Arc-Sine Law and the Libor Reform

2020 ◽  
Author(s):  
Vladimir Piterbarg
Keyword(s):  
Sine Law

scholarly journals Alternate current measurement

1908 ◽  
Vol 80 (539) ◽  
pp. 310-352
Keyword(s):  
Alternate Current ◽  
Current Theory ◽  
High Ratio ◽  
Quadratic Functions ◽  
Wave Form ◽  
Numerical Comparison ◽  
Sine Law ◽  
The Magnetic Field ◽  
Exciting Voltage ◽  
Careful Design

The lack of precision of measurements with alternate currents, as compared with those using direct currents, is mainly due to the relative sensitiveness of the instruments available for such tests. The fact that the turning moment acting on the moving system depends in one case on the square of the current and in the other on the first power of the current, readily explains the high ratio between the currents needed to cause the minimum measurable deflection in the two cases, but this ratio is, nevertheless, most striking when a numerical comparison is actually made on some fair basis. The only likely way at present of improving alternate current instruments is to use iron cored electromagnets to increase the strength of the magnetic field. I have found that the difficulties due to varying permeability and hysteresis of the iron can be avoided by exciting the electromagnet in shunt. It proves possible, with careful design, to construct an electromagnet whose flux is connected with the exciting voltage by a strict mathematical law involving no variable physical properties like permeability, etc. Such an electromagnet is eminently suited for measuring purposes. The theoretical and experimental study of instruments constructed on this principle has brought out certain novel points which are set forth in the present paper. The first part discusses the mathematical relations of cyclic quantities having a common fundamental period, and constitutes a development of a method already published. This method is the only one known to me which is independent of assumptions in regard to the wave form of the quantities dealt with. The usual methods, which are based on the erroneous assumptions of sine law wave form, are not any simpler in working, and are most unsatisfactory when the accuracy of new results has to be critically examined. All alternate current measurements refer to mean squares or to mean products, and the natural method of obtaining the connections between such squares and products is to study the properties of quadratic functions of the variables. The earliest instance of this in alternate current theory was in connection with the “three voltmeter method.” Such processes lead to a very simple form of calculus appropriate to cyclic quantities.

A Study on Incremental Forming of Vertical Wall Square Box with Small Radius Corner

2010 ◽  
Vol 139-141 ◽  
pp. 1510-1513
Author(s):  
Liu Ru Zhou
Keyword(s):  
Tool Path ◽  
Vertical Wall ◽  
Small Radius ◽  
Forming Process ◽  
Corner Radius ◽  
Single Process ◽  
Incremental Sheet Metal Forming ◽  
Metal Forming Process ◽  
Sine Law ◽  
Square Box

According to sine law, a vertical wall square box can’t be formed by NC incremental sheet metal forming process in a single process, rather, it must be formed in multi processes. A vertical wall square box can be considered to consist of corners and straight sides. Straight sides and corners affect each other and the effect is different in various square boxes. The effect depends on the ratio r/B of the corner radius r and straight side width B. The smaller r/B, the larger the effect of straight side on corner is. In this case, the deformation in the straight sides isn’t even, and the metal of the corner is compressed and gradually piled up. With the increase of r/B, the deformation becomes more uniform. The tool path with gradually reduced corner radius is adopted to overcome this question. A vertical wall square box with small corner radius is successfully formed.

scholarly journals A dynamical model of a crystal structure. III

1949 ◽  
Vol 196 (1045) ◽  
pp. 182-194 ◽  
Keyword(s):  
Repulsive Interaction ◽  
Shear Stresses ◽  
Displacement Curve ◽  
Plastic Yielding ◽  
Maximum Shear Strain ◽  
Perfect Lattice ◽  
Interaction Terms ◽  
Shear Strains ◽  
Sine Law ◽  
Small Shear

A calculation of the maximum shear strain under which a two-dimensional close-packed lattice is stable has been carried out in terms of the forces between the lattice components. Two types of force were used; those between floating bubbles, which enabled a comparison with experiments on actual rafts of bubbles to be made, and also the forces derived from a potential V = Ae β r 2 , which form has been frequently proposed as an approximation to the repulsive interaction terms between metal ions. The conclusion reached is that this maximum strain may be considerably less than that deduced from a simple sine law approximation to the shear force versus displacement curve. Detailed consideration is given to edge effects in bubble rafts, and reasonable agreement with experimental results obtained. The overall result is that the formation of dislocations and consequent plastic yielding can occur in an initially perfect lattice only at quite large shear strains. The analogy with metals is discussed, and we conclude that the low strengths of metallic single crystals are explicable only on the assumption that they are not perfect and that dislocations already exist in them and move under very small shear stresses.

A New Test Method for Determination of Spinnability of Metals

1961 ◽  
Vol 83 (2) ◽  
pp. 119-124 ◽  
Author(s):  
Richard L. Kegg
Keyword(s):  
Test Method ◽  
Reduction Of Area ◽  
Work Done ◽  
Sine Law ◽  
Shear Spinning ◽  
Qualitative Discussion

A new test method is described whereby the spinnability, i.e., the ability to undergo shear spinning deformation, without fracture, of any material may be determined. Work done to date with this test method indicates that a good nonspinning criterion for estimating spinnability is the tensile reduction of area at fracture. Results of the spinnability test suggest that a feature of this test be adopted in the design of production cone-spinning mandrels. A qualitative discussion of the effects of deviating from the sine law is presented.

scholarly journals Some Invariant Laws Related to the Arc Sine Law

1968 ◽  
Vol 39 (1) ◽  
pp. 258-260 ◽  
Author(s):  
J. P. Imhof
Keyword(s):  
Sine Law

Some independence results related to the arc-sine law

1996 ◽  
Vol 9 (2) ◽  
pp. 447-458
Author(s):  
Jean Bertoin ◽  
Marc Yor
Keyword(s):  
Independence Results ◽  
Sine Law
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