A. Wallis’s original derivation of his formula for 𝜋

2018 ◽  
pp. 131-138
Keyword(s):  
Original Derivation

scholarly journals Electrochemical Frequency Modulation: New Approach and Revision of Previous Model

2021 ◽  
Author(s):  
Shashi lalvani ◽  
Lei Kerr ◽  
Shamal Lalvani ◽  
Dominic Olaguera-Delogu
Keyword(s):  
Frequency Modulation ◽  
Corrosion Current ◽  
Peak Potential ◽  
New Approach ◽  
Harmonic Currents ◽  
Algebraic Error ◽  
Tafel Slopes ◽  
Incorrect Model ◽  
Original Derivation

Abstract A careful evaluation of the earlier model (1-2) for electrochemical frequency modulation (EFM) involving two sinusoidal applied potentials for the determination of corrosion parameters shows an algebraic error. Although the missing term in the original derivation appears to be insignificant, it is found that errors involved in corrosion current determination, and especially in evaluation of the Tafel slopes can be very significant, which is of consequence because of the rising popularity of this technique. The magnitude of error is found to be a function of the inherent corrosion characteristics (anodic and cathodic Tafel slopes) of the corroding material as well as the applied peak potential of the modulation. A corrected model with detailed steps showing the appropriate math is presented. In addition, using the experimental data available in the literature, the errors involved in estimating the corrosion parameters by the earlier EFM model of Bosch et al (1-2) are evaluated. The corrected corrosion current and the Tafel slopes can be recovered from the incorrect model without the benefit of the harmonic currents, as shown in this paper.The analysis is also presented for the case of only one applied sinusoidal frequency modulation, which offers several advantages over the multiple frequency modulation.

scholarly journals On Milne–Barbier–Unsöld relationships

2018 ◽  
Vol 27 (1) ◽  
pp. 76-79 ◽  
Author(s):  
Frédéric Paletou
Keyword(s):  
Source Function ◽  
Short Review ◽  
Specific Intensity ◽  
Original Derivation

Abstract This short review aims to clarify upon the origins of so-called Eddington-Barbier relationships, which relate the emergent specific intensity and the flux to the photospheric source function at specific optical depths. Here we discuss the assumptions behind the original derivation of Barbier (1943).We also point to the fact that Milne had already formulated these two relations in 1921.

A geometrical derivation of the shape density

1991 ◽  
Vol 23 (03) ◽  
pp. 496-514 ◽  
Author(s):  
Colin R. Goodall ◽  
Kanti V. Mardia
Keyword(s):  
Rotation Group ◽  
Shape Space ◽  
Canonical System ◽  
Single Step ◽  
Equivalence Classes ◽  
Polar Coordinates ◽  
Dimension 2 ◽  
Original Derivation ◽  
Insight Into ◽  
Original Configuration

The density for the shapes of random configurations of N independent Gaussian-distributed landmarks in the plane with unequal means was first derived by Mardia and Dryden (1989a). Kendall (1984), (1989) describes a hierarchy of spaces for landmarks, including Euclidean figure space containing the original configuration, preform space (with location removed), preshape space (with location and scale removed), and shape space. We derive the joint density of the landmark points in each of these intermediate spaces, culminating in confirmation of the Mardia–Dryden result in shape space. This three-step derivation is an appealing alternative to the single-step original derivation, and also provides strong geometrical motivation and insight into Kendall's hierarchy. Preform space and preshape space are respectively Euclidean space with dimension 2(N–1) and the sphere in that space, and thus the first two steps are reasonably familiar. The third step, from preshape space to shape space, is more interesting. The quotient by the rotation group partitions the preshape sphere into equivalence classes of preshapes with the same shape. We introduce a canonical system of preshape coordinates that include 2(N–2) polar coordinates for shape and one coordinate for rotation. Integration over the rotation coordinate gives the Mardia–Dryden result. However, the usual geometrical intuition fails because the set of preshapes keeping the rotation coordinate (however chosen) fixed is not an integrable manifold. We characterize the geometry of the quotient operation through the relationships between distances in preshape space and distances among the corresponding shapes.

scholarly journals Non-Linear Current-Potential Relations in an Axon Membrane

1961 ◽  
Vol 44 (6) ◽  
pp. 1055-1057 ◽  
Author(s):  
Kenneth S. Cole
Keyword(s):  
Membrane Potential ◽  
Giant Axon ◽  
Squid Giant Axon ◽  
External Current ◽  
Current Electrode ◽  
Axon Membrane ◽  
Linear Current ◽  
One Step ◽  
Original Derivation ◽  
Current Potential

The membrane current density, Im, in the squid giant axon has been calculated from the measured external current applied to the axon, Io, by the equation See PDF for Equation where Vm is the membrane potential under the current electrode and r1 and r2 are the external and internal longitudinal resistances. The original derivation of this equation included in one step an assumption of a linear relation between Im and Vm. It is shown that the same equation can be obtained without this restricting assumption.

Applications of the Ogilvie-Tuck Formulae to Ship Hydrodynamics

2010 ◽  
Author(s):  
Piotr J. Bandyk ◽  
Robert F. Beck ◽  
Xinshu Zhang
Keyword(s):  
Steady Flow ◽  
Degrees Of Freedom ◽  
Hydrodynamic Coefficients ◽  
Practical Applications ◽  
Classical Work ◽  
Strip Theory ◽  
The Mean ◽  
And Mathematics ◽  
Original Derivation ◽  
Ship Speed

The work of Ogilvie and Tuck [1] was critical in extending strip theory to problems with forward speed. Salvesen et al. [2] used some of those findings in their classical work to determine the hydrodynamic coefficients in five degrees-of-freedom. Other methods have also utilized the formulae with success. The original derivation makes several assumptions about the hull shape and the steady flow around it. In practice, these assumptions do not hold exactly. Other simplifications are usually made, i.e. the mean ship speed is used instead of the actual steady flow around the hull. These may violate the original assumptions, but the results are generally satisfactory. The truly elegant aspect of the Ogilvie-Tuck hydrodynamic coefficients is that they can be calculated from zero-speed results. This is a product of the approach, assumptions, and mathematics done in Appendix A of the original work [1] to derive the Ogilvie-Tuck theorem, also called Tuck’s theorem by many authors. The Ogilvie-Tuck formulae include the hydrodynamic coefficients, expressions of the mj terms, and the Ogilvie-Tuck theorem. This paper discusses the original derivation and several practical applications, including those where the assumptions may be violated. Several qualitative or quantitative statements can be made about the errors introduced by simplifications. Some computational results are presented to emphasize the significance in practical use.

Malayan Trombidiid Larvae, Part I. (Acarina: Trombidiidae)

Parasitology ◽  
1932 ◽  
Vol 24 (2) ◽  
pp. 145-174 ◽  
Author(s):  
B. A. R. Gater
Keyword(s):  
Suspected Case ◽  
Definite Evidence ◽  
Tsutsugamushi Disease ◽  
Original Derivation

In 1915 Dowden reported a suspected case of tsutsugamushi disease, or Japanese river fever, from the Federated Malay States. Twelve years later Fletcher and Field (1927) were able to examine a series of cases and established Allothrombium. This spelling is adopted by Oudemans. In the absence of definite evidence of the original derivation, it would appear, in the light of Opinion 34 of the Rules, that the original spelling should be preserved, and Trombicula is therefore used in this paper.

A geometrical derivation of the shape density

1991 ◽  
Vol 23 (3) ◽  
pp. 496-514 ◽  
Author(s):  
Colin R. Goodall ◽  
Kanti V. Mardia
Keyword(s):  
Rotation Group ◽  
Shape Space ◽  
Canonical System ◽  
Single Step ◽  
Equivalence Classes ◽  
Polar Coordinates ◽  
Dimension 2 ◽  
Original Derivation ◽  
Insight Into ◽  
Original Configuration

The density for the shapes of random configurations of N independent Gaussian-distributed landmarks in the plane with unequal means was first derived by Mardia and Dryden (1989a). Kendall (1984), (1989) describes a hierarchy of spaces for landmarks, including Euclidean figure space containing the original configuration, preform space (with location removed), preshape space (with location and scale removed), and shape space. We derive the joint density of the landmark points in each of these intermediate spaces, culminating in confirmation of the Mardia–Dryden result in shape space. This three-step derivation is an appealing alternative to the single-step original derivation, and also provides strong geometrical motivation and insight into Kendall's hierarchy. Preform space and preshape space are respectively Euclidean space with dimension 2(N–1) and the sphere in that space, and thus the first two steps are reasonably familiar. The third step, from preshape space to shape space, is more interesting. The quotient by the rotation group partitions the preshape sphere into equivalence classes of preshapes with the same shape. We introduce a canonical system of preshape coordinates that include 2(N–2) polar coordinates for shape and one coordinate for rotation. Integration over the rotation coordinate gives the Mardia–Dryden result. However, the usual geometrical intuition fails because the set of preshapes keeping the rotation coordinate (however chosen) fixed is not an integrable manifold. We characterize the geometry of the quotient operation through the relationships between distances in preshape space and distances among the corresponding shapes.

Integrated Plant, Observer, and Controller Optimization With Application to Combined Passive/Active Automotive Suspensions

2003 ◽  
Author(s):  
Hosam K. Fathy ◽  
Panos Y. Papalambros ◽  
A. Galip Ulsoy
Keyword(s):  
Optimization Problems ◽  
System Optimization ◽  
Optimization Strategy ◽  
Guarantee System ◽  
Control Optimization ◽  
Car Suspension ◽  
State Measurement ◽  
Full State ◽  
And Control ◽  
Original Derivation

The plant and control optimization problems are coupled in the sense that solving them sequentially does not guarantee system optimality. This paper extends previous studies of this coupling by relaxing their assumption of full state measurement availability. An original derivation of first-order necessary conditions for plant, observer, controller, and combined optimality furnishes coupling terms quantifying the underlying trilateral coupling. Special scenarios where the problems decouple are pinpointed, and a nested optimization strategy that guarantees system optimization strategy that guarantees system optimality is adopted otherwise. Applying these results to combined passive/active car suspension optimization produces a suspension design outperforming its passive, active, and sequentially optimized passive/active counterparts.

scholarly journals Collisions through liquid layers

1949 ◽  
Vol 45 (3) ◽  
pp. 488-488
Author(s):  
F. R. Eirich ◽  
D. Tabor
Keyword(s):  
Kinetic Energy ◽  
Liquid Film ◽  
Plug Flow ◽  
Vertical Direction ◽  
Velocity Change ◽  
Image Position ◽  
The Right ◽  
The Given ◽  
Original Derivation ◽  
Derivation Equation

The initial collision (see pp. 571–2). The instantaneous velocity change of the hammer (from V0 to V) when it first strikes the liquid film should be calculated from energy and not momentum considerations, since the hammer has momentum in the vertical direction whilst the liquid is expressed in a horizontal direction and its total momentum is, by symmetry, zero at every instant. The kinetic energy dE of an annulus of liquid of radius r will be ½mc2, where m = 2πrhρdr, and c, the radial velocity of flow, is ½rV/h, since the liquid starts moving in plug flow. Integrating from r = 0 to r = R, we find that the total kinetic energy E imparted to the liquid film is . Assuming that extraneous energy losses, including any energy imparted to the anvil, are negligible, we may equate this to the energy loss of the hammer . HenceTo a first approximation this yieldsThe term at the right-hand side of the denominator is one-half that given in the original derivation (equation (20)), so that the instantaneous decrease in the velocity of the hammer is even less marked. For the given case where M = 400g., R = 1 cm., ρ = 1.6 and h = 5 × 10−2 cm., the velocity decrease is less than 2%.Vol. 44 (1948), pp. 566–80.

scholarly journals Feynman’s propagator in Schwinger’s picture of Quantum Mechanics

2021 ◽  
pp. 2150187
Author(s):  
F. M. Ciaglia ◽  
F. Di Cosmo ◽  
A. Ibort ◽  
G. Marmo ◽  
L. Schiavone ◽  
...  
Keyword(s):  
Quantum Mechanics ◽  
Riemannian Manifold ◽  
Point Particle ◽  
Lagrangian Function ◽  
Natural Family ◽  
Quantum Propagator ◽  
Original Derivation ◽  
Classical Lagrangian

A novel derivation of Feynman’s sum-over-histories construction of the quantum propagator using the groupoidal description of Schwinger picture of Quantum Mechanics is presented. It is shown that such construction corresponds to the GNS representation of a natural family of states called Dirac–Feynman–Schwinger (DFS) states. Such states are obtained from a q-Lagrangian function [Formula: see text] on the groupoid of configurations of the system. The groupoid of histories of the system is constructed and the q-Lagrangian [Formula: see text] allows us to define a DFS state on the algebra of the groupoid. The particular instance of the groupoid of pairs of a Riemannian manifold serves to illustrate Feynman’s original derivation of the propagator for a point particle described by a classical Lagrangian L.

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