hilbert technique, still, remains an extremely important tool for modern scientists, and the areas **of** application continue to broaden, a good review may be found in [10], among others. The Riemann-Hilbert tech- nique is theoretically well developed. However, to over- come problem caused by slow evaluation and existence singularises near the integral contour some approximate methods have to be considered, which do not receive enough attention from authors. Until 2000, there does

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A nonlinear wave phenomenon is the important area **of** scientific research, which many scientists in the past have studied about mathematical models explaining the wave behavior. There are mathematical models which describe the dynamic **of** wave behaviors–for example, the KdV **equation**, the RLW **equation**, the Rosenau **equation**, and many others [1]–[10]. The KdV **equation** has been used in very wide applications, such as magnetic fluid waves, ion sound waves, and longitudinal astigmatic waves [4]–[6]. The RLW **equation**, which was first proposed by Peregrine [7], [8] provides an explanation on a different situation **of** a nonlinear dispersive wave from the more classical KdV **equation**. The RLW **equation** is one **of** models which are encountered in many areas, e.g. ion–acoustic plasma waves, magnetohydro- dynamic plasma waves, and shallow water waves. Since the case **of** wave–wave and wave–wall interactions cannot be described by the KdV **equation**, Rosenau [9], [10] proposed an **equation** for describing the dynamic **of** dense discrete systems; it is known as the Rosenue **equation**. The existence and uniqueness **of** the **solution** for the Rosenau **equation** were proved by Park [11], [12]. For the further consideration **of** the nonlinear wave, a viscous term u xxt needs to be included:

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, , , where : and : are continuously differentiable functions and , are some intervals **of** real numbers. Furthermore, a **solution** , ∞ **of** system (2) is uniquely determined by initial conditions , for , , . Along with the system (2), we consider the corresponding vector map , , , , , , , . An equilibrium point **of** system (2) is a point , that satisfies

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Abstract: The life **of** bucket teeth in shovel and dragline deployed in handling **of** overburden rock is an important contributor to the stores cost and is also responsible for the loss **of** valuable availability and utilisation time **of** these critical equipment. To ascertain the effect **of** rock type on longevity **of** bucket teeth, a study has been conducted in two large opencast mines **of** Singrauli Coalfields. The results **of** this study is presented in this paper. There was a significant variation as compared to the actual figures **of** the mine, it establish useful relationship between the type **of** mineral present in the overburden and the life **of** bucket teeth **of** shovel and dragline.

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Stratonovich sense) arises as a prototype model in understanding a wide variety **of** phenomena. It appears in modeling transport in porous media, suspended sediment transport in open-channel flows, advective transport **of** substances, waves motions in random media and from turbulent transport theory, u(x, t) be the concentration **of** a passive substance convected by a turbulent fluid, such as smoke in turbulent air.

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Moreover, the logistic **equation** was published in 1838 by Pierre Franc¸ois Verhulst to model the growth **of** world population and was based on the assessment **of** population statistics available, complementing the theory **of** exponential growth **of** Thomas Robert Malthus. It can be applied to models with time dependence and has a wide application area since the inhibiting factors are taken into consideration. Furthermore, the logistic **equation** proved applicable in a series **of** probabilistic events and related to chaos theory and industrial and business dynamics [7, 25]. Recently, the logistic **equation** has been applied to describe the growth **of** populations both in the laboratory natural habitat, limiting the growth by influence **of** competition, mortality and fertility factors. However, the logistic model does not fit very well in cases where there are more complex relationships acting as interactions within food webs or several features that occur in most cases in nature dependencies[5, 6, 7, 8, 9, 22]. Aiming to generalize the logistic **equation** and give a better description for some **of** those events, El-Sayed, at all [5], studied the corresponding fractional-order logistic by a numerical analyze.

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We have approached a theoretical description **of** the tem- poral evolution **of** the modified van der Pol oscillator with an additive Gaussian white noise in the region where birhythmic- ity (in the absence **of** noise) occurs. To get an analytical insight on this system we have used an explicit **solution** based on the phase-amplitude approximation **of** the Fokker-Planck **equation** to analytically derive the probability distributions. The activation energies associated to the switches between different attractors have been derived analytically and numeri- cally. We have found that the agreement is fairly good. The characteristics **of** the birhythmic properties in a modified van der Pol oscillator are strongly influenced by both the nonlin- ear coefficients a, b and the noise intensity D. The boundary **of** the existence **of** multi-limit-cycle solutions, in the paramet- ric (a,b)-plane, decreases with the increase **of** the noise inten- sity D. Finally, the analytic estimate **of** the stability **of** the two attractors varies with the control parameters (the dissipation a and b) in a way that resembles phase transitions: for most pa- rameters **value** the system is located around only one attractor, the other being visited with a vanishingly small probability. Only at special values **of** the control parameters the residence times are comparable, in agreement with experimental obser- vations **of** birhythmicity in Biological systems: the passage from an attractor to another only occurs by varying the exter- nal parameters and not under the influence **of** noise. 43,44

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4. Musaev V.K. Vozdeystvie prodol’noy stupenchatoy volny na podkreplennoe krugloe otverstie v uprugoy srede [Impact **of** the Longitudinal Steo-shaped Wave on a Supported Circular Hole in an Elastic Medium]. All-Union Conference “Modern Problems **of** Structural Mechanics and Strength **of** Aircrafts.” Collected abstracts. Moscow Institute **of** Aviation, 1983, p. 51.

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subsequent decreases over-time lagged inflation on the right-hand side, are significant. Columns (3) to (8) **of** Table 2 present panel estimates with time and fixed-effects for the various samples and two calendars **of** IT adoption described in Table 1. From column (3) to (4), I change from MSH’s to BS’s IT calendar – the one used by LY – to check how important it is to vary the dates **of** IT adoption. The different calendars **of** IT adoption make a quantitative **difference**, but not important enough to change the qualitative result that IT mattered . BS’s IT adoption dates reduce the IT direct impact on the average inflation level from -1.37 to -1.01 percentage points, or its implied cumulative effect from -2.28 to -1.68 percentage points. Anyway, the IT effect is still significantly important, different from LY’s irrelevance conclusion (in their Table 3, Panel A), and differences between columns (3) and (4) mainly reflect that my identification strategy **of** a one- year lag in policy effects matches better with MSH’s calendar.

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