Reply to the Discussion on the paper “Closed form solution for a nonlocal elastic bar in tension”

Since pairwise registration is a necessary step for the seamless fusion of point clouds from neighboring stations, a closed-form solution to planar feature-based registration of LiDAR (Light Detection and Ranging) point clouds is proposed in this paper. Based on the Plücker coordinate-based representation of linear features in three-dimensional space, a quad tuple-based representation of planar features is introduced, which makes it possible to directly determine the difference between any two planar features. Dual quaternions are employed to represent spatial transformation and operations between dual quaternions and the quad tuple-based representation of planar features are given, with which an error norm is constructed. Based on L2-norm-minimization, detailed derivations of the proposed solution are explained step by step. Two experiments were designed in which simulated data and real data were both used to verify the correctness and the feasibility of the proposed solution. With the simulated data, the calculated registration results were consistent with the pre-established parameters, which verifies the correctness of the presented solution. With the real data, the calculated registration results were consistent with the results calculated by iterative methods. Conclusions can be drawn from the two experiments: (1) The proposed solution does not require any initial estimates of the unknown parameters in advance, which assures the stability and robustness of the solution; (2) Using dual quaternions to represent spatial transformation greatly reduces the additional constraints in the estimation process.

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Glulam – closed form solution using statistical smearing, truncating, calibrating

Australian Journal of Structural Engineering ◽

10.1080/13287982.2021.1926062 ◽

2021 ◽

pp. 1-12

Author(s):

Henry Robert Milner

Keyword(s):

Closed Form ◽

Closed Form Solution ◽

Form Solution

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A CLOSED-FORM PRICING FORMULA FOR EUROPEAN EXCHANGE OPTIONS WITH STOCHASTIC VOLATILITY

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964820000698 ◽

2021 ◽

pp. 1-10

Author(s):

Puneet Pasricha ◽

Anubha Goel

Keyword(s):

Closed Form ◽

Stochastic Volatility ◽

Closed Form Solution ◽

Form Solution ◽

Stochastic Volatility Model ◽

Strike Price ◽

Volatility Model ◽

Exchange Options ◽

Pricing Formula ◽

Exchange Option

This article derives a closed-form pricing formula for the European exchange option in a stochastic volatility framework. Firstly, with the Feynman–Kac theorem's application, we obtain a relation between the price of the European exchange option and a European vanilla call option with unit strike price under a doubly stochastic volatility model. Then, we obtain the closed-form solution for the vanilla option using the characteristic function. A key distinguishing feature of the proposed simplified approach is that it does not require a change of numeraire in contrast with the usual methods to price exchange options. Finally, through numerical experiments, the accuracy of the newly derived formula is verified by comparing with the results obtained using Monte Carlo simulations.

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