On Murphy’s Integrated Circuit Yield Integral

1995 ◽  
Vol 117 (2) ◽  
pp. 159-164
Author(s):  
John H. Lau
Keyword(s):  
Density Distribution ◽  
Closed Form ◽  
Integrated Circuit ◽  
Defect Density ◽  
Closed Form Solution ◽  
Form Solution ◽  
Simple Equation ◽  
Multichip Module ◽  
Yield Formula ◽  
Circuit Yield

An approximated closed-form integrated circuit (IC) yield formula based on a Gaussian defect density distribution for the compounder in Murphy’s yield integral is presented. Also, a closed-form solution for the average number of faults (AD0) in an IC is obtained for a given IC yield (Y). Furthermore, based on the new IC yield formula a simple equation for determining the number of yielded chips in a wafer is given. Finally, the multichip module yield (Ym) and resultant shipped multichip module yield (Yms) based on the new IC yield formula are provided.

Natural Convection in a Vertical Annulus Containing Water Near the Density Maximum

1987 ◽  
Vol 109 (4) ◽  
pp. 899-905 ◽  
Author(s):  
D. S. Lin ◽  
M. W. Nansteel
Keyword(s):  
Heat Transfer ◽  
Natural Convection ◽  
Aspect Ratio ◽  
Density Distribution ◽  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
Vertical Flow ◽  
Density Maximum ◽  
Vertical Annulus

Steady natural convection of water near the density extremum in a vertical annulus is studied numerically. Results for flow in annuli with aspect ratio 1≤A≤8 and varying degrees of curvature are given for 103≤Ra≤105. It is shown that both the density distribution parameter R and the annulus curvature K have a strong effect on the steady flow structure and heat transfer in the annulus. A closed-form solution for the vertical flow in a very tall annulus is compared with numerical results for finite-aspect-ratio annuli.

The gravity anomaly of two‐dimensional sources with continuous density distribution and bounded by continuous surfaces

Geophysics ◽  
1992 ◽  
Vol 57 (4) ◽  
pp. 623-628 ◽  
Author(s):  
Tapio Ruotoistenmäki
Keyword(s):  
Density Distribution ◽  
Closed Form ◽  
Gravity Anomaly ◽  
Closed Form Solution ◽  
Vertical Direction ◽  
Form Solution ◽  
Two Dimensional ◽  
Layered Intrusions ◽  
Integration Interval ◽  
Closed Form Solutions

The gravity anomaly of a complicated two‐dimensional (2-D) source having arbitrary surfaces and density varying in either horizontal or vertical direction is calculated using a combination of closed form solutions and numerical integration. The surfaces and density can be defined by continuous or piecewise continuous two‐dimensional functions in the integration interval. For example, the anomalies for intrusions or folded sedimentary units, having an arbitrary density in the horizontal direction and a polynomial density distribution in the vertical direction, can be calculated using surfaces represented by functions of the horizontal dimension. When modeling dipping layered intrusions or sedimentary beds the surfaces are represented by functions of the vertical dimension in which case the density can be an arbitrary function of depth and a polynome function of horizontal coordinate. The accuracy of the method is defined by the user. The method gives simple and general equations to calculate anomalies of complicated sources which have no closed form solution, thus reducing the number of algorithms needed in interpretation programs.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

scholarly journals A Closed-Form Solution to Planar Feature-Based Registration of LiDAR Point Clouds

2021 ◽  
Vol 10 (7) ◽  
pp. 435
Author(s):  
Yongbo Wang ◽  
Nanshan Zheng ◽  
Zhengfu Bian
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Simulated Data ◽  
Real Data ◽  
Point Clouds ◽  
Form Solution ◽  
Spatial Transformation ◽  
Dual Quaternions ◽  
Feature Based ◽  
Planar Feature

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.

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

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.

Sign in / Sign up

Export Citation Format

Share Document