This paper studies the equilibrium dynamics of a growth model with public finance where two different allocations of public spending with two different elasticities are considered. Fiscal policy is part of the aggregate economy by explicitly including the public sector in the production function. This generates a potential relationship between government and production. The model analyzes the equilibrium dynamics and derives a closedform solution for the optimal shares of consumption, capital accumulation, taxes and composition of the two different public expenditures which maximize a representative household's lifetime utilities for a centralized economy. Under the simplifying assumption that the inverse of the intertemporal elasticity of substitution equals the physical capital share ( = ) , the model identifies the three main shortcomings associated with this procedure: consumption is proportional to physical capital stock, the initial physical capital stock determines the long-run balanced growth paths, and transitional dynamics for the variables in the model are partially simplified.
where α ∈ (0, 2] is the characteristic exponent, and γ is a quantity analogous to the variance called the dispersion. Unfortunately, SαS does not provide an analyticform PDF except for special cases. An approximation PDF model with less computational burden, called a simplified bi-parameter Cauchy Gaussian mixture (BCGM), is given in 
We have proposed a rigorous vectorial Gaussian beam tracing method to extract a 3D generalized analyticmodel for the VIPA pattern after the lens in a closed-form formulation. To our knowledge, this is the first VIPA demultiplexer 3D performance analysis with arbitrary polarization input Gaussian beam at an arbitrary vertical plane using a generalized imaging lens system. The comparisons with previously- published 2D results show a good agreement. The advantages and potentials of the proposed method were also investigated previously in  repeated here for convenience: 1) The source beam waists are finite in both directions that are modeled as a generalized elliptic vectorial Gaussian beam. 2) The vectorial behavior of electromagnetic fields is modeled precisely by vectorial reflection and transmission coefficients. 3) This method can handle both types of VIPA (air- filled n 2 = 1 and solid n 2 = n > 1 and moreover, the device in
The structure of the paper is as follows. The SSP model and ancillary mathematical background are described in §II, and §III defines a two-letter code classification of omni- conductors and omni-insulators, in which every alternant or non-alternant graph appears in exactly one of eight categories. This is refined in §IV to give a systematic classification of conduction/insulation behaviour of alternants in terms of a three-letter acronym (TLA) for ‘near-omni’ systems. We prove that of 81 conceivable cases, only 14 are realisable. §§V to VIII, supported by two mathematical appendices, show how this reduction is achieved, and give families of chemical examples (Figure 3 and TableVI). §IX describes the startingly simple restriction of the full classification for benzenoids, and §X states our overall conclu- sions. The end result is a complete description of ballistic conduction at the Fermi level as predicted within the SSP model.
mal or Gaussian distribution function for the temporal evolu- tion of the daily number of new cases (deaths, or alternatively infections) at time t due to the COVID-19 pandemic disease provides quantitatively correct descriptions for the monitored rates in many different countries. If applied early enough at the beginning of the pandemic wave the Gauss model (GM) makes realistic and reliable predictions for the future evolu- tion of the first wave. It has been argued that the assumption of a Gaussian time evolution is well justified by the central limit theorem of statistics, 3 an agent-based model, 4 a Taylor
Abstract. In this study, an optimal integrated vendor-buyer inventory model with defective items is proposed. Most researches on defective items assumed that an inspection process was carried out by the buyer. We consider that the vendor conducts the inspection process and disposes defective items in multiple batches. We prove that the function of annual cost is convex, and obtain closed-form expressions. A solution procedure is used to derive the optimal order quantity, the number of shipments, and the number of defective item disposals. Numerical examples are provided to illustrate our model. Setting the fraction of defective items to zero, the numerical examples indicate that the proposed model can result in the solutions to the existing models without considering defective items. Moreover, a sensitivity analysis is used to reveal the eects of cost parameters on the optimal solution. We show that when the disposal cost is relatively low, a multiple disposals strategy may perform better than a single disposal strategy.
are fitted using a simple model, and indicate that R3 is only 50% closed: the closedform is a four-helix bundle, while in the open state helix 1 is twisted out. Strikingly, a mutant of R3 that binds RIAM with an affinity similar to wild-type but more weakly to vinculin is shown to be 0.84 kJ mol -1 more stable when closed. These results demonstrate that R3 is thermodynamically
Abstract—A closedformanalytic solution is introduced for arbitrary Coupled Nonuniform Transmission Lines (CNTLs). First, the diﬀerential equations of CNTLs are written as a suitable matrix diﬀerential equation. Then, the matrix diﬀerential equation is solved to obtain the chain parameter matrix of CNTLs. Afterward, the voltage and current of lines are obtained at any point using the chain parameter matrix. The validation of the introduced solution is studied, ﬁnally.
It is convenient to start by investigating the distributions of the cosmological parameter t in the closed cos- mic model at various epochs according to Equation (17). Figure 1(a) shows no evident change of t with cosmic time until t 14.36 Gyr, then t decreases in relatively higher rate towards t 0.5 Gyr . On the other hand t exhibits a gradual change with time in the time range t 0.5 Gyr t me as seen in Figure 1(b),
Tim Murphy, Kenneth Weinstein and Eric Zwick for excellent research assistance. We are par- ticularly grateful to Fan Zhang who introduced us to Lambert’s W-function, which is needed to express our implicit solution for the refinancing diﬀerential as a closedform equation. We also thank Brent Ambrose, Ronel Elul, Xavier Gabaix, Bert Higgins, Erik Hurst, Michael LaCour-Little, Jim Papadonis, Sheridan Titman, David Weil, and participants at seminars at the NBER Summer Insti- tute and Johns Hopkins for helpful comments. Laibson acknowledges support from the NIA (P01 AG005842) and the NSF (0527516). Earlier versions of this paper with additional results circulated under the titles “When Should Borrowers Refinance Their Mortgages?” and “Mortgage Refinancing for Distracted Consumers.” The views expressed in this paper do not necessarily reflect the views of the Federal Reserve Board or the Federal Reserve Bank of Chicago.
An analytical form for the steady state average speed of motor PWM control is derived in this paper. Compared to the Simulink model used in , the analytical form significantly reduces the simulation time related to sensitivity analysis, which makes further sensitivity analysis feasible. In this paper, only the impact of each parameter on the steady state average speed is analyzed. The research work is still being carried out to fully analyze the sensitivity with all parameters and their interactions considered. The model-based analysis result can be used to predict the performance of PWM motor speed control for a given set of tolerance bands for the design parameters. It can also provide the design engineers with information to choose the tolerance band for each parameter and possible overall trade-off between the parameters and cost. For any given variance for the steady state average speed, a range for each parameter or certain constraint between the parameter can be found. Tens of thousand of tests can be conducted in the simulation environment instead of building so many actual systems. The benefit in time-to-market and cost savings can be significant. Even though the derivation of the analytical form may not apply to all other systems, tools for symbolic computations allows one to deploy the method used in this paper in many other similar problems. Future work also includes validation of the simulation results by conducting actual testing.
In another influential paper, Rudd and Whelan (2006) have argued that the Gali and Gertler HYPC is inappropriate for estimating the weights because the implied expectations are not strictly rational as they are not model consistent. Using the closedform solutions for the Gali and Gertler HYPC, to get model consistent rational expectations, they found that the weight for forward looking expectations is insignificant. Although the Gali and Gertler methodology may not be appropriate, Rudd and Whelan ’s specifications also have limitations because their reduced form weights do not sum to unity. It is necessary, therefore, to re- estimate these weights with the constraint that they should add to unity. This is the main objective of this paper and is structured as follows. Section 2 presents specifications of the HYPC. Section 3 discusses our empirical results and Section 4 concludes.
In this paper we investigate the generalisation of Wendland’s com- pactly supported radial basis functions to the case where the smoothness parameter is not assumed to be a positive integer or half-integer and the parameter ℓ, which is chosen to ensure positive deﬁniteness, need not take on the minimal value. We derive suﬃcient and necessary conditions for the generalised Wendland functions to be positive deﬁnite and deduce the native spaces that they generate. We also provide closedform repre- sentations for the generalised Wendland functions in the case when the smoothness parameter is an integer and where the parameter ℓ is any suitable value that ensures positive deﬁniteness, as well as closedform representations for the Fourier transform when the smoothness parame- ter is a positive integer or half-integer.
the hard-to-obtain graph supp(S opt ), without having to solve the GL. Furthermore, we will show that the GL problem has a simple closed-form solution that can be easily derived merely based on the thresholded sample covariance matrix, provided that its underlying graph has an acyclic structure. This result will then be generalized to obtain an approxi- mate solution for the GL in the case where the thresholded sample covariance matrix has an arbitrary sparsity structure. This closed-form solution converges to the exact solution of the GL as the length of the minimum-length cycle in the support graph of the thresholded sample covariance matrix grows. The derived closed-form solution can be used for two pur- poses: (1) as a surrogate to the exact solution of the computationally heavy GL problem, and (2) as an initial point for common numerical algorithms to numerically solve the GL (see Friedman et al. (2008); Hsieh et al. (2014)). The above results unveil fundamental properties of the GL in terms of sparsification and computational complexity. Although conic optimization problems almost never benefit from an exact or inexact explicit formula for their solutions and should be solved numerically, the formula obtained in this paper suggests that sparse GL and related graph-based conic optimization problems may fall into the category of problems with closed-form solutions (similar to least squares problems). 3. Main Results
In this note a Fredholm integral equation of the first kind with exponential expressions for the kernel and right hand side is considered. The task of finding a practically usable solution to such an equation may need more effort than following a standard procedure, even when such a procedure yields a formal solution. An apparently elegant solution as an orthogonal polynomial expansion is obtained using the standard method based on transformation to a form where the kernel is an orthogonal polynomial generating function, but this is of limited use due to slow convergence. It is shown that this can nevertheless be transformed into a closedform solution that is computationally efficient.
The analytical tractability and its maximum stability property make the generalized extreme value (GEV) distribution an attractive choice in the theo- retical and econometric modelling of unobservables in incomplete information games. This paper presents new results on conditional moments of order statistics of GEV distributed random variables. And it provides a recursive algorithm to derive the GEV density in high dimensional problems, thereby enabling simulating the Nested Multinomial Logit (NMNL) model on the basis of the Markov chain Monte Carlo protocol of McFadden (1999).
which are in the physiologically expected range. However, in their study, iterative procedures such as steepest descent algorithm were used to find minimum of the RLS cost function and a closedform solution was not provided. Bias and variance of the estimator were also estimated using some Monte-Carlo simulations.
We now compare the refinancing diﬀerentials implied by our model and those reported by Chen and Ling (1989). Chen and Ling calculate optimal diﬀerentials for a model in which the log one-period nominal interest rate follows a random walk, the time of exogenous prepayment (or the expected holding period) is known with certainty, and the real mortgage principle is allowed to decline over time because of inflation and continuous principle repayment. Chen and Ling use numerical methods to solve the resulting system of partial diﬀerential equations.
A way in which one may understand these benefits is to look deeper into the structure of the closedform solution. The closed-form solution involves an under- standing of the definitions of the system parameters and the product/s of their interactions, and the combination of this information to reach a desired conclu- sion. This is an integral process in solving physical processes analytically and an effective way in understanding their true nature as it can unveil the limitations and assumptions made. It is through this fundamental structure that it may be said closedform solutions offer an advantage over numerical solutions, as the analysis of a system and the interactions of its components is one which can yield great insight. One can examine an equation, alter it, incorporate it; as methods to see the interconnections and intricacies of the processes within it.
ABSTRACT: Steady incompressible Jeffrey fluid flow through a uniform tube with an overlapping stenosis is investigated. Assuming the stenosis to be mild, the equations governing the flow of the proposed model have been solved and closedform solutions are obtained for velocity, pressure drop, volumetric flow rate, resistance to the flow and wall shear stress. The resistance to the flow increases with the height and length of the stenosi. The effects of other parameters on the flow characteristics also have been studied.