Reply by author to Dr. LaFehr

It is difficult to include all references when dealing with a subject so well studied as the gravitational attraction of a circular disc. Although the practical usefulness of Nettleton’s paper can not be denied by anyone, it nevertheless gives no details (except for some references) of the computation of solid angles subtended by a disc from which his graphs (Geophysics, 1942, Figure 4) result. My short note deals with (in what I consider an easy way of) obtaining a closed form expression for the solid angle. For applications of the result the reader would do well to look up Nettleton’s classic paper.

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On: “Gravitational Attraction of a Circular Disc,” by Shri Krishna Singh (GEOPHYSICS, Feb. 1977, p. 111–113)

Geophysics ◽

10.1190/1.1440756 ◽

1977 ◽

Vol 42 (4) ◽

pp. 877-877

Author(s):

T. R. Lafehr

Keyword(s):

Mineral Resource ◽

Short Note ◽

Three Dimensional ◽

Salt Dome ◽

Circular Disc ◽

Gravitational Attraction ◽

Dimensional Problem ◽

Gravity And Magnetic ◽

The Subject ◽

Classic Paper

It is unfortunate that in the Short Note no mention whatever is made of Nettleton’s classic paper on the subject: “Gravity and Magnetic Calculations”. (Geophysics, 1942, p. 308). Not only did Nettleton’s paper precede Singh by 35 years but it provides a more practical approach (stacked discs) to solving a difficult three‐dimensional problem. Nettleton’s method endured long and well until modern computers rendered it less efficient several years ago; his technique has been applied countless times to thousands of salt dome and mineral resource problems.

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GRAVITATIONAL ATTRACTION OF A CIRCULAR DISC

Geophysics ◽

10.1190/1.1440704 ◽

1977 ◽

Vol 42 (1) ◽

pp. 111-113 ◽

Cited By ~ 15

Author(s):

Shri Krishna Singh

Keyword(s):

Cross Sections ◽

Solid Angle ◽

Vertical Component ◽

Circular Disk ◽

Circular Disc ◽

Gravitational Attraction ◽

Exploration Seismology ◽

Solid Angles ◽

The Cross

The vertical component of gravitational attraction [Formula: see text] of a circular disk is of some interest in geophysics since it can be used to obtain attraction of 3-D bodies whose parallel sections are circular and also since the solid angle Ω subtended by a disc at any point is proportional to [Formula: see text] at the same point (Ramsey, 1940, p. 36). Solid angles may be needed in some diffraction calculations in exploration seismology (see, e.g., Hilterman, 1975). It is clear, however, that in calculation of attraction from 3-D bodies, approximation of the cross‐sections by a polygon (Talwani and Ewing, 1960) has wider application.

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Closed-form expression for the capacitance of the shielded microstrip circular disc capacitor by dividing the capacitance approach

International Journal of Electronics ◽

10.1080/00207218508939048 ◽

1985 ◽

Vol 58 (3) ◽

pp. 509-511

Author(s):

S. S. BEDAIR

Keyword(s):

Closed Form ◽

Circular Disc ◽

Closed Form Expression ◽

Form Expression

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Synthesizing of Planar and Conformal Double-Sided Parallel-Cross Dipoles FSS Based on Closed-Form Expression

IEEE Access ◽

10.1109/access.2021.3096419 ◽

2021 ◽

pp. 1-1

Author(s):

Yassine Zouaoui ◽

Larbi Talbi ◽

Khelifa Hettak ◽

Naresh K. Darimireddy

Keyword(s):

Closed Form ◽

Closed Form Expression ◽

Form Expression

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Distribution of Consensus in a Broadcasting-based Consensus-forming Algorithm

ACM SIGMETRICS Performance Evaluation Review ◽

10.1145/3453953.3453974 ◽

2021 ◽

Vol 48 (3) ◽

pp. 91-96

Author(s):

Shigeo Shioda

Keyword(s):

Distribution Function ◽

Probability Density Function ◽

Closed Form ◽

Probability Density ◽

Infinite Number ◽

Stable Distribution ◽

Random Variable ◽

Cauchy Distribution ◽

Closed Form Expression ◽

Form Expression

The consensus achieved in the consensus-forming algorithm is not generally a constant but rather a random variable, even if the initial opinions are the same. In the present paper, we investigate the statistical properties of the consensus in a broadcasting-based consensus-forming algorithm. We focus on two extreme cases: consensus forming by two agents and consensus forming by an infinite number of agents. In the two-agent case, we derive several properties of the distribution function of the consensus. In the infinite-numberof- agents case, we show that if the initial opinions follow a stable distribution, then the consensus also follows a stable distribution. In addition, we derive a closed-form expression of the probability density function of the consensus when the initial opinions follow a Gaussian distribution, a Cauchy distribution, or a L´evy distribution.

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Colored HOMFLY-PT for hybrid weaving knot $$ {\hat{\mathrm{W}}}_3 $$(m, n)

Journal of High Energy Physics ◽

10.1007/jhep06(2021)063 ◽

2021 ◽

Vol 2021 (6) ◽

Author(s):

Vivek Kumar Singh ◽

Rama Mishra ◽

P. Ramadevi

Keyword(s):

Closed Form ◽

Topological Strings ◽

Chebyshev Polynomials ◽

Fibonacci Numbers ◽

Infinite Family ◽

Closed Form Expression ◽

Two Dimensional ◽

Laurent Polynomials ◽

Form Expression ◽

Torus Knots

Abstract Weaving knots W(p, n) of type (p, n) denote an infinite family of hyperbolic knots which have not been addressed by the knot theorists as yet. Unlike the well known (p, n) torus knots, we do not have a closed-form expression for HOMFLY-PT and the colored HOMFLY-PT for W(p, n). In this paper, we confine to a hybrid generalization of W(3, n) which we denote as $$ {\hat{W}}_3 $$ W ̂ 3 (m, n) and obtain closed form expression for HOMFLY-PT using the Reshitikhin and Turaev method involving $$ \mathrm{\mathcal{R}} $$ ℛ -matrices. Further, we also compute [r]-colored HOMFLY-PT for W(3, n). Surprisingly, we observe that trace of the product of two dimensional $$ \hat{\mathrm{\mathcal{R}}} $$ ℛ ̂ -matrices can be written in terms of infinite family of Laurent polynomials $$ {\mathcal{V}}_{n,t}\left[q\right] $$ V n , t q whose absolute coefficients has interesting relation to the Fibonacci numbers $$ {\mathrm{\mathcal{F}}}_n $$ ℱ n . We also computed reformulated invariants and the BPS integers in the context of topological strings. From our analysis, we propose that certain refined BPS integers for weaving knot W(3, n) can be explicitly derived from the coefficients of Chebyshev polynomials of first kind.

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