By Emil Grosswald, Marvin Isadore Knopp, Mark Sheingorn

Emil Grosswald used to be a mathematician of significant accomplishment and noteworthy breadth of imaginative and prescient. This quantity can pay tribute to the span of his mathematical pursuits, that is mirrored within the wide variety of papers gathered right here. With contributions by way of prime modern researchers in quantity idea, modular capabilities, combinatorics, and comparable research, this ebook will curiosity graduate scholars and experts in those fields. The top of the range of the articles and their shut connection to present learn tendencies make this quantity a needs to for any arithmetic library

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19 Table 1. Comparison of the average squared errors. ,n, (14) where ir(xa) = 1/ [l + exp{ —wT4>(xa;v)}] is the conditional probability. ,n} according to models: (a) pr(Y — \\x) = 1/ [1 +exp{-sin(27rx 1 ) - cos(27r:r2)}], (6) pr(Y = l|a:) = 1/ [1 + exp{ — exp(a:i) sin(7nE2)}] and (c) pr(Y = l\x) = 1/ [1 + exp {— x\ sin(7ra2)}], where the design points are uniformly distributed in [0,2] x [0,2]. We first fit the radial basis function network logistic regression model to a set of the 0-1 response data shown in Figure 2, which were generated from true structure (a).

It is shown that the conventional hedge does not necessarily provide the most effective constant hedge. K e y w o r d s : dynamic hedge, hedging effectiveness, futures markets. 1 Introduction The most common method for determining the hedge ratio is known as the conventional hedging strategy, whereby the ratio is determined by dividing the 32 33 (unconditional) covariance between cash and futures returns by the (unconditional) variance of the futures returns. The hedge ratio may be equivalently obtained by regressing the returns to holding the spot asset AS on the returns to holding the futures asset A F .

47 Figure 1. d - random secants Figure 2. A - random secants. 3 \i - Randomness: mean free path randomness (Isotropic Uniform Randomness) A secant is defined by its distance from the origin and the direction of its normal. For a disc, the distance and direction are independently and uniformly distributed. For general /C, a secant is generated by intersection with a /^-random secant within a disc containing K. 4 v - Randomness: interior radiator randomness (also I randomness) A secant is defined by a uniform random point in the interior of /C and an isotropic direction.